Pairing in Fermionic Systems: Basics Concepts and Modern Applications (Series on Advances in Quantum Many-Body Theory)

Series on Advances in Quantum Many-Body Theory - Vol. 8

PAIRING „

FERMIONIC SYSTEMS Basic Concepts and Modern Applications

If World Scientific

PAIRING FERMIONIC SYSTEMS Basic Concepts and Modern Applications

Series on Advances in Quantum Many-Body Theory Edited by R. F. Bishop, C. E. Campell, J. W. Clark and S. Fantoni (International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories)

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Microscopic Approaches to the Structure of Light Nuclei Ed/fed by R. F. Bishop and N. R. Walet

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Basic Concepts and Modern Applications

World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING « SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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Series on Advances in Quantum Many-Body Theory — Vol. 8 PAIRING IN FERMIONIC SYSTEMS Basic Concepts and Modern Applications Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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To the Memory of Adelchi Fabrocini

Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. Richard P. Feynman

Preface

As we approach the Golden Anniversary of the publication of the Bardeen-CooperSchrieffer (BCS) theory of superconductivity, appreciation for its conceptual fertility and ubiquity as a paradigm for the behavior of quantum matter has never been greater. The last few years have seen remarkable progress in our understanding of fermionic pairing across an impressive range of physical systems. Beyond the familiar examples of electronic systems in condensed-matter physics, these include ultra-cold atomic vapors, superfluid 3 He, finite nuclei, nucleonic matter, and quark matter. One common feature that has emerged is a tension between factors that favor standard Cooper pairing (simple attractive forces between the fermions) and factors that disfavor it (typically, conservation laws that separate the Fermi momenta of different species). Other complications arise from the state dependence and momentum-space structure of the realistic fermion-fermion interactions, medium-modification of these interactions and of consistent fermion self-energies, and retardation effects due to finite propagation times of exchange bosons. To understand the implications of such conflicts and complications, we need to extend the standard Bardeen-Cooper-Schrieffer (BCS) theory to cover a much wider class of possible fermionic pairing patterns. The topic of non-conventional fermion pairing is particularly rich in opportunities for interdisciplinary cross-fertilization, because it draws together theoretical ideas from atomic physics, condensed-matter physics, nuclear physics, particle physics, and astrophysics. These ideas have direct relevance to experiment. Pairing in cold, high-density quark matter will drastically affect the transport properties underlying the astrophysical signatures that are sought by astronomers studying neutron stars. Pairing in isospin-asymmetric nuclear systems is an important facet of the physics of nuclei far from stability being created at radioactive ion-beam facilities and in natural stellar nucleosynthesis, as well as the physics of neutron-star interiors. Experimental studies of ultra-cold, trapped atomic gases continue to energize one of the most successful and fast-growing areas of mainstream physics, at the crossroads of new thrusts in condensed-matter physics, atomic physics, quantum optics, and quantum-information science. This book collects a coherent set of expositions on fermionic pairing that grew out of lectures given in the Institute for Nuclear Theory at the University of

Vll

Vlll

Preface

Washington. The chapters cover theoretical aspects of fermion pairing in various areas of intense current research. In assembling the volume, it has been our intent to provide an accessible and useful introduction to basic principles of the theory of superfluid systems, and to their specialized application in a number of physical contexts. This aim is reflected in the arrangement of the chapters, which divide naturally into three blocks. The first block, comprising Chapters 1-5, examines fermionic pairing at the subnuclear level through explorations of color-superconducting phases of deconfined quarks at high baryon densities. The second block (Chapters 5-8) is devoted to pairing at the nuclear level, focusing on the formation of pair-condensed nucleonic superfluids below the deconfinment density in neutron-star matter and nuclear matter; some attention is given also to the higher-order clustering phenomenon of quartetting in both infinite nuclear matter and finite nuclei. The third block (Chapters 9 & 10) is concerned with pairing at the atomic level, as exemplified in dilute, ultracold systems of trapped fermionic atoms; at the same time, it addresses important theoretical issues of wider interest that arise when the pairing is between two fermionic species with unequal densities. Each block starts with an opening chapter which gives a broader perspective of the corresponding topical area. A. Sedrakian, J. W. Clark, M. Alford Tubingen, St. Louis June 2006

Contents

Preface 1.

Color Superconductivity in Dense, but not Asymptotically Dense, Quark Matter

vii

1

Mark Alford and Krishna Rajagopal 2.

Larkin-Ovchinnikov-Fulde-Ferrell Phases in QCD

37

Giuseppe Nardulli 3.

Phase Diagram of Neutral Quark Matter at Moderate Densities

63

Stefan B. Riister, Verena Werth, Michael Buballa, Igor A. Shovkovy and Dirk H. Rischke 4.

Spontaneous Nambu-Goldstone Current Generation Driven by Mismatch

91

Mei Huang 5.

The CFL Phase and ms: An Effective Field Theory Approach

109

Thomas Schdfer 6.

Nuclear Superconductivity in Compact Stars: BCS Theory and Beyond

135

Armen Sedrakian and John W. Clark 7.

Pairing Properties of Dressed Nucleons in Infinite Matter Willem H. Dickhoff and Herbert Miiiher

175

x

Contents

8.

Pairing in Higher Angular Momentum States: Spectrum of Solutions of the 3P2-3F2 Pairing Model

201

Mikhail V. Zverev, John W. Clark and Victor A. Khodel 9.

Four-Particle Condensates in Nuclear Systems

221

Gerd Ropke and Peter Schuck 10.

Realization, Characterization, and Detection of Novel Superfluid Phases with Pairing between Unbalanced Fermion Species

253

Kun Yang 11.

Phase Transition in Unbalanced Fermion Superfiuids

269

Heron Caldas Subject Index

283

Chapter 1 C o l o r S u p e r c o n d u c t i v i t y in D e n s e , b u t n o t A s y m p t o t i c a l l y D e n s e , Quark M a t t e r

Mark Alford Physics Department, Washington University Saint Louis, MO 63130, USA Krishna Rajagopal Center {or Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139 USA Nuclear Science Division, Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA At ultra-high density, matter is expected to form a degenerate Fermi gas of quarks in which there is a condensate of Cooper pairs of quarks near the Fermi surface: color superconductivity. In this chapter we review some of the underlying physics, and discuss outstanding questions about the phase structure of ultradense quark matter. We then focus on describing recent results on the crystalline color superconducting phase that may be the preferred form of cold, dense but not asymptotically dense, three-flavor quark matter. The gap parameter and free energy for this phase have recently been evaluated within a Ginzburg-Landau approximation for many candidate crystal structures. We describe the two that are most favorable. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We describe qualitative insights into what makes a crystal structure favorable which can be used to winnow the possibilities. We close with a look ahead at the calculations that remain to be done in order to make quantitative contact with observations of compact stars.

Contents 1.1. Introduction 1.2. Review of color superconductivity 1.2.1. Color superconductivity 1.2.2. Highest density: Color-flavor locking (CFL) 1.2.3. Less dense quark matter: stresses on the CFL phase 1.2.4. Kaon condensation: the CFL-K 0 phase 1.2.5. The gapless CFL phase 1.2.6. Beyond gapless CFL 1.2.7. Crystalline pairing 1.2.8. Single-flavor pairing 1

2 3 3 3 4 5 5 6 10 10

2

Color Superconductivity

in Dense Quark

1.2.9. Mixed phases 1.3. The crystallography of three-flavor quark matter 1.3.1. Introduction and context 1.3.2. Model, simplifications and ansatz 1.3.3. Ginzburg-Landau approximation 1.3.4. General results 1.3.5. Two plane wave structure 1.3.6. Multiple plane waves 1.3.7. Free energy comparisons and conclusions 1.3.8. Implications and future work 1.4. Coda Bibliography

Matter

11 11 11 14 19 22 24 25 28 30 32 33

1.1. Introduction The exploration of the phase diagram of matter at ultra-high temperature or density is an area of great interest and activity, both on the experimental and theoretical fronts. Heavy-ion colliders such as the SPS at CERN and RHIC at Brookhaven have probed the high-temperature region, creating and studying the properties of quark matter with very high energy density and very low baryon number density similar to the fluid which filled the universe for the first microseconds after the big bang. In this paper we discuss a different part of the phase diagram, the low-temperature high-density region. Here there are as yet no experimental constraints, and our goal is to understand the properties of matter predicted by QCD well enough to be able to use astronomical observations of neutron stars to learn whether these densest objects in the current universe contain quark matter in their core. We expect cold, dense, matter to exist in phases characterized by Cooper pairing of quarks, i.e. color superconductivity, driven by the Bardeen-Cooper-Schrieffer (BCS) 1 mechanism. The BCS mechanism operates when there is an attractive interaction between fermions at a Fermi surface. The QCD quark-quark interaction is strong, and is attractive in many channels, so we expect cold dense quark matter to generically exhibit color superconductivity. Moreover, quarks, unlike electrons, have color and flavor as well as spin degrees of freedom, so many different patterns of pairing are possible. This leads us to expect a rich phase structure in matter beyond nuclear density. Calculations using a variety of methods agree that at sufficiently high density, the favored phase is color-flavor-locked (CFL) color-superconducting quark matter 2 (for reviews, see Ref. 3). However, there is still uncertainty over the nature of the next phase down in density. Previous work 4,5 had suggested that when the density drops low enough so that the mass of the strange quark can no longer be neglected, there is a continuous phase transition from the CFL phase to a new gapless CFL (gCFL) phase, which could lead to observable consequences if it occurred in the cores of neutron stars. 6 However, it now appears that some of the gluons in the gCFL phase have imaginary Meissner masses, indicating an instability towards a lower-energy phase. 7-18 Analysis in the vicinity of the unstable gCFL phase cannot determine the nature of the lower-energy phase that resolves the instability. However, the instability is telling us that the system can lower its energy by turning

M. Alford and K. Rajagopal

3

on currents, suggesting that the crystalline color superconducting phase, in which the condensate is modulated in space in a way that can be thought of as a sum of counterpropagating currents, is a strong candidate.

1.2. Review of color superconductivity 1.2.1. Color

superconductivity

The fact that QCD is asymptotically free implies that at sufficiently high density and low temperature, there is a Fermi surface of weakly-interacting quarks. The interaction between these quarks is certainly attractive in some channels (quarks bind together to form baryons), so we expect the formation of a condensate of Cooper pairs. We can see this by considering the grand canonical potential Q = E — fiN, where E is the total energy of the system, fi is the chemical potential, and N is the number of quarks. The Fermi surface is defined by a Fermi energy Ep = n, at which the free energy is minimized, so adding or subtracting a single particle costs zero free energy. Now switch on a weak attractive interaction. It costs no free energy to add a pair of particles (or holes), and if they have the right quantum numbers then the attractive interaction between them will lower the free energy of the system. Many such pairs will therefore be created in the modes near the Fermi surface, and these pairs, being bosonic, will form a condensate. The ground state will be a superposition of states with all numbers of pairs, breaking the fermion number symmetry. A pair of quarks cannot be a color singlet, so the resulting condensate will break the local color symmetry SU(3)co\or. The formation of a condensate of Cooper pairs of quarks is therefore called "color superconductivity". The condensate plays the same role here as the Higgs condensate does in the standard model: the colorsuperconducting phase can be thought of as the Higgs phase of QCD.

1.2.2. Highest

density:

Color-flavor

locking

(CFL)

It is by now well-established that at sufficiently high densities, where the up, down and strange quarks can be treated on an equal footing and the disruptive effects of the strange quark mass can be neglected, quark matter is in the color-flavor locked (CFL) phase, in which quarks of all three colors and all three flavors form conventional Cooper pairs with zero total momentum, and all fermionic excitations are gapped, with the gap parameter 5$ ~ 10 — 100 MeV. 2,3 This has been confirmed by both Nambu-Jona-Lasinio (NJL) 2 ' 3 ' 20 and gluon-mediated interaction calculations. 3 ' 21 " 23 The CFL pairing pattern is 2 (q?C^)

= 50(K + l)5?5f + 60(K - l)SfSf

[Stf(3)coior] x SU(3)L x

SU(3)RXU(1)B

= <5oea/3i%;v + *>*(• • •) -*

SU(3)C+L+R

XZ2

(1.1)

4

Color Superconductivity

in Dense Quark

Matter

Color indices a, ft and flavor indices i,j run from 1 to 3, Dirac indices are suppressed, and C is the Dirac charge-conjugation matrix. The term multiplied by K corresponds to pairing in the (65,65), which although not energetically favored breaks no additional symmetries and so K is in general small but nonzero. 2 ' 21 ' 22,24 The Kronecker <5's connect color indices with flavor indices, so that the condensate is not invariant under color rotations, nor under flavor rotations, but only under simultaneous, equal and opposite, color and flavor rotations. Since color is only a vector symmetry, this condensate is only invariant under vector flavor+color rotations, and breaks chiral symmetry. The features of the CFL pattern of condensation

The color gauge group is completely broken. All eight gluons become massive. This ensures that there are no infrared divergences associated with gluon propagators. All the quark modes are gapped. The nine quasiquarks (three colors times three flavors) fall into an 8 © 1 of the unbroken global SU(3). Neglecting corrections of order K (see (1.1)) the octet and singlet quasiquarks have gap parameter 5Q and 25o respectively. A rotated electromagnetic gauge symmetry (generated by "Q") survives unbroken. The single remaining massless gauge boson is a linear combination of the original photon and one of the gluons. Two global symmetries are broken, the chiral symmetry and baryon number, so there are two gauge-invariant order parameters that distinguish the CFL phase from the QGP, and corresponding Goldstone bosons which are long-wavelength disturbances of the order parameter. When the light quark mass is nonzero it explicitly breaks the chiral symmetry and gives a mass to the chiral Goldstone octet, but the CFL phase is still a superfluid, distinguished by its spontaneous baryon number breaking. 1.2.3. Less dense quark matter:

stresses

on the CFL

phase

The CFL phase is characterized by pairing between different flavors and different colors of quarks. We can easily understand why this is to be expected. Firstly, the QCD interaction between two quarks is most attractive in the channel that is antisymmetric in color (the 3). Secondly, pairing tends to be stronger in channels that do not break rotational symmetry, 25-30 so we expect the pairs to be spin singlets, i.e. antisymmetric in spin. Finally, fermionic antisymmetry of the Cooper pair wavefunction then forces the Cooper pair to be antisymmetric in flavor. Pairing between different colors/flavors can occur easily when they all have the same chemical potentials and Fermi momenta. This is the situation at very high density, where the strange quark mass is negligible. However, even at the very center of a compact star the quark number chemical potential /i cannot be much larger than 500 MeV, meaning that the strange quark mass Ms (which is density dependent, lying somewhere between its vacuum current mass of about 100 MeV and constituent mass of about 500 MeV) cannot be neglected. Furthermore, bulk matter, as relevant for a compact star, must be in weak equilibrium and must be electrically and

M. Alford and K. Rajagopal

5

color neutral 31 " 36 (possibly via mixing of oppositely charged phases). All these factors work to separate the Fermi momenta of the three different flavors of quarks, and thus disfavor the cross-species BCS pairing that characterizes the CFL phase. If we imagine beginning at asymptotically high densities and reducing the density, and suppose that CFL pairing is disrupted by the heaviness of the strange quark before color superconducting quark matter is superseded by baryonic matter, the CFL phase must be replaced by some phase of quark matter in which there is less, and less symmetric, pairing. In the next few subsections we give a quick overview of the expected phases of real-world quark matter. We restrict our discussion to zero temperature because the critical temperatures for most of the phases that we discuss are expected to be of order 10 MeV or higher, and the core temperature of a neutron star is believed to drop below this value within minutes (if not seconds) of its creation in a supernova. 1.2.4. Kaon condensation:

the CFL-K0

phase

Bedaque and Schafer37,38 showed that when the stress is not too large (high density), it may simply modify the CFL pairing pattern by inducing a flavor rotation of the condensate which can be interpreted as a condensate of "if0" mesons, i.e. the neutral anti-strange Goldstone bosons associated with the chiral symmetry breaking. This is the "CFL-if 0 " phase, which breaks isospin. The if°-condensate can easily be suppressed by instanton effects,39 but if these are ignored then the kaon condensation occurs for Ms > m 1 / 3 ^ 2 / 3 for light (u and d) quarks of mass m. This was demonstrated using an effective theory of the Goldstone bosons, but with some effort can also be seen in an NJL calculation. 40 ' 41 1.2.5.

The gapless

CFL phase

The nature of the next significant transition has been studied in NJL model calculations which ignore the /^-condensation in the CFL phase and which assume spatial homogeneity. 4,5 ' 42 " 47 It has been found that the phase structure depends on the strength of the pairing. If the pairing is sufficiently strong (so that So ~ 100 MeV where <5o is what the CFL gap would be at /i ~ 500 MeV if Ms were zero) then the CFL phase survives all the way down to the transition to nuclear matter. For a wide range of parameter values, however, we find something more interesting. We can make a rough quantitative analysis by expanding in powers of M 2 //U 2 and 5/fi, and ignoring the fact that the effective strange quark mass may be different in different phases. 33 Such an analysis shows that, within a spatially homogeneous ansatz, as we come down in density we find a transition a t / J « ^M^/SO from CFL to another phase, the gapless CFL phase (gCFL). 4 ' 5 In this phase, quarks of all three colors and all three flavors still form ordinary Cooper pairs, with each pair having zero total momentum, but there are regions of momentum space in which certain quarks do not succeed in pairing, and these regions are bounded by momenta at which certain fermionic quasiparticles are gapless. This variation on BCS pairing — in which the same species of fermions that pair feature gapless quasiparticles — was first proposed for two flavor quark matter 48 ' 49 and in an atomic physics context. 50

6

Color Superconductivity

in Dense Quark

Matter

For M 2 /'// > 2<5o, the CFL phase has higher free energy than the gCFL phase. This follows from the energetic balance between the cost of keeping the Fermi surfaces together and the benefit of the pairing that can then occur. The leading effect of Ms is like a shift in the chemical potential of the strange quarks, so the bd and gs quarks feel "effective chemical potentials" /zff = fi — | / j 8 and ^ = n + | / i 8 — -j*-. In the CFL phase fi8 = - M 2 / ( 2 / i ) , 3 3 , 3 4 SO /iff - /j,^ = M 2 //z. The CFL phase will be stable as long as the pairing makes it energetically favorable to maintain equality of the bd and gs Fermi momenta, despite their differing chemical potentials. 4,51 It becomes unstable when the energy gained from turning a gs quark near the common Fermi momentum into a bd quark (namely M 2 //x) exceeds the cost in lost pairing energy 2S0. So the CFL phase is stable when M2 — L < 2 Jo • (1.2) /i

Including the effects of if °-condensation expands the range of M 2 / / / in which the CFL phase is stable by a few tens of percent. 38,40 ' 41 For larger M 2 / / i , the CFL phase is replaced by some new phase with unpaired bd quarks, which cannot be neutral unpaired or 2SC quark matter because the new phase and the CFL phase must have the same free energy at the critical Ms2//u = 2S0. The obvious approach to finding this phase is to perform a NJL model calculation with a general ansatz for the pairing that includes differences between the flavors, for example by allowing different pairing strengths 5UCL, Sds, Sus. This was done in Refs. 4,5, and the resultant "gCFL" phase was described in detail. In Figs. 1.1 and 1.2 we show the results. The gCFL phase takes over from CFL at M 2 //x « 2S0, and remains favored beyond the value M 2 //x sa A60 at which the CFL phase would become unfavored. 1.2.6. Beyond

gapless

CFL

The arguments above led us to the conclusion that the favored phase of quark matter at the highest densities is the CFL phase, and that as the density is decreased there is a transition to another color superconducting phase, the gapless CFL phase. However, it turns out that the gCFL phase is itself unstable, and that in the regime where the gCFL phase has a lower free energy than both the CFL phase and unpaired quark matter, there must be some other phase with lower free energy still. The nature of that phase has to date not been determined rigorously, and in subsequent subsections we shall discuss various possibilities. However, the crystalline color superconducting phases analyzed very recently in Ref. 52 whose free energies are shown in Fig. 1.2 appear particularly robust, offering the possibility that over a wide range of densities they are the ground state of dense matter. We shall therefore discuss these phases at much greater length in Section 1.3. In all the contexts in which they have been investigated (whether in atomic physics or the g2SC phase in two-flavor quark matter or the gCFL phase) the gapless paired state turns out to suffer from a "magnetic instability": it can lower its energy by the formation of counter-propagating currents. 7-18 In the atomic physics context, the resolution of the instability is phase separation into macroscopic regions

M. Alford and K. Rajagopal

30 25

> 21, 20

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M2s/\i [MeV] Fig. 1.1. Gap parameter S versus M^/fj, for the CFL and gCFL phases as well as for crystalline color superconducting phases with two different crystal structures described in Section 1.3.6. These gap parameters have all been evaluated in a NJL model with the interaction strength chosen such that the CFL gap parameter is 25 MeV at M^/fj. = 0. In the gCFL phase, 5ud > 5U3 > Sds. The calculation of the gap parameters in the crystalline phases has been done in a Ginzburg-Landau approximation. Recall that the splitting between Fermi surfaces is proportional to M^/n, and that small (large) M^/fi corresponds to high (low) density.

of two phases in one of which standard BCS pairing occurs and in the other of which no pairing occurs. 53-55 Phase separation into electrically charged but color neutral phases is also a possibility in two-flavor quark matter. 56 In three-flavor quark matter, where the instability of the gCFL phase has been established in Refs. 8, 12, phase coexistence would require coexisting components with opposite color charges, in addition to opposite electric charges, making it very unlikely that a phase separated solution can have lower energy than the gCFL phase. 5 ' 42 Furthermore, color superconducting phases which are less symmetric than the CFL phase but still involve only conventional BCS pairing, for example the much-studied 2SC phase in which only two colors of up and down quarks pair 57-59 but including also many other possibilities,60 cannot be the resolution of the gCFL instability. 33 ' 60 It seems likely, therefore, that a ground state with counter-propagating currents is required. This could take the form of a crystalline color superconductor 19 ' 52 ' 61-71 — the QCD analogue of a form of non-BCS pairing first considered by Larkin, Ovchinnikov, Fulde and Ferrell. 72 Or, given that the CFL phase itself is likely augmented by kaon condensation, 37 ' 38 ' 40 ' 41 it could take the form of a phase in which a CFL kaon condensate carries a current in one direction balanced by a counter-propagating current in the opposite direction carried by gapless quark quasiparticles. 14 ' 18 The instability of the gCFL phase appears to be related to one of its most interesting features, namely the presence of gapless fermionic excitations around the ground state. These are illustrated in Fig. 1.3, which shows that there is one mode (the bu-rs quasiparticle) with an unusual quadratic dispersion relation, which is expected to give rise to a parametrically enhanced heat capacity and neutrino emissivity and anomalous transport properties. 6 The instability manifests itself in

8

Color Superconductivity

.

0

.

,

,

in Dense Quark

i

50

,

100

150

,

,

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,

200

.

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250

M^/n [MeV] Fig. 1.2. Free energy U relative to that of neutral unpaired quark matter versus M%/n for the CFL, gCFL and crystalline phases whose gap parameters are plotted in Fig. 1.1. Recall that the gCFL phase is known to be unstable, meaning that in the regime where the gCFL phase free energy is plotted, the true ground state of three-flavor quark matter must be some phase whose free energy lies below the dashed line. We see that the three-flavor crystalline color superconducting quark matter phases with the most favorable crystal structures that we have found, namely 2Cube45z and CubeX described in (1.31) and (1.33), have sufficiently robust condensation energy (sufficiently negative fi) that they are candidates to be the ground state of matter over a wide swath of M%//Li, meaning over a wide range of densities.

imaginary Meissner masses MM for some of the gluons. MM is the low-momentum current-current two-point function, and MM/(g2S2), with g the gauge coupling, is the coefficient of the gradient term in the effective theory of small fluctuations around the ground-state condensate. The fact that we find a negative value when the quasiparticles are gapless indicates an instability towards spontaneous breaking of translational invariance. Calculations in a simple two-species model 10 show that imaginary MM is generically associated with the presence of gapless charged fermionic modes. The earliest calculations for the three-flavor case show that even a very simple ansatz for the crystal structure yields a crystalline color superconducting state that has lower free energy than gCFL in the region where the gCFL—•unpaired transition occurs. 69 ' 71 It is reasonable, based on what was found in the two-flavor case in Ref. 65, to expect that when the full space of crystal structures is explored, the crystalline color superconducting state will be preferred to gCFL over a much wider range of the stress parameter Mf/(<Sn/z), and this expectation has recently been confirmed by explicit calculation. 52 It is conceivable that the whole gCFL region is actually a crystalline region, but the results shown in Fig. 1.2 that we describe in Section 1.3 suggest that there is still room for other possibilities (like the currentcarrying meson condensate) at the highest densities in the gCFL regime. An alternative explanation of the consequences of the gCFL instability was advanced by Hong 73 (see also Ref. 49): since the instability is generically associated with the presence of gapless fermionic modes, and the BCS mechanism implies that

M. Alford and K. Rajagopal

9

Gapless CFL phase

Fig. 1.3. Dispersion relations of the lightest quasiquark excitations in the gCFL phase, at p. = 500 MeV, with Ms = 200 MeV and interaction strength such that the CFL gap parameter at Ms = 0 would be <5rj = 25 MeV. Note that in there is a gapless mode with a quadratic dispersion relation (energy reaching zero at momentum p\u2) as well as two gapless modes with more conventional linear dispersion relations.

any gapless fermionic mode is unstable to Cooper pairing in the most attractive channel, one might expect that the instability will simply be resolved by "secondary pairing". This means the formation of a (qq) condensate where q is either one of the gapless quasiparticles whose dispersion relation is shown in Fig. 1.3. After the formation of such a secondary condensate, the linear gapless dispersion relations would be modified by "rounding out" of the corner where the energy falls to zero, leaving a secondary energy gap 6S, which renders the mode gapped, and removes the instability. In the case of the quadratically gapless mode there is a greatly increased density of states at low energy (in fact, the density of states diverges as E~1/2), so Hong calculated that the secondary pairing should be much stronger than would be predicted by BCS theory, and he specifically predicted 5S oc G23 for coupling strength Gs, as compared with the standard BCS result 6 oc exp(—const/G s ). This possibility was worked out in a two-species NJL model in Ref. 74. This allowed a detailed exploration of the strength of secondary pairing. The calculation confirmed Hong's prediction that in typical secondary channels Ss oc G2S. However, in all the secondary channels that were analyzed it was found that the secondary gap, even with this enhancement, is from ten to hundreds of times smaller than the primary gap at reasonable values of the secondary coupling. This shows that that secondary pairing does not generically resolve the magnetic instability of the gapless phase, since it indicates that there is a temperature range Ss
10

Color Superconductivity

1.2.7. Crystalline

in Dense Quark

Matter

pairing

The pairing patterns discussed so far have been translationally invariant. But in the region of parameter space where cross-species pairing is excluded by stresses that pull apart the Fermi surfaces, one expects a position-dependent pairing known as the crystalline color superconducting phase. 19 ' 52 ' 61-72 This arises because one way to achieve pairing between different flavors while accommodating the tendency for the Fermi momenta to separate is to allow Cooper pairs with nonzero total momentum yielding pairing over parts of the Fermi surfaces. We describe this phase extensively in Section 1.3. As we will discuss there, this phase may resolve the gCFL phase's stability problems. 1.2.8. Single-flavor

pairing

At densities that are so low that Ms puts such a significant stress on the pairing pattern that even the crystalline phase becomes unfavored, we expect a transition to a phase with no cross-species pairing at all. (This regime will only arise if <So is so small that very large values of M^/(fiS0) can arise without fi being taken so small that nuclear matter becomes favored.) "Unpaired" quark matter. In most NJL studies, matter with no cross-species pairing at all is described as as "unpaired" quark matter. However, it is well known that there are attractive channels for a single flavor pairing, although they are much weaker than the 2SC and CFL channels. 26-30 Calculations using NJL models and single-gluon exchange agree that the favored phase in this case is the color-spinlocked (CSL) phase, 26 in which all three colors of each flavor, with each pair of colors correlated with a particular direction for the spin. This phase does not break rotational symmetry. See first panel of Fig. 1.4.

•_, red green blue

<3A'3S'1s'+XCY,) red green blue

J

red green blue

Fig. 1.4. Pictorial representation of CSL pairing, 2SC+CSL pairing, and 2SC+1SC pairing in neutral quark matter. See Ref. 75.

"Unpaired" species in 2SC quark matter. If the stress is large enough to destroy CFL pairing, and furthermore is large enough to preclude cross-species pairing in which the Cooper pairs have nonzero momentum and form a crystalline color superconducting phase, we can ask whether there is any pattern of BCS pairing which can persist in neutral quark matter with arbitrarily large splitting between Fermi surfaces. The answer to this is no, as demonstrated via an exhaustive search in Ref. 60. Although unlikely, it remains a possibility that up and down quarks with two colors could pair in the 2SC pattern.

M. Alford and K. Rajagopal

11

This does not happen in NJL models in which Ms is a parameter. However, in models in which one instead solves self-consistently for the quark masses, the 2SC phase tends to become more robust. It remains to be seen whether this relative strengthening of the 2SC phase in such models is sufficient to allow it to exist at values of M 2 //x which are larger than those where the crystalline phases shown in Fig. 1.2 and discussed at length in Section 1.3 are favored. If there is a regime in which the 2SC phase survives, this leaves the blue quarks unpaired. In that case one might expect a "2SC+CSL" pattern, illustrated in the second panel of Fig. 1.4, which would again be rotationally symmetric. However, the 2SC pattern breaks the color symmetry, and in order to maintain color neutrality, a color chemical potential is generated, which also affects the unpaired strange quarks, splitting the Fermi momentum of the blue strange quarks away from that of the red and green strange quarks. This is a small effect, but so is the CSL pairing gap, and NJL model calculations indicate that the color chemical potential typically destroys CSL pairing of the strange quarks. 75 The system falls back on the next best alternative, which is spin-1 pairing of the red and green strange quarks (third panel of Fig. 1.4). Single-flavor pairing phases have much lower critical temperatures than multiflavor phases like the CFL or crystalline phases, perhaps as large as a few MeV, more typically in the eV to many keV range, 2 6 - 3 0 so they are expected to play a role late in the life of a neutron star.

1.2.9. Mixed

phases

Another way for a system to deal with a stress on its pairing pattern is phase separation. In the context of quark matter this corresponds to relaxing the requirement of local charge neutrality, and requiring neutrality only over long distances, so we allow a mixture of a positively charged and a negatively charged phase, with a common pressure and a common value of the electron chemical potential \xe that is not equal to the neutrality value for either phase. Such a mixture of nuclear and CFL quark matter was studied in Ref. 76. In three-flavor quark matter it has been found that as long as we require local color neutrality such mixed phases are not the favored response to the stress imposed by the strange quark mass. 4 ' 5,42 Phases involving color charge separation have been studied 36 but it seems likely that the energy cost of the color-electric fields will disfavor them.

1.3. The crystallography of three-flavor quark matter In this section we review recent work by Rajagopal and Sharma. 52 Here, we focus on explaining the context, the model, the results and the implications. We do not describe the calculations.

1.3.1. Introduction

and

context

The investigation of crystalline color superconductivity in three-flavor QCD was initiated in Ref. 69. Although such phases seem to be free from magnetic insta-

12

Color Superconductivity

in Dense Quark

Matter

bility,70 it remains to be determined whether such a phase can have a lower free energy than the meson current phase, making it a possible resolution to the gCFL instability. The simplest "crystal" structures do not suffice,69'71 but experience in the two-flavor context 65 suggests that realistic crystal structures constructed from more plane waves will prove to be qualitatively more robust. The results of Ref. 52 confirm this expectation. Determining the favored crystal structure(s) in the crystalline color superconducting phase(s) of three-flavor QCD requires determining the gaps and comparing the free energies for very many candidate structures, as there are even more possibilities than the many that were investigated in the two-flavor context. 65 As there, we shall make a Ginzburg-Landau approximation. This approximation is controlled if S <S 6Q, where 5 is the gap parameter of the crystalline color superconducting phase itself and So is the gap parameter in the CFL phase that would occur if Ms were zero. We shall find that the most favored crystal structures can have S/5Q as large as ~ 1/2, meaning that we are pushing the approximation hard and so should not trust it quantitatively. Earlier work on a particularly simple one parameter family of "crystal" structures in three-flavor quark matter, 71 simple enough that the analysis could be done both with and without the Ginzburg-Landau approximation, demonstrates that the approximation works when it should and that, at least for very simple crystal structures, when it breaks down it always underestimates the gap S and the condensation energy. Furthermore, the Ginzburg-Landau approximation correctly determines which crystal structure among the one parameter family that we analyzed in Ref. 71 has the largest gap and lowest free energy. The underlying microscopic theory that we use is an NJL model in which the QCD interaction between quarks is replaced by a point-like four-quark interaction with the quantum numbers of single-gluon exchange, analyzed in mean field theory. This is not a controlled approximation. However, it suffices for our purposes: because this model has attraction in the same channels as in QCD, its high density phase is the CFL phase; and, the Fermi surface splitting effects whose qualitative consequences we wish to study can all be built into the model. Note that we shall assume throughout that 5o <^ JJL. This weak coupling assumption means that the pairing is dominated by modes near the Fermi surfaces. Quantitatively, this means that results for the gaps and condensation energies of candidate crystalline phases are independent of the cutoff in the NJL model when expressed in terms of the CFL gap 5Q\ if the cutoff is changed with the NJL coupling constant adjusted so that 5Q stays fixed, the gaps and condensation energies for the candidate crystalline phases also stay fixed. This makes the NJL model valuable for making the comparisons that are our goal. We shall consider crystal structures in which there are two condensates

(ud) ~ S3 ^2 ex P (2i
(us) ~<& 2 $3exp(2"l2-r) •

(°)

a

As in Refs. 69, 71, we neglect (ds) pairing because the d and s Fermi surfaces are twice as far apart from each other as each is from the intervening u Fermi surface.

M. Alford and K. Rajagopal

13

Were we to set S2 to zero, treating only (ud) pairing, we would recover the two-flavor Ginzburg-Landau analysis of Ref. 65. There, it was found that the best choice of crystal structure was one in which pairing occurs for a set of eight qf 's pointing at the corners of a cube in momentum space, yielding a condensate with face-centered cubic symmetry. The analyses of three-flavor crystalline color superconductivity in Refs. 69, 71 introduced nonzero 62, but made the simplifying ansatz that pairing occurs only for a single q3 and a single q2. We consider crystal structures with up to eight qg's and up to eight q^'s. We evaluate the free energy 0.(62,63) for each three-flavor crystal structure in a Ginzburg-Landau expansion in powers of the J's. We work to order 6^6^ with p + q = 6. At sextic order, we find that 0(6,6) is positive for large 6 for all the crystal structures that we investigate. 52 This is in marked contrast to the result that many two-flavor crystal structures have negative sextic terms, with free energies that are unbounded from below when the Ginzburg-Landau expansion is stopped at sextic order. 65 Because we find positive sextic terms, we are able to use our sextic Ginzburg-Landau expansion to evaluate 6 and 0(6,6) for all the structures that we analyze. We can then identify the two most favorable crystal structures, 2Cube45z and CubeX, which are described further in Section 1.3.6. To a large degree, our argument that these two structures are the most favorable relies only on two qualitative factors, which we explain in Section 1.3.5. CubeX and 2Cube45z have gap parameters that are large: a third to a half of SQ. We have already discussed the implications of this for the Ginzburg-Landau approximation in Section 1.3.1 above. The large gaps naturally lead to large condensation energies, easily 1/3 to 1/2 of that in the CFL phase with Ms = 0, which is SS^fi2/IT2. This is remarkable, given the only quarks that pair are those lying on (admittedly many) rings on the Fermi surfaces, whereas in the CFL phase with Ms = 0 pairing occurs over the entire u, d and s Fermi surfaces. The gapless CFL (gCFL) phase provides a useful comparison at nonzero Ms. For 26Q < M2/fx < 5.26o, model analyses that are restricted to isotropic phases predict a gCFL phase, 4 ' 5,44 finding this phase to have lower free energy than either the CFL phase or unpaired quark matter, as shown in Fig. 1.2. However, this phase is unstable to the formation of current-carrying condensates. 7-18 and so it cannot be the ground state. The true ground state must have lower free energy than that of the gCFL phase, and for this reason the gCFL free energy provides a useful benchmark. We find that three-flavor crystalline color superconducting quark matter with one or other of the two crystal structures that we argue are most favorable has lower free energy (greater condensation energy) than CFL quark matter, gCFL quark matter, and unpaired quark matter for M2 2.9<50 < — < 10.4<50 . (1.4) M (See Fig. 1.2 above.) This window in parameter space is in no sense narrow. Our results therefore indicate that three-flavor crystalline quark matter will occur over a wide range of densities. (That is unless the pairing between quarks is so strong, i.e. So so large making M2/6Q SO small, that quark matter is in the CFL phase all the way down to the density at which quark matter is superseded by nuclear matter.)

Color Superconductivity

14

in Dense Quark

Matter

However, our results also indicate that unless the Ginzburg-Landau approximation is underestimating the condensation energy of the crystalline phase by about a factor of two, there is a fraction of the "gCFL window" (with 2S0 < M^/fi < 2.9S0, in the Ginzburg-Landau approximation) in which no crystalline phase has lower free energy than the gCFL phase. This is thus the most likely regime in which to find the current-carrying meson condensates of Refs. 14,18.

1.3.2. Model, simplifications

and

ansatz

1.3.2.1. Neutral unpaired three-flavor quark matter The analysis of beta-equilibrated neutral three-flavor quark matter in an NJL model has been discussed in detail elsewhere (For example, Ref. 5), and here we just summarize the main points. We assume that the up and down quarks are massless. The strange quark mass is a parameter Ms. We couple a chemical potential \x to quark number, and always work at /J, = 500 MeV. To ensure neutrality under color and electromagnetism we couple a chemical (electrostatic) potential \ie to negative electric charge, and chemical (color-electrostatic) potentials ^3 and \i% to the diagonal generators of SU(3)coiOI. In full QCD these gauged chemical potentials would be the zeroth components of the corresponding gauge fields, and would naturally be forced to their neutrality values. 33 ' 77 In the NJL context, we enforce neutrality explicitly by choosing them to satisfy the neutrality conditions

an _ do. _ on dfje

dfj,3

dflS

The assumption of beta-equilibrium is built into the calculation via the fact that the only flavor-dependent chemical potential we introduce is /xe, ensuring for example that the chemical potentials of d and s quarks with the same color must be equal. The presence of a nonzero strange quark mass, combined with the requirements of equilibrium under weak interactions and charge neutrality, drives down the number of strange quarks, and introduces electrons and additional down quarks. We assume that this is a small effect, i.e. we expand in powers of /ie/M a n d Mg//j,2. Working to linear order, we can simply treat the strange quark mass as a shift of —Mg/(2fi) in the strange quark chemical potential. (The corrections from higherorder terms in M^j'y? in an NJL analysis of a two-flavor crystalline color superconductor have been evaluated and found to be small, 64 and we expect the same to be true here.) In this approximation we find the Fermi momenta of the quarks and electrons are given by d

, Me

pF = M +

Y

u

^Me

e

P> = A* - - j j -

ir free energy is 3(p£) 4 + 3 ( ? 4 ) 4 + (peF)4 , 3 [*' unpaired =

^ .4 A („« -

, .22 » / r 22\ , M MS ) +

+ ^ 1

^

PF

2

= Me

/ ,

P dp ( v V x

2 , ^- M, , BS fV - ^ \, , 2\, , 2 + ... /

l

fie\

2

+ ^

-

M

-

y

)

(1.6)

M. Alford and K. Rajagopal

15

To this order, electric neutrality requires M2 (L7) * = ^ ' 2 2 and because we are working to lowest order in M / / / we need no longer be careful about the distinction between pp's and /x's, as we can simply think of the three flavors of quarks as if they have chemical potentials M<* = /•*« + 2 4<5/x = M 2 /(2/i), 4 a criterion that is reduced somewhat by kaon condensation which acts to stabilize CFL pairing. 38,41 We are now considering quark matter at densities that are low enough (/z < M2/(2So)) that CFL pairing is not possible. The gap parameter 60 that would characterize the CFL phase if M2 and 5/i were zero is nevertheless an important scale in our problem, as it quantifies the strength of the attraction between quarks. Estimates of the magnitude of So are typically in the tens of MeV, perhaps as large as 100 MeV. 3 We shall treat Jo as a parameter, and quote results for So = 25 MeV, although our results can easily be scaled to any value of So as long as the weakcoupling approximation So
16

Color Superconductivity

in Dense Quark

Matter

1.3.2.3. Crystalline color superconductivity in two-flavor quark matter Crystalline color superconductivity can be thought of as the answer to the question: "Is there a way to pair quarks at differing Fermi surfaces without first equalizing their Fermi momenta, given that doing so exacts a cost?" The answer is "Yes, but it requires Cooper pairs with nonzero total momentum." Ordinary BCS pairing pairs quarks with momenta p and — p, meaning that if the Fermi surfaces are split at most one member of a pair can be at its Fermi surface. In the crystalline color superconducting phase, pairs with total momentum 2q condense, meaning that one member of the pair has momentum p + q and the other has momentum —p + q for some p. 1 9 ' 7 2 Suppose for a moment that only u and d quarks pair, making the analyses of a two-flavor model found in Refs. 19,61-68 (and really going back to Ref. 72) valid. We sketch the results of this analysis in this subsection. The simplest "crystalline" phase is one in which only pairs with a single q condense, yielding a condensate (ipu(x)Cj5ipd(x))

oc (5exp(2iq-r)

(1-10)

that is modulated in space like a plane wave. (Here and throughout, we shall denote by r the spatial three-vector corresponding to the Lorentz four-vector x.) Assuming that 5 1, the quarks on each Fermi surface that can pair lie on one ring on each Fermi surface, the rings having opening angle 2 C O S _ 1 ( 1 / T / ) = 67.1°. The energetic calculation that determines 77 can be thought of as balancing the gain in pairing energy as 77 is increased beyond 1, allowing quarks on larger rings to pair, against the kinetic energy cost of Cooper pairs with greater total momentum. If the S/S^L —» 0 Ginzburg-Landau limit is not assumed, the pairing rings change from circular lines on the Fermi surfaces into ribbons of thickness ~ 6 and angular extent ~ S/6^t. After solving a gap equation for S and then evaluating the free energy of the phase with condensate (1.10), one finds that this simplest "crystalline" phase is favored over two-flavor quark matter with either no pairing or BCS pairing only within a narrow window 0.707 62Sc < Sfi < 0.754 62sc ,

(1.11)

where ^ s c is the gap parameter for the two-flavor phase with 2SC (2-flavor, 2-color) BCS pairing found at Sfi = 0. At the upper boundary of this window, 5 —> 0 and one finds a second order phase transition between the crystalline and unpaired phases. At the lower boundary, there is a first order transition between the crystalline and BCS paired phases. The crystalline phase persists in the weak coupling limit only if <W<^2SC is held fixed, within the window (1.11), while the standard weak-coupling limit ^SC/A* —• 0 is taken. Looking ahead to our context, and recalling that in three-flavor quark matter 5/J, = M 2 /(8/i), we see that at high densities one finds the CFL phase (which is the three-flavor quark matter BCS phase) and in some window

M. Alford and K. Rajagopal

17

of lower densities one finds a crystalline phase. In the vicinity of the second order transition, where 5 —> 0 and in particular where S/Sfi —> 0 and, consequently given (1.11), S/52SC —> 0 a Ginzburg-Landau expansion of the free energy order by order in powers of S is controlled. The Ginzburg-Landau analysis can then be applied to more complicated crystal structures in which Cooper pairs with several different q's, all with the same length but pointing in different directions, arise. 65 This analysis indicates that a face-centered cubic structure constructed as the sum of eight plane waves with q's pointing at the corners of a cube is favored, but it does not permit a quantitative evaluation of 6(5^). The Ginzburg-Landau expansion of the free energy has terms that are quartic and sextic in 8 whose coefficients are both large in magnitude and negative. To this order, fl is not bounded from below. This means that the Ginzburg-Landau analysis predicts a strong first order phase transition between the crystalline and unpaired phase, at some 5fi significantly larger than 0.754<52SC, meaning that the crystalline phase occurs over a range of 6fi that is much wider than (1.11), but it precludes the quantitative evaluation of the Sfi at which the transition occurs, of S, or of f2. In three-flavor quark matter, all the crystalline phases analyzed in Ref. 52 have Ginzburg-Landau free energies with positive sextic coefficient, meaning that they can be used to evaluate 5, Q and the location of the transition from unpaired quark matter to the crystalline phase with a postulated crystal structure. For the most favored crystal structures, we shall see that the window in parameter space in which they occur is given by (1.4), which is in no sense narrow. 1.3.2.4. Crystalline color superconductivity in neutral three-flavor quark matter We now turn to three-flavor crystalline color superconductivity. We shall make weak coupling (namely So,Sfi -C fi) and Ginzburg-Landau (namely 5
{ipia{x)Cj5iPjP(x))

oc Y, 7=1

Yl

8je2ia^-TeIa^m

,

(1.12)

q?e{q7}

where a,(3 are color indices, i, j are flavor indices, and where q", q£ and q§ and 5x, 62 and 63 are the wave vectors and gap parameters describing pairing between the (d, s), (u, s) and (u, d) quarks respectively, whose Fermi momenta are split by 26fii, 25fi2 and 25fiz respectively. From (1.8), we see that 5/J,2 = Sfi3 = 5^i\/2 = M^/(8/i). For each I, {q^} is a set of momentum vectors that define the periodic spatial modulation of the crystalline condensate describing pairing between the quarks

18

Color Superconductivity in Dense Quark Matter

whose flavor is not / , and whose color is not I. Our goal in this chapter is to compare condensates with different choices of {qj}'s, that is with different crystal structures. To shorten expressions, we will henceforth write XL° = X^qae{q V ^ n e condensate (1.12) has the color-flavor structure of the CFL condensate (obtained by setting all q's to zero) and is the natural generalization to nontrivial crystal structures of the condensate previously analyzed in Refs. 69,71, in which each {qj} contained only a single vector. In all our results, although not in the derivation of the Ginzburg-Landau approximation itself in Ref. 52, we shall make the further simplifying assumption that <5i = 0. Given that Sfi\ is twice Sfi2 or Sfi3, it seems reasonable that 8% -C 82,83. We leave a quantitative investigation of condensates with <5i ^ 0 to future work. 1.3.2.5. NJL Model and Mean-Field

Approximation

As discussed in Section 1.3.1, we shall work in a NJL model in which the quarks interact via a point-like four-quark interaction, with the quantum numbers of singlegluon exchange, analyzed in mean field theory. By this we mean that the interaction term is interaction = - j j \ $ T A v $ ) { $ ? A S l > ) Av

Av

,

A

(1.13) A

The full expression for T is (T )ai,pj = Y{T )ai3Sij, where the T are the color Gell-Mann matrices. The NJL coupling constant A has dimension -2, meaning that an ultraviolet cutoff A must be introduced as a second parameter in order to fully specify the interaction. Defining A as the restriction that momentum integrals be restricted to a shell around the Fermi surface, fj, — A < |p| < ^ + A, the CFL gap parameter can then be evaluated: 3,65 -*-2

(1.14)

S0 = 2 3 A exp

In the limit in which which S -C SQ,S^I T + -1pT 6 (x)tjj,

(1.15)

where S(x) is related to the diquark condensate by the relations S(x) = \\YA»{WT){TAv)T

.

(1.16)

The ansatz (1.12) can now be made precise: we take S(x) = 8CF{x) ® C7 5 ,

(1.17)

M. Alford and K. Rajagopal

19

with 3

6cF(x)ai,0j

= E

Efc^'^Ja/se/y •

(1-18)

/ = i q?

Upon making the mean field approximation, the full Lagrangian is quadratic and can be written simply upon introducing two component Nambu-Gorkov spinors. Upon so doing, gap equations can easily be derived.52 Without further approximation, however, they are not tractable because without further approximation it is not consistent to choose finite sets {q/}. When several plane waves are present in the condensate, they induce an infinite tower of higher momentum condensates. 65 The reason why the Ginzburg-Landau approximation, to which we now turn, is such a simplification is that it eliminates these higher harmonics. 1.3.3. Ginzburg-Landau

approximation

The form of the Ginzburg-Landau expansion of the free energy can be derived using only general arguments. We shall only consider crystal structures in which all the vectors q" in the crystal structure {q/} are "equivalent". By this we mean that a rigid rotation of the crystal structure can be found which maps any q" to any other q* leaving the set {q/} invariant. (If we did not make this simplifying assumption, we would need to introduce different gap parameters 6j for each vector in {q/}.) As explained in Section 1.3.2.4, the chemical potentials that maintain neutrality in three-flavor crystalline color superconducting quark matter are the same as those in neutral unpaired three-flavor quark matter. Therefore, ^crystalline = ^unpaired + £1(61,62,63)

,

(1-19)

with ^unpaired given in (1.6) with (1.7), and with £1(0,0,0) = 0. Our task is to evaluate the condensation energy £1(61,62,63). Since our Lagrangian is baryon number conserving and contains no weak interactions, it is invariant under a global U(l) symmetry for each flavor. This means that £1 must be invariant under 61 —> el^16i for each J, meaning that each of the three <$/'s can only appear in the combination 6J6J. (Of course, the ground state can and does break these U(l) symmetries spontaneously; what we need in the argument we are making here is only that they are not explicitly broken in the Lagrangian.) We conclude that if we expand £1(61,62,63) in powers of the 6j,s up to sextic order, it must take the form

msi}) = % E

. 1

PICLI Wi

1 / + 3 E 7/(TO3 + E

+z ^ ( E foWi? + E fa Wjtj) \ 1

I>J

1i J J 6*I5I5*J5j6*J8j + 7123 W 2 * W a

J

(1 20)

-

where we have made various notational choices for convenience. The overall prefactor of 2/i2/7r2 is the density of states at the Fermi surface of unpaired quark matter with Ms — 0; it is convenient to define all the coefficients in the Ginzburg-Landau

20

Color Superconductivity

in Dense Quark

Matter

expansion of the free energy relative to this. We have defined Pj = dim{q/}, the number of plane waves in the crystal structure for the condensate describing pairing between quarks whose flavor and color are not I. Writing the prefactor Pj multiplying the quadratic term and writing the factors of | and | multiplying the quartic and sextic terms ensures that the aj, f3j and 7/ coefficients are defined the same way as in Ref. 65. The form of the Ginzburg-Landau expansion (1.20) is model independent, whereas the expressions for the coefficients aj, /3j, Pu, 7/, JIJJ and 7123 for a given ansatz for the crystal structure are model-dependent. In Sections IV and V of Ref. 52 these coefficients are evaluated by deriving the Ginzburg-Landau approximation to this model. We see in Eq. (1.20) that there are some coefficients — namely aj, /3j and 7/ — which multiply polynomials involving only a single 6j. Suppose that we keep a single 51 nonzero, setting the other two to zero. This reduces the problem to one with two-flavor pairing only, and the Ginzburg-Landau coefficients for this problem have been calculated for many different crystal structures in Ref. 65. We can then immediately use these coefficients, called a, (3 and 7 in Ref. 65, to determine our aj, (31 and 7/. Using aj as an example, we conclude that

+

IH

g/+fr*A _ 1 ,

(

5

lsc

C = « ( , , , W = - l t ? l « Bqj-SfiJ F T h 2 l » !g\4( b q^]-Stf) hi

. (1-21)

where 6m is the splitting between the Fermi surfaces of the quarks with the two flavors other than / and qi = |q"| is the length of the q-vectors in the set {q/}. (We shall see momentarily why all have the same length.) In (1.21), <52sc is the gap parameter in the BCS state obtained with 6m = 0 and <5/ nonzero with the other two gap parameters set to zero. Assuming that 60
(1.22)

In the Ginzburg-Landau approximation, in which the 61 are assumed to be small, we must first minimize the quadratic contribution to the free energy, before proceeding to investigate the consequences of the quartic and sextic contributions. Notice that ai depends on the cutoff A and the NJL coupling constant A only through ^2SC, and depends only on the ratios qi/Sfij and Sm/62SC- 07 is negative for Sfij/S2SC < 0.754, and for a given value of this ratio for which aj < 0, OLJ is most negative and (1.21) is minimized for19'65-72 qi =ry 8m with 77 = 1.1997,

(1.23)

where 77 is defined as the solution to

Mogf^Ul.

d-24)

2T? V7?-1, We therefore set qi = rj5m henceforth and upon so doing find that (1.21) becomes

«'"'>--Hasfc)-

(125)

-

Minimizing a/ has fixed the length of all the q-vectors in the set {q/}, thus eliminating the possibility of higher harmonics.

M. Alford and K. Rajagopal

21

It is helpful to imagine the (three) sets {q 7 } as representing the vertices of (three) polyhedra in momentum space. By minimizing a/, we have learned that each polyhedron {q^} can be inscribed in a sphere of radius rjSfij. From the quadratic contribution to the free energy, we do not learn anything about what shape polyhedra are preferable. In fact, the quadratic analysis in isolation would indicate that if aj < 0 (which happens for £/uj < 0.754 ^ s c ) then modes with arbitrarily many different q/'s should condense. It is the quartic and sextic coefficients that describe the interaction among the modes, and hence control what shape polyhedra are in fact preferable. The quartic and sextic coefficients /3j and 7/ can also be taken directly from the two-flavor results of Ref. 65. Note that these coefficients are independent of <52SC and the NJL coupling constant and cutoff, as long as the weak-coupling approximation 82SC <SC /i. is valid. 52,65 Given this, and given (1.23), the only dimensionful quantity on which they can depend is S^ij. They are therefore given by /?/ = /3/<5/u2 and 7/ = ll&\A where fii and 7/ are dimensionless quantities depending only on the directions of the vectors in the set {qj}. J3 and 7 have been evaluated for many crystal structures in Ref. 65, resulting in two qualitative conclusions. Recall that, as reviewed in Section 1.3.2.3, the presence of a condensate with some q° corresponds to pairing on a ring on each Fermi surface with opening angle 67.1°. The first qualitative conclusion is that any crystal structure in which there are two q"'s whose pairing rings intersect has very large, positive, values of both /? and 7, meaning that it is strongly disfavored. The second conclusion is that regular structures, those in which there are many ways of adding four or six q"'s to form closed figures in momentum space, are favored. Consequently, the favored crystal structure in the two-flavor case has 8 q°'s pointing towards the corners of a cube. 65 Choosing the polyhedron in momentum space to be a cube yields a face-centered cubic modulation of the condensate in position space. Because the /?/ and 7/ coefficients in our problem can be taken over directly from the two-flavor analysis, we can expect that it will be unfavorable for any of the three sets {q/} to have more than eight vectors, or to have any vectors closer together than 67.1°. It can be proved that /3/j is always positive, and in all cases investigated to date JUJ also turns out to be positive. 52 This means that we know of no exceptions to the rule that if a particular {q/} is unfavorable as a two-flavor crystal structure, then any three-flavor condensate in which this set of q-vectors describes either the Si, 62 or 63 crystal structure is also disfavored. The coefficients (3JJ and JJJJ cannot be read off from a two-flavor analysis because they multiply terms involving more than one Si and hence describe the interaction between the three different Si's. Before evaluating the expressions for the coefficients,52 we make the further simplifying assumption that Si = 0, because the separation Sfii between the d and s Fermi surfaces is twice as large as that between either and the intervening u Fermi surface. This simplifies (1.20) considerably, eliminating the 7123 term and all the /3JJ and 7/7.7 terms except ^32, 7223 and 7332- We further simplify the problem by focussing on crystal structures for which {q2J and {q3j are "exchange symmetric", meaning that there is a sequence of rigid rotations and reflections which when applied to all the vectors in {q 2 } and {q 3 } together has the effect of exchanging {q^} and {cb}. If we choose an exchange

22

Color Superconductivity

in Dense Quark

Matter

symmetric crystal structure, upon making the approximation that 5^ =
Upon making these simplifying assumptions, the only dimensionful quantity on which /?32 and 7322 can depend is Sfi, meaning that P32 = Pzi/S^2 and 7322 = 7322/^M4-52 Here, $32 and 7322 are dimensionless numbers, depending only on the shape and relative orientation of the polyhedra {q2} and {qs}, which must be evaluated for each crystal structure. Doing so requires evaluating one loop Feynman diagrams with 4 or 6 insertions of Sj's. Each insertion of Si (8j) adds (subtracts) momentum 2q° for some a, meaning that the calculation consists of a bookkeeping task (determining which combinations of 4 or 6 q"'s are allowed) that grows rapidly in complexity with the complexity of the crystal structure and a loop integration that is nontrivial because the momentum in the propagator changes after each insertion. See Ref. 52, where this calculation is carried out explicitly for 11 crystal structures in a mean-field NJL model upon making the weak coupling (So, Sfi
1.3.4. General

results

Upon assuming that Si = 0, assuming that the crystal structure is exchange symmetric, and making the approximation that S/j.2 = 8/13 = 8/J, = M 2 /(8/x), the free energy (1.20) reduces to 1_1_

Cl(S2,S3) =

^ Pa(S(i) (8l + %) + - — (p(S* + Si) + faStSl) 2S~i?

(1.26) + l ^ V Z

S

4

+ $) + 7322 (<^3 +

^D)

where /?, 7, ^32 and 7322 are the dimensionless constants that can be calculated for each crystal structure, 52 and where the J/i-dependence of a is given by Eq. (1.25). It is easy to show that extrema of f2(525 ^3) in (<52,^3)-space must either have 82 = S3 = 5, or have one of 82 and £3 vanishing.52 The latter class of extrema are two-flavor crystalline phases. We are interested in the solutions with 82 = S3 = 8. (Furthermore, the three-flavor crystalline phases with 82 = S3 = 8 are electrically neutral whereas the two-flavor solutions in which only one of the
=

^

2pam52

+

^ _

h a

+

^_%R

(1.27)

where we have defined 7eff = 27 + 27322 •

We have arrived at a familiar-looking sextic order Ginzburg-Landau free energy function, whose coefficients can be evaluated for specific crystal structures. 52

M. Alford and K. Rajagopal

23

If /?eff and 7eff are both positive, the free energy (1.27) describes a second order phase transition between the crystalline color superconducting phase and the normal phase at the <5/x at which a(J/i) changes sign. From (1.25), this critical point occurs where S/J, = 0.754 ^ s c - In plotting our results, we will take the CFL gap to be So = 25 MeV, making J 2 sc = 21/3<50 = 31.5 MeV. Recalling that S/i = M 2 /(8/i), this puts the second order phase transition at M2 —*• M

= 6 . 0 3 S2sc = 7.60 S0 = 190.0 MeV .

(1.29)

a=0

(The authors of Refs. 69, 71 neglected to notice that it is t^sc, rather than the CFL gap So, that occurs in Eqs. (1.21) and (1.25) and therefore controls the S\i at which a = 0. In analyzing the crystalline phase in isolation, this is immaterial since either SQ or &2SC could be taken as the parameter denning the strength of the interaction between quarks. However, we shall compare the free energies of the CFL, gCFL and crystalline phases, and in making this comparison it is important to take into account that #2sc = 21/3<5o-) For values of M2 / JJL that are smaller than (1.29) (that is, lower densities), a < 0 and the free energy is minimized by a nonzero 5 = Smin and thus describes a crystalline color superconducting phase. If /3eff < 0 and %g > 0, then the free energy (1.27) describes a first order phase transition between unpaired and crystalline quark matter occurring at a = a» = 3/3eff/(32P7eff). At this positive value of a, the function fl(S) has a minimum at S = 0 with ft = 0, initially rises quadratically with increasing S, and is then turned back downward by the negative quartic term before being turned back upwards again by the positive sextic term, yielding a second minimum at

S = S»M*± ,

(1.30)

V 47eff also with fi = 0, which describes a crystalline color superconducting phase. For a < a*, the crystalline phase is favored over unpaired quark matter. Eq. (1.25) determines the value of Sfi, and hence M 2 / / J , at which a = a„ and the first order phase transition occurs. If a* 0. However, if /?eff < 0 and 7eff > 0, making the transition first order, we see from (1.30) that at the first order transition itself <5 is large enough to make the quantitative application of the Ginzburg-Landau approximation marginal. This is a familiar result, coming about whenever a Ginzburg-Landau approximation predicts a first order phase transition because at the first order phase transition the quartic and sextic terms are balanced against each other. We shall find that our Ginzburg-Landau analysis predicts a first order phase transition; knowing that it is therefore at the edge of its quantitative reliability, we shall focus on qualitative conclusions.

24

Color Superconductivity

1.3.5. Two plane wave

in Dense Quark

Matter

structure

We begin with the simplest three-flavor "crystal" structure in which {q 2 } and {q 3 } each contain only a single vector, yielding a condensate in which the (us) and (ud) condensates are each plane waves. This simple condensate yields a qualitative lesson which proves extremely helpful in winnowing the space of multiple plane wave crystal structures. 52 For this simple "crystal" structure, all the coefficients in the Ginzburg-Landau free energy can be evaluated analytically.52 The terms that occur in the three-flavor case but not in the two-flavor case, namely ^32 and 7322, describe the interaction between the two condensates and hence depend on the angle between q 2 and q 3 . For any angle , both ,832 and 7322 are positive, both increase monotonically with 4>, and both diverge as cf> —> 7r.52 This tells us that within this two plane wave ansatz, the most favorable orientation is c/> = 0, namely q 2 || q3- Making this choice yields the smallest possible /3eg- and %g within this ansatz, and hence the largest possible 6 and condensation energy, again within this ansatz. The divergence at 4> —» 7r tells us that choosing q 2 and q 3 precisely antiparallel exacts an infinite free energy price in the combined Ginzburg-Landau and weak-coupling limit in which 5 -C 6(i, 60 <§C (i, meaning that in this limit if we chose <j> = -K we find 6 = 0, Away from the Ginzburg-Landau limit, when the pairing rings on the Fermi surfaces widen into bands, choosing <j> = ir exacts a finite price meaning that 6 is nonzero but smaller than that for any other choice of <j>. The high cost of choosing q 2 and q 3 precisely antiparallel can be understood qualitatively as arising from the fact that in this case the ring of states on the •u-quark Fermi surface that "want to" pair with d-quarks coincides precisely with the ring that "wants to" pair with s-quarks. 71 The simple two plane wave ansatz has been analyzed in the same NJL model that we employ upon making the weakcoupling approximation but without making a Ginzburg-Landau approximation. 71 All the qualitative lessons that we have learned from the Ginzburg-Landau approximation remain valid, including the favorability of the choice 0 = 0, but we learn further that in the two plane wave case the Ginzburg-Landau approximation always underestimates 6.71 The analysis of the simple two plane wave "crystal" structure, together with the observation that in more complicated crystal structures with more than one vector in {q 2 } and {q 3 } the Ginzburg-Landau coefficient /?32 (7322) is given in whole (in part) by a sum of many two plane wave contributions, yields an important lesson for how to construct favorable crystal structures for three-flavor crystalline color superconductivity: 52 {q 2 } and {q 3 } should be rotated with respect to each other in a way that best keeps vectors in one set away from the antipodes of vectors in the other set. Crystal structures in which any of the vectors in {q 2 } are close to antiparallel to any of the vectors in {q 3 } are strongly disfavored because if a vector in {q 2 } is antiparallel (or close to antiparallel) to one in {
M. Alford and K. Rajagopal

25

and {q3J should each be chosen to yield crystal structures which, seen as separate two-flavor crystalline phases, are as favorable as possible. In Section 1.3.2.3 we have reviewed how this should be done and the conclusion that the most favored {q 2 } or {q3} in isolation consists of eight vectors pointing at the corners of a cube. 65 Second, the new addition in the three-flavor case is the qualitative principle that {q 2 } and {013} should be rotated with respect to each other in such a way as to best keep vectors in one set away from the antipodes of vectors in the other set. 1.3.6. Multiple

plane

waves

Rajagopal and Sharma have analyzed 11 different crystal structures, 52 in each case calculating /? and 7 and ^32 and 7322, and hence /3efr and 7eff that specify the free energy as in (1.28). The 11 structures allow one to make many pairwise comparisons that test the two qualitative principles described in Section 1.3.5. There are some instances of two structures which differ only in the relative orientation of {q 2 } and {q3} and in these cases the structure in which vectors in {q 2 } get closer to the antipodes of vectors in {q 3 } are always disfavored. And, there are some instances where the smallest angle between a vector in {q 2 } and the antipodes of a vector in {q3J are the same for two different crystal structures, and in these cases the one with the more favorable two-flavor structure is more favorable. These considerations, together with explicit calculations, indicate that two structures, which we denote "2Cube45z" and "CubeX", are particularly favorable.

Fig. 1.5. The momenta that constitute the CubeX (left) and 2Cube45z (right) crystal structures. The set q3 of (ud) pairing momenta is given by the solid-tipped (red online) arrows. The set q2 of {us) pairing momenta is given by the hollow-tipped (blue online) arrows.

The 2Cube45z crystal momenta are illustrated in Fig. 1.5, right panel; {q 2 } and {qs} each contain eight vectors pointing at the corners of a cube. If we orient {q 2 } so that its vectors point in the (±1, ± 1 , ±1) directions, then {q 3 } is rotated relative to {q 2 } by 45° about the z-axis. In this crystal structure, the (ud) and (us)

26

Color Superconductivity

in Dense Quark

Matter

condensates are each face-centered cubic, the structure which in isolation is the most favored two-flavor crystal structure. 65 The relative rotation has been chosen to maximize the separation between any vector in {q2J and the nearest antipodes of a vector in {c^}. The color-flavor and position space dependence of the condensate, defined in (1.17) and (1.18), is given by

8cF{x)ai,Pj

=^2a0£2ij

25

2-K , . 2ix , cos — [x + y + z) + cos — (—x + y + z) 2TT

.

.

2?r .

+ cos — [x — y + z) + cos — {—x — y + z) a a (1.31) + e3a/3€3ij 25 COS

(V%X + z) + COS

( \Ply + Zj

+ cos — ( —y/2y + z) + cos — ( —y/2x + z J

where a and /3 (i and j) are color (flavor) indices and where \/3TT _ 4.536 q Sfi

A*

1.764M?

(1.32)

is the lattice spacing of the face-centered cubic crystal structure. For example, with Mll\± = 100,150,200 MeV the lattice spacing is a = 72,48,36 fm. Eq. (1.31) can equivalently be written as 5CF{x)ai,pj = ^2a^2ijh{^) + e3a/3€3ij53(r), with (1.31) providing the expressions for <52(r) and ^ ( r ) . A three-dimensional contour plot that can be seen as depicting either ^ ( r ) or ^ ( r ) separately can be found in Ref. 65. We have not found an informative way of depicting the entire condensate in a single contour plot. Note also that in (1.31) and below in our description of the CubeX phase, we make an arbitrary choice for the relative position of ^ ( r ) and (^(r). We show in Ref. 52 that one can be translated relative to the other at no cost in free energy. Of course, rotating one relative to the other changes the Ginzburg-Landau coefficients ,832 and 7322 and hence the free energy. We now turn to a description of the CubeX crystal structure. We arrive at this structure by reducing the number of vectors in {q 2 } and {q 3 }. This worsens the two-flavor free energy of each condensate separately, but allows vectors in {q 2 } to be kept farther away from the antipodes of vectors in {<\3}. We do not claim to have analyzed all structures obtainable in this way, but we have found one and only one which has a condensation energy comparable in magnitude to the 2Cube45z structure. The CubeX structure is illustrated in Fig. 1.5, left panel; {q 2 } and {q 3 } each contain four vectors forming a rectangle. The eight vectors together point toward the corners of a cube. The two rectangles intersect to look like an "X" if

27

M. Alford and K. Rajagopal

viewed end-on. In the CubeX phase, the color-flavor and position space dependence of the condensate is given by ScF(x)ai,/3j

=£2ap£2ij

2<J

2?T

27T

(X

0/

cos — (x + y + z) + cos — ( - x — y + z) (1.33) , . 2-K . . t cos — (—x + y + z) + cos — (x — y + z) ITS

We provide a depiction of this condensate in Fig. 1.8

0.2

0.4

z

Fig. 1.6. The CubeX crystal structure of Eq. (1.33). T h e figure extends from 0 to a / 2 in the x, y and z directions. Both <52(r) and <$3(r) vanish at the horizontal plane. S2(r) vanishes on the darker vertical planes, and 53(r) vanishes on the lighter vertical planes. On the upper (lower) dark cylinders and the lower (upper) two small corners of dark cylinders, S2(r) = +3.U (fa(r) = -3,36). On the upper (lower) lighter cylinders and the lower (upper) two small corners of lighter cylinders, S3(r) = - 3 . 3 5 (5 3 (r) = +3.35). Note that the largest value of |<5r(r)| is 4<5, occurring along lines at the centers of the cylinders. The lattice spacing is o when one takes into account the signs of the condensates; if one looks only at |<Sr(r)|, the lattice spacing is a/2, a is given in (1.32). In (1.33) and hence in this figure, we have made a particular choice for the relative position of <53(r) versus ^ M - I n f a c t > oae c a n b e translated relative to the other with no cost in free energy.

We are confident that 2Cube45z is the most favorable structure obtained by rotating one cube relative to another. We are not as confident that CubeX is the best possible structure with fewer than 8+8 vectors. However, the two most favorable structures that we have found, 2Cube45z and CubeX, are impressively robust and make the case that three-flavor crystalline color superconducting phases are the ground state of cold quark matter over a wide range of densities. If even better crystal structures can be found, this will only further strengthen this case.

28

Color Superconductivity

1.3.7. Free energy comparisons

and

in Dense Quark

Matter

conclusions

The gap parameter <5 and free energy 0(5) can be evaluated for all the crystal structures whose Ginzburg-Landau coefficients have been determined. 52 For a given crystal structure, 0,(5) is given by Eq. (1.27), with /?eff and %K taken from Table II of Ref. 52. The quadratic coefficient a is related to J/x by Eq. (1.25). Recall that we have made the approximation that Sfj,2 = S/J.^ = 8pL = M 2 /(8|u), valid up to corrections of order Mf/fi4. At any value of M 2 //x, we can evaluate a(J/x) and hence 0(6), determine 5 by minimizing 0, and finally evaluate the free energy 0 at the minimum. In Fig. 1.7, we give an example of 0,(5) for various M^/fj, for one crystal structure with a first order phase transition (CubeX), illustrating how the first order phase transition is found, and how the 5 solving the gap equations — i.e. minimizing 0 — is found. In Figs. 1.1 and 1.2, we plot 6 and 0 at the minimum versus Mil\x for the two most favorable crystal structures that we have found, namely the CubeX and 2Cube45z described in Section 1.3.6. In Figs. 1.1 and 1.2, we have chosen the interaction strength between quarks such that the CFL gap parameter at M s = 0 is 5o = 25 MeV. However, our results for both the gap parameters and the condensation energy for any of the crystalline phases can easily be scaled to any value of 5Q. Recall that the quartic and sextic coefficients in the Ginzburg-Landau free energy do not depend on Jo- And, recall from Eq. (1.25) that 60 enters a only through the combination o^sc/J/", where ^2SC = 23^o and 5^, = M 2 /(8/x). This means that if we pick a 50 ^ 25 MeV, the curves describing the gap parameters for the crystalline phases in Fig. 1.1 are precisely unchanged if we rescale both the vertical and horizontal axes proportional to Jo/25 MeV. In the case of Fig. 1.2, the vertical axis must be rescaled by (Jo/25 MeV) 2 . Of course, the weak-coupling approximation Jo "C /x, which we have used to simplify the calculation of the Ginzburg-Landau coefficients in Ref. 52

A [MeV]

Fig. 1.7. Free energy Q vs. S for the CubeX crystal structure, described in Section 1.3.6, at four values of M^/fj,. From top curve to bottom curve, as judged from the left half of the figure, the curves are M}/n = 240, 218.61, 190, and 120 MeV, corresponding to a = 0.233, 0.140, 0, -0.460. The first order phase transition occurs at M^/ti = 218.61 MeV. The values of S and Q at the minima of curves like these are what we plot in Figs. 1.1 and 1.2.

M. Alford and K. Rajagopal

29

will break down if we scale Jo to be too large. We cannot evaluate up to what Jo we can scale our results reliably without doing a calculation that goes beyond the weak-coupling limit. However, such calculations have been done for the gCFL phase in Ref. 44, where it turns out that the gaps and condensation energies plotted in Figs. 1.1 and 1.2 scale with So and <5Q to good accuracy for So < 40 MeV with /i = 500 MeV, but the scaling is significantly less accurate for So = 100 MeV. Of course, for So as large as 100 MeV, any quark matter in a compact star is likely to be in the CFL phase. Less symmetrically paired quark matter, which our results suggest is in a crystalline color superconducting phase, will occur in compact stars only if So is smaller, in the range where our results can be expected to scale well. Fig. 1.1 can be used to evaluate the validity of the Ginzburg-Landau approximation. The simplest criterion is to compare the <5's for the crystalline phases to the CFL gap parameter Jo- This is the correct criterion in the vicinity of the 2nd order phase transition point, where S/J, = Mj/(8/x) « Jo as in (1.29). Well to the left, it is more appropriate to compare the J's for the crystalline phase to 5/J, = Mg/(8/j,). By either criterion, the Ginzburg-Landau approximation is at the edge of its domain of validity, a result which we expected based on the general arguments of Section 1.3.4. Therefore, although we expect that the qualitative lessons that we have learned about the favorability of crystalline phases in three-flavor quark matter are valid, and expect that the relative favorability of the 2Cube45z and CubeX structures and the qualitative size of their S and condensation energy are trustworthy, we do not expect quantitative reliability of our results. There is therefore strong motivation to analyze crystalline color superconducting quark matter with these two crystal structures without making a Ginzburg-Landau approximation. It will be very interesting to see whether the Ginzburg-Landau approximation underestimates S and the condensation energy for the crystalline phases with CubeX and 2Cube45z crystal structures, as it does for the much simpler structure in which £2(1") and Ja(r) are each single plane waves.71 Fig. 1.2 makes it clear that the three-flavor crystalline color superconducting phases with the two most favored crystal structures that have been found are robust by any measure. Their condensation energies reach about half that of the CFL phase at Ms = 0. Correspondingly, these two crystal structures are favored over the wide range of M^/fi seen in Fig. 1.2 and given in Eq. (1.4). Taken literally, Fig. 1.2 indicates that within the regime (1.4) of the phase diagram occupied by crystalline color superconducting quark matter, the 2Cube45z phase is favored at lower densities and the CubeX phase is favored at higher densities. Although we do have qualitative arguments why 2Cube45z and CubeX are favored over other phases, we have no qualitative argument why one should be favored over the other. Also, in this context the Ginzburg-Landau approximation is not reliable enough for us to conclude that one phase is favored at higher densities while the other is favored at lower ones. The proper conclusion from these results is that 2Cube45z and CubeX are the two most favorable phases we have found, that both are robust, that the crystalline color superconducting phase of three-flavor quark matter with one crystal structure or the other occupies a wide swath of the QCD phase diagram, and that their free energies are similar enough to each other that a definitive comparison of their free energies will require a calculation that goes beyond the Ginzburg-Landau approximation that we have used here.

30

1.3.8. Implications

Color Superconductivity

and future

in Dense Quark

Matter

work

The purpose of this calculation has been to find candidates for the true ground state of quark matter in the "gCFL window", the range of values of M%//j. where calculations that are limited to isotropic phases predict that the gCFL phase is favored. Fig. 1.2 shows that, according to a Ginzburg-Landau theory keeping terms up to sextic order, in most of this range the gCFL phase can be replaced by a much more favorable three-flavor crystalline color superconducting phase and that these crystalline phases may be favored over a wide range of Mf/fi above the gCFL window also. Fig. 1.2 also indicates that at the lower end of the range of M^/fi, closest to the CFL—>gCFL transition, there is a narrow region where it is hard to find a crystalline color superconducting phase with lower free energy than the gCFL phase. This narrow region is thus the most likely place to find a ground state consisting of the gCFL phase augmented by current-carrying meson condensates described in Refs. 14, 18. Except within this window, the crystalline color superconducting phases with either the CubeX or the 2Cube45z crystal structure provide an attractive resolution to the instability of the gCFL phase. Our Ginzburg-Landau calculation provides reason to speculate that most or possibly all of the quark matter core of a compact star might be in a crystalline color superconducting phase. For example, substituting the quite reasonable values S0 = 25 MeV and Ms = 250 MeV into Eq. (1.4) we find that the crystalline phases are favored over the range 240MeV < fi < 847MeV. Obviously at the lower density end of this range all quark matter phases are replaced by nuclear matter, and the high end of this range extends far beyond the values /i ~ 500 MeV that are expected in the center of compact stars. We therefore find that with these choices for the parameters the crystalline phase covers the entire range of expected densities of stellar quark matter. Of course, if So is larger, say ~ 100 MeV, the entire quark matter core could be in the CFL phase. And, there are reasonable values of So and Ms for which the outer layer of a possible quark matter core would be in a crystalline phase while the inner core would not. We do not know So and Ms well enough to predict what phases of quark matter occur in compact stars. However, from our results it is clear that one of the possible configurations of a compact star, ultimately to be confirmed or refuted by astrophysical observations, would involve a quark matter core that is mostly or entirely in a crystalline color superconducting phase. Now that we have two well-motivated ansatze for the crystal structure of threeflavor crystalline color superconducting quark matter and a good qualitative guide to the scale of S and £1(6), we should be able to make progress toward the calculation of astrophysically observable properties of compact stars. For example, we would like to predict the effects of a crystalline color superconducting quark core on the rate at which neutron stars cool by neutrino emission and on the occurrence and phenomenology of pulsar glitches. For cooling, the relevant microphysical quantities are the specific heat and the neutrino emissivity. The specific heat of the crystalline phases will not be dramatically different from that of unpaired quark matter: it rises linearly with T because of the presence of gapless quark excitations at the boundaries of the regions in momentum space where there are unpaired quarks. 67 (In contrast, in the gCFL

M. Alford and K. Rajagopal

31

case the heat capacity is parametrically enhanced. 6 ) However, the neutrino emissivity should turn out to be significantly suppressed relative to that in unpaired quark matter. The evaluation of the phase space for direct Urea neutrino emission from the CubeX and 2Cube45z phases will be a nontrivial calculation, given that thermally excited gapless quarks occur only on patches of the Fermi surfaces, separated by the (many) pairing rings. (The direct Urea processes u + e —-> s + v and s —> u + e + v require s, u and e to all be within T of a place in momentum space where they are gapless and at the same time to have p« + p e = p s to within T. Here, T ~ keV is very small compared to all the scales relevant to the description of the crystalline phase itself.) The idea that a region of a star made of quark matter in a crystalline phase could be the place where (some) glitches originate goes back to Ref. 19, where it was suggested that a crystalline color superconductor condensate could pin rotational vortices. This leads to the possibility that the presence of crystalline quark matter in compact stars could be strongly constrained by comparing their predicted glitch phenomenology with what is actually observed. There are two key microphysical properties of crystalline quark matter that must be estimated before glitch phenomenology can be addressed: the pinning force on rotational vortices and the shear modulus of the crystal. In order for glitches to occur it is necessary for rotational vortices in some region of the star to be pinned and immobile while the spinning pulsar's angular velocity slows over years. But this can only occur if the pinning force is strong enough to lock the vortices to the crystal structure and if the structure resists deformation, i.e. has a large enough shear modulus. (The glitches themselves are then triggered by the catastrophic unpinning and motion of long-immobile vortices.) To estimate the pinning force we must analyze how the CubeX and 2Cube45z phases respond when rotated. We expect vortices to form and be pinned at the intersections of the nodal planes at which condensates vanish. This analysis faces two complications. Firstly, baryon number current can be carried by gradients in the phase of either the (us) crystalline condensate or the (ud) condensate or both, so we must determine which types of vortices are formed. Secondly, the vortex core size, 1/5, is only a factor of three to four smaller than the lattice spacing a. This means that the vortices cannot be thought of as pinned by an unchanged crystal; the vortices themselves will qualitatively deform the crystalline condensate in their vicinity. The second microphysical quantity that is required is the shear modulus of the crystal. This can be related to the coefficients in the low energy effective theory that describes the phonon modes of the crystal. 62 ' 78,79 This effective theory has been analyzed, and its coefficients calculated, for the two-flavor crystalline color superconductor with face-centered cubic symmetry. 79 Analyzing the phonons in the three-flavor crystalline color superconducting phases with the 2Cube45z and CubeX crystal structures is thus a priority for future work. The instability of the gCFL phase was a significant bend in the road to understanding the properties of dense matter, but one that is now receding behind us. With the emergence of two robust crystalline phases, possibly accounting for as much as all of the quark matter that may occur, we can glimpse on the road

32

Color Superconductivity

in Dense Quark

Matter

ahead the microphysical calculations prerequisite to identifying observations that can be used to rule out (or in) the presence of quark matter in a crystalline color superconducting phase within neutron stars.

1.4.

Coda

The project of delineating a plausible phase diagram for high-density quark matter is still not complete. We have discussed some ideas for the "non-CFL" region, but there are others such as a suggested gluon condensation in two-flavor quark matter, 13 and deformation of the Fermi surfaces (discussed so far only in non-betaequilibrated nuclear matter 8 0 ). It is very interesting to note that the problem of how a system with pairing responds to a stress that separates the chemical potentials of the pairing species is a very generic one, arising in condensed matter systems and cold atom systems as well as in quark matter. Recent work by Son and Stephanov 81 and Mannarelli, Nardulli and Ruggieri 82 on a two-species model characterized by a diluteness parameter and a splitting potential shows that between the BCS-paired region and the unpaired region in the phase diagram one should expect a translationally-broken region. In QCD this could correspond to a p-wave meson condensate or a crystalline color superconducting state. What is particularly exciting is that the technology of cold atom traps has advanced to the point where fermion superfluidity can now be seen in conditions where many of the important parameters can be manipulated, and it may soon be possible to investigate the response of the pairing to external stress under controlled experimental conditions. Pairing in a gas of ultracold fermionic atoms can never yield a complete analogue of pairing in dense quark matter, because the atoms are neutral and thus the requirement of neutrality imposes no constraints, whereas we have seen that in the quark matter context imposing both color and electric neutrality is a crucial qualitative driver for the physics, for example precluding phase separation which can easily happen in the cold atom system. However, even without yielding a complete analogue these experiments could answer qualitative questions of interest. For example, should they discover a translationally broken regime in the absence of rotation, it will be very interesting first to see what crystal structure emerges and second to study the response of such a phase to rotation.

Acknowledgments KR acknowledges the hospitality of the Nuclear Theory Group at LBNL. This research was supported in part by the Office of Nuclear Physics of the Office of Science of the U.S. Department of Energy under contracts #DE-AC02-05CH11231, #DEFG02-91ER50628, #DE-FG01-04ER0225 (O JI) and cooperative research agreement #DF-FC02-94ER40818.

M. Alford and K. Rajagopal

33

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Chapter 2 Larkin-Ovchinnikov-Fulde-Ferrell Phases in QCD

Giuseppe Nardulli Department

of Physics, University of Bari and INFN-Bari, Via E. Orabona 4, 70126, Bari, Italy E-mail: [email protected]

The Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) inhomogeneous color superconducting phase of QCD is reviewed. The case of two flavors is discussed in the hypothesis that the gap is formed by one or more plane waves. The case of three flavors is discussed in the Ginzburg-Landau limit. Contents 2.1. 2.2. 2.3. 2.4.

Introduction High density effective theory Two-species fermions with unpaired Fermi surfaces LOFF phase in QCD with two flavors: one plane wave 2.4.1. Free energy in the LOFF phase 2.4.2. Gap equation 2.5. LOFF phase of QCD with two flavors and more plane waves 2.5.1. Ginzburg Landau expansion 2.5.2. More plane waves beyond the GL expansion 2.6. LOFF phase of QCD with three flavors in the GL approximation 2.6.1. Ginzburg-Landau expansion 2.6.2. Free energy 2.6.3. Results Bibliography

2.1.

37 38 42 45 45 46 48 48 53 58 58 59 60 61

Introduction

At high baryonic density and small temperatures Quantum-Chromo-Dynamics (QCD) and the Cooper's theorem predict color superconductivity since the color antisymmetric channel between quarks is attractive, see Refs. 1-4 and the reviews. 5 - 8 For very large values of the baryonic chemical potential the energetically favored phase is the Color-Flavor-Locking (CFL) phase, 4 characterized by a homogenous spin 0 diquark condensate, antisymmetric in b o t h color and flavor. At intermediate densities the situation is more involved because the quark Fermi surfaces are unpaired, due to the strange quark mass ( M s ^ 0) and the differences 6fj, in the quark chemical potentials, induced by f3 equilibrium. Several ground states have been 37

38

G. Nardulli

considered in the literature, e.g. the 2SC phase, 2 obtained in the limit Ms —> oo, the Deformed Fermi Surface phase, 9 also with two flavors, and the gapless phases g2SC 10 and gCFL. 11 ' 12 The gapless phases are instable because the gluon Meissner masses turn out to be imaginary (for g2SC see, 13 for gCFL see 14 and also 15 ' 16 ). This paper is a review of a different color superconducting phase, the LarkinOvchinnikov-Fulde-Ferrell (LOFF) phase (for the original papers see, 17 ' 18 for applications to QCD see Refs. 19-25 and a recent review 26 ). The relevance of this phase is based on the circumstance that, for appropriate values of Sfi, pairing with nonvanishing total momentum: p i +P2 = 2q ^ 0 is favored. The phase is characterized by a gap parameter not uniform in space, which, in the simplest case, has the form of a single plane wave: A(r) = Aexp(2iq • r). As for the instability, in Ref. 27 it is shown that, with two flavors, the instability of 2SC implies that the LOFF phase is energetically favored. As to the stability of the LOFF phase in QCD, there is some debate in the literature for the case with two flavors. 28,29 A recent analysis for the three flavor case shows that, at least in the Ginzburg-Landau limit, the gluon Meissner masses are real. 30 The review is organized as follows. After a presentation, in Section 2.2, of the approximation scheme we employ, we discuss the general framework needed to discuss the case of superconductivity when the two fermionic species are unpaired in Section 2.3. Section 2.4 is devoted to the discussion of the LOFF phase of two-flavor QCD in the hypothesis that the space modulation of the gap parameter is given by a single plane wave, while the case of more plane waves is discussed in Section 2.5. While the case of two species is the snly physically interesting case in conventional superconductivity, in QCD at intermediate densities all the three quarks: u, d and s should be considered and therefore the three flavor problem should be studied. Some analyses of this problem have already appeared in the literature. 25 They are reviewed in the conclusive Section 2.6.

2.2. High density effective theory The Lagrangian for gluons and ungapped quarks with Mu — Md = 0 and Ms ^ 0 can be written as follows (color, flavor and spin indices suppressed): £ = $ (ijp-M

+ n-ro) ip

(2.1)

where M is the quark mass matrix in flavor space and /U is the matrix of chemical potentials. We will consider two cases. (1) Three flavors: In this case M potentials is given by 11 tff

=

diag(0,0, MS) and the matrix of chemical

= (VbSn - HQQV) 6a? + % ^3Tf

+ ^ T f ^ j

(2.2)

with flavor indices i,j = 1,2,3. T 3 = |diag(l, - 1 , 0 ) , T 8 = ^ d i a g ( l , 1, - 2 ) in color space and Q — diag(2/3, —1/3, —1/3) in flavor space; fiQ is the electrostatic chemical potential; fJ.s,fis a r e the color chemical potentials associated

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

39

respectively to the color charges T3 and Tg; fib is quark chemical potential which we fix to 500 MeV. (2) Two flavors: In this approximation one takes Ms —» 00, and a vanishing M: M = diag(0,0); for this case we will consider tff = (filc+ tyl3)« SaP (2-3) (i,j = 1,2 flavor indices; a, (3 = 1,3 color indices; Io = unit matrix, I3 = diag(l,-l)). In the initial part of this Section, which is devoted to an approximation scheme of QCD at high density, we shall discuss the case of three flavors. The approximation we discuss is the High Density Effective Theory (HDET), see Refs. 31-33 and the reviews 8, 34. It is an approximation based on the observation that, if fib 3> A, Sfi, the relevant modes are those near the Fermi surface. Let us therefore consider the fermion 4-momentum pM = (p°, p). Since we work near the Fermi surface, we decompose the quark momentum into a large component, proportional to Hb, and a residual small component as follows: p11 = /i(,nM + £, where n^ — (0, n). The unit vector n represents the Fermi velocity. The space part of £ has a longitudinal part £ along n, and a transverse component £± perpendicular to it, but using reparametrization invariance we can choose £±_ = 0. The time component £Q coincides with the quark energy: £0 = p°. In conclusion the residual momentum £ is given by

l=(p°,tn).

(2.4)

We introduce velocity-dependent fields ipn and ^ n by the following decomposition f dn i>(x) = J ^ e * " > - * (V>n(z) + *„(*)) ; (2-5)

J 4^

ipn and ^ n correspond to positive and negative energy solutions of the Dirac equation. Substituting the expression (2.5) in the Eq. (2.1) one gets for free quarks, at the leading order in l//i 8 in momentum space

£ = <<«(') (V • £ S^Sn + tff) 4>n,Pj (£),

(2.6)

where now the velocity-dependent fermion fields are left-handed Weyl spinors (another term with right handed spinors must be added). Moreover V11 = ( l , n ) and we shall also use the notation V^ = (1, —n) when we introduce Nambu-Gorkov fields, see below, so that: V-£=p0

V.£ = p°-£,

+ £.

(2.7)

As to the chemical potential, we have fhf = \hf - M

a /

% - ^ ° % * 3

,

(2-8)

2 fib

because we treat the strange quark mass at the leading order in the 1/fi expansion, M2 which corresponds to a shift in the chemical potential of the s quark: fis —* fis • 2 fib

40

G. Nardulli

We note that in HDET the space integrals must be treated as follows +s +

f Jh_ rf [ dn f

J (2TT)4

d£ [ °° df_

n J 4n J_6 2TT J _

X

{

2TT '

''

where 6
M?/ = M i

(2.10)

where Q is the quark electric-charge matrix. From this equation: 2 1 1 M2 M« = Mb - -Me , Md = Mb + oMe . Ms = Mb + jj "e - ^ '

i2'11)

and (2.6) becomes

£ = <*«(') (Y-e + &) J ° % Vn,«(*),

(2.12)

with /Sj = Mi - Mbit is clear that, at this order of approximation, the effect of Ms ^ 0 is to reduce the chemical potential of the strange quarks. Let us now define a new basis ipA f° r the spinor fields: 9

i>ai = Y, ( ^ ) a i V'A ,

(2-13)

A=l

where the matrices FA can be expressed by

F4t6 = T1±iT2,

F6,r = T4±iT5,

F8,9 = T6±iT7,

(2.15)

with Ta = Xa/2 the SU(3) generators and I0 the identity matrix. In this basis, ipA =

(ur,dg,Sb,dr,Ug,sr,Ub,sg,db).

Introducing the Nambu-Gorkov fields

i2M>

x*=TActX' the kinetic part of the action is

^

[dn [djPdt

I=

22JtoJl&y

where (p,)A =

* ({p°-£

XA

{

+ jiA)8AB

°

{pu,P-d,fts,P'd,pu,fia,Pu,Ps,fid)-

0

\^C217)

(P° + C-HA)6AB)

XB

(2 17)

-

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

41

Let us now consider a four fermion coupling a la Nambu-Jona Lasinio: CNJL

OC ^ y V ^ y V - .

(2-18)

Color superconductivity arises for ia(x)Cl5iP0j(x)>

? 0.

(2.19)

In the mean-field approximation one takes into account this effect by adding to the Lagrangian of Eq. (2.1) the gap term .a/3

£A = ~= Ki

1>ii,-a{*)Cll>iP,n(x)

(2.20)

+ h-C-

In the homogeneous case, such as CFL or gCFL, A"- is independent of x. In the LOFF phase with three flavors, to be treated in Section 2.6, we will take 25 A?f(x)

^AIe2i*-*eaeieijI.

=

(2.21)

i=i

Adding this term to the action gives us the result ^

fdn

fdp°d^

t

((p°-£

+ iiA)SAB + £ - p.A)

SAB

XB (2.22)

where AAB is matrix defined by AAB

= - ^

A/ e2u»-' TV [F% e7 FB c/]

(2.23)

7=1

For the case of three flavors AAB is the following 9 x 9 matrix

AAB

=

( 0 A3 A3 0 A 2 Ai 0 0 0 0 0 0 0 0 0 0 ^ 0 0

A2 Ai

0 0 0 0 0 0 0

0 0 0 0 0 0 \ 0 0 0 0 0 0 0 0 0 0 0 0 0 -A3 0 0 0 0 0 0 0 0 -A3 0 0 0 0 0 -A2 0 0 0 -A2 0 0 0 0 0 0 0 0 -Ax 0 0 0 0 -Ax 0 )

(2.24)

where we have included the r dependence in the definition of Aj. From these equations one immediately obtains the inverse fermion propagator for gapped quarks, that in momentum space is given by fnG

i\ ( A )AB

i'(V ~ (

• e + HA)SAB _A*AB (V-e-

_f(S1o1r1 AB

-AAB PA)6AB

-A*AB A„ (S0

MB

)AB (2.25)

42

G. Nardulli

Thus far we have treated the three-flavor case. We conclude this Section by writing down the analogous formulae for two flavors. In the LOFF phase with two flavors we assume P Af/(r) = A J2 e 2 i q k ' r eQ/33 eij3 , (2.26) fc=i and discuss the case of P = 1 (single plane wave) in Sections 2.4 and P > 1 in Section 2.5. From the matrix in (2.24) we can extract the submatrix relevant to the case of two flavors; in this case the matrix AAB is the 4 x 4 matrix:

AAB

=

/0 A 0 \ 0

A 0 0 \ 0 0 0 0 0 -A 0 -A 0 J

(2.27)

in the basis ipA = (ur,dg,dr,ug). This is the matrix AAB that should be included in (2.25). Therefore Eq. (2.25) holds for both 2 and 3 flavors. 2.3.

Two-species fermions with unpaired Fermi surfaces

Let us begin by discussing the case of quarks with two flavors, e.g. up and down, with different Fermi surfaces: Sfi = liu-^d ^ Q _ (2>28) Since we work in the high /i = \i\, limit,

M=^4^'

(2 29

- >

we can forget the contribution of the antiquarks, as noticed above. In this section we consider homogeneous superconductivity and, in full generality, the finite temperature case. To derive the free energy we consider Tr(lnG^o 2x2 2

InGo1) =

2

-^

/'dpdi/dx(x\p)
_ 2x2 f 2 J

4

dp (2TT)4

n

f d z ( l n d e t S ^ - IndetSo ' ) = - lndetSoV) =

detffA» det50-1(p) '

(2.30)

The factor 2 x 2 = 4 takes into account the degeneracy in spin and color; the factor 1/2 avoids the doubling of the degrees of freedom implicit in the Nambu-Gorkov procedure. While G _ 1 are matrices of rank 8, the inverse propagator S^ is a 4 x 4 matrix as follows:

sa-osc-TO-')' and [ G j ] " 1 = po - £ + 5na3 + iesignpo ,

(2 3i

-»

Larkin-Ovchinnikov-Fulde-Ferrell 1

[G 0 ]

43

Phases in QCD

= -p0 - f - (5/i(T3 - i e signpo ,

(2-32)

+

where we have made the ie prescription explicit ( e = 0 ); £ is defined by £=|P|-M

(2-33)

according to Eqs. (2.4) and (2.7). The ratio of the determinants gives the result (Ao real): det S£ (p) _ ((PQ ~ *M)2 ~ * g ) ((PQ + ^ ) 1

det So (p) ~

( ( p o _ fy)2 _ £2) ^

+

2

~ *g)

j M )2 _ £2)

(2.34)

where £ p = \je

+ Ao 2 .

(2.35)

The change of sign ±
n A (fr)_Ag

^

r d*P

<wn-i8rf + E% t n0

where g is the NJL coupling constant of dimensions mass V=

+

4 T

/(2^

2

, ujn — nT(2n + 1) and

V*fF(Sl*-t) + InM-S^-O]

(2.38)

is the free energy of the normal phase, with JF{X) being the Fermi distribution

In writing this we have extracted an infinite irrelevant constant. To get the result at T = 0 one integrates (2.30) as follows

JdpolnX(p0) = -JdpoP-?^

(2.40)

and uses the residue theorem. Therefore at T = 0 the free energy is

^

= f -|/^(l^ + ^ | + | - ^ + ^ | - 2 0 ,

and QA(S(I)

V

_ £IQ{6H)

~~

V

A| +

g

(2-41)

44

G. Nardulli

d3v ^ 3 (\EP + 8p\ + \EP - <5M| - K + M - K - M ) • (2.42)

~\l

Let us notice explicitly that we use the notation Ao = Ao(
In ^o(0)

= 0[Sfi

_ Ao(0)]arcsinh VMER ,

Ao

(2.43)

Ao

i.e. In

Ao(0)

Sfx + y/Sf

=0.

(2.44)

- A§

There are no solutions for 5/J, > Ao(0). For Sfi < Ao(0) one has two solutions. a) b)

Ao = A o (0),

(2.45)

[A 0 1 2 = 2 tyAo(0) - fAo(0)12 .

(2.46)

The first arises since for Ao = A o (0) the l.h.s. of the Eq. (2.43) is zero. But since we may have solutions only for 6/j, < Ao(0) the ^-function in Eq. (2.43) makes zero also the r.h.s.. This means that in the superconducting phase, for this solution, the gap Ao(<5/i) is actually independent of 5(i. The second solution is obtained by solving (2.44). One can note that the value of the gap parameter for the solution b is smaller than the value for the solution a. To find the ground state we must compute the free energy. First we compute the free energy for Ao = 0 from Eq. (2.41): "o(frO — — = ~y

fio(0) 2~P'

fy2

,9,7, (2 47) '

with p = Afj,2/n2 . As for Q& o n e

nas

(2.48)

> from (2.42), Q^M)

=

fioM _ £ ( _

^

2

+ Al)

(249)

In the case a this gives

n^0

=

noM_j [ _ 2 ^ a + Ag(o)]-.

(2.50)

In the other case the gap parameter to be put in (2.49) is smaller; therefore the solution b, having higher free energy, is certainly unstable. Let us concentrate on the solution a. For 5fi < Ao/\/2, Q A < ^o and therefore the BCS superconducting state is stable. 35 ' 36 At the the Clogston transition point ty = J W = ^

,

(2.51)

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

45

a transition from the superconducting (Ao ^ 0) to the normal phase (Ao = 0) occurs. This transition is first order, because for Sfi < Sfii the gap does not depend on Sfi and one passes abruptly to the normal phase. However, as we shall discuss below, the real ground state for 5/J, > 5/J.I turns out to be an anisotropic one, where the assumption of a uniform gap is not justified. This is due to the fact that the energetically favored configuration comprises Cooper pairs with a non vanishing total momentum.

2.4. LOFF phase in QCD with two flavors: one plane wave 2.4.1. Free energy in the LOFF

phase

For 6/J, larger than the Clogston limit S/j,i the interaction favors the formation of pairs with non zero total momentum. In general one expects that p

A(r)= ^ A

m

e

2 i

^-r.

(2.52)

m=l

The phase with A(r) given by (2.52) is called inhomogeneous (anisotropic, crystalline) or LOFF superconducting phase. To start with we shall assume the existence of a single q and therefore A(r) = A e 2 i q ' r . (2.53) This is the simplest hypothesis, the one considered in 18 for ordinary superconductivity and in Refs. 19-21 for QCD. The more general case represented by Eq. (2.52) is examined in ordinary superconductivity in the paper; 17 extensions to QCD are e.g. in Refs. 22,24 and in the review of Ref. 26. We will come to it below. To get the free energy in the inhomogeneous LOFF case, Q (LOFF> t we note that if the momentum of the Cooper pair is 2q, then the two fermions have momenta p + q and — p + q. Therefore we make the following substitution in the condensation energy £^5/j.= with

\p\ - A* =F <*/•* -> |±P + q| - A* T <*M ~ f ± q • v ^ fy .

(2-54)

v = n = r^- . (2.55) IPI A formal way to prove it is by a redefinition of the fields, ip —> exp{—iq • r}ip. In any case the effect is the shift Sfj, —> S/x — q • v . (2.56) The generalization of (2.37) to the LOFF case is straightforward. By the definition E'p = y/e+A? we get, at T ^ 0, instead of (2.37),

,

(2.57) 2

n{LOFF)

77 V

A2

= — 9

+^ T y> -- 4AT „£-J

, ds Ln / ,n .„ ln [4^^(2TT)3

/

- i(Sn - q • v ) ) +E'P'2 pfio —5 + ~-i?— —2—+ ...

,N2

V

(2.58)

46

G. Nardulli

Therefore at T = 0 the free energy is QiLOFF)

=

V

A

2

4 2

T-25/(^ ( M p ' q )

20

(2.59)

where |ci(p,q)| = | T ^ ± q - v + ^

±q-v + J ( e - ^ ^ l + A 2

(2.60)

are the dispersion laws of the quasi particles. 2.4.2. Gap

equation

Let us now derive the gap equation and at the same time derive the value of |q| (the direction is chosen spontaneously). We have to impose minimization with respect to the parameters A and q. Therefore the following conditions must be satisfied (2.61) The LOFF phase will be favored if its free energy is smaller than the energies of the normal state and the homogeneous superconducting phase. The gap equation at T ^ 0 is 4A 1 /F[ ^ + ,5M"q'V]_/F[^"J/X + q"Vl) 2- / g , (v^-F , (~ „" which at T = 0 becomes A A

=

1

<5/i + q

V]

+

q

V ]

(2

62)

(2

63)

- -^^ ^- - )' ff ^/(2^F( - , v - v ~ P-^^We note that, due to the presence of the step functions 6, the momentum space available for pairing in the LOFF phase is smaller in comparison with the homogeneous case: There are regions where pairing is possible and blocking regions where it is forbidden. As for the equation that fixes q we get at T ^ 0 0=

d3p

WV'q

2

A / tanh

x+ Z IT

, x-£ ' tanh ——= IT

. x E±_ tanh IT

tanh ^ 5 } IT

(2.64)

where q = q / g and x = 5fi-q-v.

(2.65)

To obtain q we can limit the analysis to the region A —> 0, i. e. the weak coupling limit, near the second order phase transition. One obtains

= dzz

o

L

I

<# $

sech —— 2T

sech -

2T

(2.66)

Larkin-Ovchinnikov-Fulde-Ferrell Phases in QCD

47

We can study this equation for T —> 0. Upon using Um^sech2^

= 6(a),

(2.67)

we obtain the equation z dz + x-xz

/:'i

(2.68)

with 12 = T " •

(2-69)

S/J,

The parameter x^ is therefore a solution of the equation 1,22 + 1 & £2 = x 2 log z 2 - 1 ,'

, 0 vnx (2.70)

x2 = — = 1.1997,

(2.71)

i. e. Zfl

which although obtained near the second order phase transition is valid almost everywhere because the range of values of Sfi where the LOFF phase with one plane wave prevails is very narrow: 5/i = (0.71,0.754) Ao, see below. In the gap equation (2.63) we can get rid of the coupling g by considering the gap parameter AQ of the homogeneous phase. As a matter of fact: 2

ln

X = f/(2^F^[-^-^ + q - v ] -^-^ + ^ - q - v l )

cm

= ££[ ^ c s i n h r p , 2 JBR 4TT A

(2.72) V

where C(9) = yjq2(zq-

cos 6»)2 - A

2

.

(2.73)

The angular integration is over the blocking region (BR) denned by q2(zq-cos6)2 > A2.

(2.74)

The complementary phase space, where the pairing is possible (pairing region) is formed by two annular regions. If tpo = 2arcos(zg) is the aperture of a cone with vertex at the origin of the Fermi spheres and axis along the direction of q, then they are roughly obtained by intersecting the pairing regions with the Fermi surfaces. Performing the integral in costf, the result can be expressed as follows: lnAo =

11

A

A

q + S^i\

fq-dfi

(2.75)

2q L" V A

where the function h(x) is defined as follows: h(x) = zarcsinhv x2 — 1 h(x) = 0

y/x2 — 1

(|a;| > 1), (N<1).

(2-76)

48

G. Nardulli

and coincides with the function G in Eq. (9) of Ref. 18 and (2.117) of Ref. 26. The reduction of the available phase space implies a reduction of the gap in comparison with the homogeneous case. Increasing Sfi increases the effect of the blocking terms; eventually a phase transition to the normal phase occurs when Sfi approaches a maximum value Sfi2. Therefore the anisotropic superconducting phase can exist in a window 6m <Sfi< Sfj.2 • (2-77) 35 36 One expects that 5fi\ is near the Clogston-Chandrasekhar ' limit Ao/\/2 because near this point the difference in energy between the isotropic superconducting and the normal phases is small. Let us determine 5fi2- For Sfi —> S/i2 the gap A —> 0. Expanding the function h(x) for i - t o o w e obtain from Eq. (2.75) , A0 „ 1 Su, q + Su 1, A2 ,„ „„, ln 1+ 1 ln (2 ?8)

X--

2f

^

- 2 i(^V)

-

which can be re-written as follows. If we define ISJI

g + S/i

1

A2,

« ( « , w = - i2+q^Uq-6fi h ^ : ->2A n 4 ( g 2-- °Sfi ^2)»

(2-79)

then Eq. (2.71) and a(1.1997<5/x2, ^ 2 ) = 0

(2.80)

determine Sfi2'6fi2 = 0.754 A 0 .

(2.81)

Since S/i2 > ^Mi ~ 0.71Ao, there exists a window of values of Sfi where LOFF pairing is possible. We will prove below, using the Ginzburg-Landau approach, that the phase transition for the one-plane wave condensate at T = 0 and Sfi = 5fi2 is indeed second-order.

2.5. LOFF phase of QCD with two flavors and more plane waves 2.5.1. Ginzburg Landau

expansion

The Ginzburg Landau (GL) expansion is valid near the second order phase transition and is obtained by performing the limit A —> 0 in the free energy and in the gap equation. The gap equation in this limit has the form 9. T r 3

-

/S(/driGo"(r'ri)A%(ri)G0(ri'r)

/'nrfr j Go(r,r 1 )A*(r 1 )G+(r 1 ,r 2 )A(r 2 )Go(r2,r3)A*(r3)C?+(r3,r) J

j=i

r

5

+ / I p r J G o (r,r 1 )A*(r 1 )G+(r 1 ,r 2 )A(r 2 )G 0 -(r 2 ,r 3 )A*(r 3 ) G+(r 3 ,r 4 ) J

j 3=1

x A(r 4 )G 0 -(r4,r 5 )A*(r 5 )G+(r5,r)) .

(2.82)

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

49

Substituting (2.52) we get A;=(^n(qk,qn)A^(qk-qn) fc J

+ ^2

( q k ' q * ' q m ' < ln) A fc A ^ A m <5 ( ( lk - q^ + q m - q n )

k,e,m

+

Yl

^(qk,q«,qmqi,qi,q„)A^A^A^A,A* (2.83)

x
(2.84)

«J(qi,q2,q3,q4) =

WP

/s/>/_:-n/.(-,f,»,

(2.85)

i=l

x

m f d™ f+s » f+°° dE

^ ( q i , q 2 , q 3 , q 4 , q 5 , q 6 ) = ~Y \ -^

\

d

i I

oo

2

-

n/i(^.^»{q}). (2-86) i = 1

with fi(E,6li,{q})

E + iesignE - 5fi + (-1)*[£ - 2 E U ( - l ) l w • q k ] moreover the condition

(2.87)

M

£(-l) f c cfc = 0

(2.88)

fc=i

holds, with M = 2,4,6 respectively for II, J and K. For 11(g) = II(q, q) one obtains

n(«z)

igp f dw f+s „ f+°° dE

igp rdw r

r

2TT ( £ + i e s i g n £ - / 2 ) 2 - £ 2 '

(2.89)

where p, = Sp, — q • w . Using the gap equation for the homogeneous pairing to get rid of the cutoff 8 we obtain the result A q + 6fi ty, = 1 - y a(q,8p) , n(g) = i + f ',1 + 1,l o g o l o g q-Sfi (2.90) with a(q,5p) introduced in (2.79). Clearly the gap equation in the GL limit, 1 = 11(g), coincides with Eq. (2.79), which was obtained within the one plane wave hypothesis. The reason is the following: since II depends only on |q|, it assumes the same value for all the crystalline configurations; therefore II does not depend on the crystalline structure of the condensate and the transition point 5p2, we have determined above, is universal. For the evaluation of J and K we have to specialize to the different LOFF condensate choices. This will be discussed below.

2

4^-vr^

G. Nardulli

50

2.5.1.1. Grand potential The difference of free energies per unit volume, fi = ( ^ A — Slo)/V is given in the GL approximation by p P

1i /

-9 ( J2 [n(qk,q„)-l]A^ATl<5q fc,n=l P

+ 9 2

J (

( lk. < l^ { lm,qn)AfcAfA^A n <5 q k _ q < + q m _ q n

^ k,l,m,n—l P

+ Q 3

x

H

ir(q k ,q^q m ,q j ,q i ,q n )A^AfA^A ; ,A*A n

k,Z,m,j,i,n=\

"qk-q«+qm-qj+qi-qn ) •

(2-91)

Let us assume that Afc = Afc = A

(for any k),

(2.92)

P ^ A 2 + ?A4 + ^A6, 2 4 6

(2.93)

so that we can rewrite (2.91) as follows: - = p where ^

= _

P

2 ™ 9P

•7(qk,q
5Z

(2.94)

k,l,m,n=l

and 2 7=

XI

K

( t lk,q«,qm,qj,qi,qn)^q k -q < +q m -qj+q 1 -q„-

(2-95)

k,£,m,j,i,n=l

We distinguish different cases: (1) (3 > 0, 7 > 0. In this case A = 0 is a maximum for O, and the minima occur at the points , A2 =

-/3+V/32-4Po^ „, v ^ L .

(2.96)

Near the transition point one has A

2

« - ^ .

(2.97)

A phase transition occurs when a = 0, i.e. at 8p, = J/i2- The transition is second order since the gap goes continuously to zero at the transition point.

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

51

(2) (3 < 0, 7 > 0. Both for a < 0 and for a > 0 A 2 in (2.96) is a minimum for fi. In the former case it is the only minimum, as A = 0 is a maximum; in the latter case it competes with the solution A = 0. Therefore the LOFF phase can persist beyond Sfi2, the limit for the single plane wave LOFF condensate, up to a maximal value J/x*. At 6/J, = S/j,* the free energy vanishes and there are degenerate minima at A2 = ^ . 47

A =0 ,

(2.98)

The critical point 6fi* is obtained by 3/32

a(q = 1.1997<5M*, «5/x*) = ^

- .

(2.99)

The phase transition from the inhomogeneous to the normal phase at 8fx* is first order. (3) P < 0, 7 < 0: In this case the GL expansion (2.93) is inadequate since Q is not bounded from below and another term (9(A 8 ) is needed. 2.5.1.2. Crystalline structures: One plane wave We now discuss in the context of the GL expansion the case we discussed already, i. e. one plane wave. First we rewrite Eq. (2.79) in the form «(*, 6,) = - l n ^ + I / 0 ( £ ) ,

(2.100)

where M-l

f0(x)=

duln(l + x u ) ,

(2.101)

so that

Then we expand a(q, 5/j.) around the transition point (q, Sfx) = (1.1997Sfi^, S^)Using (2.100) and noticing that the partial derivative of a in q vanishes due to Eq. (2.68), we get a=—^-,

(2.103)

OfJ.2

where 7] = 6fi2 - Su .

(2.104)

We observe that, for S/j, < 6/J.2, a is negative; therefore the transition at T = 0 is second order, if fi > 0, both for P — 1 and for P > 1. As to the other terms one gets for the single plane wave condensate: J

-

J

o

=

" T ^ ^ '

K

-Ko=-64[qi-5^'

(2 105)

-

G. Nardulli

52 so that, with x2 given in Eq. (2.71), M

1 Uix\{x\-V)

0.569 ^ 5fi '

2, + xl 1.637 -y = n x..4^2 ^ 3 = + -TTi- •

,„,„^ (2-106)

Since (3 > 0 the 7-term is ineffective near the transition point and Eq. (2.97) gives A 2 = Arj (xl - 1) 6H2 « 1.75777^2 •

(2.107)

We can obtain Q, from Eq. (2.93) with P = 1 and from Eqs. (2.103) and (2.106). The result is 2

fi = - ^

= -0.439p (fy - J// 2 ) 2 •

(2.108)

For generic crystals the evaluation of J and K is more complicated. We refer the reader to the original paper of Ref. 22 where an extensive analysis was performed. We limit ourselves to the case of two antipodal plane waves. 2.5.1.3. Crystalline structures: Two antipodal plane waves In this case P = 2; let the two vectors be q a , qb with q a = — q b = q and A(r) = 2 A c o s q r .

(2.109)

The integral J assumes the values Jo = ./(qa, q a , q a , q a ) ,

J* = J ( q a , q a , qb, qb) •

(2.110)

Jo has been already computed, see Eq. (2.105), while

J

«= "sfl •

<2-m)

so that one obtains 2 , , x 1 _ _ ( 2 nT J 0 + 4 J 7Tr ) = - I — (2Jo + 4J») = - J - J gp 5i4 For K we have three possibilities / 3 =

K0 = i^(qa,qa,qa,qa,qa,qa),

/ 1 ,\ 0.138 — ^ — - ! = + - _ . —-2 - 1 = + —-g \2{x22 - 1) J 6(j$ :

-ft^i = -f(qa,qa,qa,qa,qb,qb) ,

(2.112)

(2.113)

and K2 = i f ( q a , q a , q b , q b , q b , q b ) •

(2.114)

Therefore we have 7 = - — (2K0 + 12Ki + 6K2) . (2.115) gp Ko has been already computed in (2.105), whereas K\ and K2 can be evaluated using the results given in Appendix A of the review.26

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

53

2.5.1.4. Other crystalline structures An extensive analysis can be found in Ref. 22 where 23 different crystalline structures were considered. We refer the interested reader to Table I in this paper. From our previous discussion we know that the most energetically favored crystals are those which present a first order phase transition between the LOFF and the normal phase. Among the regular structures, with 7 > 0, examined in Ref. 22 the favored seems the octahedron or body-centered-cube, bcc (P = 6), with a transition point at 6fi* = 3.625Ao. Special attention, however, should be given to the face centered cube (fee). The condensate in this case is given by P

A(r) = ^

p

A fc (r) = ^

A exp(2ign k • r ) ,

(2.116)

fc=i fc=i

where P = 8 and fik are the eight unit vectors defining the vertices of the cube. The values of the parameters for this case, as computed in Ref. 22, are /3 « — 110/Sfi2 and 7 « — 459/<5/i4. Since both (3 and 7 are negative, nothing could be said about the cube and one should compute the eighth order in the GL expansion oc A 8 ; the transition would be first order if the coefficient of this term is positive. However in Ref. 22 it is argued that, given the large value of 7, this structure would necessarily dominate. Reasonable numerical examples discussed by the authors confirm this guess. 2.5.2. More plane waves beyond the GL

expansion

The study of the Larkin-Ovchinnikov-Fulde-Ferrell phase at T = 0 is a quite complicated problem and the only soluble case corresponds to the original Fulde-Ferrell phase with a single plane wave. For more plane waves we have discussed the gap equation in the Ginzburg-Landau approximation, which, however, holds only in proximity of a second order transition. The analysis of Ref. 22 shows that in some cases there is no second order transition to the normal phase. In the paper 24 a different approximation was developed in an attempt to overcome the limitations of the Ginzburg-Landau treatment. The main feature of this scheme is a weighted average, with a weight function fffl(r), over the sites defined by the crystalline structure of the condensate. The average is over a region of the size of the lattice cell. In the gap equation the relevant momenta are small with respect to the gap which is of the order of q. Therefore one may assume that the velocity dependent fields of Eq. (2.5) are slowly varying over regions of the order of the lattice size. This means that one can treat them as approximately constant, and in conclusion the average is made only on the coefficient exp{ir • a } , with «(vi, Vj, q) = 2q - /i*Vi - ^ V j ,

(2.117)

where Vj, Vj are the Fermi velocities of the two quarks forming the Cooper pair of total momentum 2q. By this average one computes 1(a) = ( exp{ir • a}gR(r)}

,

(2.118)

G. Nardulli

54

where the bracket means average over the cell. It is possible to choose fffi(r) in such a way that *3 I a \ <% ( Yq)

'(«) =

(2-119)

'

where

f1

,3,,

fOT

W ) =< *• 0

W <

^'

(2.120)

elsewhere.

An approximate expression of ffij(r) is given by -Tvqrk3 sin —jr-

**M=n

J

L

7T>"Jfc

•

(2-i2i)

fc=l

For R/n ss 1, oo limit one obtains vi = - v 2 + 0(l//i) • (2.122) The result of the calculation is the appearance of a factor 5^[/i(v • q)], where the function h was computed in Ref. 23: /i(v • q) = 1 - A T • (2.123) v •q After this average we obtain that the coefficient of the bilinear term in Eq. (2.117) is A*fl[/i(vq)] = Ae//(v-
(2.124)

If one chooses

/i^- v -qi

R=

2

2^

,

(2 . 125)

+ A2|/l(vq)|'

then Aeff

r A for (£,v)ePR =I I 0 elsewhere ,

= A6{Eu)6(Ed)

(2.126)

where Eu, Ed are the two functions €j of Eq. (2.5). This analysis holds for the one-plane-wave ansatz. In the general case of Eq. (2.116) with P plane waves, one obtains the gap equation as follows gp fdv

(dH

PA = l

Ag(v,l0)

Yj T.j 2^-^-Ai(v,4)-

(2 12?)

-

The energy integration is performed by the residue theorem and the phase space is divided into different regions according to the pole positions, defined by E = ^

2

+ A|(v,e) .

(2.128)

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

55

Therefore we get

^ l n ^ = f [ [ £ < , Ag(V'£) = f

/•/• £ g

,

fcA

,

(2.129)

where the regions Pk are defined as follows Pk = {(y,£)\AE{v,e)

= kA}

(2.130)

and we have made use of the equation — =ln|^ (2.131) 5/9 A0 relating the gap Ao of the homogeneous case to the four fermion coupling g and the density of states. The first term in the sum, corresponding to the region P i , has P equal contributions with a dispersion rule equal to the single plane wave case, i.e. E = V^ 2 + A 2 .

(2.132)

This can be interpreted as a contribution from P non interacting plane waves. In the other regions the different plane waves have an overlap. Therefore in this approximation each term in the sum corresponds to one branch of the dispersion law, i.e. to the propagation of a gapped quasi-particle with gap /cA. The corresponding region is Pk. The regions Pk do not represent a partition of the phase space since it is possible to have at the same point quasi-particles with different gaps. Let us see how this method works in some special cases. 2.5.2.1. Two antipodal plane waves For two antipodal plane waves the gap equation is

**-=[[

dV

Ag(V £)

dC

'

I f — dl—

AE(V,£)

2

Ao J JPl 4TT ^ + A|(v,e) J 7 P 2 4TT ^ The integrals over the two regions Pi and P% are as follows: f t dv J JPl 4TT

AE(ve) V '£2 + A ! ( v ) e )

{e{Ej)6{El)

=

+ 6{El)6{El)

I !<* J J 4TT 26(El)6(Eld)6(El)9(E})}A

-

y/P + A 2

(2.134)

and d v

/ / ,P2 4TT

A

jt

e(v>e)

2

^ e + A|(v,e)

J J 4*

^/e + 4A2

;

G. Nardulli

56

where the subscript 2A on the r.h.s. means that in the dispersion laws Eu'd one has to use A —> 2A. E^ d correspond to the two values €j in Eq. (2.60) with v = n i = (0,0,+l); E\d is obtained using n2=(0,0,-l). Since in Pi AE(v,e)=A,

(2.136)

then, as stressed already, Pi represents a region where the two A e / / appearing in AE have no overlap. On the other hand Pi represents the region where the two Aeff appearing in A e do overlap. In this region A|*(v, e) = 4A 2 . 2.5.2.2. Octahedron and Face Centered Cube For the octahedron (i.e. body-centered-cube, bcc) we have P = 6 in (2.116) and n1

(+1,0,0),

na = (0,+l,0),

n s = (0,0,+l),

n 4 = (-1,0,0),

n 5 = (0, - 1 , 0 ) ,

n 6 = (0,0, - 1 ) .

=

(2.137)

For the face centered cube (fee) we have P = 8 and

ni = - ^ ( + l , + l , + l ) ,

n 2 =• ^ ( + I . - I . + D

n3 = ^ ( - l , - l , + l ) ,

n 4 =• ^ ( - 1 , + 1 , + D

ns = ^ ( + l + , l , - l ) ,

n 6 =• J j ( + i . - i . - D

n7 = - L ( - l , - l , - l ) ,

n 8 =: ^ < - l , + l , - D

(2.138)

The topological structure of the different regions Pfc is now more involved because we have several overlapping regions with different dispersion laws. Before turning to the numerical results let us remark that the description one obtains by this method 24 amounts to the definition of a multi-valued gap function having P branches. Each of the branches has a gap given by kA, k — 1, • • • , P , with A the solution of the gap equation. Furthermore each non vanishing value of the gap defines a particular region in the phase space, the pairing region Pfc. The regions Pfc do not represent a partition of the phase space since it is possible to have at the same point quasi-particles with different gaps. 2.5.2.3. Numerical results and discussion We now present numerical results for four crystalline structures. Below FF corresponds to one plane wave, strip - to two antipodal waves. At the Clogston point (<5/i = 6Hi) we have • P = 1 (FF case):

zq = 0.78, — = 0.24, - ^ = - 1 . 8 x 10~ 3 , AQ pA0

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

57

2£2 PAo

/ /

0.05

/

.

0.7 0.8 0.9

./

, . / .

1.1 IZX.Z 1.4

^

-0.05 / X

Fig. 2.1. The values of the free energies of the octahedron (dashed line) and of the fee (full line) crystalline LOFF structures as a function of 5ft/Ao. The octahedron is the favored structure up to <5/x RS .95Ao; for .95Ao < Sn < 1.32Ao the fee is favored. Here, for each value of S/i, the values of zq and A are those that minimize the free energy.

• P = 2 (strip): • P = 6 (bec):

A — = 0.75,

zq9 = 1.0,

2fi — 2 = -0.08,

pAl

A0

0.90, — = 0.28,

20

= -0.11,

AQ

• P = 8 (fee):

A ,,=0.90, - = 0 . 2 1 ,

29. - ^ = -0.09.

These results show that, among the four considered cases, at Sfj, = 5/J.I the favored structure is the octahedron (P = 6). Also the case 6/J, =£ Sfii has been analyzed. 24 If 6fj,2 is the transition point to the normal phase one obtains for the four structures the following results: • P = 1 (FF):

z„ = 0.83, § ^ = 0.754, Ao

• P = 2 (strip): • P = 6 (bec):

Arj

0.81, ( 1 s t order); zq = 1.0, -p- = 0.83, Ao "'""' Ao Suo A zq = 0.95, - p == 11.22, — = 0.43, ( 1 s t order); 22

~pAQ

- > ITA Q

A0 P = 8 (fee):

nd — = 0, (2 order phase transition);

zq = 0.90, Sfj.2 = .1.32, „„

A( A

Ao

0.35, ( 1 s t order).

The values of zq and the discontinuity in A/Ao are computed at Sfi = 6^2 — 0 + . The values of zq and A are determined minimizing the free energy. One can note that for the strip, in this approximation, neither the order of the transition nor the

G. Nardulli

58

point where the transition occurs coincide with those obtained within the GinzburgLandau approximation. 17 The difference in S/j,2 is ~ 10% and in zq is ~ 17% (the value for zq is 0.83). In Fig. 2.1 we plot the free energies of the octahedron (dashed line) and of the fee (full line) phases as a function of 5/j.fAo. From this figure it follows that the octahedron is the favored structure up to Sfi « ,95Ao- For larger values of <5/i < 1.32AQ the favored structure is the fee.

2.6. LOFF phase of QCD with three flavors in the GL approximation Let us now consider the case of three flavors. We shall describe in this section some recent results obtained in the Ginzburg Landau approximation. Consider sAB

xAB

11

ft ]** = TdTZTT-. E-t + jiA'

t °^ iABU E- i r+Z T n^_fr iAr

L

(2-139)

which are the components of the free quark propagator, see Eq. (2.25). The quark propagator is obtained inverting the r.h.s of (2.25) and is given by Sll S i 2 c " *' S21 S 2 2 ,

(GA)AB =

(2-140) AB

whose components satisfy the Gorkov equations S n = S*1 + S 0 n A(r)S 2 i ,

S21 = S 0 2 2 A*(r)S u .

(2.141)

S21 is the anomalous propagator involved in the gap equation. The wave vectors qi should be derived by minimizing the free energy. We will fix the norms |qi| by a minimization procedure. As to their directions, we will limit the analysis to four structures, choosing among them the one with the smallest value of the energy. The first structure has all qi along the positive z—axis. The structures 2, 3, 4 have, respectively, q i , q2, q3 along the positive z—axis (the remaining two momenta along the negative z—axis). The gap equation in the HDET formalism for QCD with three flavors can be written as follows

A^B(r) = i*9 £ C,D=1

h

AachDbB [ - ^ J

\

[^S21(E,e)CDV^9^6ab I

,

J

(2.142) where S 2 i is given in Eq. (2.141); in the above equation hubB is a Clebsch-Gordan coefficient. It is expressed by the formula hobB = Tr[FpTbFs] in terms of the unitary matrices FA used to write the quark fields as in (2.16), i.e. in the basis A= l,---,9. 2.6.1. Ginzburg-Landau

expansion

Performing the Ginzburg-Landau expansion of the propagator S 2 i = S 0 22 A*Si 1 + S 22 A*S 0 11 AS 22 A*Si 1 + 0 ( A 5 )

(2.143)

Larkin-Ovchinnikov-Fulde-Ferrell

Phases in QCD

59

we obtain J 7 J A 7 A2, + 0 ( A 5 ) ,

A , = n , Aj + ^

7=1,2,3.

(2.144)

j

Let us comment on the functions 11/ and J\j appearing in this expansion. IIj are defined as follows: IIi = TL{qi,Sfids) , n 2 = H(q2,S(ius) , II3 = U(q3,6iJ,ud) , with 6^

= —

r

-

= y

,

SHUS = —j—

= y

- - ^

,

(2.145)

and

_ p,s~p,d

6fids = — j —

M2

= —^

•

(2.146)

H(q,5fj,) is defined by Eq. (2.90), with the gap parameter corresponding to the homogeneous case and p given by (2.48). As for Ju, we have, for the diagonal components: J n = Ji = J{q\,Snds), J22 = J2 = J(q2,SfiUs), J33 = ^3 = J{q3,SfJ-ud), with J(q,Sfi) = J given in Eq. (2.105). The off-diagonal term J12 is

J 12 = ^ ! /" *i

1

;

(2.147)

+ 0(A6)

(2 148)

7r2 y 47r (2qi • n + (is — Hd — ie) (2q2 • n + /i s - /uu - ie) ' J13 is obtained from J12 in (2.147) by the exchange q% —* q3 and /i s <-> fid', J23 from J 1 2 by q i —> q 3 and fi3 <-> ^„. 2.6.2. JVee energy The free energy per unit volume, Q, in the GL limit is

7 = y + £ ( y A/ + T A' + D ^T A'A*)

"

with

where the chemical potentials for quarks are defined in Eq. (2.11) and the coefficients are given by (p = 4/x2/7r2) 9P 9P Electric neutrality is obtained by imposing the condition

9P

which, together with the gap equations, gives, for each value of the strange quark mass, the electron chemical potential He and the gap parameters A/. Moreover one should determine qi by solving, together with the equations (2.151) and (2.144), also:

60

G. Nardulli

an

0 = dqj

H

7=1,2,3.

dqi

J=I

(2.152)

The condition (2.151) gives

Ml

(2.153) 4M This result is identical to the free fermion case, which was expected since one works near the transition point between the LOFF and the normal phase. It follows that S/J-du = fyus = #M (2.154) and Snds = 2J/x . (2.155) 2 To evaluate (2.152), it is sufficient to work at the 0(A ), which leads to q = 1.1997|fy|. As to orientation of the qj vectors, the results obtained in Ref. 25 indicate that the favored solution has Ai = 0 and therefore q i = 0. Furthermore q2 = q3 and A2 = A3. These results are consequences of the GL limit. In fact, as shown by Eqs. (2.155) and (2.154), the surface separation of d and s quarks is larger, which implies that Ai pairing is disfavored. On the other hand the surface separations of d-u and u-s quarks are equal, which implies that A2 and A3 must be almost equal. Finally q2; q3 are parallel because the pairing region on the u—quark surface is formed by two distinct rings (in the northern and southern hemisphere respectively), while for antiparallel q2, q3 the two rings overlap, which reduces the phase space available for pairing. Me

10

I

1

I

I

|

1—i

i

i—|

i

i

i

>

- 1 0-

«r -20 o -30 CO -40

-5Q

i

i

i—1

i

i

i

I

|

i—1—r—r

y

/

0

a

|

LQFF 4A —7" ^ r. /

0 <*

1

J'' /

/ :^gCFL

/

CFL/ i

25

. . . .

i

50

75 100 Ms7|LL[MeV]

125

150

Fig. 2.2. Free energy differences CILOFF — fino™ in units of 10 6 MeV 4 plotted versus Mil p. (in MeV) for various QCD phases.

Larkin-Ovchinnikov-Fulde-Ferrell

2.6.3.

Phases in QCD

61

Results

T h e numerical results are summarized in Fig. 2.2. They are obtained for fi = 500 MeV, while the value of the C F L gap for zero strange quark mass is fixed at Ao = 25 MeV. This allows a comparison between the results for t h e L O F F phase and those of Ref. 12. In order to comment Fig. 2.2 let us follow the graph for decreasing values of M^/fi. We see t h a t at about M^/fj, = 150 MeV the L O F F phase has a free energy lower t h a n the normal one. This point corresponds to a second order transition. Then the L O F F state is energetically favored till the point where it meets the gCFL line at about Mf/fi = 128 MeV. This is a first order transition since the gaps are different in the two phases. For values of M ^ / / i smaller t h a n 128 MeV, the L O F F phase in the configuration of Eq. (2.21), i.e. with one plane for each diquark condensation, has a free energy higher t h a n flgcFL a n d cannot be the true vacuum state. In particular it cannot cure the chromo-magnetic instability of the g C F L phase. It remains to be seen if some other ansatz, with more plane waves, would solve this intriguing issue.

Acknowledgments It is a pleasure to t h a n k M. Alford, J. W. Clark and A. Sedrakian for the organization of an inspiring workshop. I t h a n k R. Casalbuoni, M. Ciminale, R. G a t t o , N. Ippolito M. Mannarelli and in particular M. Ruggieri for their precious collaboration.

Bibliography 1. J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975).; B. Barrois, Nucl. Phys. B 129, 390 (1977).; S. Frautschi, Proceedings of workshop on hadronic matter at extreme density, Erice 1978; D. Bailin and A. Love,Phys. Rept. 107, 325 (1984). 2. M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B 422, 247 (1998). [arXiv:hepph/9711395]. 3. R. Rapp, T. Schafer, E. V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 8 1 , 53 (1998). [arXiv:hep-ph/9711396]. 4. M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B 537, 443 (1999). [arXiv:hep-ph/9804403]. 5. K. Rajagopal and F.Wilczek, in Handbook of QCD, M. Shifman ed. (World Scientific 2001)., [arXiv:hep-ph/0011333]; 6. M. G. Alford, Ann. Rev. Nucl. Part. Sci. 51, 131 (2001). [arXiv:hep-ph/0102047]. 7. T. Schafer, arXiv:hep-ph/0304281. 8. G.Nardulli, Riv. Nuovo Cim. 25N3, 1 (2002). [arXiv:hep-ph/0202037]. 9. H. Muther and A. Sedrakian, Phys. Rev. D 67, 085024 (2003). [arXiv:hepph/0212317]. 10. I. Shovkovy and M. Huang, Phys. Lett. B 564, 205 (2003). [arXiv:hep-ph/0302142]. 11. M. Alford, C. Kouvaris and K. Rajagopal, Phys. Rev. Lett. 92, 222001 (2004). [arXiv:hep-ph/0311286]; 12. M. Alford, C. Kouvaris and K. Rajagopal, Phys. Rev. D 71, 054009 (2005). [arXivchepph/0406137].

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13. M. Huang and I. A. Shovkovy, Phys. Rev. D 70, 051501 (2004). [arXiv:hepph/0407049]; M. Huang and I. A. Shovkovy, Phys. Rev. D 70, 094030 (2004). [arXiv:hep-ph/0408268]. 14. R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli and M. Ruggieri, Phys. Lett. B 605, 362 (2005). [Erratum-ibid. B 615, 297 (2005).] [arXiv:hep-ph/0410401]. 15. K. Pukushima, Phys. Rev. D 72, 074002 (2005). [arXiv:hep-ph/0506080]. 16. M. Alford and Q. h. Wang, J. Phys. G 31, 719 (2005). [arXiv:hep-ph/0501078]. 17. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 1964 (Sov. Phys. JETP 20, 762 (1965).). 18. P.Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). 19. M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D 63, 074016 (2001). [arXiv:hep-ph/0008208]. 20. J. A. Bowers, J. Kundu, K. Rajagopal and E. Shuster, Phys. Rev. D 64, 014024 (2001). [arXiv:hep-ph/0101067]; 21. A. K. Leibovich, K. Rajagopal and E. Shuster, Phys. Rev. D 64, 094005 (2001). [arXiv:hep-ph/0104073]. 22. J. A. Bowers and K. Rajagopal, Phys. Rev. D 66, 065002 (2002). [arXivrhepph/0204079]. 23. R. Casalbuoni, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Rev. D 66, 014006 (2002). [arXiv:hep-ph/0201059]. 24. R. Casalbuoni, M. Ciminale, M. Mannarelli, G. Nardulli, M. Ruggieri and R. Gatto, Phys. Rev. D 70, 054004 (2004). [arXiv:hep-ph/0404090]. 25. R. Casalbuoni, R. Gatto, N. Ippolito, G. Nardulli and M. Ruggieri, Phys. Lett. B 627, 89 (2005). [arXiv:hep-ph/0507247]. 26. R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004). [arXiv:hepph/0305069]. 27. I. Giannakis and H. C. Ren, Phys. Lett. B 611, 137 (2005). [arXiv:hep-ph/0412015]. 28. I. Giannakis and H. C. Ren, Nucl. Phys. B 723, 255 (2005). [arXiv:hep-th/0504053]. 29. E. V. Gorbar, M. Hashimoto and V. A. Miransky, Phys. Rev. Lett. 96, 022005 (2006). [arXiv:hep-ph/0509334]. 30. M. Ciminale, R. Gatto, G. Nardulli and M. Ruggieri, preprint Bari-TH 532/06 (2006). 31. D. K. Hong, Phys. Lett. B 473, 118 (2000). [arXiv:hep-ph/9812510]; D. K. Hong, Nucl. Phys. B 582, 451 (2000). [arXiv:hep-ph/9905523]. 32. S. R. Beane, R F. Bedaque and M. J. Savage, Phys. Lett. B 483, 131 (2000). [arXiv:hep-ph/0002209]. 33. T. Schafer, Nucl. Phys. A 728, 251 (2003). [arXiv:hep-ph/0307074]. 34. T. Schafer, eConf C030614, 038 (2003). [arXiv:hep-ph/0310176]. 35. A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962). 36. B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).

Chapter 3 Phase Diagram of Neutral Quark Matter at Moderate Densities

Stefan B . Riister Institut

fur Theoretische Physik, J.W. Goethe-Universitat, D-6O4S8 Frankfurt am Main, Germany [email protected]. uni-frankfurt. de Verena Werth

Institut

fur Kernphysik, Technische Universitdt D-64289 Darmstadt, Germany [email protected]

Darmstadt,

Michael Buballa Institut

fur Kernphysik, Technische Universitat D-64289 Darmstadt, Germany [email protected]

Darmstadt,

Igor A. Shovkovy Frankfurt

Institute for Advanced Studies, J. W. GoetheD-60438 Frankfurt am Main, Germany [email protected]. de

Universitdt,

Dirk H. Rischke Institut

fur Theoretische Physik, J. W. Goethe- Universitat, D-60438 Frankfurt am Main, Germany [email protected]. uni-frankfurt. de

We discuss the phase diagram of moderately dense, locally neutral three-flavor quark matter using the framework of an effective model of quantum chromodynamics with a local interaction. The phase diagrams in the plane of temperature and quark chemical potential as well as in the plane of temperature and leptonnumber chemical potential are discussed. Contents 3.1. Introduction 3.2. Model 3.3. Phase diagram in absence of neutrino trapping 63

64 65 68

64

S. B. Ruster et al.

3.4. Phase diagram in presence of neutrino trapping 3.4.1. General effect of neutrino trapping 3.4.2. Three-dimensional phase diagram 3.4.3. T-fi phase diagram 3.4.4. Lepton fraction YLC 3.4.5. T-p,Le phase diagram 3.5. Conclusions Bibliography

75 76 79 80 84 85 87 88

3.1. Introduction Theoretical studies suggest that baryon matter at sufficiently high density and sufficiently low temperature is a color superconductor. (For reviews on color superconductivity see, for example, Ref. 1.) In nature, the highest densities of matter are reached in central regions of compact stars. There, the density might be as large as 10po where po ~ 0.15 fm - 3 is the saturation density. It is possible that baryonic matter is deconfined under such conditions and, perhaps, it is colorsuperconducting . In compact stars, matter in the bulk is neutral with respect to the electric and color charges. Matter should also remain in (3 equilibrium. Taking these constraints consistently into account may have a strong effect on the competition between different phases of deconfined quark matter at large baryon densities. 2-6 The first attempts to obtain the phase diagram of dense, locally neutral threeflavor quark matter as a function of the strange quark mass, the quark chemical potential, and the temperature were made in Refs. 3 and 7. This was done within the framework of a Nambu-Jona-Lasinio (NJL) model. It was shown that, at zero temperature and small values of the strange quark mass, the ground state of matter corresponds to the color-flavor-locked (CFL) phase. 8,9 At some critical value of the strange quark mass, this is replaced by the gapless CFL (gCFL) phase. 6 In addition, several other phases were found at nonzero temperature. For instance, it was shown that there should exist a metallic CFL (mCFL) phase, a so-called uSC phase, 10 as well as the standard two-flavor color-superconducting (2SC) phase 11 ' 12 and the gapless 2SC (g2SC) phase. 5 In Ref. 7, the effect of the strange quark mass was incorporated only approximately through a shift of the chemical potential of strange quarks, fis —> fj,s — m 2 /(2 / u). While such an approach is certainly reliable at small values of the strange quark mass, it becomes uncontrollable with increasing the mass. The phase diagram of Ref. 7 was further developed in Refs. 13 and 14 where the shiftapproximation in dealing with the strange quark was not employed any more, although quark masses were still treated as free parameters, rather than dynamically generated quantities. In Refs. 15 and 16 the phase diagram of dense, locally neutral three-flavor quark matter was further improved by treating dynamically generated quark masses self-consistently. Some results within this approach at zero and nonzero temperatures were also obtained in Refs. 3, 17, 18 and 19. The results of Refs. 15 and 16 are presented here. Only locally neutral phases are considered. This excludes, for example, mixed 20 and crystalline 21 phases.

Phase Diagram of Neutral Quark Matter at Moderate

Densities

65

Taking them into account requires a special treatment. We also discuss the effect of a nonzero neutrino (or, more precisely, lepton-number) chemical potential on the structure of the phase diagram. 16 This is expected to have a potential relevance for the physics of protoneutron stars where neutrinos are trapped during the first few seconds of the stellar evolution. The effect of neutrino trapping on color-superconducting quark matter has been previously discussed in Ref. 3. There it was found that a nonzero neutrino chemical potential favors the 2SC phase and disfavors the CFL phase. This is not unexpected because the neutrino chemical potential is related to the conserved lepton number in the system and therefore it also favors the presence of (negatively) charged leptons. This helps 2SC-type pairing because electrical neutrality in quark matter can be achieved without inducing a very large mismatch between the Fermi surfaces of up and down quarks. The CFL phase, on the other hand, is electrically and color neutral in the absence of charged leptons when T = 0. 22 A nonzero neutrino chemical potential can only spoil CFL-type pairing. A more systematic survey of the phase diagram in the space of temperature, quark and lepton-number chemical potentials was performed in Ref. 16. In particular, this included the possibility of gapless phases, which have not been taken into account in Ref. 3. While such phases are generally unstable at zero temperature, 23 this is not always the case at nonzero temperature. 24 Keeping this in mind, we shall merely localize the "problematic" regions in the phase diagram, where unconventional pairing is unavoidable. We shall refrain, however, from speculating on various possibilities for the true ground state (see, e.g., Refs. 20, 21, 25-28) , since these are still under debate. In the application to protoneutron stars, it is of interest to cover a range of parameters that could provide a total lepton fraction in quark matter of up to about 0.4. This is the value of the lepton-to-baryon charge ratio in iron cores of progenitor stars. Because of the conservation of both lepton and baryon charges, this value is also close to the lepton fraction in protoneutron stars at early times, when the leptons did not have a chance to diffuse through dense matter and escape from the star. 29 In the model introduced in the next section, almost the whole range of possibilities will be covered by restricting the quark chemical potential to /x < 500 MeV and the neutrino chemical potential to jU„e < 400 MeV. 3.2. Model Let us start by introducing the effective model of QCD used in the analysis. This is a three-flavor quark model with a local NJL-type interaction, whose Lagrangian density is given by 8

£ = 4> (i p - m )V< + Gs ^

[(^AQV>)2 + {4>ils\ai>f

a=0

+ GDJ2[rail5e<*^eabc(ipc)bp}

[(^c)rpi75ep^erscra]

- K jdet [i> (1 + 75) i>] + det $ (1 - 75) V] ] ,

(3-1)

66

S. B. Riister et al.

where the quark spinor field ip^ carries color (a = r, g, b) and flavor (a = u, d, s) indices. The matrix of quark current masses is given by m = diagf(m u ,md,m s ). Regarding other notations, Aa with a = 1,...,8 are the Gell-Mann matrices in flavor space, and Ao = y/2/31f. The charge conjugate spinors are defined as follows: Vc = Cy>T and ipc = i>TG, where i/> — 1/^7° is the Dirac conjugate spinor and C = i7 2 7° is the charge conjugation matrix. The model in Eq. (3.1) should be viewed as an effective model of strongly interacting matter that captures at least some key features of QCD dynamics. The Lagrangian density contains three different interaction terms which are chosen to respect the symmetries of QCD. Note that we include the 't Hooft interaction whose strength is determined by the coupling constant K. This term breaks the t/(l) axial symmetry. The term in the second line of Eq. (3.1) describes a scalar diquark interaction in the color antitriplet and flavor antitriplet channel. For symmetry reasons there should also be a pseudoscalar diquark interaction with the same coupling constant. This term would be important to describe Goldstone boson condensation in the CFL phase. 30 We use the following set of model parameters: 31 mu = md = 5.5 MeV, ms = 140.7 MeV, G S A 2 = 1.835, KA5 = 12.36, A = 602.3 MeV.

(3.2a) (3.2b)

After fixing the masses of the up and down quarks at equal values, mU}Ct = 5.5 MeV, the other four parameters are chosen to reproduce the following four observables of vacuum QCD: m* = 135.0 MeV, mK = 497.7 MeV, m^ = 957.8 MeV, and fn = 92.4 MeV. 31 This parameter set gives m , = 514.8 MeV. In Ref. 31, the diquark coupling GD was not fixed by the fit of the meson spectrum in vacuum. In general, it is expected to be of the same order as the quarkantiquark coupling Gs- Here, we choose the coupling strength with Go = | G s which follows from the Fierz identity. The grand partition function, up to an irrelevant normalization constant, is given by Z

= e-nv'T

= [ VijVijj J 5* (c+^°^) ,

(3.3)

where f2 is the thermodynamic potential density, V is the volume of the three-space, and /} is a diagonal matrix of quark chemical potentials. In chemical equilibrium (which provides (3 equilibrium as a special case), the nontrivial components of this matrix are extracted from the following relation: tit

= (W a / 3 + HQQf)


M8

( T 8 ) J 6°* .

(3.4)

Here fi is the quark chemical potential (by definition, /i = / ^ B / 3 where fis is the baryon chemical potential), JIQ is the chemical potential of electric charge, while ^3 and i^s are color chemical potentials associated with two mutually commuting color charges of the SU(3)C gauge group, cf. Ref. 32. The explicit form of the electric charge matrix is Qf — diagy(|, —5, —5), a n ( i t n e explicit form of the color charge matrices is T3 = diag c (|, —1,0) and y/3T8 = diag c (i, \, —1).

Phase Diagram of Neutral Quark Matter at Moderate

Densities

67

In order to calculate the mean-field thermodynamic potential at temperature T, we first linearize the interaction in the presence of the diquark condensates A c ~ ('c)%il5eCl^Ceabc'4)bB (no sum over c) and the quark-antiquark condensates aa ~ '4>a'tlJa ( n o s u m o v e r a)- Then, integrating out the quark fields and neglecting the fluctuations of composite order parameters, we arrive at the following expression for the thermodynamic potential: 1

3

c=l

3

a=l

- AKaucrdas - — ^ In det - = - ,

(3.5)

where we also added the contribution of leptons, Q L , which will be specified later. We should note that we have restricted ourselves to field contractions corresponding to the Hartree approximation. In a more complete treatment, among others, the 't Hooft interaction term gives also rise to mixed contributions containing both diquark and quark-antiquark condensates, i.e., oc J 3 Q = 1 aa\ A a | 2 - 3 3 In this study, we neglect such terms for simplicity. While their presence may change the results quantitatively, one does not expect them to modify the qualitative structure of the phase diagram. In Eq. (3.5), S - 1 is the inverse full quark propagator in the Nambu-Gorkov representation,

^^ir,-_,),

<-)

with the diagonal elements being the inverse Dirac propagators of quarks and of charge-conjugate quarks, [G±\~l=YK»±Mo-M,

(3.7)

where K^ = (fco>k) denotes the four-momentum of the quark. At nonzero temperature, we use the Matsubara imaginary time formalism. Therefore, the energyfcois replaced with — iwn where uin = (2n + VJ-KT are the fermionic Matsubara frequencies. Accordingly, the sum over K in Eq. (3.5) should be interpreted as a sum over integer n and an integral over the three-momentum k. The constituent quark mass matrix M = diag^(M„, Md, Ms) is defined by Ma = ma- 4G S CT Q + 2ifo73<77 ,

(3.8)

where aa are the quark-antiquark condensates, and the set of indices (a, /?, 7) is a permutation of (u,d,s). The off-diagonal components of the inverse propagator (3.6) are the so-called gap matrices given in terms of three diquark condensates. The color-flavor structure of these matrices is given by

(*")«? — E ^ ^ ^ ,

(3-9)

c

and $ + = 7°($~)^7°. Here, as before, a and b refer to the color components and a and j3 refer to the flavor components. Hence, the gap parameters A i , A2,

68

S. B. Ruster et al.

and A3 correspond to the down-strange, the up-strange and the up-down diquark condensates, respectively. All three of them originate from the color-antitriplet, flavor-antitriplet diquark pairing channel. For simplicity, the color and flavor symmetric condensates are neglected in this study. They were shown to be small and not crucial for the qualitative understanding of the phase diagram. 7 The determinant of the inverse quark propagator can be decomposed as follows:15 18

n

det :

(3.10)

rp2

i=l

where e^ are eighteen independent positive energy eigenvalues. The Matsubara summation in Eq. (3.5) can then be done analytically by employing the relation 34

£l*

w2+e2

+ 21n 1 + exp

y2

-

(3.11)

Then, we arrive at the following mean-field expression for the pressure: 18

A

^ f c 2 J N + 2Tln l + e x p ( - i ^ )

p = - n = ^ 2 ^ /

3

1

+ 4K
3

|A C | 2 - 2GS J^

a

l

c*=l

+

dkk2ln

+

+ 2n^iiT' l=e,LL

j

1 +exp

Ei - em

7 + - ^Arpi T

(3.12)

x

where the contributions of electrons and muons with masses me ?» 0.511 MeV and rrifj, « 105.66 MeV, as well as the contributions of neutrinos were included. Note that muons may exist in matter in /3 equilibrium and, therefore, they are included in the model for consistency. However, being about 200 times heavier than electrons, they do not play a big role in the analysis.

3.3. Phase diagram in absence of neutrino trapping In this section, we consider the case without neutrino trapping in quark matter. This is expected to be a good approximation for matter inside a neutron star after the short deleptonization period is over. The expression for the pressure in Eq. (3.12) has a physical meaning only when the chiral and color-superconducting order parameters, aa and A c , satisfy the following set of six gap equations: dp

= 0

dp

d~Kc

= 0

(3.13)

Phase Diagram of Neutral Quark Matter at Moderate

Densities

69

To enforce the conditions of local charge neutrality in dense matter, we also require three other equations to be satisfied, n

o

^ OfJ-Q

= 0,

n

3

^ = 0 , dfi3

n

8

^ = 0 .

(3.14)

d^8

These fix the values of the three corresponding chemical potentials, [IQ, /X3 and /isAfter these are fixed, only the quark chemical potential fi is left as a free parameter. In order to obtain the phase diagram, one has to find the ground state of matter for each given set of the parameters in the model. In the case of locally neutral matter, there are two parameters that should be specified: temperature T and quark chemical potential fj,. After these are fixed, one has to compare the values of the pressure in all competing neutral phases of quark matter. The ground state corresponds to the phase with the highest pressure. By using standard numerical recipes, it is not extremely difficult to find a solution to the given set of nine nonlinear equations. Complications arise, however, due to the fact that often the solution is not unique. The existence of different solutions to the same set of equations, (3.13) and (3.14), reflects the physical fact that there could exist several competing neutral phases with different physical properties. Among these phases, all but one are unstable or metastable. In order to take this into account, one should look for the solutions of the following 8 types: (1) (2) (3) (4) (5) (6) (7) (8)

Normal quark (NQ) phase: Ai = A 2 = A 3 = 0; 2SC phase: Ai = A 2 = 0, and A 3 ^ 0; 2SCus phase: A : = A 3 = 0, and A 2 ^ 0; 2SCds phase: A 2 = A 3 = 0, and Ai ^ 0; uSC phase: A 2 ^ 0, A 3 ^ 0, and Ai = 0; dSC phase: Ai ^ 0, A 3 ^ 0, and A 2 = 0; sSC phase: Ai ^ 0, A 2 ^ 0, and A 3 = 0; CFL phase: Ai ^ 0, A 2 ^ 0, A 3 ^ 0.

Then, we calculate the values of the pressure in all nonequivalent phases, and determine the ground state as the phase with the highest pressure. After this is done, we study additionally the spectrum of low-energy quasiparticles in search for the existence of gapless modes. This allows us to refine the specific nature of the ground state. In the above definition of the eight phases in terms of A c , we have ignored the quark-antiquark condensates aa. In fact, in the chiral limit (ma = 0), the quantities aa are good order parameters and we could define additional sub-phases characterized by nonvanishing values of one or more aa. With the model parameters at hand, however, chiral symmetry is broken explicitly by the nonzero current quark masses, and the values of
70

S. B. Riister et al.

could end in a critical endpoint and there is only a smooth crossover at higher temperatures. This picture emerges from NJL-model studies, both, without 35 and with 36 diquark pairing. The numerical results for neutral quark matter are summarized in Fig. 3.1. This shows the phase diagram in the plane of temperature and quark chemical potential, obtained in the mean-field approximation in model (3.1). The corresponding dynamical quark masses, gap parameters, and three charge chemical potentials are displayed in Fig. 3.2. All quantities are plotted as functions of ^ for three different values of the temperature: T = 0, 20, 40 MeV. 60 I

320

1

1

1

340

360

380

1

1

400 420 H[MeV]

1

r

440

460

480

500

Fig. 3.1. The phase diagram of neutral quark matter in model (3.1). First-order phase boundaries are indicated by bold solid lines, whereas the thin solid lines mark second-order phase boundaries between two phases which differ by one or more nonzero diquark condensates. T h e dashed lines indicate the (dis-)appearance of gapless modes in different phases, and they do not correspond to phase transitions.

In the region of small quark chemical potentials and low temperatures, the phase diagram is dominated by the normal phase in which the approximate chiral symmetry is broken, and in which quarks have relatively large constituent masses. This is denoted by xSB in Fig. 3.1. With increasing the temperature, this phase changes smoothly into the NQ phase in which quark masses are relatively small. Because of explicit breaking of the chiral symmetry in the model at hand, there is no need for a phase transition between the two regimes. However, as pointed out above, the symmetry argument does not exclude the possibility of an "accidental" (first-order) chiral phase transition. As expected, at lower temperatures we find a line of first-order chiral phase transitions. It is located within a relatively narrow window of the quark chemical potentials (336 MeV < ix < 368 MeV) which are of the order of the vacuum values of the light-quark constituent masses. (For the parameters used in the calculation one obtains Mu = Ma =

Phase Diagram of Neutral Quark Matter at Moderate

r-

- 4 , 80 " - . 4 2 70 . -

-

43 -20

2

-

50 40

-

30 20

/

10

9U

i

i

350

400 n[MeV]

I

I

— A, 80 " - 4 2 70 - - 4 3 60 f 2 2

50

20

i

i

r~

2 a.

-40

so

-60

2.

.

-

•J •

-80 — ^Q

-100

i

•

—

-

^

\

-

^3

H8 350

400 450 n[MeV]

350

400 450 ^[MeV]

1

.*•

.'

/ /}

!

-

>'r

•

}(••

40 30

71

0

60 f 2

Densities

•

•

-

10 n

'

'

•

350

Fig. 3.2. Dependence of the quark masses, of the gap parameters, and of the electric and color charge chemical potentials on the quark chemical potential at a fixed temperature, T = 0 MeV (three upper panels), T = 20 MeV (three middle panels), and T = 40 MeV (three lower panels).

367.7 MeV and Ms = 549.5 MeV in vacuum. 31 ) At this chiral condensates, as well as the quark constituent masses, With increasing temperature, the size of the discontinuity terminates at the endpoint located at (TCT,fiCI) « (56,336)

critical line, the quark change discontinuously. decreases, and the line MeV, see Fig. 3.1.

72

5. B. Riister et al.

The location of the critical endpoint is consistent with other mean-field studies of NJL models with similar sets of parameters. 35 ' 36 This agreement does not need to be exact because, in contrast to the studies in Refs. 35, 36, here we imposed the condition of electric charge neutrality in quark matter. (Note that the color neutrality is satisfied automatically in the normal phase.) One may argue, however, that the additional constraint of neutrality is unlikely to play a big role in the vicinity of the endpoint. It is appropriate to mention here that the location of the critical endpoint might be affected very much by fluctuations of the composite chiral fields. These are not included in the mean-field studies of the NJL model. In fact, this is probably the main reason for their inability to pin down the location of the critical endpoint consistent, for example, with lattice calculations. 37 (It is fair to mention that the current lattice calculations are not very reliable at nonzero /i either.) Therefore, the predictions of this study, as well as of those in Refs. 35 and 36, regarding the critical endpoint cannot be considered as very reliable. When the quark chemical potential exceeds some critical value and the temperature is not too large, a Cooper instability with respect to diquark condensation should develop in the system. Without enforcing neutrality, i.e., if the chemical potentials of up and down quarks are equal, this happens immediately after the chiral phase transition when the density becomes nonzero. 36 In the present model, this is not the case at low temperatures. In order to understand this, one should inspect the various quantities at T — 0 which are displayed in the upper three panels of Fig. 3.2. At the chiral phase boundary, the up and down quark masses become relatively small, whereas the strange quark mass experiences only a moderate drop of about 84 MeV induced by the 't Hooft interaction. This is not sufficient to populate any strange quark states at the given chemical potential, and the system mainly consists of up and down quarks together with a small fraction of electrons, see Fig. 3.3. The electric charge chemical potential which is needed to maintain neutrality in this regime is between about —73 and —94 MeV. It turns out that the resulting splitting of the up and down quark Fermi momenta is too large for the given diquark coupling strength to enable diquark pairing and the system stays in the normal phase. At [i « 432 MeV, the chemical potential felt by the strange quarks, \i — / ^ Q / 3 , reaches the strange quark mass and the density of strange quarks becomes nonzero. At first, this density is too small to play a sizable role in neutralizing matter, or in enabling strange-nonstrange cross-flavor diquark pairing, see Fig. 3.3. The NQ phase becomes metastable against the gapless CFL (gCFL) phase at /igCFL ~ 443 MeV. This is the point of a first-order phase transition. It is marked by a drop of the strange quark mass by about 121 MeV. As a consequence, strange quarks become more abundant and pairing gets easier. Yet, in the gCFL phase, the strange quark mass is still relatively large, and the standard BCS pairing between strange and light (i.e., up and down) quarks is not possible. In contrast to the regular CFL phase, the gCFL phase requires a nonzero density of electrons to stay electrically neutral. At T = 0, therefore, one could use the value of the electron density as a formal order parameter that distinguishes these two phases. 6

Phase Diagram of Neutral Quark Matter at Moderate

300

350

400 M. [MeV]

450

Densities

73

500

Fig. 3.3. The dependence of the number densities of quarks and electrons on the quark chemical potential at T = 0 MeV. Note that the densities of all three quark flavors coincide above p. = 457 MeV. The density of muons vanishes for all values of fj..

With increasing the chemical potential further (still at T = 0), the strange quark mass decreases and the cross-flavor Cooper pairing gets stronger. Thus, the gCFL phase eventually turns into the regular CFL phase at /icFL « 457 MeV. The electron density goes to zero at this point, as it should. This is indicated by the vanishing value of fin in the CFL phase, see the upper right panel in Fig. 3.2. We remind that the CFL phase is neutral because of having equal number densities of all three quark flavors, nu = rid = ns, see Fig. 3.3. This equality is enforced by the pairing mechanism, and this is true even when the quark masses are not exactly equal. 22 Let us mention here that the same NJL model at zero temperature was studied previously in Ref. 3. The results of Ref. 3 agree with those presented here only when the quark chemical potential is larger than the critical value for the transition to the CFL phase at 457 MeV. The appearance of the gCFL phase for 443 < fi < 457 MeV was not recognized in Ref. 3, however. Instead, it was suggested that there exists a narrow (about 12 MeV wide) window of values of the quark chemical potential around fi ss 450 MeV in which the 2SC phase is the ground state. By examining the same region, we find that the 2SC phase does not appear there. This is illustrated in Fig. 3.4 where the pressure of three different solutions is displayed. Had we ignored the gCFL solution (thin solid line), the 2SC solution (dashed line) would indeed be the most favored one in the interval between \i « 445 MeV and \i w 457 MeV. After including the gCFL phase in the analysis, this is no longer the case. Now let us turn to the case of nonzero temperature. One might suggest that this should be analogous to the zero-temperature case, except that Cooper pairing is somewhat suppressed by thermal effects. In contrast to this naive expectation, the

74

S. B. Ruster et al.

0.032

0.028 r

Q.

0.024

440

450

460

H [MeV] Fig. 3.4. Pressure divided by jit4 for different neutral solutions of the gap equations at T = 0 as functions of the quark chemical potential \i: regular CFL (bold solid line), gapless CFL (thin solid line), 2SC (dashed line), normal (dotted line).

thermal distributions of quasiparticles together with the local neutrality conditions open qualitatively new possibilities that were absent at T = 0. As in the case of the two-flavor model of Ref. 5, a moderate thermal smearing of mismatched Fermi surfaces could increase the probability of creating zero-momentum Cooper pairs without running into a conflict with Pauli blocking. This leads to the appearance of several color-superconducting phases that could not exist at zero temperatures. With increasing the temperature, the first qualitatively new feature in the phase diagram appears when 5 < T < 10 MeV. In this temperature interval, the NQ phase is replaced by the uSC phase when the quark chemical potential exceeds the critical value of about 444 MeV. The corresponding transition is a first-order phase transition, see Fig. 3.1. Increasing the chemical potential further by several MeV, the uSC phase is then replaced by the gCFL phase, and the gCFL phase later turns gradually into the (m)CFL phase. (In this study, we do not distinguish between the CFL phase and the mCFL phase. 7 ) Note that, in the model at hand, the transition between the uSC and the gCFL phase is of second order in the following two temperature intervals: 5 < T < 9 MeV and T > 24 MeV. On the other hand, it is a first-order transition when 9 < T < 24 MeV. Leaving aside its unusual appearance, this is likely to be an "accidental" property in the model for a given set of parameters. The transition from the gCFL to the CFL phase is a smooth crossover at all T ^ 0. 7 ' 13 The reason is that the electron density is not a good order parameter that could be used to distinguish the gCFL from the CFL phase when the temperature is nonzero. This is also confirmed by the numerical results for the electric charge chemical potential ^Q in Fig. 3.2. While at zero temperature the value of fiQ vanishes identically in the CFL phase, this is not the case at nonzero temperatures.

Phase Diagram of Neutral Quark Matter at Moderate

Densities

75

Another new feature in the phase diagram appears when the temperature is above about 11 MeV. In this case, with increasing the quark chemical potential, the Cooper instability happens immediately after the xSB phase. The corresponding critical value of the quark chemical potential is rather low, about 365 MeV. The first color-superconducting phase is the gapless 2SC (g2SC) phase. 5 This phase is replaced with the 2SC phase in a crossover transition only when fi > 445 MeV. The 2SC is then followed by the gapless uSC (guSC) phase, by the uSC phase, by the gCFL phase and, eventually, by the CFL phase (see Fig. 3.1). In the NJL model at hand, determined by the parameters in Eq. (3.2), we do not find the dSC phase as the ground state anywhere in the phase diagram. This is similar to the conclusion of Refs. 7 and 14, but differs from that of Refs. 10 and 13. This should not be surprising because, as was noted earlier, 14 ' 19 the appearance of the dSC phase is rather sensitive to a specific choice of parameters in the NJL model. The phase diagram in Fig. 3.1 has a very specific ordering of quark phases. One might ask if this ordering is robust against the modification of the parameters of the model at hand. We can argue that some features are indeed quite robust, while others are not. 15 It should be clear that the appearance of color-superconducting phases under the stress of neutrality constraints is very sensitive to the strength of diquark coupling. In the case of two-flavor quark matter, this was demonstrated very clearly in Ref. 5 at zero as well as at nonzero temperatures. The same statement remains true in three-flavor quark matter. 15,19

3.4. Phase diagram in presence of neutrino trapping In the case of neutrino trapping, the chemical potentials of individual quark and lepton species can be expressed in terms of six chemical potentials according to their content of conserved charges. For the quarks, which carry quark number, color and electric charge, the chemical potentials were introduced in Eq. (3.4). For the neutrinos, which carry only lepton number, the chemical potentials are M*« = MLe , / V = ML„ • (3.15) while for the electrons and muons, carrying both lepton number and electric charge, the chemical potentials read He = HLC - MQ , /V = ^L„ - MQ • (3.16) In order to obtain the phase diagram, one has to determine the ground state of matter for each given set of the parameters. In the case of locally neutral matter with trapped neutrinos, there are four parameters that should be specified: the temperature T, the quark chemical potential n as well as the two lepton family chemical potentials HLC and ^L^- After these are fixed, the values of the pressure in all competing neutral phases of quark matter should be compared. The phase with the largest pressure is the ground state. For our purposes, it is sufficient to take the vanishing muon lepton-number chemical potential, i.e., fxr, = 0. This is expected to be a good approximation for matter inside a protoneutron star.

76

3.4.1. General

5. B. Riister et al.

effect of neutrino

trapping

As mentioned in the Introduction, neutrino trapping favors the 2SC phase and disfavors the CFL phase. 3 This is a consequence of the modified /3-equilibrium condition in the system. In this section, we would like to emphasize that this is a model-independent effect. In order to understand the physics behind it, it is instructive to start our consideration from a very simple toy model. Many of its qualitative features are also observed in a self-consistent numerical analysis of the NJL model. Let us first assume that strange quarks are very heavy and consider a gas of noninteracting massless up and down quarks in the normal phase at T = 0. As required by /? equilibrium, electrons and electron neutrinos are also present in the system. (Note that in this section we neglect muons and muon neutrinos for simplicity.) In the absence of Cooper pairing, the densities of quarks and leptons are given by

Expressing the chemical potentials through /i, /XQ a n < i Miei charge neutrality, one arrives at the following relation: 2(1 + \yf

- (1 - \yf

an<

- (x - y) 3 = 0 ,

i imposing electric (3.18)

where we have introduced the chemical potential ratios x = HL^/H and y = MQ//J.. The above cubic equation can be solved for y (electric chemical potential) at any given a; (lepton-number chemical potential). The result can be used to calculate the ratio of quark chemical potentials, Hd/^u = (3 — 2/)/(3 + 2y). The ratio fid/Vu as a function of ML./M i s shown in Fig. 3.5. At vanishing ^LC, one finds y « —0.219 and, thus, Hd/Hu & 1.256 (note that this value is very close to 2 1 / 3 w 1.260). This result corresponds to the following ratios of the number densities in the system: nu/nd « 0.504 and ne/nd « 0.003, reflecting that the density of electrons is tiny and the charge of the up quarks has to be balanced by approximately twice as many down quarks. At /ix,e = Mi o n the other hand, the real solution to Eq. (3.18) is y = 0, i.e., the up and down Fermi momenta become equal. This can be seen most easily if one inverts the problem and solves Eq. (3.18) for x at given y. When y = 0 one finds x = 1, meaning that /id = Hu and, in turn, suggesting that pairing between up and down quarks is unobstructed at fj,Lc = /j,. This is in contrast to the case of vanishing MLe, when the two Fermi surfaces are split by about 25%, and pairing is difficult. It is appropriate to mention that many features of the above considerations would not change much even when Cooper pairing is taken into account. The reason is that the corresponding corrections to the quark densities are parametrically suppressed by a factor of order (A//i) 2 . In order to estimate the magnitude of the effect in the case of dense matter in protoneutron stars, we indicate several typical values of the lepton fractions YLC in Fig. 3.5. As mentioned earlier, Y/, is expected to be of order 0.4 right after the

Phase Diagram of Neutral Quark Matter at Moderate

77

Densities

1.3

1.25

1.2

1-15

i •D

1.1 1.05 1 0

0.2

0.4

0.6

0.8

1

Fig. 3.5. Ratio of down and up quark chemical potentials as a function of fiLe /pi in the toy model. The crosses mark the solutions at several values of the lepton fraction.

collapse of the iron core of a progenitor star. According to Fig. 3.5, this corresponds to Hd/fJ-u ~ 1-1, i.e., while the splitting between the up and down Fermi surfaces does not disappear completely, it gets reduced considerably compared to its value in the absence of trapped neutrinos. This reduction substantially facilitates the cross-flavor pairing of up and down quarks. The effect is gradually washed out during about a dozen of seconds of the deleptonization period when the value of Yi e decreases to zero. The toy model is easily modified to the opposite extreme of three massless quark flavors. Basically, this corresponds to replacing Eq. (3.18) by 2(1 + \yf

- 2(1 -

l

-yf - (x - yf

= 0.

(3.19)

In the absence of neutrino trapping, x = 0, the only real solution to this equation is y = 0, indicating that the chemical potentials (which also coincide with the Fermi momenta) of up, down, and strange quarks are equal. This reflects the fact that the system with equal densities of up, down, and strange quarks is neutral by itself, without electrons. With increasing x oc /i/, e , the solution requires a nonzero y oc fiQ, suggesting that up-down and up-strange pairing becomes more difficult. To see this more clearly, we can go one step further in the analysis of the toy model. Let us assume that the quarks are paired in a regular, i.e., fully gapped, CFL phase at T = 0. Then, as shown in Ref. 22, the quark part of the matter is automatically electrically neutral. Hence, if we want to keep the whole system electrically and color neutral, there must be no electrons. Obviously, this is easily realized without trapped neutrinos by setting \IQ equal to zero. At non-vanishing fiLc the situation is more complicated. The quark part is still neutral by itself and therefore no electrons are admitted. Hence, the electron chemical potential

78

5. B. Rilster et al.

Me = MLe ~~ VQ m u s t vanish, and consequently /XQ should be nonzero and equal to [J,LC • It is natural to ask what should be the values of the color chemical potentials /X3 and fis in the CFL phase when /i^ e ^ 0. In order to analyze the stress on the CFL phase due to nonzero Mi e , w e follow the same approach as in Refs. 2 and 6. In this analytical consideration, we also account for the effect of the strange quark mass simply by shifting the strange quark chemical potential by —M2/(2fi). In our notation, pairing of CFL-type requires the following "common" values of the Fermi momenta of paired quarks:

( 3 - 20a )

P?™?*M = H-l£> 6^ ' P'FTZI)

= »+(f6 +2V3' T7z>

( 3 - 20b )

-common _ .. , M £ , M3 _ _M8_ PF,(ra,0„)-M+ g + 4 4 y 5

M^ 4/x

, , (3.20C)

,

common _ _ MO _ M3 _ _M8_ _ M ^ PF,( 9 .,M) " ^ 4 4/x 3 4v/3

, (

,, }

These are used to calculate the pressure in the toy model, 3

3

common

By making use of this expression, one easily derives the neutrality conditions as in Eq. (3.14). In order to solve them, it is useful to note that

Thus, it becomes obvious that charge neutrality requires fin = HLC. The other useful observation is that the expression for n^ is proportional to ^3 + (IQ . So, it is vanishing if (and only if) /X3 = —^J.Q, which means that Hz — —^Le- Finally, one can check that the third neutrality condition n& = 0 requires

The results for the charge chemical potentials [in, ^3, and /x8 imply the following magnitude of stress on pairing in the CFL phase: s

V(rd,gu) =

g 2

S^rsM) = ^ " ^

= MLe>

=^

vi - ns„ M2 * / W - ) = ^ ^ = f^-.

(3.24a)

+ -£,

(3.24b) (3.24c)

Phase Diagram of Neutral Quark Matter at Moderate

Densities

79

Note that there is no mismatch between the values of the chemical potentials of the other three quarks, /i™ = fig = fisb = \i — Ms2/(6/u). From Eq. (3.24) we see that the largest mismatch occurs in the (rs, bu) pair (for positive /U£,J. The CFL phase can withstand the stress only if the value of Sfi^rStbu^ is less than A2. A larger mismatch should drive a transition to a gapless phase exactly as in Refs. 5 and 6. Thus, the critical value of the lepton-number chemical potential is HlJ&A2-—.

(3.25)

When (J,LC > i/r"', the CFL phase turns into the gCFL' phase, which is a variant of the gCFL phase. 6 By definition, the gapless mode with a linear dispersion relation in the gCFL' phase is rs-bu instead of gs-bd as in the standard gCFL phase. (Let us remind that the mode aa-bf3 is defined by its dispersion relation which interpolates between the dispersion relations of hole-type excitations of aa-quark at small momenta, k <S /i", and particle-type excitations of 6/3-quark at large momenta,

k » nl) In order to see what this means for the physics of protoneutron stars, we should again try to relate the value of /i£ e to the lepton fraction. As we have seen, there are no electrons in the (regular) CFL phase at T = 0. Therefore, the entire lepton number is carried by neutrinos. For the baryon density we may neglect the pairing effects to first approximation and employ the ideal-gas relations. This yields

Y

-"\{^)'•

(326

»

Inserting typical numbers, n > 400 MeV and /i£ e < A < 100 MeV, one finds YLC ;$ 1 0 - 3 . Thus, there is practically no chance to find a sizable amount of leptons in the CFL phase. The constraint gets relaxed slightly at nonzero temperatures and/or in the gCFL phase, but the lepton fraction remains rather small even then (our numerical results indicate that, in general, YLC < 0.05 in the CFL phase). 3.4.2. Three-dimensional

phase

diagram

The simple toy-model considerations in the previous subsection give a qualitative understanding of the effect of neutrino trapping on the mismatch of the quark Fermi momenta and, thus, on the pairing properties of two- and three-flavor quark matter. Now, we turn to a more detailed numerical analysis of the phase diagram in the NJL model defined in Section 3.2. The general features of the phase diagram in the three-dimensional space, spanned by the quark chemical potential fi, the lepton-number chemical potential fiLc, and the temperature T, are depicted in Fig. 3.6. Because of a rather complicated structure of the diagram, only the four main phases (xSB, NQ, 2SC and CFL) are shown explicitly. Although it is not labeled, a thin slice of a fifth phase, the uSC phase, squeezed in between the 2SC and CFL phases, can also be seen. While lacking detailed information, the phase diagram in Fig. 3.6 gives a clear overall picture. Among other things, one sees, for example, that the CFL

S. B. Riister et al.

80

phase becomes strongly disfavored with increasing \xt„ and gets gradually replaced by the 2SC phase. / j L e [MeVj

1 0 0

200

T [MeV

350

400 ju [MeV]

450

500

Fig. 3.6. General structure of the phase diagram of neutral dense quark matter in the threedimensional space spanned by the quark chemical potential fi, the lepton-number chemical potential /*£, , and the temperature T,

In order to discuss the structure of the phase diagram in more detail we proceed by showing several two-dimensional slices of it. These are obtained by keeping one of the chemical potentials, /* or / i L e , fixed and varying the other two parameters.

3.4.3. T-fj, phase

diagram

We begin with presenting the phase diagrams at two fixed values of the leptonnumber chemical potential, juLc = 200 MeV (upper panel) and fiLe = 400 MeV (lower panel), in Fig. 3.7. The general effects of neutrino trapping can be understood by analyzing the similarities and differences between the diagrams with and without neutrino trapping. Here it is appropriate to note that a schematic version of the T-p, phase diagram at nLs = 200 MeV was first presented in Ref. 3, see the right panel of Fig. 4 there. If one ignores the complications due to the presence of the uSG phase and various gapless phases, the results of Ref. 3 are in qualitative agreement with the results presented here.

Phase Diagram of Neutral Quark Matter at Moderate

Densities

81

70 60

^ — ~ \

5- 40 a 5, H

NQ

\

50

^^^•~S2§g^-"""'

^

v —

30

XSB

20 •

/^TgCFL

\

/^-uSC

2SC

r

•

CFL

10 •

320

340

360

380

400 420 u[MeV]

440

460

480

500

70 60 50

NQ •

5 : 40 u 5,

\

~~lg2Sc"

g2SC'T

y<**fg2SC

t- 30 .XSB

2SC

-

20

gCFL'-f

10 320

340

360

380

400 420 u[MeV]

440

460

480

500

Fig. 3.7. The phase diagrams of neutral quark matter at fixed lepton-number chemical potentials A*Le = 200 MeV (upper panel), and /j£,e = 400 MeV (lower panel), cf. Fig. 3.1 in the absence of neutrino trapping.

In order to understand the basic characteristics of different phases in the phase diagrams in Fig. 3.7, we also present the results for the dynamical quark masses, the gap parameters, and the charge chemical potentials. These are plotted as functions of the quark chemical potential in Figs. 3.8 and 3.9, for two different values of the temperature in the case of HLC = 200 MeV and |ijr,e = 400 MeV, respectively. As in the absence of neutrino trapping, see Fig. 3.1, there are roughly four distinct regimes in the phase diagrams in Fig. 3.7. At low temperature and low quark chemical potential, there is the xSB region. Here quarks have relatively large constituent masses which are close to the vacuum values, see Figs. 3.8 and 3.9. We

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S. B. Rilster et al.

Mu Md Ms

500

2 2

E-

400

-

300

.

200

-

-

100

•

2

S 350

400 450 n[MeV]

350

400 450 |i[MeV]

350

400 450 H[MeV]

350

400 450 H[MeV]

5" 2

350

400 450 |i[MeV]

350

400 450 (i [MeV]

500

Fig. 3.8. Dependence of the quark masses, of the gap parameters, and of the electric and color charge chemical potentials on the quark chemical potential at a fixed temperature, T = 0 MeV (three upper panels) and T = 40 MeV (three lower panels). The lepton-number chemical potential is kept fixed at nLc = 200 MeV.

note that the xSB phase is rather insensitive to the presence of a nonzero leptonnumber chemical potential. With increasing HLC the phase boundary is only slightly shifted to lower values of /i. This is just another manifestation of the strengthening of the 2SC phase due to neutrino trapping. With increasing temperature, the xSB phase turns into the normal quark matter phase, whose qualitative features are little affected by the lepton-number chemical potential. The third regime is located at relatively low temperatures but at quark chemical potentials higher than in the xSB phase. In this region, the masses of the up and down quarks have already dropped to values well below their respective chemical potentials while the strange quark mass is still large, see left columns of panels in Figs. 3.8 and 3.9. As a consequence, up and down quarks are quite abundant but strange quarks are essentially absent. It turns out that the detailed phase structure in this region is very sensitive to the lepton-number chemical potential. At HLC = 0, the pairing between up and down quarks is strongly hampered by the constraints of neutrality and f3 equilibrium, see

Phase Diagram of Neutral Quark Matter at Moderate

600

l

M 500

-

u

-

i

-A, - A2 80 ' ~ *3

M , •••

/

400 •>•

\


2

300

60

-

40

-

20

-

0)

I

2

1

3"

83

Densities

200 100

v^ 350

'

400 450 n[MeV]

350

80

I

60

"

' 400 450 n[MeV)

4, A2

"

^3

100

\

•••'''

•

\

-

50 40

20

-

•

0

-

— fo

— H 350

400 450 MMeV]

. / .

350

'

400 450 H(MeV]

••• H e

350

400 n[MeV]

Fig. 3.9. Dependence of the quark masses, of the gap parameters, and of t h e electric and color charge chemical potentials on the quark chemical potential at a fixed temperature, T = 0 MeV (three upper panels), and T = 40 MeV (three lower panels). The lepton-number chemical potential is kept fixed at /t£ e = 400 MeV.

Fig. 3.1. The situation changes dramatically with increasing the value of the leptonnumber chemical potential, see Fig. 3.7. The low-temperature region of the normal phase of quark matter is replaced by the (g)2SC phase (e.g., at /i = 400 MeV, this happens at ^x,e ~ 110 MeV). With /J,LC increasing further, no qualitative changes happen in this part of the phase diagram, except that the area of the (g)2SC phase expands slightly. Finally, the region in the phase diagram at low temperatures and large quark chemical potentials corresponds to phases in which the cross-flavor strangenonstrange Cooper pairing becomes possible. In general, as the strength of pairing increases with the quark chemical potential, the system passes through regions of the gapless uSC (guSC), uSC, and gCFL phases and finally reaches the CFL phase. (Of course, the intermediate phases may not always be realized.) The effect of neutrino trapping, which grows with increasing the lepton-number chemical potential, is to push out the location of the strange-nonstrange pairing region to larger values of fx. Of course, this is in agreement with the general arguments in Section 3.4.1.

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As we mentioned earlier, the presence of the lepton-number chemical potential fiLc leads to a change of the quark Fermi momenta. This change in turn affects Cooper pairing of quarks, facilitating the appearance of some phases and suppressing others. As it turns out, there is also another qualitative effect due to a nonzero value of fiLc. In particular, we find several new variants of gapless phases which did not exist at vanishing in,c. In Figs. 3.7 and 3.11, these are denoted by the same names, g2SC or gCFL, but with one or two primes added. We define the g2SC as the gapless two-flavor color-superconducting phase in which the gapless excitations correspond to rd-gu and gd-ru modes instead of the usual ru-gd and gu-rd ones, i.e., u and d flavors are exchanged as compared to the usual g2SC phase. The g2SC phase becomes possible only when the value of (fJ-ru — Hgd)/2 =• (MQ + M3)/2 is positive and larger than A3. The other phases are defined in a similar manner. In particular, the gCFL' phase, already introduced in Section 3.4.1, is indicated by the gapless rs-bu mode, while the gCFL" phase has both gs-bd (as in the gCFL phase) and rs-bu gapless modes. The definitions of all gapless phases are summarized in the following table: Name g2SC g2SC guSC gCFL gCFL' gCFL" 3.4.4. Lepton fraction

Gapless mode(s)

e(k)~\k-kf\ ru-gd, gu-rd rd-gu, gd~ru rs-bu gs-bd rs-bu gs-bd, rs-bu

Diquark condensate (s) A3 A3

A2, Ai, Ai, Ai,

A3 A2, A3 A2, A3 A2, A3

YL^

Our numerical results for the lepton fraction YLC are shown in Fig. 3.10. The thick and thin lines correspond to two different fixed values of the lepton-number chemical potential, fiLe ~ 200 MeV and //£ e = 400 MeV, respectively. For a fixed value of MLe, w e find that the lepton fraction changes only slightly with temperature. This is concluded from the comparison of the results at T = 0 (solid lines), T = 20 MeV (dashed lines), and T = 40 MeV (dotted lines) in Fig. 3.10. As is easy to check, at T = 40 MeV, i.e., when Cooper pairing is not so strong, the fi dependence of YLC does not differ very much from the prediction in the simple two-flavor model in Section 3.4.1. By saying this, of course, one should not undermine the fact that the lepton fraction in Fig. 3.10 has a visible structure in the dependence on /x at T = 0 and T = 20 MeV. This indicates that quark Cooper pairing plays a nontrivial role in determining the value of YLC . Our numerical study shows that it is hard to achieve values of the lepton fraction more than about 0.05 in the CFL phase. Gapless versions of the CFL phases, on the other hand, could accommodate a lepton fraction up to about 0.2 or so, provided the quark and lepton-number chemical potentials are sufficiently large. From Fig. 3.10, we can also see that the value of the lepton fraction YLC ~ 0.4, i.e., the value expected at the center of the protoneutron star right after its creation, requires the lepton-number chemical potential HLC in the range somewhere between

Phase Diagram of Neutral Quark Matter at Moderate

360

380

400

420

440

460

Densities

480

85

500

M. [MeV] Fig. 3.10. Dependence of the electron family lepton fraction YLC f ° r A*Le = 200 MeV (thick lines) and fj,Le = 400 MeV (thin lines) on the quark chemical potential at a fixed temperature, T = 0 MeV (solid lines), T = 20 MeV (dashed lines), and T = 40 MeV (dotted lines).

200 MeV and 400 MeV, or slightly higher. The larger the quark chemical potential fi the larger /i£ e is needed. Then, in a realistic construction of a star, this is likely to result in a noticeable gradient of the lepton-number chemical potential at the initial time. This gradient may play an important role in the subsequent deleptonization due to neutrino diffusion through dense matter. 3.4.5. T—//i a phase

diagram

Now let us discuss the phase diagram in the plane of temperature and leptonnumber chemical potential, keeping the quark chemical potential fixed. Two such slices of the phase diagram are presented in Fig. 3.11. The upper panel corresponds to a moderate value of the quark chemical potential, /i = 400 MeV. This could be loosely termed as the "outer core" phase diagram. The lower panel in Fig. 3.11 corresponds to /i = 500 MeV, and we could associate it with the "inner core" case. Note, however, that the terms "inner core" and "outer core" should not be interpreted literally here. The central densities of (proto-)neutron stars are subject to large theoretical uncertainties and, thus, are not known very well. In the model at hand, the case p, = 400 MeV ("outer core") corresponds to a range of densities around 4/9o, while the case p, = 500 MeV ("inner core") corresponds to a range of densities around Wpo- These values are of the same order of magnitude that one typically obtains in models (see, e.g., Ref. 29).

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0

50

100

150

200 250 t k . [MeV]

300

60

—

400

"inner core"

NQ 50

350

g2SC'T'

— g2SC

40 •

(D

2 30 20

2SC

——^2-^

OOT^^uSC

•

\ ^v

•

CFL

10 ..--"" 50

100

150

200 250 H. [MeV]

300

•

•

9CFL' 350

400

Fig. 3.11. The phase diagrams of neutral quark matter in the plane of temperature and leptonnumber chemical potential at two fixed values of quark chemical potential: fj. = 400 MeV (upper panel) and fj, = 500 MeV (lower panel). The triangle in the lower panel denotes the transition point from the CFL phase to the gCFL' phase at T = 0.

At first sight, the two diagrams in Fig. 3.11 look so different that no obvious connection between them could be made. It is natural to ask, therefore, how such a dramatic change could happen with increasing the value of the quark chemical potential from ju = 400 MeV to /j, = 500 MeV. In order to understand this, it is useful to place the corresponding slices of the phase diagram in the three-dimensional diagram in Fig. 3.6. The /j, = 500 MeV diagram corresponds to the right-hand-side surface of the bounding box in Fig. 3.6. This contains almost all complicated phases with strange-

Phase Diagram of Neutral Quark Matter at Moderate Densities

87

nonstrange cross-flavor pairing. The /x = 400 MeV diagram, on the other hand, is obtained by cutting the three-dimensional diagram with a plane parallel to the bounding surface, but going through the middle of the diagram. This part of the diagram is dominated by the 2SC and NQ phases. Keeping in mind the general structure of the three-dimensional phase diagram, it is also not difficult to understand how the two diagrams in Fig. 3.11 transform into each other. Several comments are in order regarding the zero-temperature phase transition from the CFL to gCFL' phase, shown by a small black triangle in the phase diagram at /i = 500 MeV, see the lower panel in Fig. 3.11. The appearance of this transition is in agreement with the analytical result in Section 3.4.1. Moreover, the critical value of the lepton-number chemical potential also turns out to be very close to the estimate in Eq. (3.25). Indeed, by taking into account that Ms « 214 MeV and A 2 ~ 76 MeV, we obtain ^ = A 2 - M s 2 /(2/i) « 30 MeV which agrees well with the numerical value. Before concluding this subsection, we should mention that a schematic version of the phase diagram in T~HLC plane was earlier presented in Ref. 3, see the left panel in Fig. 4 there. In Ref. 3, the value of the quark chemical potential was /i = 460 MeV, and therefore a direct comparison is not straightforward. One can see, however, that the diagram of Ref. 3 fits naturally into the three-dimensional diagram in Fig. 3.6. Also, the diagram of Ref. 3 is topologically close to the \i = 500 MeV phase diagram shown in the lower panel of Fig. 3.11. The quantitative difference is not surprising: the region of the (g)CFL phase is considerably larger at /i = 500 MeV than at \x = 460 MeV.

3.5. Conclusions Here we discussed the phase diagram of neutral three-flavor quark matter in the space of three parameters: temperature T, quark chemical potential fi, and leptonnumber chemical potential /X£,e. The analysis is performed in the mean-field approximation in the phenomenologically motivated NJL model introduced in Ref. 31. Constituent quark masses are treated as dynamically generated quantities. The overall structure of the three-dimensional phase diagram is summarized in Fig. 3.6 and further detailed in several two-dimensional slices, see Figs. 3.1, 3.7 and 3.11. By making use of simple model-independent arguments, as well as detailed numerical calculations in the framework of an NJL-type model, we find that neutrino trapping helps Cooper pairing in the 2SC phase and suppresses the CFL phase. In essence, this is the consequence of satisfying the electric neutrality constraint in the quark system. In two-flavor quark matter, the (positive) lepton-number chemical potential ^£ c helps to provide extra electrons without inducing a large mismatch between the Fermi momenta of up and down quarks. With reducing the mismatch, of course, Cooper pairing gets stronger. This is in sharp contrast to the situation in the CFL phase of quark matter, which is neutral in the absence of electrons. Additional electrons due to large HLC can only put extra stress on the system. In application to protoneutron stars, the findings presented here suggest that the CFL phase is very unlikely to appear during the early stage of the stellar evolution

88

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before the deleptonization is completed. If color superconductivity occurs there, the 2SC phase is the best candidate for the ground state. In view of this, it might be quite n a t u r a l to suggest t h a t m a t t e r inside protoneutron stars contains little or no strangeness (just as the cores of the progenitor stars) during the early times of their evolution. In this connection, it is appropriate to recall t h a t neutrino trapping also suppresses the appearance of strangeness in the form of hyperonic m a t t e r and kaon condensation. 2 9 T h e situation in quark m a t t e r , therefore, is a special case of a generic property. After the deleptonization occurs, it is possible t h a t the ground state of m a t t e r at high density in the central region of the star is the C F L phase. This phase contains a large number of strange quarks. Therefore, an a b u n d a n t production of strangeness should h a p p e n right after the deleptonization in the protoneutron star. If realized in nature, in principle this scenario may have observational signatures.

Acknowledgments This review is based on work of Refs. 15 and 16. This work was supported in p a r t by the Virtual Institute of the Helmholtz Association under grant No. VHVI-041, by the Gesellschaft fur Schwerionenforschung (GSI), a n d by the Deutsche Forschungsgemeinschaft (DFG).

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Phase Diagram of Neutral Quark Matter at Moderate Densities

13. 14. 15. 16. 17. 18. 19. 20.

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24. 25.

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Chapter 4 Spontaneous Nambu-Goldstone Current Generation Driven by Mismatch Mei Huang Physics Department, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan [email protected] We review recent progress toward understanding and resolving instabilities driven by mismatch between the Fermi surfaces of the pairing quarks in a 2-flavor color superconductor. With increasing mismatch, the 2SC phase exhibits chromomagnetic instability as well as color-neutral baryon-current instability. We describe the 2SC phase in the nonlinear realization framework and show that each instability implies spontaneous generation of the corresponding pseudo NambuGoldstone current. The Nambu-Goldstone current-generation state encompasses the gluon phase as well as the one-plane-wave LOFF state. Furthermore, when the charge neutrality condition is required, there exists a narrow unstable LOFF (Us-LOFF) window within which not only off-diagonal gluons but also the diagonal 8th gluon cannot avoid magnetic instability. In this Us-LOFF window, the diagonal magnetic instability cannot be cured by an off-diagonal gluon condensate in the color-superconducting phase.

Contents 4.1. Introduction 4.2. The gauged SU(2) Nambu-Jona-Lasinio model 4.3. Chromomagnetic instabilities driven by mismatch 4.3.1. Screening masses of the gluons A = 1,2,3 4.3.2. Screening masses of diagonal gluon A — 8 4.3.3. Screening masses of the gluons with A = 4, 5,6, 7 4.4. Color-neutral baryon-current instability 4.4.1. Spontaneous baryon-current generation 4.4.2. Unstable neutral LOFF window 4.5. Spontaneous Nambu-Goldstone current generation 4.6. Conclusion and discussion Bibliography

4.1.

91 93 95 96 96 97 99 100 101 102 105 106

Introduction

Studying Q C D at finite baryon density lies within the expanded subject area of nuclear physics. In astrophysics, the behavior of Q C D a t finite baryon density and 91

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M. Huang

low temperature is central to an understanding of the structure of compact stars and conditions near the core of collapsing stars (supernovae, hypernovae). It is known that sufficiently cold and dense baryonic matter is in the color-superconducting phase. This scenario was proposed several decades ago by Frautschi, 1 and Barrois, 2 having noted that one-gluon exchange between two quarks is attractive in the colorantitriplet channel. From BCS theory, 3 we know that if there is a weak attractive interaction between fermions in a cold Fermi sea, the system is unstable with respect to formation of a particle-particle Cooper-pair condensate in momentum space. Studies of the color-superconducting phase in 1980's are discussed in Ref. 4. The topic of color superconductivity has stirred much interest in the last decade. 5-8 For recent reviews, see Ref. 9. The color-superconducting phase may exist in the central region of compact stars. To stabilize bulk matter inside compact stars, charge-neutrality as well as (3 equilibrium are necessary conditions. 10-12 These constraints induce a mismatch between the Fermi surfaces of the pairing quarks. It is clear that the Cooper pairing will eventually be destroyed when the mismatch grows sufficiently large. Without the charge-neutrality constraint, the system may exhibit a first-order phase transition from the color-superconducting phase to the normal phase as the mismatch increases. 13 It has also been found that the system can enter a spatially non-uniform LOFF (Larkin-Ovchinnikov-Fulde-Ferrell) state 1 4 , 1 5 in a certain window of moderate mismatch. The manner in which Cooper pairing is destroyed with increasing mismatch in a charge-neutral system is still not fully understood. Charge neutrality plays an essential role in determining the ground state of the system. If this condition is satisfied globally, and also the surface tension is small, a mixed phase will be favored.16 Precise calculation of the surface tension in the mixed phase is difficult. In the following discourse we shall therefore apply the charge-neutrality condition locally and focus on the homogeneous phase. It has been found that homogeneous, neutral, cold, dense quark matter can be in the gapless 2SC (g2SC) phase 17 or gapless CFL (gCFL) phase, 18 depending on the flavor structure of the system. The gapless state resembles the unstable Sarma state. 19 ' 20 However, under a natural charge-neutrality condition - i.e., only neutral matter can exist - the gapless phase is indeed a thermally stable state as shown in Refs. 17,18. The existence of thermally stable gapless color-superconducting phases was confirmed in Ref. 21 and generalized to finite temperatures in Ref. 22. Recent results based on more careful numerical calculations show that the g2SC and gCFL phases can exist at moderate baryon density in the color-superconducting phase diagram. 23 One of the most important properties of an ordinary superconductor is the Meissner effect: expulsion of the magnetic field from the superconductor. 24 In ideal color-superconducting phases, e.g., in the 2SC and CFL phases, the gauge bosons connected with the broken generators acquire masses, signaling the presence of a Meissner screening effect.25 The Meissner effect can be understood using the standard Anderson-Higgs mechanism. Unexpectedly, it was found that in the g2SC phase, the Meissner screening masses for five gluons corresponding to broken generators of SU(3)C become imaginary, which indicates a type of chromomagnetic

Spontaneous

Nambu-Goldstone

Current

93

Generation

instability in the g2SC phase. 26 ' 27 The calculations in the gCFL phase show the same type of chromomagnetic instability. 28 Recalling the discovery of a superfluiddensity instability 29 in the gapless interior-gap state, 30 it seems that the instability is an inherent property of gapless phases. (There are some exceptions: (i) It has been shown that there is no chromomagnetic instability near the critical temperature. 31 (ii) It is also found that the gapless phase in the strong-coupling region is free of any instabilities. 32 ) The chromomagnetic instability in the gapless phase still remains a puzzle. Observing that the 8th gluon's chromomagnetic instability is related to the instability with respect to the virtual net momentum of a diquark pair, Giannakis and Ren have suggested that a LOFF state might be the true ground state. 3 3 Their further calculations show that there is no chromomagnetic instability in a narrow LOFF window when the local-stability condition is satisfied. 34 ' 35 Later, in Ref. 36 it was found that a charge-neutral LOFF state cannot cure the instability of off-diagonal 4-7th gluons, whereas a gluon-condensate state 3 7 can do the job. In Ref. 38, we further pointed out that when charge neutrality is imposed, there exists another narrow unstable LOFF window in which not only off-diagonal gluons but also the diagonal 8th gluon cannot avoid magnetic instability. In a minimal model of gapless color superconductors, Hong 39 showed that the mismatch of Fermi surfaces can induce spontaneous Nambu-Goldstone current generation. In the U(l) case, the Nambu-Goldstone current-generation state resembles the one-plane-wave LOFF state or the diagonal gauge boson's condensate. We have extended the Nambu-Goldstone current-generation picture to the 2SC case in the nonlinear realization framework,38 showing that the five pseudo Nambu-Goldstone currents can be spontaneously generated by increasing the disparity between the Fermi surfaces of the pairing quarks. The Nambu-Goldstone current-generation state encompasses the gluon phase as well as the one-plane-wave LOFF state. This chapter is organized as follows. In Section 4.2, we describe the gauged SU(2) Nambu-Jona-Lasinio (gNJL) model in /3-equilibrium. We review what is known about chromomagnetic instabilities in Section 4.3. Neutral baryon-current instability and the LOFF state are studied in Section 4.4. Section 4.5 presents a general description of Nambu-Goldstone current generation in the nonlinear realization framework. Our conclusions are summarized and discussed in Section 4.6.

4.2. The gauged SU(2) Nambu-Jona-Lasinio model We adopt the gauged form of the extended Nambu-Jona-Lasinio model, 40 for which the Lagrangian density is expressed as C = q{ilp + /i7 0 ) 9 + GS[(qqf + (qij5Tq)2} + GD{(iqceebl5q)(iqeebl5qc)}, a

(4.1) a

with JDM = aM - igA*T . Here the A% are gluon fields and the T with a = 1, • • • ,8 are generators of SU(3)C gauge groups. We regard all the gauge fields as external fields that interact weakly with the system. The property of the colorsuperconducting phase characterized by the diquark gap parameter is determined

94

M. Huang

by the unknown nonperturbative gluon fields, which have been simply replaced by the four-fermion interaction in the NJL model. However, the external gluon fields do not contribute to the properties of the system; for that reason, we do not include the contribution to the Lagrangian density from the gauge-field part Cg as introduced in Ref. 37. (In Section 4.5, we will use the nonlinear realization in the gNJL model and derive a Nambu-Goldstone current state that is equivalent to the so-called gluon-condensate state.) In the Lagrangian density of Eq. (4.1), qc = CqT and q0 = qTC are chargeconjugate spinors, and C = i^y2j° is the charge-conjugation matrix. (The superscript T denotes the transposition operation.) The quark field q = qia, with i = u,d and a = r,g,b, is a flavor doublet and color triplet, as well as a four-component Dirac spinor. Further, T = (T1,T2,TZ) are Pauli matrices in flavor space, and {e)lk = e'k and (e6)Q/3 = eQ/3b are totally antisymmetric tensors in the flavor and color spaces. In /3-equilibrium, the matrix ft, of chemical potentials in color-flavor space is given in terms of the quark chemical potential //, the chemical potential for the electrical charge /i e , and the color chemical potential [i&: Ihf = (^ij

~ HeQiiW? + - ^ / i s M W •

(4-2)

In this expression, Gs and GD are the quark-antiquark coupling constant and the diquark coupling constant, respectively. In the following, we only focus on the color-superconducting phase, where (qq) = 0 and (q~l5Tq) = 0. After bosonization, one obtains the linearized version A* b A 6

C--i.sc = q{ip + h°)q -

l

-^\ifeebl5q)

-^—

- ±Ab(iqeebl5qc)

(4.3)

of the model for the 2-flavor superconducting phase, with the bosonic fields Afc ~ ifeeb-yBq,

A*b ~ iq£ebj5qc.

(4.4)

In the Nambu-Gor'kov space where

the inverse of the quark propagator is defined as

[5(P)]- 1 =f[ G o( P J 1 r 1

,l

\

A+

A

"

V

(4.6)

[G^{P)\ V

with the off-diagonal elements A - = -ie 6 £7 5 A,

A + = -iebe'y5A*,

(4.7)

and the free quark propagators GQ (P) take the form of [G±(P)]-1 = 7 ° ( P 0 ± A ) - 7 - P (4-8) The 4-momenta are denoted by capital letters, e.g., P — (po,p). We have assumed the quarks are massless in dense quark matter, and that the external gluon fields do not contribute to the quark self-energy.

Spontaneous

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95

Generation

The explicit forms of the functions Gf and ETJ read c ±

_

(fc0-E*h°A+

1

(k0

T

<5M)2 - Ef

{k0-Etr)l°~K

G ± = 2

|

(k0 +

Elh°K

(ko =F 6rf

- Ef

(fco-t-^ r )7°Afc

+

2

(k0±5tl)2-Ef,

(fco ± <5/i) - Ef

1

nr-u

'

.

1

Gf = — - ± 7°A^ + I T - ^ ^ K fc fco0- + £ 6 d fco - b&t with 2?* = Ek ± Mia, and „±

=

"12

-iA75A+

/

(4-9)

~^7 5 Afc

+

V (fco ± <*M)2 - Ef

(k0 ± <5/z)2 - Ef

t

-21

where

is an alternative set of energy projectors. The following notation has been used: E^Ek±fl, —

A* —

^,fc =

/ V

^±)2

P u r + Mdg _ Mug + (J-dr

+ A2

M-

(412)

'

Me

M8

6 3 2 2 <- — Mcig ~ AW _ Mcir ~ Mug _ rMe^ /< , o i M _ [ 2 ~~ 2 ~ 22 ' ' One sees from the dispersion relation (4.12) of the quasiparticles that when Sfi > A, there will be excitation of gapless modes in the system. The thermodynamic potential corresponding to the solution for the gapless state with Sfj, > A is a local maximum. However, under certain constraints, e.g., the charge-neutrality condition, the gapless 2SC phase can be a thermally stable state. 17

4.3. Chromomagnetic instabilities driven by mismatch In this section, we review the situation for chromomagnetic instabilities driven by mismatch in the 2SC and g2SC phases, as established in Refs. 26, 27. The polarization tensor in momentum space has the general structure n%B{P) = -y^TrD[t'XS{K)t»BS(K-PJ\

,

(4.14)

where the trace runs over the Dirac indices, and the vertices are given by T^A = diag( 5 7 "T J 4, -g^Tl) with A = 1 , . . . ,8.

96

M. Huang

The Debye masses m2D A and the Meissner masses m2M A of the gauge bosons are defined as m2D
4.3.1. Screening

masses

(4.15) 11*^(0,p).

(4.16)

of the gluons A = 1, 2, 3

Gluons A = 1,2,3 of the unbroken SU(2)C subgroup couple only to the red and green quarks. The general expression for the polarization tensor 1 1 ^ ( 0 , p) with A,B = 1,2,3 is diagonal. After performing the traces over the color, flavor, and Nambu-Gor'kov indices, the expression has the form n2T ^-^ r rlzV + rGt{K)YGt{K')

+

YG^{K)YG^{K')

+ - f ^ W ^ i K ' )

+ rztiiKh^iK')}

.

(4.17)

Making use of the definitions (4.15) and (4.16), we arrive at the result

2

4a s /i 2 J/x

mitl ^T T V;:r:\j^ W-A2'

- A)

( 4 - 18 )

for the threefold-degenerate Debye-mass squared, with as = g2/4ir. For the Meissner-mass squared we find m2M 1 — 0. The Debye screening mass in Eq. (4.18) vanishes in the gapped phase (i.e., A/ 1). As for the ideal 2SC phase, this reflects the fact that there are no gapless quasiparticles charged with respect to the unbroken SU(2)C gauge group. In the gapless 2SC phase, such quasiparticles exist and the value of the Debye screening mass is proportional to the density of states at the corresponding "effective" Fermi surfaces. The Meissner screening mass of the gluons of the unbroken SU(2)C group vanishes in the gapped and gapless 2SC phases. This is in agreement with general group-theoretical arguments.

4.3.2. Screening

masses

of diagonal gluon A = 8

The 8th gluon can probe the Cooper-paired red and green quarks, as well as the unpaired blue quarks. After the traces over the color, flavor, and Nambu-Gor'kov

Spontaneous

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Current

97

Generation

indices are performed, the polarization tensor for the 8th gluon is specified by

n£(P) = |n£(P) + fn£ 6 (P), n^s(P) = ^ +

7

E

/

(4.19)

( 0 ^ D [/G+(if)rG+(if')+7"GrW/Gr(^)

"G2+(WG+(fO +

/1 7

G 2 - ( W G j (#')

- YE^Kh^iK') ' r^ti(Kh^2(K')] , n%AP)

=^

E / ( S ^ D [7MG3+( W ^ i T )

+ 7 " G + (tf h " G + (A") + 7 ^G 4 - (K)-fG^ For the Debye screening mass one has

(4.20)

+ 7^3(X)7^3-(X')

(K')} .

(4.21)

= f^£ ,

(4.22)

and the Meissner screening mass is given by mhss = ^(l-S»9^-A)). (4.23) M 88 V ; ' 9TT \^ ^(6n)2-AiJ As easily seen from Eq. (4.23), the square of the Meissner screening mass of the 8th gluon is negative when 0 < A/Sfj, < 1, indicating a magnetic plasma instability in the gapless 2SC phase. 4.3.3. Screening

masses

of the gluons with A = 4, 5 , 6 , 7

After performing the traces over the color, flavor, and Nambu-Gor'kov indices, the diagonal components of the polarization tensor Ti^g(P) with A = B = 4,5,6,7 have the form

+ rG+(Kh»Gl(K')

+

^Gt(K)^Gt(K')

+ rG^{K)YGl{K')

+

^G^(K)YG^{K')

+ 7 "G 4 - (K)YG^ {K') + 7 "G 2 - (K)>fG; (K')] . (4.24) We note that U^(P) = U^{P) = IIg£(P) = n ^ ( P ) . Apart from the diagonal elements, there are also nonzero off-diagonal elements, n £ ( P ) = - i C ( P ) = n ^ ( P ) = - n ^ ( P ) = ifi^(P), (4.25) with n""(P) = ^

y ^3TVD[7^G+(^)7^G+(^') -7/iG+(K)7yG3+(^')

+ 7"G+(X)7-G+(^') rG+(K)YG+(K') - rG^(K)rG^(K') + YG^(K)YG^(K') - 7 ^G 4 - (K)^G^ (K') + 7 " G J ( W G 4 (K')} .

(4.26)

98

M. Huang

The physical gluon fields in the 2SC/g2SC phase are given by the linear combinations Al6 = (Al ± iA$)/y/2 and i £ 7 = (A% ± iA^)/V2. These new fields, A$fi and A? 5 , describe two pairs of massive vector particles with well-defined electromagnetic charges, Q = ± 1 . The components of the polarization tensor in the new basis read fi£(P)

= fii£(P)

= UZ(P) + n ^ ( P )

=— J

J^^b"Gi{K)YGt{K')+YG^{K)YG^{K')

+rG+(K)j»G+(K') + -fG^{K)-fG^{K')]

(4.27)

and ng£(P) = n ^ ( P ) = n ^ ( P ) - n^(P) =

4 /

^jlTrD^G+^h^+^O+T^-^h^r^')

+ y G J (^)7"G+ (K') + j»G; (K)YG^ (K')] .

(4.28)

In the static limit, all four eigenvalues of the polarization tensor are degenerate. Making use of the definition (4.15), we may derive the following result for the corresponding Debye masses: 2

1

D,A

_ 4asfi2 — ~~

A 2 + 26M2 —2A5

W ^

2

- A A^

2

^ 6{Sfl

.. ~

A)

(4.29)

Here we have assumed that fis vanishes, which is a good approximation in neutral two-flavor quark matter. The fourfold-degenerate Meissner screening mass of the gluons with A = 4, 5,6, 7 reads 2

_ 4as/iJ

2 ^ + W3gE5^_ A) A

2A 2

2

(4.30)

The results (4.29) and (4.30) for the Debye and Meissner masses interpolate between the known results in the normal phase (i.e., A/S/J, = 0) and in the ideal 2SC phase (i.e., A/S/J, = oo). The instability of off-diagonal gluons appears in the whole region of gapless 2SC phases (with 0 < A/S/i < 1) and even in some gapped 2SC phases (with 1 < A/S/J, < V2). We observe that the square of the Meissner screening mass for the off-diagonal gluons decreases monotonically to zero when the mismatch increases from zero to S/J, = A/\f2, then turns negative with further increase of the mismatch in the gapped 2SC phase. However, the behavior of the diagonal 8th gluon's Meissnermass squared is quite different, as it remains constant in the gapped 2SC phase. In the gapless 2SC phase, the Meissner-mass squared for all five of these gluons is negative.

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99

4.4. Color-neutral baryon-current instability It is not understood why gapless color-superconducting phases exhibit chromomagnetic instability. This behavior seems quite strange, especially in the g2SC phase where it is the electrical neutrality, not color neutrality, that plays the essential role. It is puzzling that the gluons can feel the instability due to the imposition of electrical neutrality on the system. In order to understand what is really going 'wrong' with the homogeneous g2SC phase, we would like to know whether there exist other instabilities besides the chromomagnetic instability. For that purpose, we probe the g2SC phase using different external sources, e.g., scalar and vector diquarks, mesons, vector currents, and so on. Here we report the most interesting result, which concerns the response of the g2SC phase to an external vector current V* = •tpj^i/j. The time and spatial components of this current correspond respectively to the baryon number density and the baryon current. From linear-response theory, the induced current and the external vector current are related through the response function

I W ) = H E ^ [t^S(K)tvS(K - P)] ,

(4.31)

K

where the trace runs over the Nambu-Gor'kov, flavor, color, and Dirac indices. The explicit form of the vertices is Ty = diag(7 M , —7M), while the vector-current response function takes the form n^(P) = n^(P) + n^(P), (4.32) with

+ rGUK)YG+(K') -

7

+

^G^(K)rG^(K')

^ r # ) / S + (K') - 7 " S + ( ^ ) 7 " S i i (K')

- 7 ^ ( ^ ) 7 ^ + (K1) - 7 " 3 + ( f f h " 3 r 2 ( f f ' ) ] , Kfi(P)

= I E /

( ^ j a ^ [ rGt(K)YGt(K')

(4-33)

+ 7 " G 3 - ( ^ ) 7 ^ 3 (K')

+ ^G+(K)YGt(K') + -fGKKWG^K')} , (4.34) the traces being performed over the Dirac space. Comparing the above formula for nvl/(P) with the formula (4.19) for the 8th gluon's self-energy Ilgg (P), it is seen that II y"(P) and Ilgg (P) are given by almost the same expression, the difference residing only in the coefficients. This can be easily understood, because the color charge and color current carried by the 8th gluon are proportional to the baryon number and baryon current, respectively. In the static long-wavelength limit (p 0 = 0 and p —> 0 ), the time and spatial components of Ily^ (P) determine the baryon-number susceptibility £ n and the baryon-current susceptibility £c, respectively, according to £„ = - lim n o v 0 ( 0 , p > m ^ , (4.35) p—*0

'

lim (9ij + ?&) n^(0,p) oc ml \ p J

P-*O

M

.

(4.36)

100

M. Huang

In the g2SC phase, m\ M becomes negative as well as £c. This means that in addition to (i) the chromomagnetic instability corresponding to broken generators of SU(3)C and (ii) the instability with respect to the net momentum of a diquark pair, the g2SC phase is also (iii) unstable with respect to an external color-neutral baryon current •tpjip. The magnetic instability of the 8th gluon, the diquark-momentum instability, and the color-neutral baryon-current instability in the g2SC phase can all be understood within a common physical picture. The g2SC phase exhibits a paramagnetic response to an external baryon current. Naturally, the color current carried by the 8th gluon, which differs from the baryon current by a color charge, also experiences the instability in the g2SC phase. The paramagnetic instability of the baryon current implies that the quark can spontaneously obtain a momentum, and because a diquark carries twice the quark momentum, it is not hard to see why the g2SC phase is also unstable with respect to a net diquark momentum. It should be noted that instability of ipjip will be induced by mismatch in all asymmetric Fermi pairing systems, including superfluid systems, in which ipjip can be interpreted as particle current. 4.4.1. Spontaneous

baryon-current

generation

The paramagnetic response to an external vector current naturally suggests that a vector current can be spontaneously generated in the system. The vector current so generated behaves as a vector potential that modifies the quark self-energy through a spatial vector condensate 7-Ey a n d breaks the rotational symmetry of the system. Moreover, it can be seen that the quasiparticles in the gapless phase spontaneously acquire a superfluid velocity, and the ground state becomes anisotropic. The quark propagator G^(P) in Eq. (4.8) is modified as G

o,v(P)}

_1

= 7°(Po ± A) " 7 " P T 7 • Vv ,

(4.37)

where the subscript V refers to the modified quark propagator. Correspondingly, the inverse [<S(P)]_1 of the quark propagator (see Eq. (4.6)) is modified as [SviP)]-^

L ° ^ v >\ A+

.

(4.38)

Goy(P)

We observe that this expression for the modified inverse quark propagator takes the same form as the inverse quark propagator in the one-plane-wave LOFF state considered in Ref. 34. The net momentum q of the diquark pair in the LOFF state 3 4 is replaced here by a spatial vector condensate S y , which breaks rotational symmetry of the system, such that the Fermi surfaces of the pairing quarks are no longer spherical. In fact, the baryon current is just one candidate for a Doppler-shifted superfluid velocity of the quarks. A spontaneously generated Nambu-Goldstone current in the minimal gapless model, 39 or a condensate of the 8th gluon's spatial component, plays the same role. All three candidates mimic the one-plane-wave LOFF state,

Spontaneous

Nambu-Goldstone

Current

Generation

101

so for the sake of simplicity in the following development, we refer to them all as a single-plane-wave LOFF state. Finding the deformed structure of the Fermi surfaces calls for self-consistent minimization of the free energy r(XV, A , ^ , fie, n&). The explicit form of the free energy can be constructed directly using the standard method. In the framework of the Nambu-Jona-Lasinio model, 12,17 one obtains

-Is/

dZP ^..nc (2TT)3

,™-l

1 T r l n Vl( [ l S VKv ( P ) r ) + - 7 ^ ,

"

'

(4-39)

4GD

where T is the temperature and GD is the coupling constant in the diquark channel. When there is no charge-neutrality condition, the ground state is determined by the thermal stability condition, i.e., the local-stability condition. The ground state is (i) in the 2SC phase when <5/x < 0.706A0 with A ~ A 0 , (ii) in the LOFF phase when 0.706A0 < Sfi < 0.754A0 and correspondingly 0 < A / A 0 < 0.242, and (hi) in the normal phase with A = 0 when the mismatch is larger than 0.754Ao- (Here, A and Ao stand for the diquark gap in the cases 5JJL ^ 0 and 5[i = 0, respectively.) 4.4.2.

Unstable

neutral LOFF

window

We now focus on the charge-neutral LOFF state and ask whether it can resolve all of the magnetic instabilities. When the charge-neutrality condition is in force, the ground state of the system is to be determined by solving the gap equations simultaneously with the chargeneutrality condition, i.e., solving dT ^ =

0

>

dT 7^

= 0

>

dT dT 75-=°' 7T-=0-

4 40

-

As shown explicitly in Ref. 36, by adjusting Ao or the coupling strength GD, the solution for the charge-neutral LOFF state can stay everywhere in the full LOFF window, including the window not protected by the local-stability condition. From the study of the charge-neutral g2SC phase, we learn the lesson that even though the neutral state is thermally stable, i.e., the thermodynamic potential is a global minimum along the neutrality line, this does not guarantee dynamical stability of the system. The question of stability of the neutral system must be further investigated in terms of the dynamical stability condition, i.e., positivity of the Meissner-mass squared. The polarization tensor for the gluons having color A = 4,5,6,7,8 should be evaluated using the modified quark propagator Sy in Eq. (4.38), i.e., n^B(P) = ± £ £ T r [ f £ S v ( 1 0 r & S v ( t f - P ) ]

,

(4.41)

K

with A, B = 4,5,6,7,8. The vertices have the explicit form T^ = diag(<j,7'1T/i, —g-f^Tj). In the LOFF state, the Meissner tensor can be decomposed into transverse and longitudinal components. In Ref. 34, these components have been evaluated explicitly in the single-plane-wave LOFF state for the off-diagonal 4-7 gluons and the diagonal 8th gluon.

102

M. Huang

According to the dynamical stability condition, i.e., positivity of the both the transverse and the longitudinal Meissner-mass squared, we can define three LOFF windows for the LOFF state: 41 (i) The stable LOFF (S-LOFF) window in the region 0 < A / A 0 < 0.39, which is free of any magnetic instability. This S-LOFF window is somewhat wider than the window 0 < A/Ao < 0.242 protected by the local-stability condition. (ii) The stable window for the diagonal gluon, characterized by the Ds-LOFF window in the region 0.39 < A / A 0 < 0.83, which is free of the diagonal 8th gluon's magnetic instability but not free of the off-diagonal gluons' magnetic instability. (iii) The unstable LOFF (Us-LOFF) window in the region 0.83 < A / A 0 < rc, with rc = A(Sfj, = A)/Ao ^ 1. In this Us-LOFF window, all the magnetic instabilities are in effect. It is important to note that the longitudinal Meissnermass squared for the 8th gluon is negative in this Us-LOFF window, while the transverse Meissner-mass squared of the 8th gluon is always zero in the full LOFF window, as guaranteed by the momentum equation. Us-LOFF is a very interesting window, as it indicates that the LOFF state cannot even cure the 8th gluon's magnetic instability. In the charge-neutral 2flavor system, it seems that the diagonal gluon's magnetic instability cannot be cured in the gluon phase, because there is no direct relation between the diagonal gluon's instability and the off-diagonal gluons' instability. (Of course, it has to be carefully checked whether all the instabilities in this Us-LOFF window can be cured by the off-diagonal gluons' condensate in the charge-neutral 2-flavor system.) It is also to be noted that in this Us-LOFF window, the mismatch is close to the diquark gap, i.e., S/J, ~ A. Therefore it will be interesting to check whether this Us-LOFF window can be stabilized by a spin-1 condensate, 42 as proposed in Ref. 39. Although it is unlikely, there is some chance to cure the diagonal instability in the charge-neutral 2SC phase by the condensation of off-diagonal gluons. It is expected that this instability will show up in some constrained Abelian asymmetric superfluid systems, e.g., in the fixed-number-density case. 43 Solving this problem would answer a significant new challenge.

4.5. Spontaneous Nambu-Goldstone current generation We have seen that besides chromomagnetic instability corresponding to broken generators of SU(3)C, the g2SC phase suffers from instability with respect to the external neutral baryon current. All of the instabilities in play are induced by increasing mismatch between the Fermi surfaces where Cooper pairing takes place. We now take up the important question of how instability is driven by mismatch. A superconductor will eventually be destroyed and enter the normal Fermiliquid state, so one natural question is: just how will an ideal BCS superconductor be destroyed by increasing mismatch? To answer this question, one must first understand what it means to be a superconductor. The superconducting phase is characterized by the order parameter A(x), which is a complex scalar field. In the example of an electronic superconductor, it has the form A(x) = |A|e 1 ¥ > ' x \ where

Spontaneous

Nambu-Goldstone

Current

Generation

103

|A| is the amplitude and

J2fa(x)Ta + —= if8(x)B

(4.42)

where ipa(a = 4, • • • ,7) and ips are five Nambu-Goldstone diquarks, and we have neglected the singular phase, which should include information on topological defects. 47 ' 48 The operator V is unitary. We next introduce a new quark field x that is connected with the original quark field q in Eq. (4.3) through a nonlinear transformation, q = vx, The charge-conjugate fields transform as

q = x^-

(4.43)

qc = V* xc , qc = Xc VT . (4.44) 47 For high-Tc superconductors, this technique is called charge-spin separation. The advantage of transforming the quark fields is that this preserves the simple structure of the terms coupling the quark fields to the diquark sources, qcA+q = xc^+X,

qA~ qc = X^~Xc-

(4.45)

104

M. Huang

In mean-field approximation, the diquark source terms have the proportionalities $ + ~ (XC X),

$~ ~ (X Xc > •

(4.46)

Introducing the new Nambu-Gor'kov spinors X = (

x

* ) ,

X = {x,xc),

(4-47)

the nonlinear realization of the original Lagrangian density Eq. (4.3) takes the form Ct^XS-fX--^-,

(4.48)

where Sni

=

{

* + [Go,-,]-1;-

( 4 4 9 )

The explicit expression for the free propagator of the new quark field is specified by [ G j , J - 1 = i ^ + A7o + 7 ^

M

(4-50)

and [GoJ-1=iPT-f^lo+l,VS.

(4.51)

Comparing with the free propagator in the original Lagrangian density, the free propagator in the nonlinear realization framework naturally takes into account the contribution from the phase fluctuations or Nambu-Goldstone currents V" = Vf (id11) V, V£ = VT (i&*)V,

(4.52)

which may be recognized as the NcNf x iVc./Vf-dimensional Maurer-Cartan one-form introduced in Ref. 46. The linear order of the Nambu-Goldstone currents V1 and VQ is given by W--Y,(#V-) a=4

Ta - -j= ( f l ^ s ) B,

V£ ~ J2 (^V-) Tr+^(d^s) a=4

(4.53)

*

BT .

(4.54)

*

The Lagrangian density Eq. (4.48) for the new quark field looks like an extension of the theory of Ref. 47 for high-Tc superconductor to a non-Abelian system, except that we have neglected the singular phase contribution from topological defects. We reiterate that the advantage of the nonlinear realization framework surrounding Eq. (4.48) is that it can naturally incorporate the contribution from the phase fluctuations or Nambu-Goldstone currents. The remaining task is to correctly solve for the ground state in the presence of phase fluctuations. The free energy T(V^, A, (i, fig, fj,e), which has the form

r^S/H^^'l-'l + iS^

(4 55)

-

Spontaneous

Nambu-Goldstone

Current

Generation

105

can be evaluated directly. Determination of the ground state corresponding to this free energy as a function of mismatch requires tedious effort and is still in progress. Here we shall briefly address salient aspects of the Nambu-Goldstone currentgeneration state, 39 the one-plane-wave LOFF state, 33 ' 34 and the gluon phase. 37 If we expand the thermodynamic potential r(V M , A,/i,/xs,/i e ) of the nonlinear realization form in terms of the Nambu-Goldstone currents, Nambu-Goldstone current generation will naturally occur with increasing mismatch, i.e., one will find (Y2a=4 ^Va) i1 0 and/or (V^g) ^ 0 at large 6/J,. The system enters an extended version of the Nambu-Goldstone current-generation state proposed in a minimal gapless model. 39 ' 49 From Eq. (4.48), we can see that V
and the CFL phase is preferred by a factor (27/4) 1 / 3 ~ 1.89. 5.2.4. Mass

terms

Mass terms modify the parameters in the effective Lagrangian. These parameters include the Fermi velocity, the effective chemical potential, the screening mass, the BCS terms and the Landau parameters. At tree level the correction to the Fermi velocity and the chemical potential are given by vF = l - - T , S^=- — . (5.19) 2pF 2pF The shift in the Fermi velocity also affects the coupling gvp of a magnetic gluon to quarks. It is important to note that at leading order in g this the only mass correction to the coupling. This is not entirely obvious, as one can imagine a process in which the quark emits a gluon, makes a transition to a virtual high energy state,

The CFL Phase and ms

R

115

R

R Y

L

> X

A

MM

+

X > M+

M

R

R

8

-*—^-x >

L

M g2MM

R

S

L

-*—^—x > M

Fig. 5.2. Mass terms in the high density effective theory. T h e first diagram shows a O ( M M t ) term that arises from integrating out the ip— field in the QCD Lagrangian. The second diagram shows a 0(M2) four-fermion operator which arises from integrating out t/>_ and hard gluon exchanges.

and then couples back to a low energy state by a mass insertion. This process would give an 0(m/fi) correction to g, but it vanishes in the forward direction. 26 Quark masses modify quark-quark scattering amplitudes and the corresponding Landau and BCS type four-fermion operators. Consider quark-quark scattering in the forward direction, v + v' —> v + v'. At tree level in QCD this process receives contribution from the direct and exchange graph. In the effective theory the direct term is reproduced by the collinear interaction while the exchange terms has to be matched against a four-fermion operator. The spin-color-flavor symmetric part of the exchange amplitude is given by

"<"•*••'>-^y&l'-fe}'

<520)

'

where Cp = (JV2 — l)/(2iV c ) and x = v • v' is the scattering angle. We observe that the amplitude is independent of x in the limit m —> 0. Mass corrections are singular as x —> 1. The means that the Landau coefficients Fi contain logarithms of the cutoff. We note that there is one linear combination of Landau coefficients, •fo — Fi/3, which is cutoff independent. Equations (5.19-5.20) are valid for Nf > 1 degenerate flavors. Spin and color anti-symmetric BCS amplitudes require at least two different flavors. Consider BCS scattering v + (-v) —> v' + (-t>') in the helicity flip channel L + L —• R+ R. The color-anti-triplet amplitude is given by M(v,-v;v,-v) = —-^-—^-, 4

PF

(5.21) PF

where mi and m 2 are the masses of the two quarks and CA = {Nc + 1)/(2NC). We observe that the scattering amplitude is independent of the scattering angle. This means that at leading order in g and m only the s-wave potential VQ is non-zero.

116

T. Schdfer

In order to match Green functions in the high density effective theory to an effective chiral theory of the CFL phase we need to generalize our results to a complex mass matrix of the form C = —$LMI})R — ^RM^IPL, see Fig. 5.2. The 5fi term is (5.22)

and the four-fermion operator in the BCS channel is C = ^T(^Wf)(i>cRC4,°){Tr

[XAM{XD)TXBM{Xcf - ^Tr [XAM(XD)T]

Tr [A B M(A c ) r ] }.

Here, we have introduced the CFL eigenstates ipA denned by tpf = A = 0,...,8. 5.2.5. Normal

quark

(5.23)

ipA(XA)ai/y/2,

matter

Before we consider quark mass effects in the superconducting phase we would like to review mass corrections in the normal phase of quark matter. We will consider three flavor quark matter with massless up and down quarks and massive strange quarks. There are some simple mass effects that we can directly deduce from Eq. (5.19). The strange quark chemical potential is shifted by S/J,S = m2/(2pF) with respect to the strange quark Fermi momentum, and the strange quark Fermi velocity is smaller than one. It would seem that perturbative corrections to these results are sensitive to momenta far away from the Fermi surface and cannot be computed in the framework of an effective field theory for modes near pp. Landau showed, however, that Galilei invariance leads to a relation between the chemical potential and the Fermi momentum that only involves the interaction on the Fermi surface. This relation was generalized to relativistic systems by Baym and Chin. 27 They showed that fidfi

=

PF + 4 fr? - frlS M

(5.24)

dpF,

where F^ are color-flavor-spin symmetric Landau coefficients. We observe that this relation involves precisely the combination of Landau coefficients that does not depend on the EFT cutoff. This implies that we can integrate Eq. (5.24) as in ordinary Landau Fermi Liquid theory. The result can be used to determine to thermodynamic potential to 0(g2). We get 28 ' 29

n=

127T 2

+

MW(M

as{Nl - 1) 167T 3

2

--m2) + -m4lnf

3 ( m

2

l n ^

—

fiu

2u4

(5.25)

with u = \/IJL2 — m2. This result corresponds to a momentum space subtraction scheme. The thermodynamic potential in the MS scheme was computed by Fraga

The CFL Phase and m,

117

and Romatschke. 30 Perturbative corrections reduce the pressure of a quark gas. The same is true for mass corrections to the pressure in the non-interacting system. The 0{asm2) term, however, increases the pressure. This is related to the fact that the exchange energy in a degenerate quark gas changes sign in going from the nonrelativistic limit, dominated by the Coulomb interaction, to the relativistic system in which magnetic interactions are more important. In applications to neutron stars we have to include weak interactions and enforce electric charge neutrality. Weak interactions can convert strange quarks into up quarks, electrons and neutrinos. We will assume that neutrinos can leave the system. Under these assumptions the system is characterized by a baryon chemical potential (j, and an electron chemical potential fie. The electron chemical potential is fixed by the condition of electric charge neutrality, (d£l)/(dfj,e) = 0. To leading order in m2s/pF the electron chemical potential is given by

^^f.U-^U**)).

(5.26)

ApF V 71" \ms )) At tree level we find /i e = A*s /2 and the Fermi surfaces are split symmetrically pF=pF

+ fis/2,

pF=pF-fis/2.

(5.27)

Perturbative corrections reduce the magnitude of/i e . In fact, since the 0(as) term is enhanced by a large logarithm log(p^/m), the electron chemical potential can become negative. This result is probably not reliable. In particular, the 0(as) term is modified if the MS mass is used.

5.3. Chiral theory of the CFL phase 5.3.1.

Introduction

The main topic of this review is the effect of the strange quark mass on the CFL phase. In principle this problem can be studied by solving a gap equations which includes mass corrections to the chemical potential, the Fermi velocity, and the quark-quark interaction. In full QCD this has not been attempted yet, but there are many calculations of this type that are based on effective four-fermion models of QCD. We will compare our results with some of these calculations in Section 5.3.8. There are several difficulties with microscopic calculations of this kind. The first is that they typically require an ansatz for the gap parameter. We will see that both flavor symmetry and isospin are broken, and that the gaps for the left and right handed fermions acquire a relative phase. This means that the number of independent gap parameters is quite large. The second difficulty is that while the ideal CFL state is automatically neutral with respect to all charges, this is no longer the case once the state is perturbed by a finite strange quark mass. This means that we have to introduce flavor chemical potentials and gluonic background fields. Finally, we know that chiral symmetry is broken in the CFL phase and this implies that physical observables are non-analytic in the quark mass. These non-analyticities are related to Goldstone bosons. This suggests that we should study the effects of a non-zero strange quark mass using an effective

T.

118

Schafer

R

L

L

R

R

L

L

R

Fig. 5.3. Contribution of the 0(M2) the CFL phase.

BCS four-fermion operator to the condensation energy in

field theory for the Goldstone modes. Indeed, it is well known that chiral symmetry places important constraints on the mass dependence of QCD observables, and that these constraints are most easily implemented by using an effective Lagrangian. 5.3.2. Chiral effective

field

theory

For excitation energies smaller than the gap the only relevant degrees of freedom are the Goldstone modes associated with the breaking of chiral symmetry and baryon number. Since the pattern of chiral symmetry breaking is identical to the one at T = \i = 0 the effective Lagrangian has the same structure as chiral perturbation theory. The main difference is that Lorentz-invariance is broken and only rotational invariance is a good symmetry. The effective Lagrangian for the Goldstone modes is given by 31 Ceff = ^ T r [VoSVoSt - vld&d&]

+ [BTr(M^)

+ h.c]

+ [AiTr(MS t )Tr(MS t ) + i 4 2 T r ( M S t M E t ) + A 3 T r ( M S t ) T r ( M t S ) + h.c] +....

(5.28)

a a

Here £ = exp(i<j) X / fw) is the chiral field, fn is the pion decay constant and M is a complex mass matrix. The chiral field and the mass matrix transform a s E - > L'ER^ and M —» LMItf under chiral transformations (L, R) e SU(3)L x SU(3)R. We have suppressed the singlet fields associated with the breaking of the exact U(l)v and approximate U(1)A symmetries. At low density the coefficients f„, B,Ai,... are non-perturbative quantities that have to be extracted from experiment or measured on the lattice. At large density, on the other hand, the chiral coefficients can be calculated in perturbative QCD. For the derivative terms this is most easily done by matching the two-point functions of flavor currents between QCD and the chiral theory. The mass terms can be computed by matching the mass dependence of the vacuum energy. At leading order in as the Goldstone boson decay constant and velocity are 32

Mass terms are determined by the operators studied in Section 5.2.4. We observe that both Eq. (5.22) and (5.23) are quadratic in M. This implies that B = 0 in

The CFL Phase and ms

119

perturbative QCD. B receives non-perturbative contributions from instantons, but these effects are small if the density is large, see Section 5.3.7. We observe that XL = M M V ( 2 P F ) and XR = M^M/(2pF) in Eq. (5.22) act as effective chemical potentials for left and right-handed fermions, respectively. Formally, the effective Lagrangian has an 5C/(3)^ x SU(3)R gauge symmetry under which XL,R transform as the temporal components of non-abelian gauge fields. We can implement this approximate gauge symmetry in the CFL chiral theory by promoting time derivatives to covariant derivatives, 33

The BCS four-fermion operator in Eq. (5.23) contributes to to the condensation energy in the CFL phase, see Fig. 5.3. We find 26 ' 32

A£ = - ^

j(TV[M])2-Tr[M2l j + f M ^ M t ) .

(5.31)

This term can be matched against the Ai terms in the effective Lagrangian. The result is 26 - 32 3A 2 M = -A2 = ^ j , A3 = 0. (5.32) The vacuum energy also receives contributions from Landau-type four-fermion operators, but these terms are proportional to Tr [MM*] and do not depend on the chiral field S. We can now summarize the structure of the chiral expansion in the CFL phase. The effective Lagrangian has the form k

-**(£) !

©"(^)'MW

(-,

Loop graphs in the effective theory are suppressed by powers of 9/(4^/^). Since the pion decay constant scales as fn ~ pp Goldstone boson loops are suppressed compared to higher order contact terms. We also note that the quark mass expansion is controlled by m2/(ppA). This is means that the chiral expansion breaks down if m2 ~ pFA. This is the same scale at which BCS calculations find a transition from the CFL phase to a less symmetric state. 5.3.3. Kaon

condensation

Using the chiral effective Lagrangian we can now determine the dependence of the order parameter on the quark masses. We will focus on the physically relevant case ms > mu = md- Because the main expansion parameter is m 2 /(pirA) increasing the quark mass is roughly equivalent to lowering the density. The effective potential for the order parameter is Veff = ^ T r [2XLZXRtf

-X\-

X\\

- Ax [(Tr(ME f )) 2 - Tr ((MS f ) 2 ) (5.34)

120

T. Schafer

CFLK

m 72n (MeV)

Fig. 5.4. Phase structure of CFL matter as a function of the effective chemical potential fis = s / ( 2 p F ) and the lepton chemical potential fiQ, from Kaplan & Reddy (2001). A typical value of fis in a neutron star is 10 MeV. Tn

The first term contains the effective chemical potential ^is = m2s/{2pF) and favors states with a deficit of strange quarks (with strangeness S = — 1). The second term favors the neutral ground state S = 1. The lightest excitation with positive strangeness is the K° meson. We therefore consider the ansatz £ = exp(iaA4) which allows the order parameter to rotate in the K° direction. The vacuum energy is

sin(a)2 + ( r 0 2 ( c o s ( a ) - l )

V(a)

(5.35)

where {m°K)2 = (4Ai jf%)m(m + ms). Minimizing the vacuum energy we obtain fis < muK

cos(o;)

(<)' ~i%~ fis >m°K

(5.36)

The hypercharge density is nY = UHs

1 -

(™0K)4

/4

(5.37)

This result has the same structure as the charge density of a weakly interacting Bose condensate. Using the perturbative result for A\ we can get an estimate of the critical strange quark mass. We find ms(crit)

=3.03 •

m]/3A2/3,

(5.38)

from which we obtain ms(crit) ~ 70 MeV for A ~ 50 MeV. This result suggests that strange quark matter at densities that can be achieved in neutron stars is kaon condensed. We also note that the difference in condensation energy between the CFL phase and the kaon condensed state is not necessarily small. For /i s —> A we have sin(a) —• 1 and V(a) -> / 2 A 2 / 2 . Since / 2 is of order /z2/(27r2) this is comparable to the condensation energy in the CFL phase.

The CFL Phase and ms

121

m

ms Fig. 5.5. Phase structure of CFL matter as a function of the light quark mass m and the strange quark mass ms, from Kryjevski, Kaplan &c Schafer (2005).

The strange quark mass breaks the SU(3) flavor symmetry to SU(2)i x U{l)yIn the kaon condensed phase this symmetry is spontaneously broken to U(1)Q. If isospin is an exact symmetry there are two exactly massless Goldstone modes, 34 the K° and the K+. Isospin breaking leads to a small mass for the K+. The phase structure as a function of the strange quark mass and non-zero lepton chemical potentials was studied by Kaplan and Reddy, 35 see Fig. 5.4. We observe that if the lepton chemical potential is non-zero charged kaon and pion condensates are also possible. 5.3.4. Eta meson

condensation

The CFL phase also contains a very light 5 = 0 mode which can potentially become unstable. This mode is a linear combination of the 77 and 77' and its mass is proportional to mumd. Because this mode has zero strangeness it is not affected by the Us term in the effective potential. However, since mu,md -C ms this state is sensitive to perturbative asm23 corrections. The relevant contribution to the effective Lagrangian is 36 6C = -SA

[(TrMEt) 2 + Tr{M^)2]

+ h.c.

(5.39)

with

"=£**>

A2

„ (ln2)2

A2

(5.40)

Here, A6 is a color-flavor symmetric gap parameter which is generated by perturbative corrections to the dominant, color-flavor anti-symmetric gap. 2 Since the onegluon exchange interaction in the color-symmetric channel is repulsive the 0(asm2s) contribution to the mass of the 77 — 77' mode tends to cancel the 0{mumd) term. When the two terms become equal the eta condenses. The resulting phase diagram is shown in Fig. 5.5. The precise value of the tetra-critical point (m*,m*s) depends sensitively on the value of the coupling constant. At very high density m* is extremely small, but at moderate density 771* can become as large as 5 MeV, comparable to the physical values of the up and down quark mass.

122

T. Schdfer

40

60

80

100

2

m s/(2PF) [MeV] Fig. 5.6. This figure shows the fermion spectrum in the CFL phase. For ms = 0 there are eight fermions with gap A and one fermion with gap 2A (not shown). Without kaon condensation gapless fermion modes appear at /i s = A (dashed lines). With kaon condensation gapless modes appear at fis = 4 A / 3 .

5.3.5. Fermions

in the CFL

phase

So far we have only studied Goldstone modes in the CFL phase. However, as the strange quark mass is increased it is possible that some of the fermion modes become light or even gapless. 37 In order to study this question we have to include fermions in the effective field theory. The effective Lagrangian for fermions in the CFL phase is 38,39

£ = TV (NUifDpN) + |

- DTi ( J V V 7 5 {A», N}) - FTr ( ^ " 7 5

{ (Tr (NLNL)

- [Tr (NL)]2) - (L <- R) + h.c.} .

[A^N]) (5.41)

NL,R are left and right handed baryon fields in the adjoint representation of flavor SU(3). The baryon fields originate from quark-hadron complementarity. 40 We can think of N as describing a quark which is surrounded by a diquark cloud, NL ~ QL(QL9L)- The covariant derivative of the nucleon field is given by D^N = d^N + i[Vfj,, N}. The vector and axial-vector currents are

V» = -l-{l;d^

+ ?d^},

4* = -^(V/,St)£,

( 5.4 2 )

where £ is defined by £2 = E. It follows that £ transforms as £ —> L£,U{x)^ = U{x)£W with U(x) £ SU(3)V. For pure SU{3) flavor transformations L = R = V we have U(x) = V. F and D are low energy constants that determine the baryon axial coupling. In perturbative QCD we find D = F — 1/2. The effective theory given in Eq. (5.41) can be derived from QCD in the weak coupling limit. However, the structure of the theory is completely determined by chiral symmetry, even if the coupling is not weak. In particular, there are no free parameters in the baryon coupling to the vector current. Mass terms are also strongly

The CFL Phase and ms

123

Fig. 5.7. Left panel: Energy density as a function of the current JK for several different values of fis = f"s/(2pF) close to the phase transition. Right panel: Ground state energy density as a function of fi3. We show the CFL phase, the kaon condensed CFL (KCFL) phase, and the supercurrent state (curKCFL).

constrained by chiral symmetry. The effective chemical potentials (XL , XR) appear as left and right-handed gauge potentials in the covariant derivative of the nucleon field. We have D0N = doN + i\r0,N], (5.43)

r 0 = - \ {i {do + ixR) £f + £t (5o + ixL) £}, where XL = MM*/(2pF) and XR = M^M/(2pF) as before. (XL,XR) covariant derivatives also appears in the axial vector current given in Eq. (5.42). We can now study how the fermion spectrum depends on the quark mass. In the CFL state we have £ = 1. For fis = 0 the baryon octet has an energy gap A and the singlet has gap 2A. As a function of fis the excitation energy of the proton and neutron is lowered, wp
±3U

A±W,

fA±|Ms, W

«^°A=

A,

(5.44)

4

{ 2A. Numerical results for the eigenvalues are shown in Fig. 5.6. We observe that mixing within the charged and neutral baryon sectors leads to level repulsion. There are two modes that become light in the CFL window /i s < 2A. One mode is a linear combination of proton and S+ particles, as well as S~ and E"~ holes, and the other mode is a linear combination of the neutral baryons (n, S ° , S ° , A8, A 0 ). 5.3.6. Meson

supercurrent

state

Recently, several groups have shown that gapless fermion modes lead to instabilities in the current-current correlation function. 41,42 Motivated by these results we have

124

T. Schafer

examined the stability of the kaon condensed phase against the formation of a nonzero current. 43 ' 44 Consider a spatially varying U(l)y rotation of the maximal kaon condensate (I U(x)£K0U\x)

=

0 0 \ 0 1/V2 ie^W/V^ . VOie-^^Vv^ l/\/2 /

(5.45)

This state is characterized by non-zero currents V = ^[V<j>K) I 0 1 0 | ,

A=\ 2

(vK) ( 0 0 \Qie-i4,K

-ie'** J . 0 /

(5.46)

In the following we compute the vacuum energy as a function of the kaon current = VK- The meson part of the effective Lagrangian gives a positive contribution

3K

£ = \vllh\-

(5-47)

A negative contribution can arise from gapless fermions. In order to determine this contribution we have to calculate the fermion spectrum in the presence of a non-zero current. The relevant part of the effective Lagrangian is C = TV (NUv^D^N)

+ TV ( i V S (pA + v • AJ N^j

+ ^-{Tr(NN)-Tr(N)Tr(N)

+ h.c.},

(5.48)

where we have used D = F = 1/2. The covariant derivative is DQN = doN + i[pv, N] and DtN = dtN + iv • [V, N] with V, A given in Eq. (5.46) and

^ = 2^ ± € Vr

(5 49)

-

The vector potential py and the vector current V are diagonal in flavor space while the axial potential p& and the axial current A lead to mixing. The fermion spectrum is quite complicated. The dispersion relation of the lowest mode is approximately given by Ul

= A+ ^

^

- \»s - \v • j K ,

(5.50)

where I = v-p — pp and we have expanded u>i near its minimum IQ = (/zs + U - J K ' ) / 4 . Equation (5.50) shows that there is a gapless mode if p,s > 4A/3 — JK/3- The contribution of the gapless mode to the vacuum energy is

£/*/£

UJI6(-LJI),

(5.51)

where dfl is an integral over the Fermi surface. The integral in Eq. (5.51) receives contributions from one of the pole caps on the Fermi surface. The result has exactly the same structure as the energy functional of a non-relativistic two-component

125

The CFL Phase and ms

curCFLK m\

Ms 55

60

65

70

75

80

85

gCFLK

-3

Fig. 5.8. Screening mass of flavor gauge fields in the CFL phase. The two curves show the second derivative of the effective potential with respect to the current at the origin and at the minimum of the potential.

Fermi liquid with non-zero polarization, see Section 5.4. Introducing dimensionless variables _ ill. - 4 A

JK

aA' we can write £ = cAffh(x)

with (h + xf'2Q(h

fh(x) =x2

(5.52)

aA

x)5/2G(h

+ x)-{h-

- x)

(5.53)

X

We have defined the constants 2 C

~ 15 4 C > 6 '

V-"2*

'V

7T2 '

a

2

152C>4'

(5.54)

where cw = (21 — 81og(2))/36 is the numerical coefficient that appears in the weak coupling result for f„. According to the analysis in 45 the function fh{x) develops a non-trivial minimum if hi < h < hi with h\ ~ —0.067 and hi ~ 0.502. In perturbation theory we find a = 0.43 and the kaon condensed ground state becomes unstable for (A - 3 ^ / 4 ) < 0.007A. The energy density as a function of the current and the groundstate energy density as a function of \xa are shown in Fig. 5.7. In these plots we have included the contribution of a baryon current JB, as suggested in. 44 In this case we have to minimize the energy with respect to two currents. The solution is of the form JB ~ JK- The figure shows the dependence on JK for the optimum value of JB- We have not properly implemented electric charge neutrality. Since the gapless mode is charged, enforcing electric neutrality will significantly suppress the magnitude of the current. 46 We have also not included the possibility that the neutral mode becomes gapless. This will happen at somewhat larger values of /Us. We note that the ground state has no net current. This is clear from the fact that the ground state satisfies 8£/5{S74>K) = 0 . As a consequence the meson current

126

T. Schafer

Fig. 5.9.

Instanton contribution to the BCS four-fermion operator in QCD with three flavors.

is canceled by an equal but opposite contribution from gapless fermions. We also expect that the ground state has no chromomagnetic instabilities. Prom the effective Lagrangian we can compute the screening length of a SU(3)p gauge field. In the CFL phase the isospin and hypercharge screening masses are 2

d2£

mv = —^

VK

^-'''H'-iTK™' 1

(5 55)

'

which shows the magnetic instability for h > 0 and has the characteristic square root singularity observed in microscopic calculations. 41 In Fig. 5.8 we show the screening mass in the kaon condensed CFL phase and the supercurrent phase as a function of fj,s. We observe that there is an instability in the homogeneous phase, but the instability disappears in the supercurrent state. 5.3.7. Instanton

effects

The results discussed in Sects. 5.3.3-5.3.6 are based on an effective chiral theory of the CFL phase. The coefficients of the effective theory are determined by matching to perturbative QCD calculations. We find that the chiral coefficients are "natural", i.e. their numerical value agrees with simple dimensional estimates. For example, fl is given, up to a coefficient of order one, by the density of states on the Fermi surface. This suggest that many of our results are valid even if the QCD coupling is not weak. A possible exception is the linear mass term in the effective chiral theory. The coefficient B in Eq. (5.28) vanishes to all orders in perturbation theory, but B receives non-perturbative contributions from instantons. In QCD with three flavors instantons induce a four-fermion operator 47 C=

f . N J 2(27r/3)4/o3 _, I n(p,n)dp eflf2f3egig2g3Mhg3 2 _ --^{i>R,f1L,gi)(i>RJ2
f2Nc-l.T T I —^—{VR,hVL, a i ){VR,hVL, a 2 ) + (Af <-> M\ L •-+ R)\ , (5.56)

see Fig. 5.9. Here, fi,gi are flavor indices and <7M„ = f [7^,7^]- The instanton size distribution n(p, fi) is given by

127

The CFL Phase and ms

n{p, fj.) = CJV

/8TT 2

2iVc

8TT 2

exp

\92

)2

exp \—Nfp2p2

0.466 exp(-1.679iV c )1.34 JV, CN

(Nc

- 1)!(JV C -

(5.57) (5.58)

2)!

At zero density, the p integral in Eq. (5.56) diverges for large p. This is the wellknown infrared problem of the semi-classical approximation in QCD. At large chemical potential, however, large instantons are suppressed and the typical instanton size is p ~ pTx
h

{p,p)dp—(iip)

V

V27T 9

2 2

A(H

^

\2n\

Tr [M + M f ] .

(5.59)

This result can be matched against the O(M) term in the effective chiral Lagrangian. We find 8?r4 r ( 6 )

B = CN-

3

36

r

^ 3V2n

9

f. p?~ . -i 2 8 T T 2 \ 6 \2*2

fAQCD\

y-jrj

12

A

^c'

(5 60)

-

where we have performed the integral over the instanton size p using the oneloop beta function. The coefficient B is related to the quark-anti-quark condensate, (ipi}>) = — 2B. The linear mass terms contributes to the mass of the kaon, 5m?K ~ (4B/f%)mm a , and tends to inhibit kaon condensation. At moderate density this contribution is quite uncertain because the result is very sensitive to the value of the strong coupling constant. Using the one-loop running coupling and A Q C D — 200 MeV gives values as large as Sm^ ~ 100 MeV. This is almost certainly an overestimate because the main contribution comes from large instantons with size p ~ 0.5 fm. If the average instanton size is constrained to be less than the phenomenological value at zero density, p = 0.35 fm, then we find Smx < 10 MeV. 5.3.8. Microscopic

models

In this section we shall compare some of our results with microscopic calculations based on Nambu-Jona-Lasinio (NJL) models, see for example. 37 ' 48 A natural choice is to model the interaction between quarks by a local four-fermion interaction with the quantum numbers of one-gluon exchange £ = ${ip + pl0-M)ij

+ G (ij^\ai>)2.

(5.61)

This Lagrangian has the symmetries of QCD except that the SU(3) color symmetry is global rather than local, and the axial U(1)A is not anomalous. The effects of the anomaly can be taken into account by adding the instanton induced four-fermion interaction given in Eq. (5.56). Since color is a global symmetry the NJL model does not exhibit a Higgs mechanism and color superconductivity leads to an extra octet of colored Goldstone bosons. The pairing ansatz is usually taken to be l {CCTBV',-) ~ A i e a 6 1 e' i j l +' A * 2e°-ah2~e - ij2 + A3e

a63

< •ij3-

(5.62)

128

T. Schafer Table 5.1. Effective chemical potential for the different quark modes. The quark modes are labeled by their color (rgb) and flavor (uds). We also show the Q charge and the corresponding baryon mode. The leading order result in the neutral CFL phase corresponds to \ie = 113 = 0 and us = —fis • mode

Q

effective chemical potential

leading order

ru

0

~l^ e + k^3 + l^ 8

Mo

gd

0

+ |Me - ^M3 + 5M8

Mo

bs

0

+ 5^e - §M8 - Ps

Mo

rd

-1

+ jMe + §M3 +

|M8

Mo

£-

gu

1

- § M e - ^M3 + jM8

MO

E+

baryon mode

(Ao.As.So)

rs

-1

+ 5/V + 5 M + §M8 - Ms

Mo — Ms

S~

bu

1

- § M e - §M8

Mo + pis

P

gs

0

+ jMe - §M3 + 5M8 - Ms

MO -

bd

0

+ j M e - §M8

Mo +M«

=0

/J-s

n

This ansatz reduces to the CFL ansatz for At = A 2 = A3 and allows for the breaking of hypercharge and isospin. The ansatz is not sufficiently rich to allow for kaon condensation. The chiral field E can be defined as S = XY\

(5.63)

where X, Y are related to the left and right handed condensates

(W'L) W ^ V abc

6

{^R)tC^R)))e eijk

^ * ~ XI

(5.64)

c

(5.65)

~ Yk .

In order to study kaon condensation we have to allow the left and right handed order parameters to be independent. This problem was studied in two papers by Buballa and Forbes. 49 ' 50 Color and electric charge neutrality are enforced by adding chemical potentials (fi3,/j,s,fie) for color isospin, color hypercharge and electric charge. In most microscopic calculations the effect of the strange quark mass is taken into account by considering an effective chemical potential fis = m2s/(2pF) for the strange quark. In Table 5.1 we list the chemical potentials for the different quark states. The quarks are labeled by color (rgb) and flavor (uds). They are grouped into the main pairing channels in the CFL phase. We also show the leading order result in the charge neutral CFL phase. In this case we find /i e = ^3 = 0 and ^8 = —Ms a n d the dominant pair breaking stress occurs in the (rs) — (bu) and (gs) — (bd) sector. We can associate the quark states in the microscopic theory with the baryon fields in the effective theory by computing their quantum numbers under the unbroken SU(3)F symmetry. The effective theory is formulated in terms of gauge invariant fields and does not require color chemical potentials. In weak coupling we can derive the effective theory by integrating out gluonic degrees of freedom. In

The CFL Phase and ms

129

this case the equations of motion for the color gauge potential automatically enforce color neutrality. We observe that the leading order results in the NJL calculation agree with the results derived from the effective field theory. The comparison was extended to the kaon condensed phase by Forbes. 50 He finds that the bu (proton) and bd (neutron) states split, and that the pair breaking stress on both states is reduced. The critical /i s for gapless states is /j,s = 1.2A, in qualitative agreement with the leading order EFT result /xs = 4A/3. Microscopic calculations show that in the CFL phase the gap parameters A i ^ a are very similar even if the pair breaking stress /xs is close to the value of the gap. 37 The main effect is a reduction of the average gap which is presumably related to the decrease in the common Fermi momentum that occurs if /J,S is increased at constant jtzjg. In the gapless CFL phase, on the other hand, the splitting between the different gap parameters becomes significant. Neither one of these two effects is included in the EFT calculations discussed in Sects. 5.3.3-5.3.6. The response of the gap to the trace part of MM^/pF can be taken into account by adding a higher order operator to the effective Lagrangian Eq. (5.41), but the coefficient of this term is model dependent. Indeed, in perturbative QCD the gap will likely increase if pp is lowered. The splitting between the gaps in the gapless CFL is due to the fact that gapless regions on the Fermi surface no longer contribute to pairing. In the EFT treatment this effect is not included because the current is assumed to be much smaller than the gap. Clearly, in the regime where the number of gapless states is large the back-reaction on the gap has to be taken into account.

5.4. Cold atomic systems A nice system in which stressed pairing can be studied in the laboratory is a cold dilute gas of fermionic atoms. Using Feshbach resonances it is possible to tune the interaction between the Bose-Einstein (BEC) limit of tightly bound diatomic molecules and the Bardeen-Cooper-Schrieffer (BCS) limit of weakly correlated Cooper pairs. The most interesting part of the phase diagram is the crossover regime where the two-body scattering length diverges. A conjectured (and, most likely, oversimplified) phase diagram for a polarized gas is shown in Fig. 5.10. In the BEC limit the gas consists of tightly bound spin singlet molecules. Adding an extra up or down spin requires energy A. For \Sfi\ = |/if — /ijj > A the system is a homogeneous mixture of a Bose condensate and a fully polarized Fermi gas. One can show that in the dilute limit this mixture is stable with regard to phase separation. 51 The Bose-Fermi-gas mixture is a gapless superfiuid. We can ask, along the lines of Section 5.3.6, whether the system is stable with respect to the formation of a non-zero supercurrent. We can study this question using the effective Lagrangian

£ = V+ (ido - e(-id) - i{d
(5.66)

130

T. Schafer

8n

1

homogenoeous superfluid 1

*-

1/a*

1/a

Fig. 5.10. Conjectured phase diagram for a polarized cold atomic Fermi gas as a function of the scattering length a and the difference in the chemical potentials 5fi = y.-^ — fJ-i, from Son & Stephanov (2005).

Here, ip describes a gapless fermion with dispersion law e(p) and ip is the superfluid Goldstone mode. The low energy parameters ft and / are related to the density and the velocity of sound. Similar to the chiral theory discussed in Section 5.3.6 the p-wave coupling of the fermions to the Goldstone boson is governed by the U(l) symmetry of the theory. Setting up a current va = dip/m requires energy f2m2v2/2. The contribution from fermions can be computed using the fermion dispersion law in the presence of a non-zero current ev(p)=e(p)

+ vs-p-Sfi.

(5.67)

The total free energy is F(v3) = -nmv2

+J - ^

ev(p)Q (-ev(p)),

(5.68)

where n is the density and we have used f2 = n/m. Son and Stephanov noticed that the stability of the gapless phase depends crucially on the nature of the dispersion law e(p). For small momenta we can write e(p) ~ eo + ocp2 + /3p4. In the BEC limit a > 0 and the minimum of the dispersion curve is at p = 0 while in the BCS limit a < 0 and the minimum is at p ^ 0. In the latter case the density of states on the Fermi surface is finite and the system is unstable with respect to the formation of a non-zero current. The free energy functional is of exactly the same type as the one given in Eq. (5.53). On the other hand, if the minimum of the dispersion curve is at zero, then the density of states vanishes and there is no instability. As a consequence there is a critical point along the BEC-BCS line at which the instability will set in. We can also analyze the system in the BCS limit. This analysis goes back to the work of Larkin, Ovchinnikov, Fulde and Ferell, 52 ' 53 see the review. 54 First consider homogeneous solutions to the BCS gap equation for Sfi ^ 0. In the regime 5^ < AQ

131

The CFL Phase and ms

X homog. BCS

v\

/

X LOFF gapless / \ .

BCS

!

"X

*8u

Fig. 5.11. Schematic behavior of the gap parameter and the quasi-particle energies near the onset of the supercurrent state (left panel) and in the BCS limit (right panel).

where Ao = A(<5/x = 0) the gap equation has a solution with gap parameter A The free energy of this solution is N 2 2

F = -^(A -2ty )

Ao. (5.69)

where N is the density of states on the Fermi surface. For S/J, > Ao/2 there is a second solution with A = (2 5p, > Ao/\/2 the gap equation has a non-trivial solution, but the free energy is higher than the free energy of the normal phase and the solution is only meta-stable. LOFF studied whether it is possible to find a stable solution in which the gap has a spatially varying phase A(f) = Ae 2 '**.

(5.70)

This solution exists in the LOFF window Sfj-i < 5y. < 5^2 with 5p,i = A 0 / \ / 2 ~ 0.71Ao and S/J.2 — 0.754Ao- The LOFF momentum q depends on S/i. Near 5^2 we have qvp ~ 1.2Sfi. The gap A goes to zero near 6fi2 and reaches A ~ 0.25Ao at 6 fii. Clearly, the LOFF solution is of the same type as the supercurrent state. The U(l) of baryon number is spontaneously broken and the phase of the condensate has a non-zero gradient. The difference is that in the supercurrent state the current is much smaller than the gap, i^(V) > A (and vp(V(p) ;§> A near 6^2)- I n the supercurrent state the Fermi surface is mostly gapped but a small shell near one of the pole caps is ungapped. In the weakly coupled LOFF state there are gapless excitations over most of the Fermi surface but pairing takes place near two rings on the northern and southern hemisphere. In the LOFF state it is energetically favorable to take linear superpositions of Eq. (5.70) with more than one plane wave. As a consequence, the LOFF state has a crystalline structure and there are planes on which the order parameter has a node. In the supercurrent state the current is much smaller than the gap and the

132

T.

Schafer

formation of nodes is not favored. This implies that there will be at least one phase transition (not shown in Fig. 5.10) that separates the supercurrent state from the LOFF phase.

5.5. Outlook Does QCD with three flavors more closely resemble the supercurrent state or the LOFF state of the cold atomic system? In the CFL phase gapless excitations first appear in a regime where the paired state is stable with respect to the normal phase or other homogeneous phases. The resulting instability can be resolved by the formation of a Goldstone boson current j < A. Whether or not the current is not only numerically but also parametrically small compared to the gap depends on certain details that need to be studied more carefully. In the kaon condensed phase the lowest mode is charged and the current is suppressed by charge neutrality. If the neutral mode becomes gapless, too, or if kaons are not condensed the current is no longer suppressed. Once the current becomes comparable to the gap it may be more appropriate to characterize the system as a LOFF state. 5 6 In particular, it is possible that nodes in the condensate appear and quark matter turns into a crystal. 55 Ultimately we would like to obtain observational evidence for exotic phases of dense matter. Polarized atomic Fermi gases have been created in the laboratory but so far neither the supercurrent state nor the LOFF state have been observed. In the case of quark matter we need to identify unique signatures of the possible phases that can be compared to observational evidence. Much work in this direction remains to be done.

Acknowledgments The work presented in this review was performed in collaboration with P. Bedaque, D. Kaplan, A. Kryjevski, and K. Schwenzer. I would like to thank M. Alford, J. Clark, and A. Sedrakian for organizing the workshop on "Pairing in fermionic systems: Beyond the BCS theory" at the INT (Seattle). This work is supported in part by the US Department of Energy grant DE-FG-88ER40388.

Bibliography 1. M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B 537, 443 (1999) [hepph/9804403]. 2. T. Schafer, Nucl. Phys. B 575, 269 (2000) [hep-ph/9909574]. 3. N. Evans, J. Hormuzdiar, S. D. Hsu and M. Schwetz, Nucl. Phys. B 581, 391 (2000) [hep-ph/9910313]. 4. M. Alford, J. Berges and K. Rajagopal, Nucl. Phys. B 558, 219 (1999) [hepph/9903502]. 5. T. Schafer and F. Wilczek, Phys. Rev. D 60, 074014 (1999) [hep-ph/9903503].

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133

6. M. Alford and K. Rajagopal, JHEP 0206, 031 (2002) [hep-ph/0204001]. 7. F. Neumann, M. Buballa and M. Oertel, Nucl. Phys. A 714, 481 (2003) [hepph/0210078]. 8. A. W. Steiner, S. Reddy and M. Prakash, Phys. Rev. D 66, 094007 (2002) [hepph/0205201]. 9. D. K. Hong, Phys. Lett. B 473, 118 (2000) [hep-ph/981251]. 10. D. K. Hong, Nucl. Phys. B 582, 451 (2000) [hep-ph/9905523]. 11. S. R. Beane, P. F. Bedaque and M. J. Savage, Phys. Lett. B 483, 131 (2000) [hepph/0002209]. 12. T. Schafer and K. Schwenzer, Phys. Rev. D 70, 054007 (2004) [hep-ph/0405053]. 13. E. Braaten and R. D. Pisarski, Phys. Rev. D 45, 1827 (1992). 14. T. Schafer and K. Schwenzer, preprint, hep-ph/0512309. 15. B. Vanderheyden and J. Y. Ollitrault, Phys. Rev. D 56, 5108 (1997). 16. C. Manuel, Phys. Rev. D 62, 076009 (2000). 17. W. E. Brown, J. T. Liu and H. Ren, Phys. Rev. D 62, 054013 (2000). 18. A. Ipp, A. Gerhold and A. Rebhan, Phys. Rev. D 69, 011901 (2004). 19. D. T. Son, Phys. Rev. D 59, 094019 (1999) [hep-ph/9812287]. 20. T. Schafer and F. Wilczek, Phys. Rev. D 60, 114033 (1999) [hep-ph/9906512]. 21. R. D. Pisarski and D. H. Rischke, Phys. Rev. D 61, 074017 (2000) [nucl-th/9910056]. 22. D. K. Hong, V. A. Miransky, I. A. Shovkovy and L. C. Wijewardhana, Phys. Rev. D 61, 056001 (2000) [hep-ph/9906478]. 23. W. E. Brown, J. T. Liu and H. c. Ren, Phys. Rev. D 61, 114012 (2000) [hepph/9908248]. 24. Q. Wang and D. H. Rischke, Phys. Rev. D 65, 054005 (2002) [nucl-th/0110016]. 25. T. Schafer, Nucl. Phys. A 728, 251 (2003) [hep-ph/0307074]. 26. T. Schafer, Phys. Rev. D 65, 074006 (2002) [hep-ph/0109052]. 27. G. Baym and S. A. Chin, Nucl. Phys. A 262, 527 (1976). 28. B. A. Freedman and L. D. McLerran, Phys. Rev. D 16, 1169 (1977). 29. V. Baluni, Phys. Rev. D 17, 2092 (1978). 30. E. S. Fraga and P. Romatschke, Phys. Rev. D 71, 105014 (2005) [hep-ph/0412298]. 31. R. Casalbuoni and D. Gatto, Phys. Lett. B 464, 111 (1999) [hep-ph/9908227]. 32. D. T. Son and M. Stephanov, Phys. Rev. D 61, 074012 (2000) [hep-ph/9910491], erratum: hep-ph/0004095. 33. P. F. Bedaque and T. Schafer, Nucl. Phys. A 697, 802 (2002) [hep-ph/0105150]. 34. T. Schafer, D. T. Son, M. A. Stephanov, D. Toublan and J. J. M. Verbaarschot, Phys. Lett. B 522, 67 (2001) [hep-ph/0108210]. 35. D. B. Kaplan and S. Reddy, Phys. Rev. D 65, 054042 (2002) [hep-ph/0107265]. 36. A. Kryjevski, D. B. Kaplan and T. Schafer, Phys. Rev. D 71, 034004 (2005) [hepph/0404290]. 37. M. Alford, C. Kouvaris and K. Rajagopal, Phys. Rev. Lett. 92, 222001 (2004) [hepph/0311286]. 38. A. Kryjevski and T. Schafer, Phys. Lett. B 606, 52 (2005) [hep-ph/0407329]. 39. A. Kryjevski and D. Yamada, Phys. Rev. D 71, 014011 (2005) [hep-ph/0407350]. 40. T. Schafer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999) [hep-ph/9811473]. 41. M. Huang and I. A. Shovkovy, Phys. Rev. D 70, 051501 (2004) [hep-ph/0407049]. 42. R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli and M. Ruggieri, Phys. Lett. B 605, 362 (2005) [hep-ph/0410401]. 43. T. Schafer, Phys. Rev. Lett. 96, 012305 (2006) [hep-ph/0508190]. 44. A. Kryjevski, preprint, hep-ph/0508180. 45. D. T. Son and M. A. Stephanov, preprint, cond-mat/0507586.

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46. A. Kryjevski and T. Schafer, in preparation. 47. T. Schafer, Phys. Rev. D 65, 094033 (2002) [hep-ph/0201189]. 48. S. B. Ruster, I. A. Shovkovy and D. H. Rischke, Nucl. Phys. A 743, 127 (2004) [hep-ph/0405170]. 49. M. Buballa, Phys. Lett. B 609, 57 (2005) [hep-ph/0410397]. 50. M. M. Forbes, Phys. Rev. D 72, 094032 (2005) [hep-ph/0411001]. 51. L. Viverit, C. J. Pethick, H. Smith, Phys. Rev. A 61 053605 (2000) [condmat/9911080]. 52. A. I. Larkin and Yu. N. Ovchinikov, Zh. Eksp. Theor. Fiz. 47, 1136 (1964); engl. translation: Sov. Phys. JETP 20, 762 (1965). 53. P. Fulde and A. Ferrell, Phys. Rev. 145, A550 (1964). 54. R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004) [hep-ph/0305069]. 55. M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D 63, 074016 (2001) [hep-ph/0008208]. 56. R. Casalbuoni, R. Gatto, N. Ippolito, G. Nardulli and M. Ruggieri, Phys. Lett. B 627, 89 (2005) [hep-ph/0507247].

Chapter 6 Nuclear Superconductivity in Compact Stars: BCS Theory and Beyond Armen Sedrakian Institute

for

Theoretical

Physics,

Tubingen

University,

72076

Tubingen,

Germany

John W. Clark Department

of Physics,

Washington

University,

St.

Louis,

Missouri

63130,

USA

T h i s c h a p t e r provides a review of microscopic theories of pairing in nuclear syst e m s a n d n e u t r o n s t a r s . Special a t t e n t i o n is given t o t h e mean-field B C S t h e o r y a n d its extensions t o include effects of polarization of t h e m e d i u m a n d r e t a r d a t i o n of t h e interactions. Superfluidity in nuclear systems t h a t exhibit isospin a s y m m e t r y is studied. W e further address t h e crossover from t h e weak-coupling B C S description t o t h e strong-coupling B E C limit in dilute nuclear s y s t e m s . Finally, w i t h i n t h e observational c o n t e x t of r o t a t i o n a l anomalies of pulsars, we discuss models of t h e v o r t e x s t a t e in superfluid n e u t r o n s t a r s a n d of t h e m u t u a l friction between superfluid and n o r m a l c o m p o n e n t s , along w i t h t h e possibility of t y p e - I s u p e r c o n d u c t i v i t y of t h e p r o t o n s u b s y s t e m .

Contents 6.1. Introduction 6.2. Many-body theories of pairing 6.2.1. Propagators 6.2.2. Mean-field BCS theory 6.2.3. Polarization effects 6.2.4. Non-adiabatic superconductivity 6.2.5. The method of correlated basis functions (CBF) 6.3. Pairing in asymmetric nuclear systems 6.3.1. Phases with broken space symmetries 6.4. Crossover from BCS pairing to Bose-Einstein condensation 6.5. Vortex states in compact stars 6.5.1. Currents and quantized circulation 6.5.2. Constraints placed by neutron-star precession on the mutual friction between superfluid and normal-fluid components 6.5.3. Type-I superconductivity in neutron stars 6.6. Concluding remarks Acknowledgments Bibliography

135

136 138 138 141 143 146 149 153 156 159 162 162 165 167 169 170 170

136

A. Sedrakian and J. W. Clark

6.1. Introduction Neutron stars represent one of the densest concentrations of matter in our universe. These compact stellar objects are born in the gravitational collapse of luminous stars with masses exceeding the Chandrasekhar mass limit. The observational phenomena characteristic of neutron stars, such as the pulsed radio emission, thermal X-ray radiation from their surfaces, and gravity waves emitted in isolation or from binaries, provide information on their structure, composition, and dynamics. The properties of superdense matter are fundamental to our understanding of nature at small distances characteristic of nuclear forces and of the underlying theory of strong interactions - QCD. In fact, neutron stars (NS) provide a unique setting in which all of the known forces - strong, electroweak, and gravitational - play essential roles in determining observable properties. Except for the very early stages of their evolution, neutron stars are extremely cold, highly degenerate objects. Their interior temperatures are typically a few hundreds of keV, far below the characteristic Fermi energies of the constituent fermions, which run to tens or hundreds of MeV. Although very repulsive at short distances, the strong interaction between nucleons is sufficiently attractive to induce a pairing phase transition to a superfluid state of neutron-star matter. As will be discussed in this chapter, the existence of superfluid components within neutron stars has far-reaching consequences for their observational manifestations. Historically, the first observational evidence for superfluidity in neutron-star interiors was provided by the timing of radio emissions of pulsars, the first class of neutron stars, discovered in 1967 by Jocelyn Bell. The pulsed emissions, with a typical periodicity of seconds or less, are locked to the rotation period of the star. Although pulsars are nearly prefect clocks, their periods increase gradually over time, corresponding to a secular loss of rotational energy. Significantly, some pulsars are found to exhibit deviations from this impressive regularity. The pulsar timing anomalies divide roughly into three types, (i) Glitches or Macrojumps. These are distinguished by abrupt increases in the rotation and spin-down rates of pulsars by amounts Aft/ft ~ 1 0 - 6 - 1 0 - 8 and Afi/fi ~ 1(T 3 . After a glitch, Afi/fi and Afi/fl slowly relax toward their pre-glitch values, on a time scale of order weeks to years, in some cases with permanent hysteresis effects.1'2 Such behavior is attributed to a component within the star that is only weakly coupled to the rigidly rotating normal component responsible for the emission of pulsed radiation - an interpretation supported by fits of the measured rotation under different modeling assumptions, (ii) Timing Noise or Microjumps. These represent irregular, stochastic deviations in the spin and spin-down rates that are superimposed on the nearperfect periodic rotation of the star. The origin of microjumps remains unclear, but they could be evidence of stochastic coupling between the superfluid and normal components. 2 (iii) Long-Term Periodic Variabilities. Observed in the timing of few pulsars, most notably PSR B1828-11, these deviations strongly constrain theories of superfluid friction inside NS, if their periodicities are interpreted in terms of NS precession.3 X-ray observations from orbiting spacecraft, which yield estimates of surface temperature for a half-dozen or so young neutron stars, further reinforce the

Nuclear Superconductivity

E

LAB WeV]

in Compact

Stars

137

,!

[MeV]

Fig. 6.1. Left panel. Dependence of experimental scattering phase shifts in 3Si, 3P2, 3 £ , 2 , and D\ partial waves on laboratory energy. Right panel. Dependence of critical temperatures of superfluid phase transitions in attractive channels on chemical potential. The corresponding densities are indicated by arrows. 3

picture of NS with superfiuid content. 4 " 7 At the stellar ages involved, neutrino emission from the dense interior dominates thermal evolution, with nucleonic superfluidity acting to suppress the main emission mechanisms. The existing measurements of surface temperatures indicate that superfiuid hadronic components must be present in some NS, since otherwise they would cool to temperatures below the empirical estimates on very short time scales. Finally, since NS with their huge gravitational fields are expected to be major sources of gravitational wave radiation, it is believed that observation of gravitational waves from oscillating neutron stars can provide further information on the state of matter in their interiors. In particular, the eigenfrequencies and damping rates of gravity waves may carry imprints of dissipation processes in the superfiuid phases. 8-10 The existence of neutron-star superfluidity, first envisioned by Migdal 11 in 1959, is broadly consistent with microscopic theories of nucleonic matter in NS. Shortly after the advent of the Bardeen-Cooper-Schrieffer (BCS) theory in 1957, BCS pairing of nucleons in nuclei and infinite nuclear matter was suggested and studied. 12 ' 13 With the discovery of pulsars, the implications of nucleonic pairing for neutronstar properties were explored, 14 and viable microscopic calculations of pairing gaps began to appear soon thereafter. 15-19 Partial-wave analysis of the nucleon-nucleon (NN) scattering data yields information on the dominant pairing channel in nuclear- and neutron-matter problems in a given range of density (see Fig. 6.1). At high densities, corresponding to laboratory energies above 250 MeV, the most attractive pairing channel is the tensor-coupled 3 P2~3-p2 channel, 20 ' 21 whenever isospin symmetry is even slightly broken. This condition holds inside neutron stars, with the partial densities of neutron and proton fluids differing quite significantly, except in special meson-condensed phases where

138

A. Sedrakian and J. W. Clark

the nucleonic matter is isospin-symmetric. In such a case the phase-shift analysis predicts that the most attractive pairing interaction is in the 3Z>2 wave. 22,23 At low density, isospin-symmetric nuclear matter exhibits pairing due to the attractive interaction in the 3S\-3Di partial wave, a tensor component of the force again being responsible for the coupling of the S and D waves.24^27 This interaction channel is distinguished by its ability to support a two-body bound state in free space - the deuteron. However, under the highly isospin-asymmetric condition typical in neutron stars, neutron-proton pair condensation is quenched by the large discrepancy between the neutron and proton Fermi momenta (see Section 6.3). This review is devoted to several aspects of nucleonic superfluidity in neutron stars that are of major current interest. Section 6.2 provides an overview of the many-body theories of pairing in neutron stars, with a special focus on the Green's function description of pairing, and the effects of self-energies and vertex corrections. The key ideas of the correlated-basis (or "CBF") approach to superfluid states of Fermi systems are also presented. Section 6.3 is concerned with the possibility of pairing between particles lying on different Fermi surfaces, in particular, between protons and neutrons in asymmetric nuclear matter. Here we determine the critical value of isospin asymmetry at which a transition from 3S\-3Di to 1So pairing occurs. (This threshold value is small compared to p — n asymmetries typical of neutron-star matter). More generally, we consider several competing phases that could exist in asymmetric mixtures of fermion species. There exist systems of this kind, notably dilute, ultracold atomic gases and baryonic matter as described by QCD at high density, where "asymmetric pairing" is enforced. (In the case of dilute atomic gases, such conditions can be "tuned in" by external fields, while in the QCD problem, the conditions of charge neutrality and /? equilibrium together with the heaviness of the strange quark lead naturally to asymmetric pairing among light quarks.) In Section 6.4 we turn to the phenomenon of crossover from BCS superconductivity to Bose-Einstein condensation, as it occurs in fermionic systems that support a two-body bound state in free space. Section 6.5 is devoted to the physics of superfluids at the "mesoscopic" scale, with discussions of flux quantization, neutron vorticity, the electrodynamics of superconducting protons, and their implications for modeling the dynamics of rotational anomalies in pulsar timing. Our conclusions and related perspectives are summarized in Section 6.6. Two other recent reviews 28 ' 29 offer complementary information and perspectives on nuclear pairing and nucleonic superfluidity.

6.2. 6.2.1.

Many-body theories of pairing Propagators

This subsection outlines the Green's functions method for the treatment of superfluid systems. The original formulations of this approach are due to Gor'kov and Nambu, who employed thermodynamic Green's functions. 30 Herein we consider the real-time, finite-temperature formalism, which is suited to studies of both equilibrium and non-equilibrium systems. Our discussion is restricted to equilibrium

Nuclear Superconductivity

in Compact

Stars

139

systems, and we shall work with the retarded components of the full Green's function of the non-equilibrium theory. Superfiuid systems are described in terms of 2 x 2 matrices of propagators (known as Nambu-Gor'kov matrix propagators) that are defined as r

{ >\ - ( Gap(x,x') ' -\-Fl0(x,x')GaP(x,x')J

Faf3(x,x')\

yap[X X)

'-i{Tipa(x)i>l(x')) (T^Ux)^(x'))

{T^a{x)^p{x'))

'

(6.1)

-i{Ti>t{x)Mx')),

where ipa(x) are the baryon field operators, x is the space-time coordinate, the indices a and (3 stand for the internal (discrete) degrees of freedom, and T and T denote time-ordering and inverse time-ordering of operators, respectively. The 2 x 2 matrix Green's function (6.1) satisfies the Schwinger-Dyson equation Gap{x,x') = G°ap(x,x') + YJJdAx"dix",goai{x,x'")ni5{x"l,x")g50{x",x'),

(6.2)

where the free-propagator matrices Q%p{x,x') are diagonal in the Nambu-Gor'kov space. The matrix structure of the self-energy Qap(x, x') is identical to that of the propagators: the on-diagonal elements are £(p) and £(— p), and the off-diagonal elements are A(p) and A^(p). Fourier transforming Eq. (6.2) with respect to the relative coordinate x — x', one obtains the Dyson equation in the momentum representation. In this representation, the components of the Nambu-Gor'kov matrix obey the coupled Dyson equations Gaf3{p) = G0af3(P) + G0ai{p) \piS[p)GSp{jp) + A7s(p)F^spip)} F*ap{p) = G0ai(-P)

[A\g(p)Gsp(p)

+ ^s(-p)Fhp(p)]

•

,

(6.3) (6.4)

Here Gap{p) and Goap(p) are the full and free normal propagators, F^ap(p) and Fap(p) are the anomalous propagators, and Y,ap(p) and Aap(p) are the normal and anomalous self-energies. The Greek subscripts are the spin/isospin indices, and summation over repeated indices is understood. For systems with time-reversal symmetry, it is sufficient to solve Eqs. (6.3) and (6.4), since this symmetry implies that Aap(p) = [A'ap(p)]*. It is instructive to rewrite Eqs. (6.3)-(6.4) in terms of auxiliary Green's functions G^{P) = G0af3(p) + G ^ ( p ) E 7 4 ( p ) G 0 ^ ( p )

(6.5)

describing the unpaired state. The solution of this equation is G^ (p) = 5ap [u — e(p)] - 1 , where e{p) = ep + S(p) and ep is the free single-particle spectrum. (N.B. Assuming that the forces conserve spin and isospin, the self-energy S(p) is diagonal in spin and isospin spaces). Combining Eqs. (6.3), (6.4), and (6.5), we derive an alternative but equivalent form of the Schwinger-Dyson equations, namely Gaf3(p) = G^{p)

\sif3 + A^s(p)F}p(p)]

,

Flp{p) =

G^(-p)A\s(p)Gsp(P),

A. Sedrakian and J. W. Clark

140

which can be solved to obtain Ga0(P) = <W

F\{p)

»-BA(P)

+ ES
(

=

^ ^ . (6.7) [W - EA(p)}2 - Es{v)2 - A 2 ( P ) Here we have made the substitution A(p)A t (p) = —A2(p) and denned symmetric and antisymmetric parts of the single-particle spectrum in the normal state, ES/A = [s(p) ± e(—p)} /2. For isotropic systems, the self-energy is invariant under reflections in space (i.e. the self-energy is even under p —> —p); furthermore, for systems that are time-reversal invariant, the self-energies are even under the transformation u> —> —UJ. Hence the antisymmetric piece of the spectrum EA must be absent when both conditions are met. The poles of the propagators (6.6) determine the excitation spectrum of the superfluid system, given by apyy>

uj± = EA(p)±y/Es(p)2

+ A*(P).

(6.8)

Here one sees that there is a finite energy cost ~ 2A for creating an excitation from the ground state of the system when EA = 0, a property that leads to the existence of superflow or supercurrent in paired fermionic systems. The solutions (6.6) and (6.7) are completely general, all functions being dependent on the three-momentum and the energy. Superfluid systems are often treated in the quasiparticle approximation, in which the self-energies are approximated by their on-mass-shell counterparts. The rationale behind such an approach is that the nuclear system in its ground state can then be described in terms of Fermi-liquid theory, if one neglects pair correlations. Switching on the pair correlations precipitates a rearrangement of the Fermi-surface, but it is assumed that the quasiparticle concept remains intact. Within this framework, the wave-function renormalization is defined by expanding the normal self-energy as S(w) = S(e p ) + dulT,(uj)\u,=ep(uj — ep), where £P = ep + ReS(e p ) is the on-mass-shell single-particle spectrum in the normal state that solves Eq. (6.5). Now, it is seen that the propagators (6.6) and (6.7) retain their form if they are renormalized as G(p) = Z(p)G(w + i8,p),

Fi(p) = Z(p)F\uj

+ i6,p),

A 2 (p) =

Z(p)2A2(p), (6.9)

where Z(p) = [l — dul'E(uj)\hJ=ev] is the wave-function renormalization and the tilde identifies renormalized quantities. An additional feature of the renormalized propagators is that the quasiparticle spectrum e(p) is now constrained to the mass shell. (N.B. For time-local interactions the gap function is energy-independent, so there is no need to expand A(w) around its on-shell value. We shall return to the off-shellness of the self-energies in Subsec. 6.2.4.) Renormalization of propagators within the quasiparticle picture suggests that the probability of finding an excitation with given momentum p is strongly peaked at the value ep. The wave-function renormalization takes into account corrections that are linear in the departure from this value. From the computational point of view,

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such corrections require a knowledge of the off-shell normal self-energies. The dependence of the imaginary part of the self-energy (quasiparticle damping) on frequency follows from Fermi-liquid theory, being given by 2Im£(w) = a 1 + (UJ/2TTT) , where a is a density-dependent constant. The real part of the self-energy, ReE(o;), can be computed from ImE(w) via the Kramers-Kronig dispersion relation only if the latter function is known for all frequencies u> £ [—00,00]. Accordingly, the foregoing result from Fermi-liquid theory should be supplemented by a model of the high-energy tail of the quasiparticle damping. The momentum dependence of propagators can be approximated by introducing an effective quasiparticle mass. For nonrelativistic particles, expansion of the normal self-energy around the Fermi momentum leads to -1

(6.10) , — m* m PF where /i* = — e(pp) + \i — ReE(pir). A closed system of equations determining the properties of the superfluid system is obtained by specifying the self-energies in terms of the propagators and interactions, as discussed below. = 1 -I

£

(P) = —-KP-PF)-V

6.2.2. Mean-field

BCS

dpReT,(p)\p=PF

theory

The BCS-type theory of superconductivity as applied to nuclear systems is predicated on a mean-field approximation to the anomalous self-energy. Specifically, the anomalous self-energy is expressed through the four-point vertex function T(p,p') in the form A{P)

=

-2/^r(P,P')ImiV)/(^'),

(6.11)

where /(w) = [1 + exp(/3w)] _1 and f3 is the inverse temperature. We observe that the gap is energy-independent when the interactions are local in time, corresponding to no retardation, as in the case where the effective interaction T(p,p') is replaced by the bare interaction V(p,p'). In this case, carrying out the renormalization according to Eq. (6.9) and integrating over the energy, we arrive at 3 1 - 3 3 A(p) = Z{p) | ^

¥

y

( p

, p ' ) ^ ( p ' ) ^ ^ [/("+) " / ( " - ) ] ,

(6.12)

where u>± = ± 1 / ^ + A 2 [cf. Eq. (6.8)]. Further progress requires partial-wave decomposition of the interaction in Eq. (6.12). To avoid excessive notation, our consideration focuses on a single, uncoupled channel, for which we obtain a onedimensional gap equation

A(p) = Z(p) J ^Vip^Zip1)^^

[/(*+) - /(*_)] ,

(6.13)

where V(p,p') is the interaction in the given partial wave. The gap equation is supplemented by the equation for the density of the system,

» = - » / | i > - w M - j / ^ r (i+£)'(*>•(614) 1

— i )

142

A. Sedrakian and J. W. Clark

which determines the chemical potential in a self-consistent manner. Here g is the isospin degeneracy factor. Eqs. (6.11)-(6.14) define mean-field BCS theory. In subsequent discussions based on this theory, the tilde notation will be suppressed. Assuming that the interaction V(p,p') and the single-particle spectrum ep in the unpaired state are known, Eqs. (6.13) and (6.14) form a closed system for the gap and chemical potential. Within this formulation, the presence of a hard core in the interaction causes no overt problems, for one can readily show that at the critical temperature, Eq. (6.13) transforms into an integral equation that sums the particleparticle ladder series to all orders. In the special case in which the interaction is momentum-independent, Eq. (6.13) leads to the familiar weak-coupling formula A(p F ) ~ V

exp (-

)

-)

,

(6.15)

where the density of states is specified by V(J>F) = m*pFZ2(pF)/2Tr2 (for one direction of isospin). The weak-coupling formula is often used to estimate the magnitude of the gap. However, because realistic pairing interactions are momentumdependent, the density dependence of the gap predicted by Eq. (6.13) may deviate markedly from that given by the weak-coupling formula (6.15). Moreover, as argued in Ref. 34, if the pairing interaction acquires a strong momentum dependence due to a short-range repulsive core, the weak-coupling formula may well produce a meaningless or useless estimate of the gap, since its derivation requires that V(p,p') take a negative value on the Fermi surface. The normal-state self-energy is written as S(P) = - 2 y ~^T

(p,p';p + p')A ImG(p')/(«'),

(6.16)

where the subscript A indicates antisymmetrization of the final states and the contribution oc ImT is omitted for simplicity. For nuclear systems, the amplitude T is often approximated by the scattering T-matrix, which sums up the ladder diagrams and is generally nonlocal. An alternative is to replace the T-matrix by an effective time-local interaction that is fitted to properties of finite nuclei (e.g. a Skyrme or Gogny force), in which case Eq. (6.16) reduces to a mean-field Hartree-Fock approximation. However, one must be aware that the normal-state spectrum itself must depend on the anomalous self-energy A(p). The common replacement of G(p) by GN (p) when computing the normal-state spectrum is an approximation (sometimes called the "decoupling approximation"), which is justified when the pairing effects can be viewed as a perturbation to the normal state, but must be made with care. Fig. 6.2 shows the 1So pairing gap in neutron matter and symmetrical nuclear matter for the high-precision phase-shift-equivalent Nijmegen potential and the effective Gogny DSl force, for different approximations to the single-particle spectrum. 35 In the case of smooth effective forces such as the the Gogny interaction, a Hartree-Fock approximation to the normal self-energy is suitable and was adopted; for the (realistic) Nijmegen potential, the T-matrix was calculated in Brueckner theory. In both cases, the full momentum-dependent self-energies ReE(p) were used in the gap equation. The momentum renormahzation yields an effective mass m*/m

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i

0.2

i

i

0.4

i

i

0.6

i

i

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i

0.8 1 k F [fm']

i

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i

1.2

1.4

.

143

i

1.6

i

i

1.8

Fig. 6.2. Singlet S-wave ( 1 5o) pairing gap in neutron matter and nuclear matter versus Fermi momentum. The heavy and light lines correspond respectively to the phase-shift-equivalent Nijmegen interaction and the Gogny effective interaction DSl. Solid lines and dashed-dotted lines label results for pure neutron matter using the free-space single-particle spectrum and implementing single-particle renormalization, respectively, while the dashed line refers to symmetrical nuclear matter with single-particle renormalization.

less than unity, thus reducing the density of states v(pp) on the Fermi surface and hence also reducing the size of the gap. The wave-function renormalization factor Z(p) is also less than one, leading to an additional suppression of the gap. However, the the magnitude of this effect is yet to be established. 32 ' 33

6.2.3. Polarization

effects

An improvement upon the mean-field BCS approximation to fermion pairing is achieved in theories that take into account the modifications of the pairing interaction due to the background medium. In diagrammatic language, the class of modifications known as "polarization effects" or "screening" arise from the particle-hole bubble diagrams, ideally summed to all orders starting from the bare interaction as the driving term. Consider the following integral equation describing the fourfermion scattering process pi + P2 —> Pz + Pi'-

T(p,p\ q) = U(p,p', q) - i J ^

U(p,p", q)GN{p" + q/2)GN(p" - q/2)T{p",p', q) , (6.17)

where q = p\ —pi is the momentum transfer, p = p\ +P3, and p' = P2+P4- Eq. (6.17) sums the particle-hole diagrams to all orders. To avoid double summation in the gap equation, the driving term U(p,p',q) must be devoid of blocks that contain particle-particle ladders. This driving interaction depends in general on the spin and isospin and can be decomposed as Uq = fq + gq{* • a') + [fq + g'q{a • or')} (r • r ' ) ,

(6.18)

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A. Sedrakian and J. W. Clark

where cr and T are the vectors formed from the Pauli matrices in the spin and isospin spaces. We assume here that the interaction block U depends only on the momentum transfer. For illustrative purposes, the tensor part of the interaction and the spin-orbit terms are ignored. (However, see Subsec. 6.2.4, where the tensor component is included by means of pion exchange.) Solution of the integral equation (6.17) then takes the form Ga v(PF)Tq -(a-a') l + A{q)Fq + l + A(q)G, F' GL (6.19) + l + A{q)Fq + l+A(q)G'q-(a-a') (T-T>), where Fa v(PF)fq, Ga -- v{PF)gq, Fq = v{pF)f'q, and G'q v(PF)g'a yq !- while A(q) = v{pF

d4p"

•7

(2TT)<

GN(p" +

q/2)GN(p"-q/2)

(6.20)

is the (dimensionless) Lindhard function (or polarization tensor). We are tacitly assuming that the system is in a state characterized by a well-defined Fermi sphere. Then, if the momenta of both particles lie on the Fermi surface, the momentum transfer is related to the scattering angle via q = 2pFsin6/2, and the parameters F, F', G, and G' can be expanded in spherical harmonics with respect to the scattering angle, according to Pi (cos 6>),

{G(q))-2^{Gl

(6.21)

and similarly for F'(q) and G'(q). The Landau parameters Fi, Gi Fi and G depend only on the density. The isospin degeneracy of neutron matter, reflected in T-T' = 1, implies that the number of independent Landau parameters for each q or I reduces from four to two, defined by Fn = F + F' and Gn = G + G'. Commonly, only the lowest-order harmonics in the expansion (6.21) are needed. For a singlet pairing state, in which the total spin of the pair is S = 0 and cr • a' = —3, the pairing interaction is given by A(q)G% A(g)lff (6.22) -3G£ 1 v(pF)Tq = iff 1 1 + A(g)F0" l + A(q)G% In general, the polarization tensor A(q) is complex-valued. However, in the limit of zero energy transfer (at fixed momentum), it is real and becomes simply

A(q) = - l + f

1-

4j&

In

2pF2pF + q

(6.23)

in the zero-temperature limit. Eq. (6.22) contains two distinct contributions: the direct part generated by the terms 1 inside the square brackets, and the remaining, induced part that accounts for density and spin-density fluctuations. If the Landau parameters are known - either by inferring them from experiment or by computing them within an ab initio many-body scheme - the effect of polarization can be assessed by defining an averaged interaction r

= A fPFdqqT(q). *PF Jo

(6.24)

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T

0I

0

4-^

1 0.5

1

i 1

1

i-^J 1.5

—

1

k F [fm"']

Fig. 6.3. Singlet S-wave (1S'o) pairing gaps in neutron matter versus Fermi momentum, as obtained from microscopic calculations that attempt to account for medium-modification of the pairing interaction and self-energies. The curves are labeled as: A - Wambach et al. [42], B - Chen et al. [48], C - Chen et al. [49], D - Schwenk et al. [52], E - Schulze et al. [51], F - Fabrocini et al. [50].

The effect of density fluctuations oc FQ is to enhance the attraction in the pairing interaction 36 and therefore increase the gap, while the spin-density fluctuations oc GQ tend to reduce the attraction and decrease the gap. 37 At densities typical of the inner crust of a neutron star, the values of microscopically derived Landau parameters imply that the suppression of pairing via spin-density fluctuations is the dominant effect.37 The Landau parameters of neutron matter and symmetrical nuclear matter have been studied extensively within complex many-body schemes38^40 whose description is beyond the scope of this chapter. It should be noted, however, that the matrix elements of the pairing interactions derived in some of these schemes, including the Babu-Brown approach 38 ' 39 and its extensions such as polarization-potential theory, 41-42 have been used in conjunction with the weakcoupling approximation (6.15), which - as indicated above - is generally inadequate in the nuclear/neutron-matter context. The initial microscopic calculations of the effects of medium polarization on pairing were carried out within an alternative many-body approach, the method of Correlated Basis Functions (CBF) 4 3 - 4 9 (to be described in Subsec. 6.2.5). At a qualitative level, the findings of the CBF studies of Chen et al. 48 ' 49 are consistent with much of the later work based on Green's functions and Fermi-liquid theories. Until recently, there was broad agreement that the screening reduces the singlet 5-wave gap by factor of 3 or so (c.f. Fig. 6.2). However, the density profiles of the calculated gaps differ considerably. This is illustrated in Fig. 6.3, which presents a composite plot of theoretical predictions for the dependence of the 1SQ pairing gap at the

A. Sedrakian and J. W. Clark

146

Fermi surface, A(kp), upon the Fermi wave number kp = PF/fi = (•in2p)1^3. The six curves in the plot correspond to various microscopic approaches that include a screening correction. The results of Wambach et al., 42 Schulze et al., 51 and Schwenk et al. 52 are based on microscopic treatments rooted in Landau/Fermi-liquid theory, with polarization-type diagrams summed to all orders. The results of Chen et al. 48 ' 49 and Fabrocini et al. 50 were obtained within two different implementations of CBF theory. Further assessment of the status of quantitative microscopic evaluation of the singlet-5 gap is deferred until Subsec. 6.2.5, where the elements of CBF approaches to the pairing problem are reviewed. We may already remark, however, that explicit comparison of the pairing matrices constructed in the different theories could help to eliminate discrepancies introduced by use of the weak-coupling approximation in some of the theoretical treatments. Another important consideration is consistent inclusion of medium effects on both the pairing interaction and the self-energies. 6.2.4. Non-adiabatic

superconductivity

Since mesons propagate in nuclear matter at finite speed, the interactions among nucleons are necessarily retarded in character. As a consequence, the self-energies (and in particular the gap function) must depend on energy or frequency. Within the meson-exchange picture of nuclear interactions, the lightest mesons - pions - should be the main source of nonlocality in time. This suggests that it may be fruitful to consider a pairing model in which the interactions are modeled in terms of pion exchanges, plus contact terms that can be approximated by Landau parameters. Thus one assumes an interaction structure VNN = -^-(tr-V)(T-4>)

+ U{q)

(6.25)

in which <j> is the pseudoscalar isovector pion field satisfying the Klein-Gordon equation, fn is the pion-nucleon coupling constant, and m„- is the pion mass. Here (f> is the pseudoscalar isovector pion field satisfying the Klein-Gordon equation, / w is the pion-nucleon coupling constant, and m^ is the pion mass. The operator structure of the term U(q) is like that of Eq. (6.18), but with constants differing numerically since the tensor one-pion exchange is treated separately. For static pions, the one-pion-exchange two-nucleon interaction in momentum space is given by

[ffl ffa + Sia(n)1

^) = n4^ "

^'

(626)

-

where q is the momentum transfer, n = q/q, and S12(n) is the tensor operator. This interaction is known to reproduce the low-energy phase shifts and, to a large extent, the deuteron properties. 53 Below, however, the pairing correlations are evaluated from the diagrams that contain dynamical pions, with full account of the frequency dependence of the pion propagators. The static results can be recovered, and a relation to the phase shifts established, only in the limit u> —> 0 in the pion propagator. The dynamics at intermediate and short range is dominated in turn

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a

a

Fig. 6.4. Top panel. Baryon Fock-exchange self-energies for normal (left graph) and anomalous (right graph) sectors. The solid lines correspond to fermions; and the wavy lines, to pions. The blobs symbolize RPA-renormalized vertices; and the dots, bare pion-nucleon vertices. Bottom panel. Normal and anomalous Hartree diagrams. The dots stand for contact Landau interactions; the straight lines are shown for clarity.

by the correlated two-pion, /o-meson, and heavier-meson exchanges. Short-range correlations are crucial for a realistic description of low-energy phenomena, being necessary to achieve nuclear saturation. Moreover, the response functions calculated from one-pion exchange alone would already precipitate a pion-condensation instability in nuclear matter at an unrealistically low density. The interaction (6.25) leads to a time-nonlocal formulation of nuclear superconductivity in neutron-star matter. 54 The Dyson-Schwinger equation (6.2) provides the starting point. Since the formulation will incorporate the full frequency dependence of the self-energies, it is useful (and also conventional 55 ) to define the wave-function renormalization differently than in Subsec. 6.2.2. Thus we set Z(p) = 1 — W~1T,A(P), the retarded self-energy being decomposed into components even (S) and odd (A) in w, i.e. S(p) = £s(p) + IU(p). The single-particle energy is then renormalized as Es = ep + T,s(Es,p). Accordingly, the propagators now take the forms G(p) = F(p) =

ujZ(p) + Es(p) (w + ir))2Z(p)2 - Es(p)2 - A(p) 2 ' A(p)

(6.27)

iri)*Z(p)*-Es(p)2-AW

(u, +

(6.28)

where AA* = —A2. The self-energies of the theory are shown in Fig. 6.4. The analytical counterparts of the Fock self-energies are

£ F o c W ) = -2j-^J^E0(q)lmG(e,p-q)C(uJ,e,q)E(q),

(6.29)

AFock(u,,p) = - 2 J-^L

(6.30)

J'X^-Eo(q)ImF(e,p-q)C(uJ,e,q)E(q),

where

C(«,,e,q) = J0°^B{w',q.

m

+ g(u>')

£ — U)' — UJ — IT]

+i-f(e)

+ g(w')

£ + U)' — OJ — 17]

(6.31)

A. Sedrakian and J. W. Clark

148

a [1/fm]

Fig. 6.5. Pion spectral function in neutron matter as a function of energy and momentum transfer. The density corresponds to kp = 0.55 f m _ 1 .

Here B{q) is the pion spectral function, while S°(g) and S(q) are the bare and renormalized pion-neutron vertices. One remarkable feature of Eqs. (6.29)-(6.31) is that the energy and momentum dependence of the self-energies is determined by the dynamical features of the meson (here pion) field. Another salient feature is that the normal and anomalous sectors are coupled, in contrast to the BCS case, where the unpaired single-particle energy is unaffected by the pairing. The most important contribution to the pion spectral function comes from the coupling to virtual particle-hole states, which are described by the (retarded) particle-hole polarization tensor II(u;,p). Specifically, one finds -2lm.UR{q)

(6.32) R 2 R 2 rat ReU {q)] + [lmU {q)} ' The spectral function of pions in neutron matter is illustrated in Fig. 6.5 for the momentum transfer range 0 < q < 2kp, where kp = 0.55 fm - 1 . It is seen that (i) the spectral function has a substantial weight for finite energy transfer, the maximum being determined by the pion dispersion relation u>2 = q2 + m\ + Ren(w, q) and (ii) the spectral function is substantially broadened due to the excitations of particle-hole pairs, which are treated in the random-phase approximation (RPA). 53 In addition to the Fock self-energies we need to include the Hartree contribution, which reduces to B{q)

.BCS

2

2

[u> — q

p) = (">,p) = -2(F? -

3GS) J 0^^F(co

+ OJ',P + p').

(6.33)

Solutions of the self-consistent equations (6.30) and (6.31) are shown in Fig. 6.6 at zero temperature and for densities specified by the indicated Fermi momenta. At small energy transfers, the imaginary components of the gap and wave-function renormalization vanish, and one recovers the BCS limit. For finite energy transfers these functions develop substantial structure that reflects the features of the pion spectral function [the driving term in the kernel of integral equations for A(q) and

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100 co [MeV]

Fig. 6.6. Top panel. Frequency dependence of real (heavy lines) and imaginary (light lines) parts of the gap function, Ai(u>) and A2(w), respectively, for kp = 0.4 (solid), 0.5 (long-dashed), 0.55 (dashed), and 0.6 (dashed-dotted lines) f m _ 1 . The on-shell values of the pairing gap are 0.1, 0.7, 1.4, and 3.7 MeV for kp = 0.4, 0.5, 0.55, and 0.6 f m - 1 . Bottom panel. Frequency dependence of real (left panel) and imaginary (right panel) parts of the wave-function renormalization, Z\ (a>) and ZiiyS). Labeling is the same as in the top panel.

Z(q)}. Note that the actual value of the gap on the mass shell does depend on the detailed structure of these functions far from the mass shell. However, it is possible to renormalize the pion spectral function such that the high-energy tails are eliminated while the on-mass-shell physics is unchanged. Theories that explicitly include the light mesons - pions or kaons - in the computational scheme have the advantage that they embody the precursor phenomena associated with the softening of the pion (kaon) modes close to the threshold for condensation. In the case of P-wave pairing, an enhancement of the pairing correlation has been predicted. 56 6.2.5. The method of correlated

basis functions

(CBF)

The variational formulation of BCS theory is based on the trial wave function (BCS state)

*BCS = I ]

[(1 - M 1/2 + hj/Vj^ipJ |0>

(6.34)

where the real function hp, which gives the occupation probability of the pair state (P T>~P ! ) , is subject to variation. For this trial state one may compute the anomalous density XP = ( * B C 3 | ^ 1 P J * B C S ) = h]!2{l - hp)1'2

,

(6.35)

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A. Sedrakian and J. W. Clark

whose Fourier image specifies the spatial structure of a Cooper pair. The standard coupled BCS equations for the energy gap, the fermionic density, and the quasiparticle energy at zero temperature, namely

A

^ = -Jwr>VM^mA{p'h 3

dp f d*p J

(2TT)3^

p2dp _ f fp^ J

\

-

£(P) - »

(6.37)

2E(p)

(2TT)2

E(p) = ^[e{p)-tf

(6 36)

+ ^(p),

(6.38)

are generated naturally upon (i) evaluating the expectation value of the grandcanonical Hamiltonian for the BCS state (6.34), and (ii) performing a variational minimization of this functional with respect to hp, under the constraint that the expectation value of the particle-number density coincides with the prescribed density. Treatment of strongly interacting fermionic systems (including nuclear problems) within a variational framework calls for improved superfluid trial states. The systems of interest are characterized by a bare two-body interaction containing a strong inner repulsive core along with longer-range attractive components. To obtain a reasonable energy expectation value, the trial function must adequately describe the short-range geometric correlations induced by the repulsion, which inhibits the close approach of a pair of particles. 43 " 46 The simplest choice involves the Jastrow correlation factor N

Fj = T[ fifij), J-J. i<j

lim / ( r ) - 0 , r—»0

lim f(r) -» 1,

(6.39)

r—+oo

which is suited to efficient description of state-independent two-body correlations, especially the short-range repulsive effects. As a bonus, with proper optimization of the two-body function f(r), the Jastrow factor can also incorporate effects of virtual phonon excitations and indeed can reproduce the correct asymptotic behavior of long-range correlations. A substantially improved trial superfluid state of definite particle number may be formed by applying the Jastrow operator (6.39) to an iV-particle projection of the BCS trial state, expressed in the configuration-space representation as _1_ *BCS = -7T7 M(ri,r2),#r3,r4)...

7JJ-

^-i,r

N

)} .

(6.40)

Here, (ri,rj) is the antisymmetrized Fourier image of \p given by Eq. (6.35), times a spin function, A is an antisymmetrization operator acting on particle pairs in different <j) factors, and Af is a normalization constant. In spirit, this is the approach adopted in the early work of Yang and Clark, 15 ' 16 although in practice they determined the one-body, two-body, etc. density matrices required for their cluster-expansion treatment of the short-range correlations from the BCS state (6.34), rather than from its ./V-particle projection (6.40).

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Following up on the work of Refs. 15, 16, a formal variational theory 58 of the superfiuid ground state of uniform, infinite nucleonic systems was developed for a trial correlated BCS state constructed in Fock space,

I*-) = E E ^WX^IBCS), N

(6.41)

m(JV)

where JPW is an unspecified correlation operator meeting certain minimal conditions and {|3?m )} is a complete set of Fermi-gas Slater determinants, both referred to the ./V-particle Hilbert space. Thus, the correlated normal states { ^ " ' I f t ')} are superposed with the same amplitudes as the model states |$m ') have in the corresponding grand-canonical representation of the original BCS state (6.34). Repetition of steps (i) and (ii) above for this correlated superfiuid Ansatz yields a theory having the same structure as ordinary BCS theory, when the "decoupling approximation" is applied. In this approximation, only one Cooper pair at a time is considered, while treating the background as normal. Formally, the expectation value of the grand-canonical Hamiltonian is expanded in terms of the deviations of the Bogolyubov amplitudes up = (1 — hp)1^2 and vp = hp about their normalstate values, retaining deviant terms at most of first order in v^, — 8(p) and second order in upvp, where 8{p) is the Fermi step. Within this framework, the gap equation and density constraint maintain the same mean-field forms as obtained for the bare BCS state, except for the attachment of renormalization factors z~l to the gap function A(p) when it appears in quasiparticle energy denominators. The presence of correlations introduced by the operator F^N^ is otherwise reflected only in the replacement of the pairing matrix elements V(p,p') and single-particle energies s(p) derived from the bare interaction based on the BCS trial state (6.34) and variational steps (i)-(ii), by effective pairing matrix elements V(p,p') and correlation-dressed single-particle energies £(p) built from combinations of diagonal and off-diagonal matrix elements of the Hamiltonian and unit operators in the correlated normal bases { F ^ l ^ ) } . When the stateindependent Jastrow choice Fj of Eq. (6.39) is assumed for the correlation operator F, the dressed quantities V(p,p') and £(p) can be evaluated by Fermi hypernettedchain (FHNC) methods developed in Ref. 57, with results for neutron matter and liquid 3 He reported in Ref. 58. An important advance in the CBF approach to pairing was made in Ref. 47, where the variational description was extended to create a correlated-basis perturbation theory for the exact superfiuid ground state. Again imposing the decoupling approximation, a sequence of approximations to the grand-canonical energy may be defined, each preserving under variation the standard form of the gap equation, but with successive improvements on the effective pairing matrix elements and dressed self-energies. (A convenient modification of the trial correlated ground state (6.41) was made by inserting the normalization factor ($m '\F^ F(N^\$m >)~1/2 inside the summations. This eliminates the renormalization factors z~l mentioned above.) Making the Jastrow choice for the correlation operator F, the leading perturbative corrections to the variational results for V(p,p') and £(p) were generated, represented in diagrammatic form, and evaluated. These corrections include the leading

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contribution from medium polarization within the CBF framework. Here we should point out that the terms in the CBF perturbation expansions of the various quantities are not easily interpreted in terms of conventional Goldstone or Feynman diagrams, although they may have similar appearance. A given order in the CBF expansion, including the "zeroth-order" variational term, will contain pieces belonging to any number of perturbative orders in the conventional sense. In general there will be terms accounting for the nonorthogonality of the basis, terms that correct the average-propagator approximation inherent (for example) in the Jastrow description of F-correlations, terms that correct for non-optimality of the chosen F-correlations, etc. In microscopic studies of nucleonic systems, it is generally imperative to include the effects of state-dependent correlations arising from realistic NN interactions which contain, separately in each spin and isospin channel, contributions of central, tensor, and spin-orbit character. In principle, the CBF perturbation expansions provide for systematic correction of the Jastrow Ansatz for the correlation operator F , so as to incorporate these state-dependent effects. However, it is clearly preferable to take account of state dependence already in the choice of F , thus reducing the need to correct the variational treatment with CBF perturbation theory. A suitably general correlation Ansatz, within the class containing only two-body correlation factors, is given by N

F = S

I[/fa')> i<j

n

/fa) = E /«fa>afa),

(6-42)

a=l

where S is a symmetrizing operator and f(ij) contains terms for the same operators oa(ij) as are present in the assumed realistic NN interaction (e.g., the Argonne vis model 59 ), or an adequate subset of them. Profound difficulties arise in the implementation of this choice, due to non-commutativity among the oa(ij) operators. The analog of FHNC resummation being still beyond our reach for such state-dependent correlations, existing calculations proceed with straightforward cluster or power-series expansions, perhaps with vertex corrections. 49 ' 50 It is important to appreciate that the extended Jastrow form (6.42) of the F-operator is equipped to include the lion's share of the polarization corrections (just as the simple state-independent Jastrow form is capable of capturing the major effects of density-density fluctuations). Chen et al. 48 applied CBF pairing theory as developed in Ref. 47 to superfluid neutron matter in the 1 5o phase, assuming state-independent Jastrow correlations and taking account of the leading CBF perturbation corrections to the variational treatment. The polarization and other corrections produced a very substantial suppression from the variational estimate of the gap A(fcf), the peak value being reduced by a factor ~ 4 and situated at lower density. This treatment was updated in Ref. 49. Major improvements were made in the choice of the variational two-body correlation functions. Significantly larger gap values were obtained, but again the perturbative corrections were estimated to suppress the peak value by a factor ~ 4 and shift its location to a lower density. Quantitatively, the results of the later of the two perturbative CBF calculations are the more reliable. Shortly after the work of Krotscheck and Clark, 58 an independent approach to CBF description of pairing based on the trial superfluid state (6.41) was launched

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by Fantoni. 60 With immediate specialization of the correlation operator F to the state-independent Jastrow form (6.39), it proved feasible to extend the diagrammatic techniques of standard Fermi hypernetted-chain theory 44 ' 61 and thereby enable practical evaluation of the one-body density matrix and radial distribution function associated with the correlated BCS state (6.41). Derivation of the corresponding gap equation and density constraint was achieved without resorting to the decoupling approximation. Quite recently,50 Fantoni's CBF approach has been generalized - insofar as practicable - to include state-dependent correlations of the form (6.42). The results for the ^ o neutron gap lie distinctly higher than the results of earlier work designed to include nontrivial medium effects on pairing (see Fig. 6.3). Thus, they conflict with the general consensus that these effects lead, on balance, to a strong suppression of the gap value. Concurrent estimates 50 of the gap based on an auxiliary-field diffusion Monte Carlo (AFDMC) calculation lie even higher than the new CBF estimates (by roughly 0.5 MeV, with a peak value of more than 2.5 MeV at kF ~ 0.6 far1). A recent numerical study of 1So pairing in neutron matter within the self-consistent Green's function (SCGF) method 27 gives results in essential agreement with the AFDMC estimates. One might conclude from this agreement that the mediumpolarization effects, arising from the exchange of spin-density fluctuations and other virtual processes, 37 are less important than previously imagined. On the other hand, the SCGF calculation, by construction, neglects such collective correlations of longer range, while the AFDMC stochastic estimates, obtained for relatively small samples of neutrons, might also fall short in their inclusion of these effects. At any rate, the latest computational results continue to highlight the extreme sensitivity of the 1 5o pairing gap to the assumptions made in pursuing its evaluation by microscopic methods. The quantitative situation for pairing in spin-triplet T = 1 states is even less clear. 56 ' 62

6.3. Pairing in asymmetric nuclear systems The isospin asymmetries characteristic of neutron-star cores, with proton fractions ~ 5%, are too large to permit isospin-singlet (neutron-proton) pairing. A possible exception involves Bose-Einstein condensation of kaons at densities several times nuclear saturation density, in which case the matter is approximately isospinsymmetric. In high-density isospin-symmetric nuclear matter, neutron-proton pairs form in the 3Z?2 partial wave. 22 ' 23 However, once the isospin symmetry is slightly broken, 3I?2 pairing is suppressed and isospin-triplet neutron-neutron and protonproton pairs are formed. Due to their large partial density, neutrons pair in the 3 i-2-3-f2 tensor-coupled channel, 20,21 while the less abundant protons pair in the ^ o state. 17 - 63 We have seen in Subsec. (6.2.1) that for fermionic systems which are invariant under reversal of time and reflections of space, the quasiparticle spectrum is symmetric under p —» — p, and consequently the antisymmetric piece EA in Eq. (6.8) vanishes. Depending on the system, these symmetries could be broken either by the presence of external gauge fields or due to intrinsic properties such as the mass dif-

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ference in mixtures of gases. At any rate, we now focus on systems having EA 7^ 0, and the pairing in question is between fermions that lie on different Fermi surfaces. We shall call such systems asymmetric superconductors (hereafter ASC). Initial studies of ASC where carried out in the early sixties when, shortly after development of the BCS theory of superconductivity, metallic superconductors with paramagnetic impurities were studied experimentally. 64-67 Since collisions with impurities can flip the spins of electrons, an imbalance between spin-up and spindown electron populations is created. This effect can be mimicked by introducing an average, effective magnetic field that lifts the electron-spin degeneracy due to its interaction with the electron magnetic moment. The novel aspect of the studies of ASC (apart from the new context) is the realization that a correct interpretation of the results requires a self-consistent solution of the gap and density equations, even in the weak-coupling limit where the changes in the value of the chemical potential due to pairing are small. To avoid undue complications in describing ASC, we will operate within the framework of conventional BCS theory, in the sense that effects of wave-function renormalization and medium polarization are neglected. We shall also suppress the additional complication of the 3Si~3Di tensor coupling. This aspect is not essential for the present discussion; see Refs. 24-26 for the relevant details. Thus, the equations underlying the theory of ASC are taken to be (6.13) and (6.14), the spectrum being given by Eq. (6.8) with EA 7^ 0. If the spatial symmetries are unbroken, then EA = Sp = (pn — pp)/2, where pn and pp are the neutron and proton chemical potentials. In general, Eqs. (6.13) and (6.14) must be solved selfconsistently. Consider first the procedure in which Eq. (6.13) is solved by parametrizing the asymmetry in terms of the difference in the chemical potentials, and the densities of the species are computed after the gap equation is solved. Such an analysis predicts 64-66 a double-valued character of the gap as a function of Sp. On the first branch, the gap has a constant value A((fyi) = A(0) over the asymmetry range 0 < Sp < A(0) and vanishes beyond the point Sp = A(0). The second branch exists in the range A(0)/2 < Sp < A(0), with the gap increasing from zero at the lower limit to A(0) at the upper limit. Only the portion 5p < A(0)/V2 of the upper branch is stable, i.e., it is only in this range of asymmetries that the superconducting state lowers the grand thermodynamic potential from that of the normal state. 6 6 Thus, the dependence of the superconducting state on the shift in the Fermi surfaces is characterized by a constant value of the gap, which vanishes at the Chandrasekhar-Clogston 64,65 limit 6p! = A(0)/V2. A different picture emerges from an alternative treatment of the problem in which particle-number conservation is incorporated explicitly by solving Eqs. (6.13) and (6.14) self-consistently.68'69 These studies find a single-valued gap as a function of the isospin asymmetry a = (p n — pP)/(pn + PP)- Minimizing the free energy of an asymmetric superconductor at fixed density and temperature leads to globally stable solutions over the entire region of density asymmetries where non-trivial solutions of the gap equation exist. 68 ' 69 This can be seen in Fig. 6.7, where the temperature and asymmetry dependence of the pairing gap and the free-energy of a homogeneous asymmetric superconductor are shown. In particular, we see that

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a=0.0 a=0.05 = o.07

« = o,i

\. *,

• " ~~~

""""'•V

2J

\

\

\

\

T[MeV]

\ \ \ \ \ \ \ \ v

T[MeV]

Fig. 6.7. Left panel. Temperature dependence of pairing gap for density asymmetries a = 0.0 (solid), 0.05 (dashed-dotted), 0.07 (dashed), and 0.1 (dotted). Right panel. Temperature dependence of free energy. Labeling of asymmetries is as in the left panel.

for a fixed temperature, the gap and the free energy are single-valued functions of the density asymmetry a in the particle-number-conserving scheme - in contrast to what is found in the non-conserving scheme, where double-valued solutions appear. At large asymmetries, the dependence of the gap on the temperature shows a "re-entrance" phenomenon. As the temperature is increased from low values at which the asymmetry is too large to sustain a gap, a critical temperature is reached at which pairing correlations take hold. (For example, this is seen for the a = 0.1 case in Fig. 6.7). This behavior can be attributed to the increase of phase-space overlap between the quasiparticles that pair, due to the thermal smearing of the Fermi surfaces. Further increase of temperature suppresses the pairing gap at a higher critical temperature due to thermal excitation of the system, in much the same way as in the symmetric superconductors. Clearly, in this scenario the pairing gap has a maximum at some intermediate temperature. The values of the two critical temperatures are controlled by different mechanisms. The superconductivity (or superfluidity) is destroyed with decreasing temperature at the lower critical temperature when the smearing of the Fermi surfaces becomes insufficient to maintain the required phase-space coherence. The upper critical temperature is the analog of the BCS critical temperature and corresponds to a transition to (re-entrance of) the normal state because of thermal excitation. Another aspect of the asymmetric superconducting state is the gapless nature of the excitations. 70,71 One may draw an analogy to the non-ideal Bose gas, for which only a fraction of the particles are in the zero-momentum ground state at temperatures below the critical value for Bose condensation. The dynamical properties of gapless superconductors, such as response to electroweak probes and transport, depend on the ratio £ = A/5/x in an essential way: for C > 1> the response of the system is similar to that of an ordinary superconductor; in the opposite limit ( < 1, the system's behavior is essentially non-superconducting (see e.g. Ref. 72). These features are easily understood by examining the excitation spectrum in both limits. While the homogeneous ASC phase is globally stable, it could become locally unstable in a certain temperature-asymmetry domain. The local stability requires that the free energy is a convex function of the appropriate variables. 71 ' 73-79 For

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large enough asymmetries the low-temperature domain is locally unstable against spontaneous generation of superfluid current, which manifests itself in the negative superfluid density and purely imaginary Meissner mass. 74 ' 76 ' 79 6.3.1. Phases

with broken space

symmetries

We now turn to a special class of ASC characterized by broken global symmetries - translational, rotational, or both. In 1964, Larkin and Ovchinnikov80 and, independently, Fulde and Ferrell 81 (LOFF) discovered that the superconducting state can sustain asymmetries beyond the Chandrasekhar-Clogston limit if electrons pair with nonzero center-of-mass (hereafter CM) momentum. The weak-coupling result for the critical shift in the Fermi surfaces for onset of the LOFF phase is Sfi2 = 0.755 A(0) [> Sfii = 0.707 A(0)]. Since the condensate wave function depends on the CM momentum of the pair, its Fourier transform will vary in configuration space, giving rise to a lattice structure. The Fulde-Ferrell state is predicated on a plane-wave form A(r) = Aexp(—iP • r) for the gap function. Larkin and Ovchinnikov considered a number of lattice types and concluded that the body-centeredcubic (bbc) lattice is the most stable configuration near the critical temperature. Recent studies of this problem in the vicinity of the critical temperature employing Ginzburg-Landau theory show that the face-centered-cubic (fee) structure is favored.82 For illustrative purposes, let us consider the Fulde-Ferrell state, in which case the quasiparticle spectrum is given by

where the upper sign corresponds to neutrons and the lower sign to protons. The spectrum (6.43) is obtained by applying the following transformation to Eq. (6.8): Es - • Es + (P2 + p2)/2m and EA -» 5 p. ± p • P. Onset of the LOFF phase entails a positive increase in the quasiparticle kinetic energy oc Q2, which disfavors the Fulde-Ferrell state relative to the BCS state. However, the anisotropic term <x Pp, which can be interpreted as a dipole deformation of the isotropic spectrum, modifies the phase-space overlap of the fermions and promotes pairing. The LOFF phase becomes stable when the increase in the kinetic energy required to move the condensate is smaller than the reduction in potential energy made possible by the increase in the phase-space overlap. The magnitude of the total momentum serves as a variational parameter for minimization of the ground-state energy of the system. The pairing gap and the free energy of ASC with finite momentum are shown in Fig. 6.8. It is assumed that the gap function depends parametrically on the magnitude of the CM momentum, but is independent of its direction. 83 For such an Ansatz the anisotropy of the spectrum appears only in the Fermi functions in the kernels of Eqs. (6.13) and (6.14) and is averaged through the phase-space integration. It is seen in Fig. 6.8 that an ASC-LOFF state arises for arbitrary finite momentum of the condensate below some critical value. For large enough asymmetries the minimum of the free energy moves from P = 0 to intermediate values of P, i.e., the ground state of the system corresponds to a condensate with

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Fig. 6.8. Left panel. Dependence of pairing gap A in the LOFF phase on density asymmetry a and total momentum P of the condensate, relative to pairing gap Aoo in the limit P = 0 = a. Right panel. Dependence of free energy of the LOFF phase on the same quantities as in the left panel.

nonzero CM momentum of Cooper pairs. Note that for the near-critical range of asymmetries, the condensate exists only in the LOFF state and its dependence on the total momentum exhibits the re-entrance behavior found in the temperature dependence of the homogeneous ASC. The order of the phase transition from the LOFF to the normal state is a complex issue that depends on the preferred lattice structure, among other things (see Ref. 84 and work cited therein). Due to the Pauli exclusion principle, a noninteracting fermionic gas fills an isotropic Fermi sphere; similarly, if there are two types of noninteracting fermions, each species fills an isotropic Fermi sphere. Consider now a strongly interacting system that is a Fermi liquid rather than a Fermi gas. According to Fermi-liquid theory, the states of the interacting system are reached by switching the interaction on adiabatically. Driven by this process, the noninteracting gas evolves into a strongly-interacting liquid, in which the dressed single-particle degrees of freedom - the quasiparticles - once again fill a spherical shell isotropically. However, this simple Fermi-liquid picture may not hold in two-component (or multi-component) fermionic systems in which the fermions of differing species interact via strong pairing forces. Indeed, there can exist a stable superconducting phase that sustains ellipsoidal deformations of the Fermi-surfaces, a phase hereafter referred to as deformed Fermi-surface superconductivity 85 ' 86 (DFS). The quadrupole deformations of the Fermi surfaces are described by expanding the quasiparticle spectrum in spherical harmonics and keeping the I = 2 contributions, 85 ' 86 w± («) = w±(g) + e2t±P2{x), (6.44) where w± (q) is the spectrum of the homogeneous ASC and the coefficients e2 describe the deformations of the Fermi surfaces that break the rotational 0(3) symmetry down to 0(2). The 0(2) symmetry axis is chosen spontaneously. Thus, the quasiparticle spectrum of the DFS phase is obtained from the spectrum of homogeneous ASC by the transformations Es -> Es + (e 2 , + + £ 2 , - ) / 2 M , EA-*EA + (e2,+ - e2,-)/2fi. (6.45)

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Fig. 6.9. Left panel. Dependence of pairing gap in the DFS phase on density asymmetry and total momentum of the condensate. Right panel. Dependences of free energy of the DFS phase for the same input parameters as in Fig. 5.

We observe that the leading harmonic term responsible for deformation of either Fermi surface is that for / = 2, not 1 = 1, since the latter corresponds to a translation of one Fermi sphere relative to the other, without deformation. The deformations are deemed to be stable if they lower the free energy of the system relative to its value in the undeformed state. The deformation parameters e^± are determined by minimization of the free energy of the system, as was done for the Cooper-pair momentum parameter P in the case of broken translational invariance (LOFF states). Three-dimensional plots of the dependence of the pairing gap and free energy of the DFS phase on asymmetry and the relative deformation Se = (e2,+ — e2,_)/2^A are provided in Fig. 6.9. Fig. 6.10 shows a typical deformed Fermi-surface configuration that lowers the expected ground-state energy below that of the non-deformed state. At a = 0, the critical deformation for which pairing ceases is the same for prolate and oblate deformations. At finite a and in the positive range of Se, the maximum value of the gap is attained for constant Se; at negative Se the maximum increases as a function of deformation and saturates for Se ~ 1. The re-entrance phenomenon sets in for large asymmetries as Se is increased from zero to finite values. (N.B. The essential difference between LOFF and DFS phases is that in the latter, the translational symmetry of the superconductor remains unbroken.) To complete our discussion of pairing states in nonrelativistic asymmetric superconductors, we briefly mention some of the alternatives to the LOFF and DFS phases. One possibility is that the system prefers a phase separation of the superconducting and normal phases in real space, such that the superconducting phase contains particles with matching chemical potentials, i.e. is symmetric, while the normal phase remains asymmetric. 87 Equal-spin (-isospin, -flavor) pairing is another option, if the interaction between like-spin particles is attractive. 88 ' 89 Since the separation of the Fermi surfaces does not affect spin-1 pairing on each Fermi surface, an asymmetric superconductor may evolve into a spin-1 superconducting state (rather than a non-superconducting state)

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2

1.5

0.5

*

0

0.5

•1.5

•2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

Fig. 6.10. Projection of the Fermi surfaces on a plane parallel to the axis of symmetry breaking. T h e concentric circles correspond to the two populations of spin(isospin)-up and spin(isospin)down fermions in a spherically symmetric state (5e = 0), while the deformed symbols correspond to the state with relative deformation 5e = 0.64. The spin(isospin)-density asymmetry is a = 0.35.

as the asymmetry is increased. Therefore spin-1 pairing becomes the limiting case for very large asymmetries. If the single-particle states defining the different Fermi surfaces are characterized by spin (as is the case in the metallic superconductors), the pairing interaction in a spin-1 state should be P-wave and the transition is from 5-wave to P-wave pairing. If the fermions are characterized by one or more additional discrete quantum numbers (say isospin as well as spin), the transition may occur between different S-w&ve phases (e.g. from isospin-singlet to isospintriplet in the case of nuclear matter). The possibilities become especially rich in dense quark matter.

6.4. Crossover from B C S pairing to Bose-Einstein condensation A crossover from BCS superconductivity to Bose-Einstein condensation (BEC) is exhibited in fermionic systems with attractive interactions under sufficient decrease of the density and/or sufficient increase of the interaction strength. The transition from large overlapping Cooper pairs to tightly bound non-overlapping bosons can be described entirely within the ordinary BCS theory, if the effects of fluctuations are ignored (mean-field approximation). Early studies of this type of transition were carried out in the contexts of ordinary superconductors, 90 excitonic superconductivity in semiconductors, 91 and, at finite temperature, an attractive fermion gas. 92 Although the BCS and BEC limits are physically quite different, the transition between them is found to be smooth within ordinary BCS theory.

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— na=1.26

2-

— na' = 0.63 — na3 = 0.32

1=

'-

na' = 0.21

•••• na' = 0.13

0^

0.5

1.5

2 T[MeV]

2.5

3.5

Fig. 6.11. Dependence of pairing gap (upper panel) and chemical potential (lower panel) on temperature for fixed values of the ratio / = n o / n , where n is the baryon density and no = 0.16 f m - 3 is the saturation density of symmetrical nuclear matter. Values of the diluteness parameter na3 assume a scattering length a = 5.4 fm.

Several authors have considered the BCS-to-BEC transition in the nuclear context. In isospin-symmetric nuclear matter, neutron-proton (np) pairing undergoes a smooth transition leading from an assembly of np Cooper pairs at higher densities to a gas of Bose-condensed deuterons as the nucleon density is reduced to extremely low values. 93-97 This transition may be relevant - and could then yield valuable information on np correlations - in low-density nuclear systems (especially the nuclear surface), in expanding nuclear matter from heavy-ion collisions, and in supernova matter. The underlying equations of the theory are (6.13) and (6.14) with EA = 0; we shall address the effects of asymmetry at a later point. The top panel of Fig. 6.11 shows the dependence of the gap function on temperature for several densities n, given in terms of the ratio / = no/n, where no = 0.16 f m - 3 is the saturation density of symmetrical nuclear matter. The bottom panel shows the associated chemical potentials fi computed self-consistently from Eq. (6.14). The low- and high-temperature asymptotics of the gap function are well described by the BCS relations A(T -> 0) = A(0) - [27raA(0)T] 1 / 2 exp(-A(0)/T) and A(T -> Tc2) = 3.06/3[Tc2(Tc2 - T)] 1 / 2 , respectively, where Tc2 is the critical temperature of the phase transition. However, the BCS weak-coupling values a = 1 = (3 must be replaced with a ~ 0.2 and /3 ~ 0.9. As a consequence, the ratio of the zero-temperature gap to the critical temperature deviates from the familiar BCS result A(0)/TC2 = 1.76. Deviations from the original BCS theory are understandable in that (i) the system is in the strong-coupling regime, and (ii) the pairing is in a spin-triplet rather than a spin-singlet state. 9 7

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One measure of coupling strength is the ratio A(0)/|/i| of the zero-temperature energy gap to the magnitude of the chemical potential. It is seen from Fig. 6.11 that the strong-coupling regime is realized for / > 40 (i.e., A ^> /x). At / = 20 the system is in a transitional regime (A ~ fi). Another measure of coupling is the diluteness parameter n\a\3, where a is the scattering length. In agreement with the first criterion, the matter is in the dilute (or strong-coupling) regime for / > 40, since this range corresponds to na3 = 0.63 < 1 when a is taken as the triplet n — p scattering length, 5.4 fm. A signature of the crossover from weak to strong coupling is the change of sign of the chemical potential, which occurs for / w 80 (Fig. 6.11), slightly below the crossover density between weak-coupling and strong-coupling regimes. In the limit of vanishing density, / —» oo, the value of the chemical potential at T = 0 tends to ^(oo) = —1.1 MeV, which is half the binding energy of the deuteron in free space. Indeed, in this limit the gap equation reduces to the Schrodinger equation for a two-body bound state, with the chemical potential assuming the role of the energy eigenvalue.90 Thus, the BCS condensate of Cooper pairs in the 3 S\-3D\ state evolves into a Bose-Einstein condensate of deuterons as the system crosses over from the weak- to the strong-coupling regime. The crossover is smooth, taking place without change of symmetry of the many-body wave function. How does isospin asymmetry affect the transition? As the system is diluted, the critical asymmetry at which the pairing disappears increases from small values of the order of 0.1 up to the asymptotic value a = 1. The reason for this behavior is that in the low-density matter, the excess neutrons do not appreciably change the wave functions of protons, which are bound into pairs. 96 At asymptotically small densities, the chemical potential of protons tends to iip(oo) = —2.2 MeV, which is just the binding energy of the system per half the number of particles bound into deuterons. The chemical potential of neutrons is determined by the excess particles in the continuum and goes asymptotically as /i n (oo) —> 0 (i.e. there is ultimately no energy cost in adding a neutron to the system). Note that the asymptotic behavior described is independent of the degree of isospin asymmetry. In closing this section, we call attention to the remarkable progress achieved during the last few years in trapping and manipulating ultracold fermion gases. The strength of the two-body interaction between the constituent fermionic atoms can be tuned using the Feshbach resonance mechanism, by varying the external magnetic field;98"100 thus, the entire range from weak to strong couplings can be probed. Recent experiments on ultracold atomic gases have begun to explore their properties in cases where pairing occurs between atoms in different hyperfine states, which are unequally populated. 101 ' 102 Systems of this kind are also subject to intensive theoretical study, with specific attention to homogeneous ASC phases, 7 3 - 7 9 ' 1 0 3 phases with broken symmetries 104 and their realization in finite trap geometries. 105 The universal features of ASC revealed by this effort should contribute significantly to our understanding of nucleonic pairing under isospin-asymmetric conditions.

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6.5. V o r t e x s t a t e s in compact s t a r s 6.5.1. Currents

and quantized

circulation

The macroscopic physics of neutron-star rotation and its anomalies observed in the timing of pulsars can be described within the hydrodynamic theory of superfiuids suitably extended to multifiuid systems. 106-112 The elementary constituents of a Fermi superfiuid - the Cooper pairs - are characterized by a coherence length £. On length scales L » ( , the condensate of Cooper pairs can be described by a single wave function ip, and the condensate forms a macroscopically coherent state. At an intermediate or "mesoscopic" scale, stellar rotation and the presence of a magnetic field lead to the formation of vortices, macroscopic quantum objects whose distinctive property is the quantization of circulation around a path encircling the vortex core. Since the condensate wave function must be single-valued at each point of the condensate, the circulation is quantized in units of 2nh. On writing ip = V'oe1*, the gauge-invariant superfiuid velocities can be expressed through the gradient of the phase of the superfiuid order parameter \ a n < i the value of the vector potential A: vT = ;^-Vxr-—A. (6.46) 2mT mTc In this expression, eT = (e, 0) specifies the electric charge of protons (p) and neutrons (n) respectively, mT is their bare mass, and r stands for n or p. Applying the curl operator to Eq. (6.46) and implementing quantization of the circulation (with the phase of the superfiuid order parameter changing by 2ir around a closed path), one finds curltv = —vr mT

Y" S^fx *-^ j

- xTJ) - —B mTc

= uT ,

(6.47)

where irh/mT is the quantum of circulation, vT = U}T/LJT is a unit vector along a given vortex line, xTj defines the position of a vortex line in the plane orthogonal to the vector vT, 5^ is a two-dimensional Dirac delta function in this plane, and B = curl A is the magnetic-field induction. The index j is summed over the sites of vortex lines. Eq. (6.47) treats the vortex cores as singularities in the plane orthogonal to uT\ this simplification is justified on scales larger than the coherence length of the condensate. For a single vortex, the integral of Eq. (6.47) completely determines the superfiuid pattern. Since this equation is linear, the superfiuid pattern created by a larger number of vortices is formed by superposition of the flows induced by each vortex. Obviously, the resulting net flow depends on the arrangement of the vortices. The condensate wave function can be written as ip(x) = f(r)el9 in cylindrical polar coordinates (r, 0,z). Upon integrating Eq. (6.47), the neutron and proton superfiuid velocities then become

^

=2^<

*>u = ^K>Q§>

(6 48)

-

where Ki is the Bessel function of imaginary argument. The divergence of the neutron-vortex velocity vn(r) as r —> 0 is regularized by a cutoff A ~ £ n . The

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long-range nature of vn(r) results in slow falloff of a density perturbation in the condensate. In a proton vortex, the supercurrent is screened exponentially on length scales of the order of the penetration depth A. Thus, for r 3> A, K\{r/\) ~ exp(—r/A). On global, hydrodynamic scales, transition to a continuum vortex distribution can be carried through on the right-hand side of Eq. (6.47) by defining vortex densities n T = V • 6^(x - xTj). Since the curl of vn is simply 2fi for rigid-body rotations, the number density of vortices in the neutron superfluid is related to the macroscopic angular velocity of the neutron condensate by the Feynman formula nn =

—- .

(6.49)

For typical pulsar periods P in the range 0.05 < P < 0.5 s, one has nn ~ 6.3. x 10 3 P _ 1 ~ 10 4 -10 5 per cm 2 . In the case of a charged superfluid, Eq. (6.47) can be transformed to a contour integral over a path along which vp = 0, since the supercurrent is screened beyond the magnetic field penetration depth A. If the proton superfluid is a type-II superconductor (i.e., A/£ p > l / v 2 ) , the continuum vortex limit leads to the estimate np = — ~ 5 x 10 18 c m - 2 ,

(6.50)

where $o = i^hc/e is the flux quantum. We note that the number of proton vortices per neutron vortex is np/nn ~ 10 13 — 10 14 , independently of their arrangement. The energy of a bundle of neutron or proton vortices is minimized by a triangular lattice with a unit cell area n~l = (\/3/2) d 2 . The lengths of "basis vectors" of the lattices in the neutron and proton condensates (the inter-vortex distances) are

dn =

{7^Ji)

'

dp=

{v^)

'

(6 51)

-

where B is the mean magnetic-field induction. Using the estimates given in Eqs. (6.49) and (6.50), one finds that the neutron and proton inter-vortex distances are dn ~ 1 0 - 2 — 10~ 3 cm and dp ~ 1 0 - 9 cm, respectively. For typical values of the microscopic parameters, the penetration depth is of the order 100 fm = 1 0 - 1 1 cm. Therefore the conditions £„
\P21 Pll)

\V2j

(6 . 52) ^

V

where 1 and 2 label the isospin projections. The off-diagonal elements, which would vanish in the noninteracting limit, are evidently responsible for the entrainment effect. One fundamental consequence of this effect is that the neutron vortex carries a non-quantized magnetic flux 106 ' 113 of the same order of magnitude as the flux

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quantum 3>o- If the proton fluid forms a type-II superconductor, the number of proton vortices (sometimes called flux tubes) is 10 12 — 10 13 per neutron vortex [see Eq. (6.50)]. Accordingly, the neutron-vortex motion (dynamics) is likely to be affected by the proton-vortex array. The arrangement of the proton-vortex lattice is a complex issue, and there exist several models for its configuration. (i) A class of (flux-tube) models envisions the proton-vortex array to be spatially homogeneous, the vorticity vector being inclined by some angle with respect to the spin vector. 114 Further, it is assumed that the flux tubes act as extended pinning centers for neutron vortices. In such models, a change in the neutronvortex distribution is achieved by vortex creep of neutron vortices through the array of flux tubes. (iia) Vortex cluster models predict clustering of proton vortices over about 10% of the area occupied by a neutron vortex. One particular model generates a bundle of proton vortices coaxial with the neutron vortex, through the entrainment currents induced by neutron-vortex circulation. 115 This gives rise to an average axisymmetrical magnetic field whose magnitude is compatible with pulsar observations. (iib) The homogeneous distribution of proton vortices could be generically unstable towards phase separation between a phase containing dense mesh of proton vortices and a phase devoid of vortices. A necessary condition is that the vortex lattice is sufficiently dilute, the mean intervortex distance being much larger than the penetration depth. 116 (iic) Proto-neutron stars are likely to possess natal magnetic fields; the nucleation of such a field will be associated with a first-order normal-superconducting phase transition, squeezing the field into bubbles of superconducting regions with high B ~ 10 14 G, and forming stable protonic vortex arrays which again cover about 10% of the total area. 117 The dynamics of the neutron-vortex array in vortexcluster models is controlled by the electromagnetic scattering of electrons off a vortex cluster. Current models of BCS pairing of protons do not exclude the possibility that there is a transition from type-II to type-I superconductivity of protons as the density is increased. 118 Type-I superconducting protons will have domain structures analogous to those observed in laboratory experiments on terrestrial superconducting materials. The electrodynamics of the proton domain structures in NS can be treated by adapting the theories developed for laboratory superconductors, in which the magnetic fields are generated by normal currents driven around a cylindrical cavity by temperature gradients. 119 A recent theoretical study 120 examined the effect of interactions between neutron and proton Cooper pairs on the status of proton superconductivity. The results suggest that type-I superconductivity can be enforced throughout the entire stellar core if the strength of the interaction between Cooper pairs is significant. However, within the mean-field BCS theory, the Cooper pairs are noninteracting entities, and any deviations from this picture must be due to fluctuations. Alford et al. 121 estimated the strength of the interaction between neutron and proton Cooper pairs due to fluctuations and found it to be

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too small to account for the interactions assumed in Ref. 120. Nevertheless, type-I superconductivity of protons is not excluded by current calculations of 1S'o pairing of protons, and we shall address below its potential implications for the macroscopic manifestations of superfluidity in neutron stars. 6.5.2. Constraints placed by neutron-star precession friction between superfluid and normal-fluid

on the mutual components

Since most of the inertia of a NS is carried by the neutron superfluids in the core and in the crust, the key to an understanding of NS rotational anomalies lies in the transfer of angular momentum from the superfluid to the normal (unpaired) component of the star, whose rotation is observed through the magnetospheric emission. At the local hydrodynamical scale, the rate of angular momentum transfer between the superfluid and normal components is determined by the equation of motion of a superfluid neutron vortex line. In the approximation that the inertial mass of the vortex is neglected, the equation of motion is UJS{VS

- vL) x v + C(vL - vN) + ('{vL - vN) x i / = 0,

(6.53)

where Vs and VJV are the superfluid and normal fluid velocities, VL is the velocity of the vortex, u is a unit vector along the vortex line, ws is the unit of circulation, and £, C are (dimensionless) friction coefficients, also known as the drag-to-lift ratios. These coefficients encode the essential information on the microscopic processes of interaction of vortices with the ambient unpaired fluid. Microscopic calculations commonly indicate C ~ 0, and one is left with a single parameter £. In the NS crust, neutron vortices are embedded in a lattice of neutron-rich nuclei, and the £ coefficient is determined by the interaction of the vortices with the nuclei and the electron plasma. (In some models, neutron vortices are localized - pinned to the nuclei or situated in between them. 122,123 If the pinning is strong, Eq. (6.53) is not valid, since the forces acting on the vortex are not linear functions of velocities. However, the regime of perfect pinning can be identified with the £ —+ oo limit.) In the core of the star, the friction is controlled by the interaction of neutron vortices with the ambient electron-proton plasma; Eq. (6.53) is valid under these conditions. Initial studies of the dynamical coupling between the superfluid and the normal fluid focused on interpretation of the observed post-glitch relaxation of pulsar rotational periods. It turns out that such interpretation is fraught with ambiguity, because the long relaxation times can be obtained in the two opposite limits of weak (C —> 0) and strong (C —» oo) couplings. Recent observation 3 of long-term periodic variabilities in PSR B1828-11, if attributed to precession of this pulsar, challenges existing theories of vortex dynamics in NS. 1 2 4 - 1 2 6 The importance of the inferred precession mode stems from the fact that it involves non-axisymmetric perturbations of the rotational state, removing the degeneracy with respect to £ that is inherent in the interpretation of post-glitch dynamics. In the frictionless limit, a star must precess at the classical frequency efi, where e is the eccentricity and Q is the rotation frequency. Clearly, then, there must exist a crossover from damped to free precession as £ is decreased. The crossover is determined by the dimensionless parameters (IS/IN)P and (IS/IN)P'Here,

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/3 = C/[(l - C')2 + C2], 13' = 1-/3(1(')/(, Is is the moment of inertia of the superfluid, and IN is the moment of inertia of the crust plus any component coupled to it on time scales much shorter than the precession time scale. The precession frequency is 125 flp = eCls

i+

'£) + *&

(6.54)

where Qs is the spin frequency and e is the eccentricity. A no-go theorem 125 states that Eulerian precession in a superfluid neutron star is impossible if (IS/IN)C >1 (assuming as before £' —> 0). There is a subtlety in this result: the precession is impossible because the precession mode, apart from being damped, is renormalized by the non-dissipative component of the superfluid/normal-fluid interaction (oc /3'). In effect, the value of the precession eigenfrequency drops below the damping frequency for any ( larger than the crossover value. This counterintuitive result cannot be obtained from arguments based solely on dissipation. In fact, according to Eq. (6.54), the damping time scale for precession increases linearly with £, and in the limit (, —> oo one would (wrongly) predict undamped precession. If a neutron star contains multiple layers of superfluids, the picture is more complex, but the generic features of the crossover are the same. 125 While it is common to study perturbations from the state of uniform rotation, the precessional state may actually correspond to the local energy minimum of an inclined rotor if there is a large enough magnetic stress on the star's core. 127,128 In the core of a neutron star, the quantized neutron-vortex array is embedded in an electron-proton plasma, with the protons in a superconducting state. Electrons will scatter off the anomalous magnetic moments of (ungapped) neutron quasiparticles localized in the core of a neutron vortex. 129 Because of the 3 P2 spin-1 nature of the order parameter of the neutron superfluid in the core, the 3 P2 vortex core has an additional magnetization that scatters electrons more effectively.130 An even more efficient scattering mechanism comes into play due to the flux ~ $o induced by the proton supercurrent on the neutron vortex via the entrainment effect.113 If the protons are non-superconducting in some regions of the core, then the strong nuclear interaction between protons and neutron quasiparticles localized within a vortex core leads to an efficient coupling of the electron-proton plasma and the neutron superfluid.131 The above models belong to the class of weak-coupling theories, i.e. ( < 1, and, according to the no-go theorems, are compatible with free precession of the neutron star. However, these theories of mutual friction assume (unrealistically) that the proton-vortex lattice has no effect whatsoever on neutron-vortex dynamics in the core. The mechanism underlying mutual friction in the flux-tube models is the slow motion of neutron vortices through the pinning barriers (here, flux tubes) via thermally activated creep. Since, in general, the creep models presume that the vortex lattice closely follows the rotation of the pinning centers (N.B. the case of perfect pinning corresponds to £ —> oo), the effective friction in these models is large, 132 ( > 1. The kelvon-vortex coupling in the core provides another interaction channel, leading again to 1 2 6 £ » 1. Similarly, for vortex-cluster models, in which the neutron-vortex lattice and the associated proton-vortex cluster move coherently,

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electron scattering by proton-vortex clusters also gives 109 C, ~> 1. Accordingly, these theories are incompatible with free precession of a neutron star. This conclusion has been stressed by Link, 126 and it has been argued that type-I superconductivity could be an alternative. In the crust of a neutron star, the neutron-vortex lattice is embedded in a lattice of nuclei and the charge-neutralizing background created by an almost homogeneous electron sea. In the vortex-creep models, the neutron-vortex lattice maintains rotational equilibrium via thermal and quantal creep through the pinning barriers (nuclei). 122,123 Hence these models imply C > 1 and are incompatible with free precession. If the pinning is absent, either because re-pinning cannot be achieved in post-jump equilibrium 133 and/or because of mutual cancellation of the forces from different pinning centers, the freely flowing neutron-vortex lattice interacts with the electron-phonon component of the crust. These interactions are weak and lead to 1 3 4 ' 1 3 5 ( < C l . Note that the above estimates assume that the ratio IS/IN is roughly of order unity. While it is difficult to estimate this ratio precisely, it is unlikely to differ from unity by many orders of magnitude. 6.5.3. Type-I superconductivity

in neutron

stars

The equilibrium structure of alternating superconducting and normal domains in a type-I superconductor is a complicated problem that depends on, among other things, the nucleation history of the superfluid phase. By flux conservation, the ratio of the sizes of the superfluid and normal domains is given by the relation ds/dN = y/Hcm/B ~ 10, where B ~ 10 12 G is the average value of the magnetic induction and Hcm ~ 10 14 G is the thermodynamic critical magnetic field. We first examine a model 118 in which the magnetic field generated by the entrainment effect supports the formation of domains coaxial with the neutron vortex. Consider a vortex that moves at a constant velocity VL and carries a coaxial normal domain of protonic fluid relative to the background electron liquid. Continuity of the electrochemical potentials of the superfluid and normal phases across the boundary between them entails the existence of a constant transverse electric field mtvLVv(a) E=

p

pK

' (6.55) ea across the normal domain (see Fig. 6.12). The energy dissipated per unit length of a vortex is W = aE2 (a/6) 2 , where a is the electrical conductivity and the factor (a/6) 2 is the fractional area occupied by the domain. Combining this relation with Eq. (6.55), we obtain an alternative expression W = r\v\ for the dissipation, which identifies the friction coefficient as 1 3 6 b

<= — = - ^ (AV

f?In (-) coth ( -^) - if ,

(6.56)

Ps^s ps^sc \2nabJ [X \aj \ a } J where <J>i = (pi2/pn)$o- The zero-field conductivity of ultra-relativistic electrons is (To = ne^cTc/peF, where pef is the electron Fermi momentum, ne is the electron number density, and TC is the relaxation time for Coulomb scattering of electrons off protons in the normal domains. 14 The conductivity a = o"o/(wcrc)2 entering

A. Sedrakian and J. W. Clark

\y

Fig. 6.12. The structure of a rotational vortex placed in a type-I superconductor. The vortex velocity field is indicated by the concentric circles. T h e non-superconducting domain (shaded region) of radius a is coaxial with t h e vortex and carries a magnetic field HCTa ~ 10 1 4 G. T h e vortex motion along t h e i-axis generates a transverse electric field, which drives t h e electron current through the domain and causes Ohmic dissipation.

Eq. (6.56) includes the effect of the bending of electron trajectories in the magnetic field. In this formula, o»c = eHcm/pF is the electron cyclotron frequency, which is proportional to the thermodynamic critical field Hcm. For the typical density range p = (7.9 — 8.6) x 10 14 g/cm 3 , the friction coefficient has the order of magnitude C, ~ 10~ s at the temperature T = 108 K and scales with temperature as £ oc T2. The drag-to-lift ratio satisfies the condition £ -C 1 for all T below the critical temperature of the superfluid phase transition. We turn next to the friction coefficient as predicted by the model of Buckley et al., 120 in which the normal domains contain a large number N ~ 10 neutron vortices. In this case, the damping of the differential rotation between the electronproton plasma and the neutron superfluid is due to the interaction of domain (i.e., non-superconducting) protons with the core quasiparticles confined in the neutronvortex core. The relaxation process is therefore the same as in the case where the proton fluid is non-superconducting over the entire bulk of the core. However, the result needs to be rescaled by the ratio of the areas occupied by the normal and superconducting layers. The relaxation time per single vortex is 1 3 1

(6.57) T „ = 6 (Hay i ^«p(( U B-ALV \PFnJ hm*pTanv \ eFnTJ where pFp and ppn are the Fermi momenta of protons and neutrons, fi* = rn Tn p n/(Tnj> + mn) i s t n e reduced effective mass in terms of the proton and neutron effective masses, anp is the total in-medium neutron-proton scattering cross section, A n is the gap in the neutron quasiparticle spectrum, and epn is the neutron Fermi energy. [Eq. (6.57) differs from the analogous expression in Ref. 131 by the factor Amn/hP; here P is the pulsar period and mn is the free-space neutron mass.]

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In the relaxation-time approximation and zero-temperature limit, the friction is written as PS^S

PS^SCTnp

'

where np is the proton number density. For proton densities in the range pp = (4 - 8) x 10 14 g/cm 3 and temperatures T ~ 1 0 7 - 8 , one finds £ < 0.1. For a given model of the type-I superconducting structure, the friction coefficient £ must be rescaled by a factor (d^/ds)2 ~ 0.01. Now we are in a position to discuss the implications of the friction mechanisms described above for neutron-star precession. The condition (IS/IN)C < 1 seems to fulfilled unless Is/IN > 1- The magnitude of the ratio Is/IN depends on the superfluid/normal-fluid friction within all superfluid regions of the neutron star and is hard to assess. Glitches and post-glitch relaxation provide a model-independent lower bound, IS/IN > 0.1. On the other hand, an upper bound is difficult to set. The deep interior of the star, if superfluid, could be decoupled from the observable parts of the star on evolutionary time scales, without any effect on short-time-scale physics (although one does require (, —> 0, rather than £ —> oo, to prevent damping of the precession). At any rate, it is rather unlikely that Is/IN exceeds unity by many orders of magnitude, and appealing to the lower bound on the ratio of the moments of inertia, one can conclude that the precession is undamped for both dissipation mechanisms considered.

6.6. Concluding remarks This review has covered a number of aspects of nucleonic superfluidity, ranging from microscopic theories of pairing in nuclear systems and neutron stars to mesoscopic frictional processes in superfluids and rotational anomalies in pulsars. Our survey of this important subfield is by no means complete. While the selection of topics is naturally biased toward the primary interests of the authors, we have chosen topics and problems with the intent of elucidating (i) the fascinating relationships between the physics involved at different scales and (ii) the richness of the contributions from diverse subfields of physics. We close by listing a number of issues and problems that call for further clarification and concerted effort within the general framework of our discussions. • The pairing problem at the level of mean-field BCS theory, with the pairing driven directly by in-vacuum nuclear interactions, is essentially solved within the density range over which these interactions are constrained by experiment. On the other hand, issues such as the screening of nuclear interactions, renormalization of the single-particle spectrum, and off-shell energy behavior of the pairing gap still defy quantitative resolution. Broadly speaking, extensions beyond the BCS theory are needed that incorporate fluctuation corrections while providing a consistent treatment of short-range correlations. • Superfluid phases with broken space-time symmetries have received much attention from theorists in recent years, while experimental realization of asymmetric

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superfluids opens the possibility of testing the predictions of theory. Importantly, relevant experiments are now probing the B C S - B E C crossover via the Feshbach resonance mechanism. Given a broad effort, there is the prospect of mapping out the superfluid phase diagrams of interesting fermionic systems, b o t h experimentally a n d theoretically, in the space of coupling strength, spin/isospin asymmetries, t e m p e r a t u r e , etc. • T h e rotational anomalies observed in neutron stars continue to provide useful constraints on the state of the superfluid m a t t e r in neutron-star interiors. Further theoretical studies of vortex dynamics, combined with pulsar timing observations, may be expected to shed new light on the internal structure of the superfluid phases of neutron stars, especially on the question whether protons form a type-I or type-II superconductor.

Acknowledgments We acknowledge useful interactions with A. Bulgac, J. M. Cordes, W . H. Dickhoff, J. Dukelsky, V. A. Khodel, T. T . S. Kuo, B. Link, U. Lombardo, H. Miither, A. Polls, P. Schuck, H.-J. Schulze, I. Wasserman, D. N. Voskresensky, a n d M. V. Zverev. AS acknowledges research support through a Grant from the SFB 382 of the Deutsche Forschungsgemeinschaft; J W C , through Grant No. PHY-0140316 from the U.S. National Science Foundation.

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Chapter 7 P a i r i n g P r o p e r t i e s of Dressed Nucleons in Infinite M a t t e r

Willem H. Dickhoff Department of Physics, Washington University, St. Louis, Missouri 63130, USA Herbert Miither Institut

fur Theoretische Physik, D-72076 Tubingen,

Universitat Germany

Tubingen,

A comprehensive survey of the pairing properties of nucleonic matter is presented that includes the complete off-shell propagation associated with short-range and tensor correlations. For this purpose, the gap equation has been solved in its most general form employing the full energy and momentum dependence of the normal self-energy contributions. The latter correlations include the self-consistent calculation of the nucleon self-energy that is generated by the summation of ladder diagrams. A huge reduction in the strength and density range of Si- D\ pairing is obtained for symmetric nuclear matter as compared to the standard BCS treatment. Similar dramatic results pertain to 1So pairing of neutrons in neutron matter.

Contents 7.1. Introduction 7.2. Role of correlations in low-energy nucleon propagation 7.3. Short-range correlations and nucleon propagation in infinite matter 7.4. Pairing results for dressed nucleons 7.5. Conclusions Acknowledgments Bibliography

7.1.

175 177 180 187 195 197 197

Introduction

T h e description of pairing in infinite nuclear systems has a long history. For a recent review we refer t o Ref. 1. In the development of a detailed understanding of pairing phenomena involving nucleons in symmetric or pure neutron m a t t e r several obstacles have been encountered. Some of t h e associated puzzles will be addressed in t h e present discussion. Straightforward goals of pairing studies of infinite nucleonic m a t t e r include t h e t e m p e r a t u r e and density range of superfluidity with t h e accom175

W. H. Dickhoff and H. Miither

176

20

> u 15

|

10

& 2 5

0

0

1

2

kF (frrf1)

3

Fig. 7.1. Minimum gap between sp states at kp necessary to avoid a pairing instability in the ladder equation as a function of kp. For the 3 5 i - 3 D i case the limit of zero density yields half the binding energy of the deuteron.

panying magnitude of the gap as a function of these variables. Studies with realistic nucleon-nucleon (AW) interactions that describe experimental AW scattering data up to pion production threshold, have been mostly confined to the solution of the gap equation that employs a description of nucleons as mean-field particles. Such calculations for symmetric nuclear matter lead to gaps for the 3Si-3D\ coupled channel of the order of 10 MeV at densities around saturation. 2- * An illustration of these results is provided in Fig. 7.1. The value of the minimum gap to avoid a pairing instability when solving the ladder equation (summation of all particleparticle (pp) and hole-hole (hh) diagrams) in nuclear matter is plotted as a function of kp for the 3Si-3D\ and 1SQ channels. 3 The puzzle associated with these results is the absence of empirical information in finite nuclei that can be reconciled with such huge gaps for the proton-neutron pairing in the 3Si-3Di channels. For the 1 S , 0 channel the situation is distinctly different. Typical gap sizes around normal density have quite similar magnitudes to those encountered in nuclei for like-nucleon pairing, i.e., with 1So quantum numbers. Three main issues need to be addressed before these early results can be considered definitive. The first is to clarify in what way results from nuclear and isospin polarized matter can be related to pairing phenomena in finite nuclei. We briefly return to this issue in the conclusions. Secondly, and related to the first, it is necessary to assess the relevance and the importance of the influence of polarization effects. We will employ the convention to use this term to represent the presence of additional contributions to the irreducible interaction in the gap equation beyond the bare AW interaction. The exchange of random phase approximation (RPA) phonons between particles participating in pairing qualifies for this inclusion, hence the referral to polarization contributions. For neutron matter such effects must be accurately incorporated before final conclusions about pairing phenomena in that

Pairing Properties of Dressed Nucleons in Infinite Matter

177

system can be drawn. For nuclear matter this is less obvious, since the relevance of volume polarization phenomena for the understanding of pairing phenomena in finite nuclei is limited, whereas surface polarization effects in finite systems must be deemed different and of course more relevant. A third critical issue is related to the treatment of modifications of nucleon propagation in the nuclear medium when pairing correlations are studied. This subject will be the main topic of this chapter. We start in Section 7.2 with a review of both experimental and theoretical evidence that clarifies the influence of long-range (LRC) and short-range correlations (SRC) on the propagation properties of nucleons in nuclei. The emerging picture suggests that both correlation effects are substantial and play a significant role for particles considered as candidates to participate in pairing phenomena. Since only SRC can be comfortably studied in infinite matter with the confidence that such studies are relevant for finite systems, we proceed in Section 7.3 to describe how such correlations can be incorporated in nuclear and neutron matter. In particular, we emphasize the change in the single-particle (sp) strength distribution when such effects are included. The extension of this description to the solution of the gap equation is discussed in Section 7.4 together with the corresponding results. Concluding remarks are presented in Section 7.5.

7.2. Role of correlations in low-energy nucleon propagation Advances in the understanding of the sp properties of nucleons in nuclei and nuclear matter 7 demonstrate the dominant influence of short-range and tensor correlations 3

A

>

/ / ' 1J

2

1.

h i;

/

/ s

o

/

/I / ' / i

\ \ \ \ \ \ \

\

\ ~A

i

i

\ \ ^ \ ^V \ \

/ '

''

\ 10"'

\ \

^

/

o/

V/./

/ 10°

101

r(fm) Fig. 7.2. Wave function in coordinate space for the 1So interaction at an energy corresponding to fco = 0.25 f m _ 1 . The solid line is the free wave function, given by sin(fcor)/fcor. The correlated wave function is given by the long-dashed line and clearly exhibits the suppression, at short distances, expected for the repulsive core. The latter is indicated by the dashed line and divided by a factor 100.

178

W. H. Dickhoff and H. Miither

in generating the distribution of the spectral strength associated with the addition or removal of protons from the ground states of closed-shell nuclei. As a reminder of the typical features associated with SRC, Fig. 7.2 illustrates the reduction of the correlated relative scattering wave function inside the 1SQ core of the NNinteraction, which is also displayed (reduced by a factor of 100). The reduction of the relative wave function in coordinate space is accompanied by related admixture of highmomentum components. The latter lead to the presence of faster nucleons in the ground state of a many-nucleon system than can be expected on the basis of the Pauli principle alone, i.e., in an infinite system it will lead to the occupation of states with momentum above kp. A similar occupation of high-momentum states beyond the typical values associated with confining nucleons to a finite volume, is predicted for finite nuclei.8 Associated with this occupation of high-momentum states is the depletion of states that are occupied in the noninteracting system. A depletion of states with momentum less than kp is therefore inevitable for the case of the infinite Fermi system. Similar depletion effects have been predicted for finite nuclei.9 Such a global depletion of Fermi sea due to these correlations, has recently been observed in an (e,e'p) experiment from NIKHEF. The analysis of this experiment puts the depletion of the proton Fermi sea in 2 0 8 Pb at a little less than 20% 10 in accordance with earlier nuclear matter calculations. 9 These results are summarized in Fig. 7.3. The experimental occupation numbers of the various proton shells that would be completely filled by protons in the independent-particle model relevant for this

(MeV) Fig. 7.3. Occupation numbers from the 2 0 8 P b ( e , e'p) reaction. A global depletion for all the protons that are fully occupied in the independent-particle model, is clearly visible. Data have been obtained at the NIKHEF accelerator in Amsterdam. 1 0

Pairing Properties of Dressed Nucleons in Infinite

o

Matter

179

High-energy strength due to SRC and tensor force

high momenta

>15%

continuum

100 MeV

O/

10% Coupling to surface phonons and Giant Resonances

A / •

a 73

i

ofF ' \ / \ / \ / \ / \ / \ J ' ' _ _

3 C

^

65% quasihole strength 10%

i

«

Location of high-momentum components due to SRC at high missing energy

Coupling to surface phonons and Giant Resonances

Spectral strength for a correlated nucleus

Fig. 7.4. The distribution of single-particle strength in a nucleus like 2 0 8 P b . The present summary is a synthesis of experimental and theoretical work discussed in this section. A slight reduction (from 15% to 10%) of the depletion effect due to short-range correlations (SRC) must be considered for light nuclei like l s O .

nucleus, are plotted as a function of the binding energy of these sp levels. The distribution of occupation numbers is strikingly similar to the one for nuclear matter and exhibits the effect of SRC as a global depletion, essentially independent of the nature of sp orbit. Superimposed on this global depletion is an enhanced depletion for those orbits that reside in the vicinity of the Fermi surface. The latter depletion is associated with the coupling of such particles with low-energy (mostly) surface excitations 11 and can therefore be regarded as due to LRC. As discussed above, the other predicted consequence of the presence of SRC is the appearance of high-momentum components in the ground state to compensate

180

W. H. Dickhoff and H. Miither

for the depleted strength of the mean field. Recent JLab experiments 12 indicate that the amount and location of this strength is consistent with earlier predictions for finite nuclei8 and studies of infinite nuclear matter 1 3 Of particular relevance for the discussion of pairing in this chapter is the observed and calculated energy distribution of the sp strength of the high-momentum nucleons that is located at energies far removed from the Fermi energy. The situation is similar to the case of nuclear matter 14 and even more fragmented distributions are obtained above the Fermi energy. Such sp strength distributions lead to substantial modifications of the calculated properties of low-energy phenomena, like pairing, as compared to a those generated by a traditional mean-field treatment. The main details of the nucleon sp strength distribution are summarized in Fig. 7.4. Several generic diagrams are identified which have unique physical consequences for the redistribution of the sp strength when they are taken into account in the solution of the Dyson equation. The middle column of the figure characterizes the mean-field picture that is used as a starting point of the theoretical description. The right column identifies the location of the sp strength of the orbit just below the Fermi energy when correlations are included. The physical mechanisms responsible for the strength distribution are also identified. In the left column the diagram that is responsible for the admixture of high-momentum components in the ground state is depicted. The energy domain of these high-momentum nucleons is at large missing energies. The distribution of the sp strength can be adequately described by Green's function calculations as discussed in more detail in Refs. 7-15. From these results it becomes clear that configuration mixing in nuclei must be incorporated to the highest energy scale associated with the core of the AW interaction. Furthermore, it appears that so far only in nuclear physics it is possible to identify the detailed properties of strongly interacting constituent particles. As discussed above, details of the nucleon sp strength distribution have become well established experimentally and understood theoretically. In particular, the crucial effects of short-range and tensor correlations on the distribution of the sp strength have recently been treated at a very sophisticated level. Since these correlations have demonstrated relevance for understanding finite nuclei, it is important that their consequences for pairing are studied in detail.

7.3. Short-range correlations and nucleon propagation in infinite matter In the present chapter we focus on the study of pairing correlations with a proper treatment of the nucleon propagator that accounts for the effects of short-range and tensor correlations. The effects of short-range and tensor correlations on the distribution of the sp strength has been studied at a very sophisticated level by different groups. Techniques have been developed to evaluate the sp strength from realistic NN interactions, like the Argonne V18 16 or the CDBonn 17 interaction, within the self-consistent Green's function (SCGF) approach. 18,25 A recent reference for the Green's function method in bookform is also available. 15 Such calculations reproduce the energy- and momentum-distribution of the sp strength correspond-

Pairing Properties of Dressed Nucleons in Infinite

Matter

181

ing to high-momentum nucleons as observed in experiments 12 and account for the depletion of the mean-field strength obtained in. 10 In these calculations the scattering equations for two nucleons in the medium are solved employing dressed sp propagators that contain the complete information about the energy- and momentum-distribution of the sp strength. The resulting scattering matrix T is then used to calculate the nucleon self-energy. The solution of the Dyson equation, employing this complex and energy dependent selfenergy, yields the sp propagator that enters the T matrix equation to close the self-consistency cycle. In determining the two-nucleon Green's function or the corresponding scattering matrix T, one has to face the problem of the so-called pairing instabilities that reflect the existence of NN bound state solutions in the scattering equation. The presence of such solutions provides a major numerical obstacle for a self-consistent evaluation of one- and two-body propagator within the normal Green's function approach at densities where such instabilities can occur. For this reason, recent SCGF calculations have been performed for temperatures above the critical temperature for a possible phase transition to pairing condensation. 18 ' 26 The key quantity within the SCGF approach is the sp Green's function, which can be defined in a grand-canonical formulation for both real and imaginary times t, t':27 , r , . , ,,, _ tr{exp[-/3(g - yN)m(xtW(x't')}} il*(x,t,x ,t) tr{exp[-/3(ff - pN)}}

'

(

, '

where T is the time ordering operator, /3 the inverse temperature, and /i the chemical potential of the system. The one-particle Green's function obeys a quasi-periodicity condition and can therefore be expressed for a homogeneous system in terms of the Fourier coefficients G(k,zv), where zv = -Kv/{—ij3) are the (fermion) Matsubara frequencies with odd integers v. Since these are related to the spectral function A(k,u) by J-oo

27T

ZV

- W

G(k, z„) can be continued analytically to all non-real z. On the other hand, the spectral function is related to the imaginary part of the retarded propagator G(k, UJ+ \rj) by A(k,w) = -2ImG(k,uj

+ ir]).

(7.3)

In the limit of the mean-field or quasi-particle approximation the spectral function is represented by a ^-function and takes the simple form A(k, u) = 2TT6{LJ + n -

£fc )

= 27r<5(w - Xk),

(7.4)

with the quasi-particle energy Ek for a particle with momentum k and Xk = £fc — yNote, that for convenience we define the energy variable relative to the chemical potential fi. The sp Green's function is obtained as a solution of the Dyson equation, which, for a translationally invariant system, is a simple algebraic equation

k2

G(fc,«) = l ,

(7.5)

182

W. H. Dickhoff and H. Miither

where E(/c,w) denotes the complex self-energy. By expanding the self-energy in terms of one-particle Green's functions it can be demonstrated that it inherits all analytic properties of G. It it thus possible to write S(fc,U) = S ^ ( f c ) - 1 / + ° ° do/ I m S ( f c ' " '

+ i7?)

.

(7.6)

The self-energy can be calculated in terms of the in-medium two-body scattering T matrix. It is possible to express Im£(/c, LJ+'VIJ) in terms of the retarded T matrix 26,27 1 f H3A-' r+°° di >' ImE(fc, u, + iV) = ±j ±-^ J ^ (kk'\lmT(uj

+ oJ' + irj)\kk')

x[f(w') + b(w + w')]A(k',u;').

(7.7)

Here and in the following 1

/H

eP" + 1 '

denote the Fermi and Bose distribution functions, respectively. The pole in the Bose function 6(17) at fi = 0 is compensated by a corresponding zero in the T matrix 28 ' 29 such that the integrand remains finite as long as the T matrix does not acquire a pole at this energy. Such a pole may occur below a critical temperature Tc, a phenomenon that is often referred to as a pairing instability. We will come back to this problem below. The scattering matrix T is to be determined as a solution of the integral equation (fcfc'|T(fi + \7])\pp') = 3

(kk'\V\pp')

3

fd qd q' + / ^ / (kk'\V\
G

ii(9<7'>" + iv) W

W

+ iv)\pp') ,

(7-9)

where

<*<*••»••"+*> - G£¥^-)^.-o^~J%

<-»)

stands for the two-particle Green's function of two non-interacting but dressed nucleons. Practical applications employ an angle-averaging procedure so that the twoparticle Green's function can be written as a function of the length the total and relative momentum two-vectors, P and q, only. This approximation leads to a decoupling of partial waves with different total angular momentum J. Therefore the integral equation (7.9) reduces to an integral equation in only one dimension of the form telT^P.fi

+ Mtflg') = (qlVj^W)

+ T-

Twfc'2(q\v^T\k')G°U(P,n

i,„ * Jo

(7.11)

+ Hk>){k'\Ttf,T(P,n + iv)\q').

Pairing Properties of Dressed Nucleons in Infinite

183

Matter

The summation of the partial waves, (kk'\lmT(n

+ ir))\kk') = -?- V

(2J+1)(2T +1)

x (q(k, k')\ I m ^ f T (P(fc, fc'), fi + ir/) |g(fc,

fc')>,

(7.12)

yields the T matrix in the form that is needed in Eq. (7.7). Finally, the Hartree-Fock contribution has to be added to the real part of E E /

" » = f

£ "" (JST)l

(2-/ + l)(2T +

l)/^Mfc,fc')|Vi/sr|9(fc,fc'))n(fc'), J

^

'

(7.13) where n(k) is the correlated momentum distribution — f(w)A(k,w).

(7.14)

Note, that Eq. (7.13) corresponds to a generalized Hartree-Fock contribution, since the full one-particle spectral function is employed. The solution of the scattering equation in Eq. (7.11) at temperature T = 0 employing fully dressed (but not self-consistent) sp propagators 30 already allowed to anticipate that dressing could have substantial effects on pairing properties. The phase shifts obtained from such calculations for the most important partial wave channels are summarized in Fig. 7.5. A comparison is made between phase shifts for free particles (solid line), mean-field particles at kp = 1.36 fm~ (dashed line), and dressed particles (short-dashed line) at the same density for the 1 5o, 3Si, ZP\, and 3 Di channels (corresponding to the different panels in Fig. 7.5) as a function of the on-shell wave vector. In general, one finds that the dressed phase shifts suggest weaker interactions, since they are either less repulsive or less attractive than in the mean-field. For the two 5-wave channels the most striking feature of the dressed phase shift is the disappearance of the pairing signature for the 1 So channel and the enormous reduction of the signal in the 3S\ partial wave. This signal is identified by the tendency of the phase shifts to 7r at an energy corresponding to twice the Fermi energy (corresponding to kp in the figure). While the dressed 1SQ phase shift is essentially zero at kp, it is still clearly attractive at this wave vector for the 3S\ channel suggesting a much weakened pairing strength. An illustration of the effect of self-consistency on the sp strength distribution at zero temperature but at a density at which pairing apparently does not occur is presented in Fig. 7.6 for the Reid soft-core potential. 31 The spectral strength in Fig. 7.6 is essentially identical at large negative energies for all momenta. It reflects a similar energy distribution of the imaginary part of the self-energy for these momenta. While this distribution doesn't exhibit the huge energy range found for the sp strength above the Fermi energy, 9 ' 14 it is qualitatively different from these earlier calculations with attendant implications for the binding energy per particle. 23 The results in Fig. 7.6 were obtained at a density corresponding to kp = 1.36 fm _ . 18 The resulting momentum distribution is compared to a first-generation (not selfconsistent) calculation in Fig. 7.7. The similarity of the occupation of momenta

184

W. H. Dickhoff and H. Miither

k (for1)

k (far1)

Fig. 7.5. Comparison of phase shifts for free particles (solid), mean-field particles (dashed), and dressed particles (short-dashed lines) for different partial waves. The density of the medium corresponds to kp = 1.36 fm .

below kp between the two calculations is clear. Indeed, only a slight increase in the occupation of at most 3% is observed, when self-consistency is achieved. This implies that the corresponding depletion due to short-range correlations is still about 15% for nucleons deep in the Fermi sea as was predicted in Ref. 9 and confirmed in Ref. 10. In particular, the occupation at k = 0 appears to characterize the strength of the core of the NN interaction. More modern potentials generate slightly smaller depletions at k = 0, as discussed below. The main message from these self-consistent results is that pairing has disappeared at this density when self-consistency is taken into account, since the calculations of Ref. 18 do not exhibit a pairing instability. As the gap for the 3S\ —3 D\ channel is quite large at this density for mean-field sp propagators, we can anticipate substantial changes in the pairing properties of dressed nucleons as compared to those of mean-field ones. In order to allow a much wider study of the density and temperature dependence of pairing, we will now present results for fully self-consistent calculations above the critical temperature Tc for pairing. An iterative procedure is employed to solve Eqs. (7.11)-(7.14) until a self-consistent solution is obtained. Some details of this procedure can be found in Refs. 26 and 32. The results discussed below have been reported in Ref. 33. The imaginary part of the nucleon self-energy is displayed in the upper panel of Fig. 7.8 for a temperature of 5 MeV. The purpose of this figure is to visualize some

Pairing Properties of Dressed Nucleons in Infinite

Matter

185

10" 10"' > 10"' 10"

//

C3 10

io"! 10"

-350

-250

-150

-50

50

150

250

350

E-eF(MeV) Fig. 7.6. Self-consistent spectral functions for three different momenta at kjr = 1.36 fm sponding to 0 (full), 1.36 (dotted), and 2.1 f m - 1 (dashed) as a function of E — ep-

0.8 •

1

corre-

~-=^

0.6

zF

0.4

-

0.2 •

0.5

1.5

k (fm"1)

2

2.5

Fig. 7.7. Occupation probability for nuclear matter at equilibrium density calculated by integrating hole spectral functions obtained with mean-field propagators as input (dotted line). Selfconsistent determination of the sp propagators yields the solid line.

differences between various models of the NN interaction and between symmetric nuclear matter and pure neutron matter. Therefore we consider a large interval for the energy variable w. The imaginary parts of the self-energy derived from the CDBonn interaction 17 and the Argonne V18 (ArVl8) interaction 16 are very similar at energies around u> = 0. At those energies the ArV18 yields a slightly weaker imaginary part than CDBonn. The differences get larger at positive values of u, where the imaginary part of the self-energy derived from ArV18 reaches a minimum

186

W. H. Dickhoff and H. Miither

J

0

i

I

i

500

I

1000

i

l__

1500

co [MeV] Fig. 7.8. Imaginary (upper panel) and real part (lower panel) of the retarded self-energy for nucleons with momentum A; = 225 MeV/c in symmetric nuclear matter at the empirical saturation density (p = 0.16 f m - 3 ) . Results obtained for CDBonn interaction are compared to those resulting from SCGF calculations using ArV18. Also included are results for neutrons with the same momentum in neutron matter at p = 0.08 f m - 3 . The temperature in all these calculations was fixed at T = 5 MeV.

of around —100 MeV at an energy LJ around 1.7 GeV. The minimum for the CDBonn interaction is only about —35 MeV and occurs at energies w around 0.5 GeV. The ArV18 is a stiffer interaction than CDBonn, which is, e.g., illustrated by considering the Hartree-Fock result for nuclear matter at the empirical saturation density. Such a calculation yields a total energy 30 MeV per nucleon for the ArVl8, while CDBonn generates 5 MeV per nucleon.34 This implies that the generalized Hartree-Fock contribution to the self-energy in Eq. (7.6), defined in Eq. (7.13), is more repulsive for ArV18, and a larger part of the attraction is provided by the energy-dependent contribution to the real part of the self-energy. This is immediately obvious, since the energy-dependent contribution to the real part of E is connected to the imaginary part by a dispersion relation. The lower panel of Fig. 7.8, which displays results for the real part of the self-energy, illustrates this observation: The energy dependence is larger for ArV18 as compared to CDBonn. The weaker attraction of the self-energy derived from CDBonn (for most values of w) reflects the less repulsive contribution of the generalized Hartree-Fock contribution. Figure 7.8 also displays results for the real and imaginary part of the self-energy for neutrons with the same momentum (k = 225 MeV) in pure neutron matter. The density of neutron matter considered in this figure is one half of the empirical

Pairing Properties of Dressed Nucleons in Infinite

Matter

187

saturation density of nuclear matter, which implies that these systems have the same Fermi momentum. The imaginary part of the self-energy in neutron matter is weaker than the corresponding one for symmetric nuclear matter, reflecting the dominance of proton-neutron correlations. For both interactions a minimum is obtained around 1.7 GeV. At these high energies the absolute value for the imaginary part of the self-energy is about a factor three larger for ArV18 than for CDBonn. This means that the distribution of sp strength to high energies due to central short-range correlations is much stronger for the local ArV18 interaction than for the non-local meson-exchange model CDBonn. The results for the lower energies are closer to each other. The differences in the amount of NN correlations is also reflected in the occupation probability n{k) denned in Eq. (7.14). For symmetric nuclear matter at saturation density we obtain for n(k = 0) the values 0.89 and 0.87 for CDBonn and ArV18, respectively. The corresponding value for neutron matter are n(k = 0) = 0.968 and 0.963. These numbers for nuclear matter together with the result of 0.85 for the Reid potential, discussed above, suggest that there is a direct correlation with the strength of the core of the interaction and the depletion at k = 0. Since this depletion can now be experimentally scrutinized and is found to be substantial, it is clear that a much softer interaction is ruled out by such data. Examples for spectral functions A(k,u>) are displayed in Fig. 7.9. Again we consider the case of symmetric nuclear matter at the empirical saturation density and use the CDBonn interaction. As examples we consider two momenta k = 255 MeV/c and k = 277 MeV/c, which are below and above the Fermi-momentum kp, respectively. For the momentum k < kp one finds the dominant peak at an energy below the Fermi energy CJ = 0 and a much smaller maximum at w larger than zero. For momenta k > kp the dominant quasi-particle peak is located at positive values of u! and a second maximum occurs at w < 0. Since this feature of two maxima in the spectral function is reminiscent of the two poles that are present in the BCS approximation to the Green's function, it has been discussed as the formation of a pseudo-gap or as a precursor phenomenon to a pairing condensate. 35 We note, however, that our calculations only exhibit this feature of two pronounced maxima in the spectral function, if we consider rather low temperatures, in particular T < Tc. The examples displayed in Fig. 7.9 originate from an extrapolation of the self-energy to T = 0. This is different from results obtained with simplified interactions, as they are used, for example, in Ref. 35. The interaction employed by Bozek yields an imaginary part of the self-energy, which is different from zero in a much smaller energy interval than the realistic calculations considered here. Due to this difference spectral functions with two maxima are obtained also at temperatures above Tc for this model interaction.

7.4. Pairing results for dressed nucleons At temperatures below the critical temperature for a transition to a superfluid one has to supplement the evaluation of the normal Green's function G(k, u>) with the anomalous Green's function F(k,uj). While the self-consistent inclusion of ladder diagrams has reached quite a sophistication, it remains to fully account for the pos-

W. H. Dickhoff and H. Miither

188

> u 3"

0.01 - 0 w [MeV]

k=255 MeV/c k=277MeV/c 20

Fig. 7.9. Spectral function for nucleons in symmetric nuclear matter at the empirical saturation density. Results of SCGF calculations have been extrapolated to T — 0. The CDBonn interaction has been used.

sibility of a pairing solution in such calculations for realistic NN interactions. A first step towards such a complete scheme is to include the full self-consistent dressing due to normal self-energy terms generated by ladder diagrams, in the calculation of the anomalous self-energy and the solution of the corresponding generalized gap equation. The inclusion of the anomalous Green's function F(k, u) yields a modification of the normal Green's function in the superfluid phase that can be written as 1 5 , 3 6 ' 3 8 Gs(k,w + iri) = G(k,uj + irf) - G(k,u + ir))A(k)F(k,uj + irj) F(k,u) + irj) = G{-k,-u - irf)Gs(k,Lj + ir])A(k). (7.15) under the assumption that the anomalous part of self-energy A does not depend on the energy. Therefore the full Green's function can be obtained as Gs(k,w + ir]) = —— , . , . . , A2/t.\nf i. ^T( 7 - 16 ) x z G(k,u + IT]) + A (k)G{—k, — UJ — irj) These equations must be supplemented with the definition of the anomalous selfenergy Sk (p\V\k) 2ImF(fc, w + M7)/(w). (7.17) A(p)

m

(2nf

If one employs Eq. (7.15) and uses the representation of the Green's function G in terms of the spectral function in Eqs. (7.2) and (7.3), supplemented by a corresponding definition of a spectral function for the total Green's function As(k,u) = -2ImGs(k,L0 + ir]), (7.18)

Pairing Properties of Dressed Nucleons in Infinite

Matter

189

the expression for the self-energy A can be rewritten 35 in a partial wave expansion

x#,^,(fc,o;')1"

/ H

~ f K ) A^ S T (fc). (7.19) —u^ — CJ If we ignore for a moment the difference between the spectral functions A and As, we see that this equation for the self-energy A corresponds to the homogeneous scattering equation for the T-matrix in (7.9) at energy Q = 0 and center-of-mass momentum P = 0. This means that a non-trivial solution of Eq. (7.19) is obtained if and only if the scattering matrix T generates a pole at energy fi = 0, which reflects a bound two-particle state. This is precisely the condition for the pairing instability discussed above, demonstrating that this treatment of pairing correlations is compatible with the T-matrix approximation in the non-superfluid regime discussed in Section 7.3. We may also consider Eq. (7.19) in the limit in which we approximate the spectral functions A(k,u>) and Aa(k,u>) by the corresponding mean-field and BCS approximation. The expression for the normal spectral function has been presented already in Eq. (7.4). The BCS approximation for the spectral function yields Ek + Xk x, „ v , Ek - Xk x( . T? s d(u> Ek) H —=—o(w + Ek) 2Ek 2Ek with the quasi-particle energy Aa(k,w) = 2n

Ek = ^xl

+ ^{k).

(7.20)

(7.21)

Inserting these approximations for the spectral function into Eq. (7.19) and taking the limit T = 0 reduces to the usual BCS gap equation

A,JST(P) = T,lfdkk2

(PI^OT|*> Z^ A " S T ( f c )'

(7-22)

Therefore we can consider Eq. (7.19) as a generalization of the usual gap equation. It accounts for the spreading of sp strength leading to a generalization of the form

It may be useful to emphasize that the iteration of the anomalous self-energy generates the sum of ladder diagrams for total momentum P = 0 and energy fi = 0. This feature can be obtained by considering the diagrammatic content of the anomalous self-energy given in Fig. 7.10. Diagrammatically inserting the solution of the anomalous propagator, 15 generates the contribution of ladder diagrams for the stated energy and momentum values. This observation is particularly valuable in the context of the self-consistent inclusion of ladder diagrams in the normal nucleon self-energy contribution, showing that there is a complete consistency associated with ladder diagrams and the treatment of pairing correlations. 39 As a first step towards the study of pairing correlations, we consider the usual BCS approach. This means that we solve the gap equation (7.22) assuming a

190

W. H. Dickhoff and H. Miither

*> £ ' ^ S = S 5 .

kE± Fig. 7.10.

-k'-E'

±-k-E

First-order diagram including the bare interaction for the anomalous self-energy.

spectrum of sp energies e(k) = \k + A4, which we determine from the quasi-particle energies k2 e(fc) = — + Re£(fc, e{k) - /*). (7.24) In this equation Re£ denotes the result of a SCGF calculation extrapolated to T = 0. Such spectra of quasi-particle energies are rather similar to the sp spectra used in other work. The corresponding BCS calculations therefore involve the usual procedure as it has been applied, e.g., in Refs. 1-6. Results for the gap-functions |Az(/c)| are displayed in Fig. 7.11. The upper panel of this figure shows results for symmetric nuclear matter at saturation density. The partial wave that yields the largest value for A and is therefore the relevant one in this case, is the 3Si-3Di channel describing the proton-neutron interaction. We therefore display the absolute values of the gap-functions for t = 0 and i = 2 (Ao and A2) as well as the total gap-function A = y/A2 + A\ as a function of the momentum k. Below we will mainly consider the value of the gap-function A at the Fermi momentum kp. We also compare in this figure the results obtained from CDBonn with those from ArV18. For smaller values of k the CDBonn yields larger values for the gap function, while ArV18 leads to larger gap values for momenta larger than k = 400 MeV/c. This is true for the £ — 0 component as well as the 1 = 2 component and consequently also for the total result. This feature at large values of k is in line with our observation made above, that ArV18 tends to produce a larger amount of correlations at high momenta and large energies. For lower momenta, however, CDBonn yields larger gap-functions. Therefore the gap (at the Fermi momentum) resulting from a BCS calculation, which uses CDBonn (8.6 MeV) is larger than the corresponding value calculated for ArV18 (7.6 MeV), although ArV18 tends to produce more short-range correlations than CDBonn. The situation is quite similar for the case of neutron-neutron pairing in pure neutron matter, which is displayed in the lower part of Fig. 7.11. In this case the pairing effects are dominated by the 1SQ> partial wave. At high momenta we obtain larger values for the gap function using ArV18, whereas CDBonn yields larger values for A(k) at low momenta. Therefore the value A(kp) is larger for CDBonn (1.4 MeV) than for ArV18 (1.1 MeV). These values for the neutron-neutron pairing gap are, however, much lower than the corresponding values for protonneutron pairing at the same Fermi momentum. On one hand this sounds natural, as we know that the proton-neutron interaction is stronger than the neutron-neutron interaction, leading to a bound deuteron and to more correlations (see above). On the other hand, however, one observes the effects of proton-proton and neutronneutron pairing in finite nuclei, while there is hardly any trace of proton-neutron pairing effects in nuclei.

Pairing Properties of Dressed Nucleons in Infinite

200

j. }

Matter

191

400 600 Momentum k [MeV/c]

Fig. 7.11. Results for the gap-functions |Aj(fc)| for symmetric nuclear matter (p = 0.16 f m - 3 , upper panel) and pure neutron matter (p = 0.08 f m - 3 , lower panel) obtained from a solution of the BCS equation (7.22) using the CDBonn and ArV18 interactions at T = 0. The dotted line identifies the Fermi-momentum kp.

As a next step, we now try to consider the effects of temperature and short-range correlations in the solution of the gap equation. For that purpose we will reconsider the two-particle propagator of Eq. (7.23) but replace the spectral function of the superfluid phase, As(k,cj'), by the corresponding one for the normal phase, A(k,uj'). This leads to a definition of an average energy denominator xk of the form r+°° doj f+°° du/

,1

-2xk

/ H - /("')

(7.25)

U!'

If we consider this propagator in the limit of the mean-field approximation (A(k,ui) — S(u> — £*;)) at T = 0, it reduces to an energy denominator of the form 1 "2Xfc

mf,T=0

(7.26) -2|Xfc|

This means that the energy Xk has been defined in this equation to exhibit the effects of finite temperature and correlations on the two-particle propagator. Figure 7.12 displays results for this quantity Xk, the inverse of this propagator multiplied by -2, and compares it with \xk\ using the corresponding quasi-particle energies. The dashed-dotted line represents the effects of temperature, i.e., the propagator has been calculated using the quasi-particle approach for the spectral function and the Fermi function for the temperature under consideration, while the solid line accounts for finite temperature and correlation effects. The finite temperature yields an enhancement of the effective sp energy Xk for momenta around the Fermi-momentum, only. Including in addition the effects of correlations in the propagator, we obtain larger values for Xk for all momenta. This corresponds to the well known feature that a finite temperature yields a depletion of the occupation probability of sp

192

W. H. Dickhoff and H. Miither

Momentum k [MeV] Fig. 7.12. The quantity Xfc defined in Eq. (7.25), which represents the energy denominator for the propagator of two nucleons in the medium. Results are displayed for the quasi-particle approximation in the limit T = 0 (QP, T = 0 ) , the quasi-particle approximation for finite temperature T = 5 MeV (QP, T = 5 MeV) and the dressed propagator resulting from SCGF calculations (full prop., T = 5 MeV). All results displayed in this figure were obtained for symmetric nuclear matter at density p = 0.16 f m - 3 , using CDBonn interaction.

states only for momenta just below the Fermi momentum, while strong short-range correlation provide such a depletion for all momenta of the Fermi sea. In the literature several attempts have been reported to represent the depopulation of the sp strength due to SRC by a renormalization factor Z.41'44 Our investigation of this issue demonstrates 33 that there is no simple prescription based on quasiparticle type approximations to accomplish this feat. The required replacement can of course be obtained from - I T - (T, dressed) = %

^

(T, Q P ) ,

(7.27)

which means that one determines Ze/f, e.g., from the ratio of the results displayed by the solid and the dashed-dotted line in Fig. 7.12. However, this requires the full construction of the convolution given in Eq. (7.25). Results for the gap parameter A(kp) in symmetric nuclear matter of various densities are presented in Fig. 7.13 as a function of the temperature T. We will first discuss the results obtained within the usual BCS approximation (see discussion above) in the 3 5i- 3 Z?i partial wave. At the empirical saturation density the CDBonn interaction yields a gap parameter A(/cj?) at temperature T = 0 of 8.6 MeV (see above), which decreases with increasing temperature until it vanishes at T = 5.2 MeV. At p = 0.08 fm - 3 , which is about half the empirical density, the value of the gap parameter at T = 0 is even larger (A(fc^) = 10.6 MeV) and the gap

Pairing Properties of Dressed Nucleons in Infinite

1

1

'

1

X

S N

_

\ *

S

\

5 -^ \ \

\A\ \ \ \ \

~

- - p = 0.08fm 3 - - p = 0.04fm"3 — p = 0.08, dressed" — p = 0.04, dressed

---..^ -._ ~""~"~^^">-^ X N SN *"~«.„ ^ ^N

.

\

-V

\

\ \

\

\ * ^

* *

\\\ *Nx \»* \

I \ 1

,1 1

l

1' 1' 1 '

1

,

193

' 1 p = 0.16fm"3

-10

Matter

1

,.'.

» 1 I

<

-

II

T [MeV] Fig. 7.13. Gap parameter A(fcj-) in symmetric nuclear matter as a function of temperature T. Results are presented for various densities, with and without taking into account the dressing of the sp propagator due to short-range correlations. The pairing gap disappears at p=0.16 f m - 3 , if dressed propagators are considered. All results displayed in this figure were obtained using the CDBonn interaction.

calculated within the usual BCS approach disappears only at a temperature of 5.9 MeV. This increase of the pairing gap with decreasing density can be related to the momentum-dependence of the pairing gap A(fc) as displayed in Fig. 7.11: The gap function increases with decreasing momentum. Therefore, as the Fermi momentum decreases with density, the value A(kp) tends to decrease with density. At even lower densities, however, this effect is more than compensated by the feature, that the phase-space of two-hole configurations decreases with density, so that ultimately the gap parameter will approach the binding energy of the deuteron in the limit of p —* 0. This explains the decrease of the gap parameter going from p = 0.08 f m - 3 to p = 0.04 fm" 3 . If we take the effects of short-range correlations into account, the generalized gap equation of Eq. (7.19) does not give a non-trivial solution for symmetrical nuclear matter at p = 0.16 fm - 3 . This means that the present treatment of correlation effects in nuclear matter at normal density yields a disappearance of the proton-neutron pairing predicted by the usual BCS approach. The effects of short-range correlations tend to decrease with density. As a consequence we obtain non-vanishing gaps for proton-neutron pairing at lower densities (see solid lines in Fig. 7.13). This also leads to an increase of the critical temperature and the value of A(kF) at T = 0 going from p = 0.08 f m - 3 to p = 0.04 fm~ 3 . Note, that these functions are qualitatively different from the corresponding BCS predictions.

194

W. H. Dickhoff and H. Miither

1

'

1 -— -—

"•*-

10

-..__ *•».

' 1 CDBonn, BCS CDBonn, dressed ArV18,BCS _ ArV 18, dressed

>

s <

W \ v

i

\ \ \ V

\» \\ \\

^

1

,

1

,

\ V \ I I \ \ 1 \ 1 1 1 1 II

T [MeV] Fig. 7.14. (Color online) Gap parameter A(kp) in symmetric nuclear matter at p=0.08 f m - 3 as a function of temperature T using the ArV18 and the CDBonn interaction.

Differences associated with the various interactions are displayed in Fig. 7.14. For nuclear matter with a density of p = 0.08 f m - 3 the gap parameter is presented as a function of temperature T using the BCS approximation and the generalized gap equation with dressed propagators. As has already been discussed above, the ArV18 interaction yields smaller values for the gap parameter and the critical temperature than the CDBonn interaction. Effects of pairing correlations on the spectral function are visualized in Fig. 7.15. As an example we consider nuclear matter at p = 0.08 f m - 3 and show results for the spectral function without (A(k,u>)) and with inclusion of pairing correlations (As(k,uj), see Eq. (7.18)). The momentum considered for this figure, k = 193 MeV/c, is slightly below the Fermi momentum kp = 208 MeV/c. One observes that the inclusion of pairing correlations enhances the maximum of the spectral distribution at positive values of u> considerably and shifts the quasi-particle peak to more negative values of w. The pairing correlations modify the spectral distribution into the direction which is obtained in the simple BCS approximation for A3 (k, u) in Eq. (7.20). Also note, that these modifications of the spectral function As(k, w) as compared to A(k, w) is limited to a small interval of energies around w = 0 and to momenta close to the Fermi momentum. Finally, we consider the case of neutron-neutron pairing in pure neutron matter. We will focus the attention to densities, where the pairing correlations in the lSo partial wave are dominating. Results for the gap parameter A(kp) as a function of temperature are displayed in Fig. 7.16. Using the BCS approximation with sp energies derived from the quasi-particle energies of SCGF calculations we obtain

Pairing Properties of Dressed Nucleons in Infinite Matter

195

0) [MeV] Fig. 7.15. Spectral function for nucleons with momentum k = 193 MeV/c with (solid line) and without (dashed line) inclusion of pairing correlations. Results are presented for nuclear matter of p = 0.08 fm~ 3 at a temperature T = 0.5 MeV.

a gap at T = 0, which, for the range of densities considered, increases with decreasing density. This is in agreement with results of similar calculations, which are summarized, e.g., in Ref. 1. The effects of short-range correlations are weaker in neutron matter than in nuclear matter. This has been discussed already above in connection with the results displayed in Fig. 7.8. This can also be seen in a comparison of the dressed two-particle propagator and the effective strength factor Z e / / defined in Eq. (7.27). While a calculation of Zeff for symmetric nuclear matter at p = 0.16 f m - 3 yields values for Zeff, which are typically around 0.8, corresponding values for Zeff in neutron matter at p = 0.08 f m - 3 are around 0.9. Nevertheless, also these weaker effects of short-range correlations in neutron matter are sufficient to suppress the formation of a pairing gap in neutron matter at p = 0.08 f m - 3 . Such a suppression of pairing correlations is also observed at smaller densities. In this case, however, the inclusion of the correlation effects just leads to a reduction of the gap parameter at a given temperature and a reduction of the critical temperature (see Fig. 7.16). Recent results for neutron matter obtained with state of the art Monte Carlo and CBF techniques 45 indicate that in the density range where our results overlap, there is complete agreement for the gap at T = 0 with the Monte Carlo results.

7.5. Conclusions An attempt has been made to treat the effects of short-range and pairing correlations in a consistent way within the T-matrix approach of the self-consistent Green's function (SCGF) method. The pairing effects are determined from a generalized

196

W. H. Dickhoff and H. Miither

1

-

"•*

' * -

~~

T

-•-». "*s.

^ - .

3

V

--

p = 0.08fm"3

--

p = 0.04fm"3 " dressed p = 0.02fm" 3 dressed

^ ^V N v

-\ s

>


\

\ \ \\ ^ \ \ \

1—1 s 2 2-

-

<]

--o

w -

i

\ \ \ \ -

°(>

il 0.5

1 \ 1 \ 1 \ 1 ! i l

»

1

> 1

1 1,

1

• 1 1 11

1.5

2

Fig. 7.16. Gap parameter A(kj?) in neutron matter as a function of temperature T. Results are presented for the usual BCS approximation (dashed lines) and the solution of the generalized gap equation (7.19) in the 1 5o partial wave using the CDBonn interaction.

gap equation that employs sp propagators fully dressed by short-range and tensor correlations. This equation is directly linked to the homogeneous solution of the Tmatrix equation of NN scattering in the medium, which is one of the basic equations of the SCGF approach at temperatures above the critical temperature for a phase transition to pairing condensation. While short-range and tensor correlations yield a redistribution of sp strength over a wide range of energies, the effects of pairing correlations on the spectral function are limited in nuclear matter to a relatively small interval in energy and momentum around the Fermi surface. The formation of a pairing gap is very sensitive to the quasi-particle energies and strength distribution at the Fermi surface and can be suppressed by moderate temperatures. The formation of short-range correlations are sensitive to a larger range of energies and momenta. So we observe, that the non-local CDBonn interaction is softer with respect to the formation of short-range correlations but yields larger pairing gaps compared to the local ArV18 model for the NN interaction. From this sensitivity to different areas in momentum and energy one may conclude that the features of short-range correlations should be rather similar in studies of nuclear matter and finite nuclei. The investigation of pairing phenomena, however, is rather sensitive, e.g., to the energy spectrum around the Fermi energy. Therefore the shell effects of finite nuclei may lead to quite different results for pairing properties than corresponding studies in infinite matter. The redistribution of sp strength due to the short-range correlations has a significant effect on the formation of a pairing gap. While the usual BCS approach

Pairing Properties of Dressed Nucleons in Infinite Matter

197

predicts a gap for proton-neutron pairing in nuclear matter at saturation density as large as 8 MeV, the inclusion of short-range correlations suppresses this gap completely. Correlation effects are weaker at smaller densities, but still lead to a significant quenching of the proton-neutron pairing gap and to a reduction of the critical temperature for the phase transition. Compared to symmetric nuclear matter correlation effects are weaker in neutron matter. Nevertheless, the inclusion of correlations suppresses the formation of a gap for neutron-neutron pairing at p = 0.08 f m - 3 completely and yields a significant quenching at lower densities. The effects of dressed sp propagator in the generalized gap equation could be described in terms of an effective strength factor Zeff (k), which has been considered in the literature before. 40 ' 41,44 Unfortunately, we have not been able to derive the value of this strength factor from bulk properties of the self-energy. Although the effects of pairing correlations on the sp Green's function is weak and limited to a small range in energy and momentum, these modifications are very important to extend SCGF calculations to densities and temperatures that suffer from the so-called pairing instability. The present study is a first step towards a consistent treatment of pairing and short-range correlations. While our results now appear consistent with empirical information on pairing of both isoscalar (pn) and isovector (pp or nn) kind, we note that our calculations do not include polarization contributions to the pairing interaction. This issue is sometimes identified by the term vertex corrections. Such a medium dependence of the NN interaction to be used in solving the gap equation has been studied in Refs. 46-49. Using effective interactions like the Gogny force50 Shen et al.48'49 observe indeed a significant effect, which is larger for nuclear matter than for neutron matter. The effects of the so-called induced interaction could indeed affect the lowenergy spectroscopy of nuclear matter in a significant way. Before conclusions can be drawn, calculations employing realistic interactions should be performed, which, unfortunately, are very difficult.51 Also one should be aware that a consistent treatment of dressed propagators and vertex corrections is required to obtain an approach, which guarantees the conservation of symmetries. 52 ' 53 In addition, it is possible that polarization effects are different in nuclear matter and finite nuclei on account of the difference of the sp spectra discussed above.

Acknowledgments This work is supported by the U.S. National Science Foundation under Grant No. PHY-0140316 and the "Landesforschungsschwerpunkt Quasiteilchen" of the state of Baden Wiirttemberg.

Bibliography 1. D.J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003). 2. T. Aim, G. Ropke, and M. Schmidt, Z. Phys. A 337, 355 (1990). 3. B.E. Vonderfecht, C.C. Gearhart, W.H. Dickhoff, A. Polls, and A. Ramos, Phys. Lett. B 253, 1 (1991). 4. M. Baldo, I. Bombaci, and U. Lombardo, Phys. Lett. B 283, 8 (1992).

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5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

W. H. Dickhoff and H. Miither

T. Takatsuka and R. Tamagaki, Suppl. Prog. Theor. Phys. 112, 27 (1993). M. Baldo, U. Lombardo, and P. Schuck, Phys. Rev. C 52, 975 (1995). W.H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004). H. Miither and W.H. Dickhoff, Phys. Rev. C 49, R17 (1994). B.E. Vonderfecht, W.H. Dickhoff, A. Polls, and A. Ramos, Phys. Rev. C 44, R1265 (1991). M.F. van Batenburg, Ph.D. Thesis, University of Utrecht (2001). M. G. E. Brand et al., Nucl. Phys. A 531, 253 (1991). D. Rohe, et al., Phys. Rev. Lett. 9 3 , 182501 (2004). T. Frick, Kh.S.A. Hassaneen, D. Rohe, and H. Miither, Phys. Rev. C 70, 054308 (2004). B.E. Vonderfecht, W.H. Dickhoff, A. Polls, and A. Ramos, Nucl. Phys. A 555, 1 (1993). W.H. Dickhoff and D. Van Neck, Many-Body Theory Exposed! (World Scientific, Singapore, 2005). R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, R1483 (1996). W. H. Dickhoff and E. P. Roth, Acta Phys. Pol. B 33, 65 (2002); E. P. Roth, Ph.D. thesis, Washington University in St. Louis (2000). Y. Dewulf, D. Van Neck, and M. Waroquier, Phys. Rev. C 65, 054316 (2002). P. Bozek and P. Czerski, Eur. Phys. J. A 11, 271 (2001). P. Bozek, Phys. Rev. C 65, 054306 (2002). P. Bozek, Eur. Phys. J. A 15, 325 (2002). Y. Dewulf, W.H. DickhofLD. Van Neck, E.R. Stoddard, and M. Waroquier, Phys. Rev. Lett. 90, 152501 (2003). A.E.L. Dieperink, Y. Dewulf, D. Van Neck, and M. Waroquier, Phys. Rev. C 68, 064307 (2003). T. Frick, H. Miither, A. Rios, A. Polls, and A. Ramos, Phys. Rev. C 71, 014313 (2005). T. Frick and H. Miither, Phys. Rev. C 68, 034310 (2003). L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962). T. Aim, B. L. Friman, G. Ropke and H. Schulz, Nucl. Phys. A 551, 45 (1993). T. Aim, G. Ropke, A. Schnell, N.H. Kwong, and H.S. Kohler, Phys. Rev. C 53, 2181 (1995). W.H. Dickhoff, C.C. Gearhart, E.P. Roth, A. Polls, and A. Ramos, Phys. Rev. C 60, 064319 (1999). R. V. Reid, Ann. Phys. 50, 411 (1968). T. Frick, Ph.D. Thesis, University of Tubingen (2004). H. Miither and W. H. Dickhoff, Phys. Rev. C 72, 054313 (2005). H. Miither and A. Polls, Prog. Pari. Nucl. Phys. 45, 243 (2000). P. Bozek, Nucl. Phys. A 657, 187 (1999). A.B. Migdal, Theory of Finite Fermi Systems (Wiley, New York, 1967). J.R. Schrieffer, Theory of Superconductivity (Benjamin, Massachusetts, 1964). G.D. Mahan, Many-Particle Physics (Plenum Press, New York, 1981). W.H. Dickhoff, Phys. Lett. B 210, 15 (1988). P. Bozek, Phys. Rev. C 62, 054316 (2000). P. Bozek, Phys. Lett. B 551, 93 (2003). M. Baldo and A. Grasso, Phys. Lett. B 485, 115 (2000). U. Lombardo, P. Schuck, and W. Zuo, Phys. Rev. C 64, 021301 (R) (2001).

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44. M. Baldo, U. Lombardo, H.-J. Schulze, and Zuo Wei, Phys. Rev. C 66, 054304 (2002). 45. A. Fabrocini, S. Fantoni, A. Yu. Illarionov, and K. E. Schmidt, Phys. Rev. Lett. 95, 192501 (2005). 46. W. Zuo, U. Lombardo, H.-J. Schulze, and C.W. Shen, Phys. Rev. C 66, 037303 (2002). 47. A. Sedrakian, Phys. Rev. C 68, 065805 (2003). 48. Caiwan Shen, U. Lombardo, P. Schuck, W. Zuo, and N. Sandulescu, Phys. Rev. C 67, 061302 (2003). 49. Caiwan Shen, U. Lombardo, and P. Schuck, Phys. Rev. C 71, 054301 (2005). 50. J. Decharge and D. Gogny, Phys. Rev. C 21, 1568 (1980). 51. W.H. Dickhoff and H. Muther, Nucl. Phys. A 473, 394 (1987). 52. G. Baym and L. Kadanoff, Phys. Rev. 124, 287 (1961). 53. P. Bozek, J. Margueron, and H. Muther, Ann. Phys. 318, 245 (2005).

Chapter 8 Pairing in Higher Angular Momentum States: Spectrum of Solutions of the 3 P 2 - 3 ^ 2 Pairing Model Mikhail V. Zverev Russian

Research

Center Kurchatov

Institute,

Moscow,

123182,

Russia

J o h n W. Clark McDonnell Center for the Space Sciences and Department of Washington University, St. Louis, MO 63130, USA

Physics

Victor A. Khodel Russian Research Center Kurchatov Institute, Moscow, 123182, Russia McDonnell Center for the Space Sciences and Department of Physics Washington University, St. Louis, MO 63130, USA The system of BCS gap equations of the 3Pz-3F2 pairing model, commonly used to describe nucleonic superfluidity in the dense interior of a neutron star, is solved with the aid of the separation method. This method, developed originally for quantitative study of S-wave pairing in the presence of strong short-range repulsions, allows one to reduce the system of coupled, singular, nonlinear BCS integral equations to a set of coupled algebraic equations. Adopting a perturbative strategy to account for the coupling between 3P<2 and 3 F2 channels, we identify and characterize the full spectrum of real solutions of the 3i-,2-3^?2 model. Remarkably, incisive and robust results are obtained solely on the basis of analytic arguments. Unlike the traditional Ginzburg-Landau approach, the analysis is not restricted to the immediate vicinity of the critical temperature, but is equally reliable at zero temperature.

Contents

8.1. Introduction 8.2. Separation method for singlet ,S-wave pairing 8.3. Generalized BCS gap equation and the 3P2-3i<2 model 8.4. Real solutions of the 3P2-3F2 problem 8.5. Conclusions Bibliography

201

202 204 206 211 218 220

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M. V. Zverev, J. W. Clark and V. A. Khodel

8.1. Introduction The original BCS theory was concerned with the simplest type of pairing, in which fermions pair-condense in a singlet 5-wave state. Accordingly, one important dimension to explore "beyond BCS theory" is that of pairing in higher angular momentum states. Pairing in triplet spin states is prominently exemplified in two strongly interacting homogeneous Fermi systems: (i) liquid 3 He, when cooled below 2.6 x 1 0 - 3 K, and (ii) the neutron subsystem of the fluid interior of a neutron star, for temperatures below ~ 109 K. In liquid 3 He, the ratio of the distance between atoms to the radius of the repulsive core in the 3 He- 3 He interaction is of order of unity, and consequently the effective in-medium interaction differs quite significantly from the vacuum interaction between the atoms. This is an underlying reason for the manifestation of P-pairing in liquid 3 He, although the vacuum scattering length has the sign appropriate for the occurrence of 5-pairing. A similar situation prevails in the dense interior region of a neutron star: the strong short-range repulsion in the nucleon-nucleon (AW) interaction suppresses : 5o pairing, which dominates in the low density neutron fluid embedded in the star's crust. In both of these Fermi liquids, 3 He and neutron matter, the effective in-medium interaction contains a component that mixes two-body states having orbital angular momenta L ± 2. Specifically, pairing channels with L = 1 and L = 3 are coupled by the magnetic dipole force in the case of liquid 3 He, and, in neutron matter, by the tensor force arising from pion exchange. However, the effect is of minuscule importance in the 3 He problem, since the magnetic dipole component of the interaction is smaller in magnitude than the dominant central part by a factor 1 ~ 1 0 - 7 . By contrast, the magnitude of the dominant spin-orbit force in the noncentral part of the vacuum AW interaction is only a few times larger than the tensor component, 2 so the effects of the tensor force should not be neglected. Moreover, pion exchange is responsible for the most powerful fluctuations in dense neutron matter. 3 ' 4 Thus, a quantitative treatment of triplet pairing in neutron matter must take account of the 3 P2 _ 3 ^2 channel coupling. Determination of the complete superfluid phase diagram of the neutron liquid has two facets. First, one must identify and characterize the set of admissible solutions of the system of BCS gap equations. Second, one must evaluate the relative stability of the different solutions under variation of density, temperature, and other relevant parameters, so as to uncover the possible phase transitions and map out the actual phase diagram. Unfortunately, the mathematical difficulties associated with this program are much more challenging than in the case of Spairing. For that reason, the majority of studies of the triplet pairing problem have been limited to the construction of a phenomenological free-energy functional and solution of the corresponding Ginzburg-Landau equations. This approach, while applicable and effective at temperatures near the critical temperature Tc, fails at temperatures much lower than Tc. Serious obstacles are encountered when one seeks a full solution of the triplet pairing problem within BCS theory at arbitrary temperatures. The number of coupled, nonlinear BCS gap equations coming into play for pairing in higher angular

Pairing in Higher Angular Momentum

States

203

momentum states grows rapidly with increasing orbital angular momentum L. The greater the number of equations to be solved, the slower is the rate of convergence of iterative procedures routinely employed to calculate the energy gap in superfluid systems with single-component, 5-wave pairing. The limited accuracy implied by this slow convergence can preclude correct evaluation of the energy splitting between different superfluid phases found in the calculations. Furthermore, due to the highly nonlinear nature of the problem, the solution reached by the iteration process may depend sensitively on the arbitrary choice made for the initial values, even if numerical accuracy is high and convergence is rapid. Because such behavior is indeed observed, the effort required to determine the full list of solutions of triplet pairing problems in this manner and construct the superfluid phase diagram may be prohibitive. In the present chapter, it will be demonstrated that the difficulties we have described can be overcome (or greatly reduced) by application of a recently developed separation method5'6 for solving the system of BCS gap equations associated with pairing states of arbitrary angular-momentum content. The separation method is a kind of "divide and conquer" approach. It recasts the BCS system in such a way as to isolate the troublesome logarithmically divergent contributions to the pairing effect, so that they may be treated separately from the remaining features of the problem, which are quite insensitive to the presence of the gap and its particular value. The original system of BCS gap equations is replaced, identically, by a coupled set of equations: (i) a nonsingular and essentially linear integral equation for the dimensionless gap function \(p) defined by A(p) = AFX(P) a n d (ii) a nonlinear algebraic equation for the gap magnitude Ap = A(pp) at the Fermi surface. (For simplicity of explanation, we write the gap equation in single-component form, although it general it will have multicomponent structure, corresponding to the different magnetic substates that enter.) This reformulation admits a robust and rapidly convergent iteration procedure for the determination of the gap function. In effect, the problem is reduced to solution of a set of coupled algebraic equations. The separation approach is equally reliable and precise in the limiting temperature regimes T —> Tc and T —* 0, and in between. As will be seen, it provides a fruitful basis for analysis as well as an effective numerical tool. The work on the 3P2-3-p2 pairing model 7-11 to be reviewed here is exhaustive in its assembly of the set of solutions whose structural expression involves only real numbers. The implementation of this program of classification requires no numerical computations; remarkably, everything can be done analytically. In principle, the same method can be readily applied to the determination of complex solutions. However, this more complicated problem inevitably entails some computational effort. In principle, and presumably also in practice, the same approach can be used to complete the theory of superfluid phases of liquid 3 He. The chapter is organized as follows: In Section 8.2, we set the stage for the more elaborate treatment of triplet pairing in dense neutron matter by applying the separation method to solution of the S-wave BCS gap equation at T = 0. In Section 8.3 we employ the separation approach to develop a full set of coupled equations of the 3P2~3-p2 model, and present an incisive perturbative treatment of the mixture of the zP
204

M. V. Zverev, J. W. Clark and V. A. Khodel

analytical solution of these equations. The nature of these solutions is discussed, and a convenient and succinct catalog of the solutions is provided.

8.2. Separation method for singlet S-wave pairing In this section, we illustrate the separation procedure for the simple case of 1SQ pairing at zero temperature. The BCS equation for the gap function A(p) is

A(p) = -Jvip,P')~^-)A(p')dr'

,

(8.1)

where E2(p) = e2(p) + A2(p) is the single-particle spectrum of the superfluid liquid, e(p) = PF{P — PF)/M* is the normal-state single-particle spectrum of the system measured from the Fermi surface, M* is the effective mass, and dr = p2dp/(2TT)2. A minimal prerequisite for application of the separation method is that the singlet S'-wave effective particle-particle interaction V(p,p') has a nonzero value on the Fermi surface, i.e. V(PF,PF) 7^ 0- Otherwise the problem is obviated, since the logarithmic divergence inherent in the kernel of the BCS gap equation disappears. The key step of the method consists of decomposing the effective interaction V(p,p') into a separable part and a remainder W(p,p') that vanishes when either argument is on the Fermi surface: V(P,p') = VF^p)cf>(p') + W(p,p') . (8.2) The choice 4>(p) = V(P,PF)/VF, with VF = V(PF,PF) ^ 0, meets the required conditions W(PF,P') = W(p,pp) = 0 for all p,p'. The gap equation (8.1) then takes the form

A(p) + VFHP) J(p')^jdr' + JW(P>P')£^dT' = ° •

(8-3)

The next step is to define a shape function x(p) through A(p) = Apx(p), with Ap = A(pp) and X(PF) = 4>(PF) = 1- Setting p = pp in Eq. (3) and invoking W(pp,p) = 0, we obtain the equation

for the gap amplitude A ^ at the Fermi surface, a result conveniently expressed as 1 + VF[

(p)x(p) *(P)*(P)

2

w9
A2Fx2(p)}1/2

2[e (p) +

(8.5)

On the other hand, dividing Eq. (8.3) by A ^ and using Eq. (8.4), we arrive at the equation X(P) + ( W(P,P')-

J

2

^v

2[e (p>) +

^

d T

A2FX2(p>)]V2

' = *W

(8 6)

-

for determination of the shape factor x{p)In these steps, the original gap equation (1) has been replaced, without approximation, by the two equivalent equations (8.5) and (8.6). The essential effect of the substitution (8.2) is that for p' —> pp the integrand in the second term of Eq. (8.6)

Pairing in Higher Angular Momentum

States

205

vanishes because W(p,pp) = 0 identically, and the near-singular situation is circumvented. The corresponding integral then becomes insensitive to any reasonable variation of A(p') = Apxip') within the E(p') denominator. 5 In other words, the shape function \(p) turns out to be practically independent of the choice made for A(p') within the integral term of Eq. (8.6). As a result, the term proportional to Ap in the denominator of Eq. (8.6) can be neglected to a very good approximation, and this equation then becomes a linear integral equation for x(p); amenable to solution by standard numerical methods. Knowing the shape function x{p)> the algebraic equation (8.5) for the gap amplitude AJF is readily solved as well. If high precision is required, repeated iteration back and forth between the amplitude and shape equations (8.5) and (8.6) converges very rapidly. An analogous decomposition of the gap problem has been carried through in higher partial waves and arbitrary temperatures. 7 ' 10 ' 11 This generalization is presented and applied in the following sections. We emphasize that the separation method is not merely a trick leading to easier and more accurate numerical solution of gap equations. It also provides a sound basis for insightful analysis of the mathematical nature of superfluid solutions of many-fermion problems, with findings that may well have generic applicability. Indeed, this aspect of the method figures prominently in the comprehensive study of 1 S'o pairing carried out by Khodel et al. 5 In particular, their analysis reveals the key role played by the first node p = po > 0 of the gap function A(p) in the existence of a nontrivial solution of the gap equation, for pairing interactions that contain a strong short-range repulsion. Moreover, they were able to establish that the pairing gap at the Fermi surface, Ap, vanishes exponentially not only in the low-density limit pp —• 0, but also as the Fermi momentum pp rises and approaches the upper critical value ppc defined by pp = PFC = PO{PF), beyond which there exists no nontrivial solution of the gap equation. The numerical results for the singlet S-wave gap function in neutron matter display a remarkable universality of structure, visible especially in the stability of po under variation of density and phase-shift-equivalent interaction. Upon renormalizing the gap equation in terms of the vacuum S-wave scattering amplitude, this behavior is seen to be a manifestation of the resonant nature of the neutron-neutron interaction at low energy, which leads to a scattering amplitude of nearly separable form.5 The separation strategy has also facilitated a recent theoretical analysis 12 of experimentally measured departures of thermodynamic properties of superfluid 3 He from the standard predictions of BCS theory. In this case, triplet P-wave pairing is in play, and the quantities in question are the ratios A(T = 0)/T c and [C3(TC) — Cn(Tc)} /Cn(Tc), where Cs and Cn are the superfluid and normal specific heats. The deviations of these ratios from their BCS values can be consistently explained in terms of the structure of the gap function in momentum space, with A(p) exhibiting a pair of nodes that closely embrace the Fermi surface. Such a situation arises if, as expected, the effective particle-particle interaction in liquid 3 He features a strong short-range repulsion surrounded by a weak attractive well, with the consequence that the relevant (P-wave) pairing interaction V, evaluated on the Fermi surface, has a sign opposite to that envisioned in BCS theory (i.e., positive rather than negative).

206

M. V. Zverev, J. W. Clark and V. A. Khodel

Both of these examples serve to highlight the fact that in the past, too little attention has been paid to the momentum-space structure of the pairing interaction in strongly interacting fermion systems. This caveat should be kept mind during further explorations of pairing phenomena beyond BCS. 8.3. Generalized BCS gap equation and the ZP^—ZF^ model In the dense nucleonic quantum-fluid interior of a neutron star, the spin and isospin of the preferred neutron pairing state is S = 1 and T = 1 (triplet-triplet), while the orbital momentum L is uncertain. Our task in this section is threefold: (i) Formulate the general any-channel gap equation for given S and T; (ii) motivate the reduction of the general formulation to the 3 P 2 - 3 -F2 triplet-pairing model of the neutron superfluid at baryon densities ~ 2po, where po is the saturation density of symmetrical nuclear matter; and (iii) use the separation method to simplify the treatment of the coupled equations for the gap components corresponding to different magnetic substates. We begin with the generalized BCS gap equation in the 2 x 2 spin-matrix form

Aa/3(p) = - J V ^ ( p , P i ) t a P h ^ 1 i ) ) / 2 T ) ^ 1 , s ( p i ) d v l , where dv = d3p/(2n)3.

(8.7)

Defining spin-angle matrices

(G£S(n)) Q/3 = J2 C\fafiCtfMaMLYLML{n), Ms

(8.8)

ML

we introduce a partial-wave decomposition

AQ/3(p) = £

AJLM(p) (G&(n)) a/J

(8.9)

J,L,M

of the gap matrix and a partial-wave expansion

V(P,PI)=

£

(-i^WViMG&wlGZAm))*

(8.io)

LL'JM

of the particle-particle-irreducible interaction block V. Straightforward manipulations then serve to recast the matrix equation (8.7) as a set of coupled multicomponent BCS gap equations

AiM(P) =

£ x

(-i) 1+ ^

{wiM^"1^)

tanh(E(p,)/2T)Ari(pi)^

(g n )

It is important already to note that the spin traces ^

J l M l

( n ) = Tr[(GiM(n))*G^Ml(n)]

(8.12)

obey the orthogonality relation f SJL^JlMl

(n)dn =

6LLI5JJISMM1

•

(8.13)

Pairing in Higher Angular Momentum

207

States

The quasiparticle (qp) energy E(p) = >/e2(p) + I> 2 (p)

(8.14) 1

entering Eqs. (8.11) has a more complicated form than in the case of S'o pairing, although the single-particle spectrum e(p) of the normal Fermi liquid, measured relative to the Fermi surface, is still conventionally parameterized with an effective mass M*. However, the part of E(p) due to the pairing effect, D2(P) = \

£

(AJLM(p)Y

Ai\^(p)SJL^J^(n),

(8.15)

LJML1JiMi

which one would like to identify as the energy gap in the qp spectrum, has acquired an angle dependence through the spin traces SJLL 1 ^ n ) . Because of the angular variation exhibited by the function D(p), it has become customary to adopt the quantity W(kF))1/2

(8.16)

as a representative measure of the energy gap, where the overbar signifies an average over all directions. Close inspection reveals that complete solution of the set of equations (8.11) presents formidable difficulties. The squared-gap quantity D2(p) entering the quasiparticle energy E(p) not only introduces angle dependence into the problem. According to the definition (8.15), it also couples the components A ^ M to their counterparts with generic angular momentum quantum numbers L\, J\, and M\. Fortunately, the job of solving the system (8.11) is greatly facilitated by exploiting the fact that the angular dependence only comes into play near the Fermi surface; hence it can be ignored in those integral contributions to the r.h.s. of Eq. (8.11) in which the region around the Fermi surface is suppressed. 7,9 In practice, then, the gap equations approximately decouple in the variables U, Li, and Ji when one makes use of the orthogonality property (8.13). We next take advantage of specific properties of the free-space NN interaction that is ultimately responsible for pairing. This two-body interaction has the salient feature that its net central component in a relative P-state of two neutrons is quite weak, as reflected in the behavior of the experimental P-wave phase shifts at energies relevant to the pairing problem in neutron matter. Consequently, the spin-orbit component assumes a crucial role in promoting the 3P2 pairing channel. 9,14-16 The dominance of the spin-orbit force implies that the contributions to triplet pairing from so-called "nondiagonal" terms with L', L\ ^ 1 or J\ ^ 2 appearing on the right in equations (8.11) can be evaluated perturbatively, in terms of the set of principal gap components A 2 M (p), with M = 0, ± 1 , ±2. It is important to note at this point that the spin-orbit interaction loses its dominant role in triplet pairing as the density approaches the critical value for the pion-condensation phase transition. 3,4 However, even when the pion fluctuations become strong, perturbative treatment of the 3 P2- 3 P2 coupling associated with nondiagonal contributions entering Eq. (8.11) remains applicable. Focusing on these contributions, we observe that two are of leading significance. The first involves

208

M. V. Zverev, J. W. Clark and V. A. Khodel

the integral of the product V^S^™1 A2M\ and the second, the integral of the product V^S2^2MlAJMl. The minor components A\M are related to principle components A1 * through the tensor part of the pairing interaction. The relevant coupling parameter is the ratio 77 = — (pp\V23\PF)/{PF\Vh \PF}- The further analysis is carried out in terms of the set of three major gap amplitudes A2M(p), with M running from 0 to 2 by virtue of the relations between AfM(p) and A2'~M(p) imposed by time-reversal invariance. The above simplifications define the 3 P2~ 3 i r 2 pairing model that we intend to solve. The generalized BCS system (8.11) assumes the specific form A?W(P)

+ £

=E

MJ

f(p\V2u\Pl)S2rM>

( m ) t a n h g ( P l } / 2 r ) A f * (pOdui,

/
+ s/
/
^ y

2&o ( P I )

(8.17) which will now be subjected to analysis. In writing the right-hand sides of this set of equations, we have replaced the quasiparticle energy E(p;rj) by Eo(p;r] = 1 In

0) = [e2{p) + D2(p)Y'\ where D 0 (p) is the gap function of the much-studied 3 P2 pairing model in which the tensor coupling between F and P states is ignored. This substitution is justified by the relatively small sizes of the quantities (p|Vi 3 |pi), Al^faJ.andfrlVfilPi). A well-known feature of the 3P2 pairing model 13 15 is the high parametric degeneracy of the spectrum of its solutions. 17 This property has been investigated in detail in Refs. 7,9 in terms of the two ratios Xx = D2lj&/D\a = -D^Ve/D20 22 20 2 and A2 = D VE/D = D\~ y/6/Df. The degeneracy is reflected in the existence of a set of curves Ai(A2) in the A1-A2 plane, along which all the BCS equations of the 3 P2 problem are satisfied. As we shall see, this degeneracy is lifted in the 3 p 2 3 P2 pairing model where 77 ^ 0. A finite set of points (Ai, A2), dependent on the value of 77, replaces the set of solution curves Ai(A2) of the 3 p2 model. Our objective is to identify the different solutions of the system (8.17) and to establish their structure in the case of small 77. The analysis is aided by the fact that the parameters Ai = /i(?7) and A2 = f2{v) a r e continuous functions of the coupling constant 77. This property implies that the number of solutions of the 3 P2~ 3 P2 pairing problem, as well as their structure, remains the same no matter how small 77 is. Consequently, implementation of our program reduces to determination of the functions f\ and / 2 at 77 = 0, i.e., Ai(77 = 0) and A2(?7 = 0). It then becomes apparent that the quasiparticle energy £(77) may be replaced by EQ on the left in Eqs. (8.17) as well as on the right, since taking into account the difference between E(rj) and EQ within (8.17) cannot, in itself, lift the parametric degeneracy. This conclusion is confirmed in the numerical calculations. 11

Pairing in Higher Angular Momentum

209

States

The "nondiagonal" integrals on the right in Eqs. (8.17) are rapidly convergent, with their overwhelming contributions coming from momenta adjacent to the Fermi surface. This feature greatly expedites application of the perturbation strategy. For E(p) significantly in excess of the energy gap A.F of Eq. (8.16), the energies E(p) and \e(p)\ are coincident to high precision, such that the angular integration in Eq. (8.17) yields a null result. Thus, when treating the nondiagonal contributions it is sufficient to know the minor gap components A§ M (p) at the point p = pF, which may be efficiently evaluated in terms of the coefficients D\M = AlM(pp) with Mi = 0 , 1 , 2 . In this process, we retain, on the r.h.s. of the last of Eqs. (8.17), only the dominant contribution containing a large logarithmic factor L = (27T2)-1 ln(eF/&F), where eF is the Fermi energy. This factor is angleindependent; hence the respective angular integral is easily evaluated, leading to the simple connection &1M(P = PF) = -L{pF\V213\PF)DlM

= VVFLDlM

~ r,D\M ,

(8.18)

where VF = ( P F I V I I I P F ) - In arriving at this relation we have employed the equality 1 = VFL, which holds when one keeps only logarithmic contributions. Substitution of the result (8.18) into the first of Eqs. (8.17) leads to the closed system of equations

E/ [ 5 3 r M > i ) + ^ 2 M l ( n i ) ]

^ ^ ^ ^ ^ ( p O c f a i

(8.19) for finding the set of three gap functions AfM(p) with M = 0,1,2. At this point we invoke the separation method and assert the decomposition 7 ' 9 A 2 M (p) = D\MX{P)

(8.20)

of the gap component into a "universal" shape factor x(p) that is independent of the magnetic quantum number M, and a numerical coefficient D\M that embodies the dependence on M. The function x(p)> normalized by X{PF) = 1, is the solution of a linear integral equation. As argued in Refs. 7, 9, this decomposition holds to high accuracy in the problem domain under consideration. Accordingly, our problem reduces to the determination of the three key coefficients D\M, which obey a set of coupled algebraic equations obtained by setting p — pF in Eqs. (8.19). With M = 0, 1, and 2, these equations read D?" + V „ X > i M l / /*(P)

tanh

2 E o ( p ) / 2 r ) S™2Ml

= r,vF J2 D™1 [ [ S 3 T M 1 (n) + ^ 2 3 M 2 M l (n)l K0(n)dn, Mi

with VFcj>{p) =

^P>2drdn (8.21)

^


and

Ko(a)-_J{plv^}^^gmx(p)l±3.

(8,2)

210

M. V. Zverev, J. W. Clark and V. A. Khodel

The search for solutions will be limited to those with real coefficients D\M. The structure of the superfluid phases having complex coefficients D\M can be explored and established in much the same way, although the analysis is considerably more complicated. To streamline the task of finding solutions, it is helpful to rewrite Eqs. (8.21) in terms of the ratios Ai and A2. This step ensures coincidence between the left-hand sides of these equations and their counterparts in the model of pure 3 P 2 pairing solved in Ref. 7. Substitution of the explicit form of S™2 * (n) into Eqs. (8.21), followed by straightforward algebra, gives a system of three equations, 10 ' 11 A2 + VF [A2(Jo + J5) - Ai Ji - J 3 ] = T}VFT2 Ai + vF [-(A2 + l ) J i + Ai(J 0 + 4J 5 + 2J 3 )/4] = rjvpn 1 + vp [-(A2 J3 + Ai J i ) / 3 + J 5 ] = r)vFr0 •

(8.23)

for these ratios and the gap value AF. Here, as before,7 tanh£0(p)/2T p2dpdn x(p)- 47T 2£o(p)

Ji = jj w,
(8.24)

with /o = 1 - 3z 2 , /1 = 3 ^ / 2 , / 3 = 3(2a;2 + z2 - l ) / 2 , and / 5 = (1 + 3z 2 )/2, and z = costf, x = sin •& cos if, and y = sin $ sin if. Among the integrals Ji (i = 1, • • • 5), only J5 contains a singular principal term going like ln(eF/AF)All the other Ji converge close to the Fermi surface, where the quasiparticle energy can be approximated by the formula Eo(p) — [e2(p) + -Co(n)] with #F 1+ Dl(n) = 2[1 + (A? + AD/3] L

A2 3 cos2 6 + A2, sin2 6 + Ml

+ cos2 6)

2Ai(l -f A2) cos6sin6cosip + -(A 2 — 4A 2 )sin 0cos2y>

(8.25)

In terms of the coefficients Ai, A2, the right-hand sides of Eqs. (8.23) read r 2 (Ai, A2) = A 2 s 22 + Ais 2 i + V&S20 , ^i(Ai,A 2 ) = A 2 s 12 + Aisn +

V6sw,

ro(Ai, A2) = ~^ s o2 + ~/F S OI + soo ,

(8.26)

where «M2

J [ s 2 f 2 2 (n) + 5 2 3 M22 (n) + 5 3 2 1 M2 '" 2 (n) + S 2 3 M2 >- 2 (n)] K0(n)dn

SMI

J [sir'H

+ 5 2 3 M21 (n) - S™2'~\n)

% o = J [5 3 T 2 0 (n) + S 2 3 M20 (n)] K0(n)dn

.

- ^ ^ ( n ) ] K0(n)dn

, , (8.27)

Pairing in Higher Angular Momentum

States

211

8.4. Real solutions of the 3 P 2 - 3 F 2 problem Equations (8.23) have three familiar 13-15 one-component solutions with definite magnetic quantum numbers M = 0, 1, and 2. To uncover the structure and the spectrum of the multicomponent, mixed-M solutions of the perturbed problem, a two-step transformation is applied to the system (8.23). The integral J5 introduces the gap value Air into the description, but it is irrelevant to the phase structure. As a first step, we combine Eqs. (8.23) so as to eliminate terms involving J5 from the first pair and, at the same time, reduce the number of Ji integrals in each equation to two. The resulting equations are (A2 + l)[3Ai(A2 + 1) Jo - 2(A2 - 2A2 + 6) J x ] =

vBi,

2

(A2 + 1)[(A - 4A2) Ji + A!(A2 + 1) J 3 ] = r,B2 ,

(8.28)

with Bi = 2Ai(2A2 + 3)r 2 (Ai, A2) - A(X22 - 3)r 1 (A 1 , A2) - 6Ai(A2 + 2)r 0 (Ai, A2) , B2 = -A 1 r 2 (A 1 ,A 2 )+4A 2 r 1 (A 1 ,A 2 )-3A 1 A 2 r 0 (A 1 ,A 2 ) . (8.29) Two cases are to be distinguished. The first corresponds to A2 = —1, a particular solution of the pure 3 P 2 pairing problem. 7 In this case, the left-hand sides of Eqs. (8.28) vanish identically, and hence so must their right-hand sides, leading to the single restriction Air 2 (Ai; - 1 ) + 4 n ( A i ; - 1 ) - 3Air 0 (Ai; - 1 ) = 0.

(8.30)

As will be seen, this condition is satisfied at any Ai. Therefore the particular solution A2 = — 1 found in the pure 3 P 2 problem survives intact when the 3 P 2 coupling is switched on. Let us now assume A2 ^ — 1 and proceed to the second step. Following Ref. 7, we perform a rotation z = £cos/3 + usin/3, x = — £ sin/? +-u cos/3,

*• '

with the objective of removing the integral Ji from Eqs. (8.28). To this end, the parameter £ = tan f3 is taken as a root of the quadratic equation c 2 + 3 - A 2 c _ i = 0 A2

The integrals Jj transform according to Jo -» Jo (cos2 j3 - - sin2 (3) - J 3 sin2 (3, Ji -> - - Jo sin2 (3 + J3 ("cos2 P+-

sin2 @\ ,

J\ -> ( 7 Jo + 2 J a ) sin/3 cos/3.

(8.33)

Substitution of the transformed integrals into Eqs. (8.28) yields (A2 + l)[A 1 J 0 + A 2 J 3 ] = r ? P 1 , (A2 + l)[i4i Jo + A2 J 3 ] = -2»7P 2 ,

(8.34)

'

212

M. V. Zverev, J. W. Clark and V. A. Khodel

with Ax = \\X{1 + A2)(2 - C2) - |(A? - 2A2 + 6)C, A2 = -3Ai(l + A2)C2 - (A? - 2A^ + 6)C •

(8.35)

(Some details of this step are provided in Refs. 6,7.) The left-hand members of the two equations (8.34) are seen to be identical. In fact, the universalities of pure 3 P 2 pairing revealed in Ref. 7 stem from this key property. The solutions of the restricted problem derived from Eqs. (8.34) at rj = 0 fall into two groups composed of states that are degenerate in energy. This remarkable feature is independent of temperature, density, and details of the in-medium interaction. There is an upper (i.e., higher-energy) group consisting of states whose angle-dependent order parameters have nodes and a lower group without nodes (cf. Ref. 17). In addition to the energy degeneracies, the multicomponent pairing solutions, which obey the relation 7 (A? - 2A2. - 6A2)(A? + 2 - 2A2) = 0,

(8.36)

manifest a parametric degeneracy with respect to the coefficient ratios Ai and A2, as they in general define curves rather than points in the (Ai, A2) plane. Among solutions of the 3 P 2 - 3 i ; l 2 pairing model there exist some for which B\ and S 2 in Eq. (8.34) vanish simultaneously at certain sets of the parameters Ai and A2 satisfying Eq. (8.36). Indeed, consider the sets corresponding to Ai=0;

A2 = ± 1 , ± 3 ,

(8.37) 3

which determine two-component solutions of the pure P 2 pairing problem. 7 In these cases, it may be observed from Eqs. (8.29) and (8.26) that 5i(Ai = 0, A2) ~ B 2 (Ai = 0, A2) ~ n ( 0 , A2) = A 2 si 2 (0, A2) + V6sw(0, A 2 ), (8.38) where «i2(Ai = 0, A2) and Sio(Ai = 0, A2) are defined by Eq. (8.27). The quantities D%(\i = 0, A2; n) and Ko(Xi = 0, A2; n) are even functions 7 ' 15 of cos<^, while both Sfi a 2 (n) + S23122(n) + ^3i , - 2 ( n ) + S 2 3 12 ' _2 (n) and S 3 T°(n) + Sw 2 0 (n) are odd in costp (for details, see Ref. 11). As a result, both the matrix elements si 2 (Ai = 0, A2) and sio(Ai = 0, A2) vanish identically when the integration over

(8.39)

the quantities rjf being defined by Eq. (8.26). Inserting the explicit expressions for the rM, this auxiliary condition can be rewritten as G(Ai,A2)-|^0(^y,z;A1,A2)*(a:,2/,z;A2)«5(l-a:2-y2-Z2)^^=0, (8.40)

Pairing in Higher Angular Momentum

States

where $(x, z; \\,A2) = Aii?2(a:,.z) - (A2 -3)Ri(x,z) -3XiRo(x,z). Rk have been evaluated in Ref. 11, with the results RQ{X,Z)

213

The quantities

= -j=So2{x,z) + —j=S0i(x,z) + Soo{x,z) = J^L{ 7 [2A 2 z 2 + 6(A2 + 2)z2 -

10X2x2z2

- 5(A2 + 3)z 4 - A2 - l] + 2X1(16xz - 35a:z 3 )} , Ri(x,z)

= A2S'i2(a:,z) + XiSn(x,

z) +

(8.41)

V6Sw(x,z)

= - ^ L { [48(A2 + l)xz - 35(A2 + 3)a;z3 - 70A2z3.z] + 7A!(z2 + x2 - 10x 2 z 2 )} , R2(x,z)

(8.42)

= A 2 5 22 (x,z) + Ai52i(a;, z) + \Z6S2o(x, z) = ^y=J7[-20A 2 a; 4 + 2(10A2 + 3)x 2 - 10(2A2 + 3)x 2 z 2 + 2(4A2 + 9)z2 - 5(A2 + 3)z 4 - 3A2 - 3] + 2Ai (48xz - 35xz3 - 70x3z) \ .

(8.43)

Substituting for the Rk, we obtain V(x,z;\i,\2)

= 14Ai [ 5 ( A 2 + 3 ) z 4 - 1 0 A 2 x 4 + 1 5 ( A 2 - 3 ) x 2 z 2 +6(A2 + \){x2 - z2)} + 70 [(A2 - 3)(A2 + 3) + 2A2] xz3 +70 [2A2(A2 - 3) - 2A2] xzz - 96(A2 - 3)(A2 + \)xz.

(8.44)

The relation (8.40) supplements the spectral condition (8.36). As a direct consequence, the strong parametric degeneracy inherent in pure 3 P 2 pairing is lifted in the case of 3 .P 2 - 3 .F 2 pairing. With the exception of the straight-line solution A2 = 1 noted above, the solutions of the problem are now represented by a set of isolated points in the (Aj, A2) plane. The system formed by Eqs. (8.36) and (8.39) is amenable to analytic solution. We begin the search for solutions of Eq. (8.40) with the particular solution (Ai, A2 = — 1) of the pure 3 P 2 pairing problem. In this case, *(ar, y, z\ Ai, A2 = - 1 ) ~ z 4 + x4 - 6x2z2 = 8z 4 - 3(1 - y 2 ) 2 - 4(1 - y2)(2z2 - 1 + y2), while, as seen from Eq. (8.25), the gap function the single variable y: D2(x,y,z;X1,X2

= -l)~hx2

-DQ(A2

(8.45)

= — l ; n ) depends only on

+ z2) = l(l-y2).

(8.46)

M. V. Zverev, J. W. Clark and V. A. Khodel

214

Since the integrals n-v2

I

8 z 4 - 3 ( l - y 2 2)\ 2 N/I

- v2 -

dz

and

2z2

/ -Vi-y2

i + y2 dz y/l-y2z2

(8.47)

both vanish, the integral G(Ai,A 2 — —1) also vanishes at any Ai. Accordingly, the degenerate solution A2 = — 1 of the uncoupled 3 P 2 pairing problem survives when the 3i<2 channel is involved. In search of other solutions, we again make use of the transformation (8.31), applying it now to the whole integrand of Eq. (8.40). As found in Ref. 7, the gap function Do reduces to a function of the single variable t under such a transformation; hence the same property holds for the factor KQ in Eq. (8.40), which is denned as a functional of Do by Eq. (8.22). To ascertain how the factor \& is transformed, let us write down the results of the transformation for simple terms entering this function. In implementing the transformation we omit odd-power terms u and w3, which do not contribute to the integral (8.40). We obtain x4 -> t4 sin4 f3 + u4 cos4 (3 + 6i V sin2 (3 cos2 /3, z* -> t4 cos4 /3 + u4 sin4 (3 + 6t2u2 sin2 (3 cos2 (3, x2z2 t4 sin2 /3 cos2 P + u4 sin 2 f3 cos2 (3 + t 2 u 2 (sin 4 /3 + cos4 /3 - 4 sin2 (3 cos2 /?), u4 sin 3 (3 cos (3 - t4 cos3 /? sin /? + 3i V (cos3 f3 sin (3 - sin 3 /3 cos (3), xz 3 xz u4 cos3 f3 sin (3 - t4 sin 3 /? cos (3 - 3t2u2 (cos3 (3 sin (3 - sin 3 (3 cos /?), 2 X

2 —Z

{t2 - u2) (cos2 (3 - sin2 (3).

(8.48)

After inserting these relations into the formula (8.44) for ^ , simple algebra yields *(t, u; Ax, A2) = (1 + C 2 )" 2 [Uu4 + Tt4 + V{u2 - t2) + WuH2] ,

(8.49)

where U(X1:

A2) = 7 0 J A ! ( A 2 + 3)C4 + [(A2 - 3)(A2 + 3) + 2A2]

C3

+ 3A!(A2 - 3)C2 + [2A2(A2 - 3) - 2A2] C - 2A!A 2 |, (8.50)

T(Ai, A2) = 70J-2A!A 2 C 4 - [2A2(A2 - 3) - 2A2]C3 + 3Aj(A2 - 3)C2 - [(A2 - 3)(A2 + 3) + 2A2]C + Ai(A2 + 3 ) 1 ,

(8.51)

V(A 1 ,A 2 ) = 24 7A1(A2 + 1)(1 + C 2 )(1-C 2 ) 8(A 2 -3)(A 2 + 1)(1 + C2)C

(8.52)

Pairing in Higher Angular Momentum

States

215

W(XU A2) = 210 J2A!(A2 + 3)C2 - 4AiA2C2 + Ai(A2 - 3)(C4 + 1 - 4( 2 ) + [(A 2 -3)(A 2 + 3) + 2A 2 ](C-C 3 ) (8.53)

+ [2A2(A2-3)-2A2](C3-0} • These results can be simplified slightly by employing the connection Ai(C2 — 1) = (A2 — 3)£, and we finally arrive at 4 2 = 70 Ai(A2 + 3)C + 5Ai(A2 - 3)C - 2AiA2

U(\i,X2)

+ (A 2 -3)(A 2 + 3)C3 + 2A 2 (A 2 -3)C

(8.54)

2 4 T(A l t A a ) = 70 Ai(A2 + 3) + 5AX(A2 - 3)C - 2AxA2<

- ( A 2 - 3 ) ( A 2 + 3)C-2A2(A2-3)C3 ^(A 1 ,A 2 ) = 180A1(A2 + 1 ) ( 1 - C 4 ) , = 420A!(A2 - 3)(C4 - 6C2 + 1).

W{X1,X2)

(8.55) (8.56) (8.57)

The auxiliary integrals

I

h

I

du y / l - t

2

-

u2du y/l-t2-

U2

=

7T

7T(1 -

t2

J

(8.58)

and V I —i

/

u^du y/l-t 2

3 7 r ( l - t 22\2 )

J

(8.59)

are helpful in completing the evaluation of G(Ai,A 2 ) by integration over the new variables. The integrals Io, J 2 , and I4 are related by h = l{l-t2),

J2 = j j ( l - t 2 ) 2 I 0 )

J2 = I ( l - t V o .

(8-60)

Using these connections, it is easy to verify that integration of the combinations 8u 4 — 3(1 — t2)2 and 2u 2 — 1 + t2 over u gives zero. It follows then that if we make the following replacements ,2^i(l-*2),

-4-|(l-*2)

2

(8.61)

M. V. Zverev, J. W. Clark and V. A. Khodel

216

in Eq. (8.44), the new function

*'(*,*!, A2) =

8(1

1 C 2 )2 {[ 3 ^(Ai, A2) + 8T(A!, A2) - AW(XUX2)} t4

- [6t/(Al5 A2) + 12V(Ai, A2) - 4W(XU A2)] t2 + 3l/(A 1 ,A 2 )+4V(A 1 ,A 2 )}

(8.62)

will guarantee the same result as given by \t upon integration of Eq. (8.40). For this integral to vanish identically, with the function Ko(t) regarded as arbitrary, the coefficients of all powers of t in the function \I>' must be zero: 3t/(Ax, A2) + 8T(A1) A2) - 4W(X1,X2) = 0, 3f/(A1, A2) + 6^(Ai, A2) - 2W(X1, A2) = 0, 3tf(Ai,A2) + 4V(Ai,A 2 ) = 0.

(8.63)

This system reduces to the chain of equations T(Ai,A 2 ) = V(Ai,A 2 ) = W(XUX2)

= -^U(X1,X2),

(8.64)

where Aj and A2 are to satisfy the relation (8.36). It can be proved that to determine all solutions of this set it is sufficient to solve the equation V(A 1 ,A 2 ) = W(A 1 ) A 2 ),

(8.65)

3(A2 + 1)(1 - C4) = 7(A2 - 3)(C4 - 6C2 + 1),

(8.66)

which has the explicit form

and then to verify that the other equalities in (8.64) hold for these solutions. To solve Eq. (8.66), consider the first branch of Eq. (8.36), A? = 2A^ + 6A2 ,

(8.67)

2A £ = tan/3=—-.

(8.68)

( A 2 - 3 ) ( 5 A ^ + 2 4 A 2 - 9 ) = 0.

(8.69)

for which Ai

Then Eq. (8.66) is recast to

One root of this equation is obviously A2 = 3, which corresponds to Ai = 6. Another pair of roots is given by

A2 = | (±V5T - 4) ,

(8.70)

yielding respectively A1 = | A / 2 ( 1 7 T 3 V ^ T )

5v Now consider the second branch of Eq. (8.36), A2 = ^ + l ,

.

(8.71)

(8.72)

Pairing in Higher Angular Momentum

States

217

with (8.73) In this case, Eq. (8.66) becomes (A2-3)(A2-26A2 + 2 9 ) = 0 .

(8.74)

The system has three different roots (A2,Ai) = (3,2)

(13T2V/35,2^6TV/35) .

(8.75)

The set of solutions of the 3P2~3F2 pairing problem revealed by the above analysis is depicted in Fig. 1 and cataloged in Table 1. Figure 1 represents solutions by their coordinates (Ai, A 2 )in the two-dimensional parameter space. Only the right half of the (Ai, A2) plane is plotted, because the pairing energies are independent of the sign of Ai. The most important message of this figure is that (with the exception previously noted) the solution curves that represent parametrically degenerate solutions of the 3 P 2 problem shrink to discrete points as the degeneracy is lifted by the perturbation that admixes the 3 i ? 2 channel. The solutions (which correspond to pairing states and ultimately to superfluid phases) divide into two categories: those whose order parameters contain nodes and those whose order parameters are nodeless. It is convenient to identify the nodal states with the symbol X and the nodeless states with the symbol O. First there are the three well-known "one-component" solutions, belonging to magnetic quantum numbers M = ±2, M = ± 1 , and M = 0, respectively. In addition, we have established the existence of ten multicomponent solutions that mix states with different values of \M\. These are comprised of five nodeless solutions and five having nodes. The five nodeless multicomponent solutions Ofc include: (i) Two two-component solutions denoted O-1-3, which are identical to the "particular" solutions found in the pure 3 P 2 case having Ai = 0 and A2 = ± 3 . (ii) Three three-component solutions associated with the branch (8.67) of Eq. (8.36). Two of these, denoted 0\ and O4, derive respectively from the upper root of the pair (8.70) and the root A2 = 3. The third, named 0 2 , derives from the lower root of the pair (8.70). The five nodal solutions Xfc consist of: (i) Two two-component solutions X±i that are identical to the "particular" solutions found in the pure 3 P 2 problem having Ai = 0 and A2 = ± 1 . Three three-component solutions associated with the branch (8.72) of Eq. (8.36). The solutions X 2 and X4 derive respectively from the lower and upper roots of the pair (8.75), while X3 derives from the root A2 = 3. The full collection of complex solutions can also be obtained along the same lines, but the calculations become much more cumbersome.

M. V. Zverev, J. W. Clark and V. A. Khodel

218

J

I

I

L

Fig. 8.1. Parameter sets (Ai,A2) defining the multicomponent solutions of the 3 P2- 3 i<2 pairing problem. Solution curves or points of the pure, uncoupled 3 P2 problem are identified with an open circle (or a filled circle) according as their order parameters are nodeless (or display nodes).

8.5. Conclusions In the present chapter, we have employed the separation method of Ref. 7 to find all the real solutions of the 3P2~3F2 pairing problem in the regime of small 77, where 77 is a dimensionless parameter measuring the importance of contributions associated with the tensor component of the pairing interaction. This regime is considered to be relevant to triplet neutron pairing in neutron-star matter. Accordingly, it has been established that the superfiuid phase diagram of dense neutron matter can exhibit numerous triplet superfiuid phases (at least 13!). A salient feature of the results of the analysis is their remarkable independence of the temperature T and of the details of the pairing interaction V. Appealing to continuity in the parameter 77, it may be expected that the general features of the spectrum of solutions delineated by the analysis will persist even when 77 is not especially small. Let us compare our method with Ginzburg-Landau (GL) theory, which is generally regarded as the standard technique for mapping the spectrum of the phases of systems with triplet pairing. In the GL method, the search for diverse phases is based on the construction of a suitable free-energy functional up to terms of fourth

Pairing in Higher Angular Momentum

States

219

Table 8.1. Identification of the thirteen solutions (or superfluid phases) of the 3 P 2 - 3 i r 2 pairing model, in terms of the parameters Ai and A2 defining their magnetic content, and in terms of their nodeless (states labeled O) or nodal (states labeled X) character. Phase Ai A2 M =0 OM=O M =1 XM=I M =2 XM=2 3 -3 1 -1

03 0_3 Xi X_i

0 0 0 0

Oi

|A/2(17-3V/2T)

o2 04

x2 x3 x4

/

|v/2(17+3v 2l) 6 2\/6-v/35 2 2V6+V/35

§(^21-4)

-§(V2l+4) 3 13-2v/35 3 13+2\/35

(or even sixth) power in the gap value A. This approach provides for simultaneous evaluation of the splitting between the different phases and efficient determination of the phase diagram. Another advantage of the GL procedure lies in the facility of including strong-coupling corrections 18 arising from the dependence of the effective interaction V on the gap value, an effect that becomes important close to the critical temperature Tc. Unfortunately, the GL method fails when the temperature T is significantly different from T c . Our method is free of this shortcoming. It is equally reliable at T = 0, close to Tc, and in between. The incorporation of strong-coupling corrections reduces to the insertion of new terms in V that depend on A itself; if this dependence is appropriately specified, no additional hurdles must be overcome to fully elucidate the triplet superfluid phase diagram in the narrow vicinity of the critical temperature Tc. As we have already indicated, the status of the 3 P2- 3 p2 pairing model, in which contributions from 3 P 2 —>3 Po or 3 P 2 —>3 Pi transitions are assumed to be unimportant, is vulnerable. This assumption fails in the denser core region of a neutron star, where the amplification of the tensor force due to the pion-exchange renormalization 3 becomes pronounced. The enhancement grows more powerful with increasing density and eventually overwhelms the spin-orbit component in the effective pairing interaction, whose strength depends only mildly on density. This effect, stemming from pionic degrees of freedom, promotes the formation of a superfluid state analogous to the B-state 19 of liquid 3 He. The procedure applied here may in principle be extended to count all possible solutions of the set of nine BCS equations for the gap function of superfluid 3 He. Since 3 P2, 3Pi, a n d 3 Po states all enter the picture, one will be faced with nine nonlinear equations for the relevant gap components (in contrast to the five encountered in the pure 3 Pa case). The task is simplified, however, by the fact that

220

M. V. Zverev, J. W. Clark and V. A. Khodel

in this problem the effective pairing interaction contains only a central component. Again the separation method will facilitate the derivation of results for the spect r u m of phases independently of the details of the interaction. T h e process may begin by severing the couplings between 3 P 2 , 3 P i , and 3 Po components to obtain a reference analytic solution.

Acknowledgments T h e research leading to many of the results described herein was supported in p a r t by the U.S. National Science Foundation under G r a n t Nos. PHY-9900173 and P H Y 0140316 ( J W C & VAK), and by the McDonnell Center for the Space Sciences (VAK & MVZ). We t h a n k A. Sedrakian, 1.1. Strakovsky, G. E. Volovik, and D. N. Voskresensky for discussions.

Bibliography 1. D. Vollhardt and P. Wolfle, The Superfluid Phases of Helium 3 (Taylor &: Francis, London, 1990). 2. R. A. Arndt, C. H. Oh, I. I. Strakovsky, R. L. Workman, and F. Dohrmann, Phys. Rev. C 56, 3005 (1997). 3. A. B. Migdal, Rev. Mod. Phys. 50, 107 (1978). 4. A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998). 5. V. A. Khodel, V. V. Khodel, and J. W. Clark, Nucl. Phys. A 598, 390 (1996). 6. V. A. Khodel, Phys. At. Nucl. 60, 1033 (1997). 7. V. A. Khodel, V. V. Khodel, and J. W. Clark, Phys. Rev. Lett. 8 1 , 3828 (1998). 8. V. A. Khodel, Phys. At. Nucl. 64, 393 (2001). 9. V. V. Khodel, V. A. Khodel, and J. W. Clark, Nucl. Phys. A 679, 827 (2001). 10. V. A. Khodel, J. W. Clark, and M. V. Zverev, Phys. Rev. Lett. 87, 031103 (2001). 11. M. V. Zverev, J. W. Clark, and V. A. Khodel, Nucl. Phys. A 720, 20 (2003). 12. M. V. Zverev, V. A. Khodel, and J. W. Clark, cond-mat/0605631. 13. T. Takatsuka and R. Tamagaki, Prog. Theor. Phys. 46, 114 (1971). 14. T. Takatsuka, Prog. Theor. Phys. 48, 1517 (1972). 15. L. Amundsen and E. 0stgaard, Nucl. Phys. A 442, 163 (1985). 16. M. Baldo, 0 . Elgar0y, L. Engvik, M. Hjorth-Jensen, and H.-J. Schulze, Phys. Rev. C 58, 1921 (1998). 17. R. W. Richardson, Phys. Rev. D 5, 1883 (1972). 18. J. A. Sauls and J. W. Serene, Phys. Rev. B 24, 183 (1981). 19. V. A. Khodel, J. W. Clark, M. Takano and M. V. Zverev, Phys. Rev. Lett. 93, 151101 (2001).

Chapter 9 Four-Particle Condensates in Nuclear Systems

Gerd Ropke Institut

fur Physik,

Rostock

University, Universitatsplatz Germany

1, 18051

Rostock,

Peter Schuck Institut

de physique

Nucleaire, IN2P3-CNRS, Universite Orsay Cedex, France

Paris-Sud,

F-91^06

Quantum condensates in nuclear matter are treated beyond the mean-field approximation, with the inclusion of cluster formation. The occurrence of a separate binding pole in the four-particle propagator in nuclear matter is investigated with respect to the formation of a condensate of a-like particles, which is dependent on temperature and density. Due to Pauli blocking, the formation of an a-like condensate is limited to the low-density region. Consequences for finite nuclei are considered. In particular, excitations of self-conjugate even-Z-even-iV nuclei near the Q-breakup threshold are candidates for the formation of such a condensate. We review some results and discuss their consequences. Exploratory calculations are performed for the density dependence of the a condensate fraction at zero temperature to address the suppression of the four-particle condensate below nuclear-matter density.

Contents

9.1. Introduction: Mean-field approximation in nuclear matter and BCS condensates 9.2. Cluster expansion of many-particle properties 9.3. Cluster mean-field approximation 9.4. Two-particle condensates at low temperatures 9.5. Four-particle condensates and quartetting in nuclear matter 9.6. Alpha-particle condensate states in light nuclei 9.7. Results for finite nuclei 9.8. Reduction of the a condensate with increasing density 9.9. Conclusions Bibliography

221

. . . 222 224 226 230 234 238 242 246 250 251

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9.1. Introduction: Mean-field approximation in nuclear matter and BCS condensates One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. At present, the formation of condensates is of particular interest in strongly coupled fermion systems in which the crossover from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) may be investigated. Among very different quantum systems such as the electron-holeexciton system in excited semiconductors, atoms in traps at extremely low temperatures, etc., nuclear matter is especially well suited to the study of correlation effects in a quantum liquid. Nuclear matter at medium densities and temperatures is treated as a manynucleon problem in which the particles interact via effective potentials. These potentials are either phenomenological, i.e., fitted to the bulk properties of nuclei (a prominent example being the Gogny 1 force), or realistic in the sense that they fit the elastic scattering phase shifts and the binding energy of the deuteron (with popular examples including the Paris, Argonne, Bonn, and Nijmegen potentials and their separable counterparts 2 ). Phase transitions in nuclear matter are of great current interest. The liquid-gas-like phase transition in nuclear matter has been extensively discussed, and a large number of experiments have been carried out to identify the fingerprints of this phenomenon. Superfluidity in nuclear matter has not been studied as intensively so far, at least from the experimental standpoint. Superfluidity in nuclear matter is a well-known phenomenon. Pairing in the singlet channel is considered in describing the binding energies of nuclei. 3 A BCS state is assumed to be adequate for the description of finite nuclei. 4 In neutron stars, superfluidity occurs when the star cools down below5 Tc « 1 — 2 MeV. The critical temperature Tc for two-particle pairing as a function of the respective chemical potentials HN of protons N = p and neutrons N = n, is obtained from the pole of the two-particle T-matrix (Thouless criterion). If this pole coincides with the two-particle chemical potential, there arises a singularity in the density. 6 The Gor'kov equation, which determines the superfluid phase transition temperature, is obtained in the mean-field approximation by considering the two-particle Green function in ladder approximation and using a factorization near the singularity. One finds [£ H F (1) + £ H F ( 2 ) -

m

- /i2] V(12) + £ [ 1 - / ( l ) - f(2)]V(U, 1'2')V(1'2') = 0, 1'2'

(9.1) where the single-particle nucleon state {1} = {pi,0"i,Ti} denotes linear momentum, spin, and species (isospin). In mean-field approximation, the influence of the medium is represented by the Hartree-Fock (HF) self-energy shift £ H F ( 1 ) = p 2 / 2 m i + £ 2 V(U, 12) ex /(2) and the Pauli blocking term, where / ( l ) = [exp(.E(l) — fJ-i)/T + l ] - 1 is the Fermi distribution function. The solution of this equation becomes especially simple in the case of a separable two-nucleon interaction potential V(12,1'2'). Separable representations of realistic interaction potentials such as the Paris interaction have been given in Ref. 2.

Four-Particle

Condensate

in Nuclear

Systems

223

The interaction in the triplet channel is stronger than that in the singlet channel, such that a bound state (the deuteron d) can be formed. Therefore it is expected that the superfluid phase transition in the triplet channel should occur in symmetric nuclear matter at relatively high temperatures. An evaluation using the Gor'kov equation 7 yields a critical temperature around 5-6 MeV. It is clear that in the zero temperature limit a large gap should exist. It should be mentioned that in Eq. (9.1), the medium contributions introduced by the Fermi distribution functions can be neglected in the low-density limit. This equation then coincides with the Schrodinger equation for the deuteron if the chemical potentials pp + pn are replaced by the energy eigenvalue. An interesting feature of the triplet-pairing case is the crossover from Bose-Einstein condensation of deuterons at low densities to BCS pairing at high densities. A similar phenomenon has been described independently in semiconductor physics. 8 In that example, the strength of the potential V was considered as variable, so that the formation of bound states (excitons) becomes possible in the case of strong attraction. In the nuclear matter case considered here, the effective strength of the interaction is changed because of Pauli blocking. In analogy to semiconductor physics, the bound states disappear with increasing density (the so-called Mott effect6'9). As the chemical potential is increased, the critical temperature Tc rises from zero to a finite value at the point where the two-particle chemical potential pp + pn coincides with the deuteron binding energy Ed = —2.225 MeV. If we choose to regard the density as the independent variable instead of the chemical potential, we need to solve the equation of state p = p(T, p) including the formation of bound states. 6 ' 8 - 10 The problem of treating the medium effects in a consistent manner is a difficult one. The Fermi distribution function refers only to quasiparticle states. However, in the medium there also occur correlated states that cannot be interpreted as quasiparticles (e.g. resonances or particles with finite lifetime); in particular, bound states may appear. All these constituents participate in phase-space occupation and Pauli blocking. As we show in the following development, it is possible to formulate a self-consistent generalization of the mean-field concept that includes correlations not encompassed by the simple mean-field description. At the same time, we will also allow for the formation of clusters of higher mass number such as a particles. Thus, we will account for the formation of a quantum condensate involving correlations of higher order in the number of fermionic constituents than the pairwise correlations that characterize BCS theory. More specifically, the BoseEinstein condensation (BEC) of a particles will be described in the low-density limit. It should be noted that until recently, the inclusion of cluster formation in the region where quantum condensates exist, i.e., for temperatures below the critical temperature T c , has remained largely an open problem in many-body theory. 11 We now proceed to outline a theory which, in important ways, goes beyond standard mean-field theories based on individual nucleonic degrees of freedom. Our approach utilizes the method of thermodynamic Green functions to treat multiparticle effects in dense matter. In addition to the single-particle Green function, we will also consider the Green functions of ^4-nucleon clusters. Most specifically,

224

G. Ropke and P. Schuck

we are concerned with the case A = 4, since a-particles are strongly bound and therefore should play an important role in low-density nuclear matter. Thus our approach will feature the concept of a-matter as a special case that has been discussed over several decades. We will first give a general treatment that applies to any cluster size A. Light clusters, with A < 4, are of current interest in experimental and theoretical studies of hot, dense hadronic matter as produced in heavy-ion collisions. For the formation of quantum condensates, the bosonic two-particle and four-particle correlations are of relevance and will be investigated below. We shall focus on the occurrence of superfluidity due to four-particle correlations rather than two-particle pairing. As is well known, the a-particle is strongly bound compared with the deuteron. Therefore, as the temperature decreases from relatively high values, the expected superfluid phase transition in symmetric nuclear matter due to triplet pairing in the deuteron channel may be preceded by the formation of a condensate in the four-particle channel before the critical temperature for the triplet pairing transition is reached. This feature of higher-cluster superfluidity is also of interest in other systems (e.g. biexcitons, two-pion states, etc.). If we include the a cluster in our considerations of the equation of state (EOS), the low-density limit of the EOS for symmetric nuclear matter will be governed by a mass-action law, with the a particles as the dominant contribution in the lowtemperature limit. We argue that the inclusion of other clusters besides a and d is not essential to an informative initial microscopic investigation of clustering effects in nuclear matter. Most of the pertinent physics can be explained without including the clusters with A=3 (triton t and helion h = 3 He), or those with 4 < A < 12 (Carbon), all of which are weakly bound compared with the a particle. On the other hand, heavier clusters starting with Carbon are of potential relevance to the equation of state. It is possible to generalize the current approach such that the a particles are combined into clusters with higher mass numbers, but this extension of the theory will not be considered here.

9.2. Cluster expansion of many-particle properties Green functions theory provides a systematic approach to the properties of manyparticle systems. Few-body correlations, and in particular the formation of bound states (clusters), are described by A-particle Green functions obeying a BetheSalpeter equation. In the low-density limit, the interaction kernel of the BetheSalpeter equation can be approximated by the bare nucleon-nucleon interaction. The sum of ladder diagrams is obtained by solving the corresponding A-particle Schrodinger equation, which yields possible bound states as well as scattering states. The bound states (both the ground state and excited states) are of relevance in the low-density, low-temperature region of the phase diagram. Within the so-called chemical picture (cf. Ref. 12), single-particle states and bound multiparticle states are treated on the same footing within a quantum statistical approach. We are interested in the equation of state of symmetric nuclear matter expressing the nucleon density p(T,/j.) as a function of temperature T = l/fcs/3 and chemical

Four-Particle

Condensate

in Nuclear

Systems

225

potential /x. In the Green-functions approach, the density of the system is given by

^M) = E / S ^ T ) T T 5 ( 1 ' W ) '

(9 2)

-

where the spectral function 5(1, u) is related to the single-nucleon self-energy E(l, to) according to 2ImE(l,a,-*0) S(lu)) = K K ' ' (w - p?/2mi - Re£(l, w)) 2 + (ImS(l, w - iO))2 ' ' ' The self-energy can be expressed through the scattering matrices via a cluster decomposition 12 illustrated in Fig. 9.1.

Fig. 9.1.

Cluster decomposition of the self-energy E ( l , w ) .

The T^-matrices are related to the A-particle Green functions by T A (1 • • • A, 1 ' . . . A',z) = VA(1... A, 1" ... A") xGA{l" ... A", 1'"... A'", z)VA{\"'... A"', I1... A') (9.4) through the potential VA{1... A, 1 ' . . . A') = ^ V{ij, i'f) Uk^j 8ktk,. When the T^-matrices are substituted into the self-energy, one must subtract out those diagrams leading to double counting. The solution of the A-particle propagator in the low-density limit is given by ^P

Z

EA n P

~

''

in terms of the eigenvalues EAtTl}p and wave functions ipA,n,p(l • • • A) of the Aparticle Schrodinger equation, where P denotes the total momentum, and the internal quantum number n covers bound as well as scattering states. The evaluation of the equation of state in the low-density limit is straightforward. Considering only the bound-state contributions, we have the result i (9.6) P{P,P) / - ^ eP{EA,n,P-Ati) _ (_1)A '

sr

A,n,P

v

'

which is an ideal mixture of components obeying Fermi or Bose statistics.

226

G. Ropke and P. Schuck

In the classical limit, the integrals over P can be carried out, and one obtains the mass-action law that determines the matter composition at given temperature and total particle density. At low densities, quantum effects become relevant. The most dramatic is Bose-Einstein condensation (BEC), which occurs for the channels with even A when EA,n,p — A/j, = 0. As the temperature is decreased from relatively high values toward zero, BEC occurs first for those clusters with maximum binding energy per nucleon. If we consider the problem of clustering in nuclear matter, the two-particle (deuteron) binding energy per nucleon is —1.11 MeV, while the four-particle (alpha) binding energy is —7 MeV. One therefore anticipates that a quantum condensate of a particles is formed first. The inclusion of scattering states 6 would slightly change the composition as well as the equation of state. An important problem is the modification of the single-particle and bound-state properties at higher nucleon densities when medium effects have to be taken into account. The change of the cluster energies due to self-energy and Pauli-bio eking contributions will be investigated in the following section. Starting from the equation of state p(f3, p.), all thermodynamic properties can be calculated after evaluating the thermodynamic potentials f(T, p) = ff /u(/3, p')dp' or p(T, p) = J^ p((3, p')dp'. Thus, the technique of cluster expansion can be used for the evaluation of any thermodynamic property. It should be mentioned that cluster expansions can also be developed for other quantities such as the polarization function.12

9.3. Cluster mean-field approximation With increasing density, the role of the medium in modifying the properties of nuclear matter becomes increasingly important. The self-energy of an A-particle cluster can in principal be deduced from contributions describing the single-particle self-energies as well as medium modifications of the interaction and the vertices. A guiding principle in incorporating medium effects is the construction of consistent ("conserving") approximations, which treat medium corrections in the self-energy and in the interaction vertex at the same level of accuracy. This can be achieved in a systematic way using the Green functions formalism. At the mean-field level, we have only the Hartree-Fock self-energy A H F = YI2 V{12,12)ex/(2) together with the Pauli blocking factors, which modify the interaction from V{12,1'2') to V(12,1'2')[1 — / ( l ) — /(2)]. The resulting effective wave equation, which includes the mean-field corrections for A = 2 clusters, reads [£ H F (1) + £ H F ( 2 ) - £ 2 , n , P ] V2,n,p(12) + Y, [1 " / ( I ) " /(2)] ^(12, l , 2 ' ) ^ , n , p ( l , 2 ' ) = 0.

(9.7)

1'2'

In this framework, the Gor'kov equation (9.1) is reproduced when the binding energy Ed,p=o coincides with 2p. Similar equations have been derived from the Green-functions approach for A = 3 and A = 4, describing triton/helion nuclei as well as a-particles in nuclear matter. The effective wave equation contains in mean-field approximation the Hartree-Fock

Four-Particle

1

u

^J -\*

Condensate in Nuclear

1

.r

T

1

1

1

.* *

/

i~-n

£-10 -

P

m

TT—l

1— 1

l

y

.•**

%

111

Systems

/ 'y

/

/

/

/

y

/

/ y

/ y

y

y

y

y

s*

'

y

•

— -

yx y

_

yy

S-20

—

P3

p0/10

QA

1

0

L

.

•

•

y

y

•

i

-

1

1

1 V 1

1

1

1

.

.

1

.

0.02 0.03 0.01 nuclear density p[fm ]

Fig. 9.2. Shift of binding energy of the light clusters (d - dash dotted, t / h - dotted, and a dashed: perturbation theory, full line: non-perturbative Faddeev-Yakubovski equation) in symmetric nuclear matter as a function of density for given temperature T = 10 MeV.

self-energy shift of the single-particle energies as well as the Pauli blocking of the interaction. The effective wave equation for A = 4 is [£ H F (1) + £ H F ( 2 ) + £ H F ( 3 ) + E H F (4) - £ 4 , n , P ] V4,„,p(12)

+E

E

%<j 1 ' 2 ' 3 ' 4 '

t1 - /(*) ~ fUW{ij,i'j')

I I Sk,k>1>4,n,pW3'4') = 0• (9.8) k^i,j

The effective wave equation has been solved using separable potentials (i) for A — 1 by integration and (ii) for A = 3,4 by means of the Faddeev approach. 13 The shifts of binding energy can also be calculated approximately via perturbation theory. In Fig. 9.2 we show the shift of the binding energy of the light clusters (d, t/h, and a) in symmetric nuclear matter as a function of density for temperature T = 10 MeV. It is found that the cluster binding energy decreases with increasing density. Finally, at the Mott density PA°™P{T) the bound state is dissolved. The clusters are not present at higher densities, having been absorbed into the nucleonic medium. For a given cluster type characterized by A, n, we can also introduce the Mott momentum f^J° tt (p, T) in terms of the ambient temperature T and nucleon density p, such that the bound states exist only for P > Pjf° u (p,T). Up to this point the medium effects, which modify the single-particle and cluster states in dense nuclear matter, have been described at the mean-field level. However, we have included some correlations in the description of the medium so that already on the mean-field level we are able to account for cluster formation. The cluster

228

G. Rdpke and P. Schuck

mean-field approach presented here requires a self-consistent solution which is not currently within reach. While this approach is still based on the mean-field concept, terms of higher than first order in the interaction between clusters (A\, AQ) are taken into account in the corresponding combined cluster [A\ + A2). Within the Green-functions approach, the self-consistent random-phase approximation (RPA) has been formulated 14 for cluster states a with creation operator Aj,, where a is specified by the number of particles in the cluster, its internal quantum state, and its center-of-mass momentum. The equation of motion for the cluster Green function G^' = (TAa(t),A^(t')) reads ih

§-tG«7' = *(* ~ 0U4*. 4l±> + E fdttTC^p1', 7

J

(9.9)

where the quantity

7

+ nra;-t'

= n%s(t-t')

(9.10)

in the driving term contains an instantaneous part "HQaa5{t — t') and a dynamical part 'H'^p~ • The instantaneous part can be interpreted as an effective Hamiltonian, and the corresponding Schrodinger equation provides the means to introduce an optimal basis of eigenstates. The dynamical part will be neglected in the current development. A similar, or parallel, approach can be formulated by specifying the Feynman diagrams that are taken into account when treating A-particle cluster propagation. ls The corresponding A-particle cluster self-energy is treated to first order in the interaction with the single particles as well as with the B-particle cluster states in the medium, but with full anti-symmetrization between both clusters A and B. For the A-particle problem, the effective wave equation reads [E{1) + ... E{A) - EAnP}4>AnP(l

+ E

...A)

E^(l--.Al'...^)^np(l'...^)

V...A' i<j

+ E

y

™ H F U ...A,l'...

A')AnP{l' ...A')

= 0,

(9.11)

V...A'

with V£(1...A,1'...A') = V(12,1'2')<533' • • • 6AA,. The effective potential Kim HF (l • • -A,l'... A') describes the influence of the nuclear medium on the cluster bound states and has the form

V^HF(l...A,l'...A')

= J2M^iv...SAAI+Y^AV^(l...A,l'...A'),

(9.12)

Four-Particle Condensate in Nuclear Systems

229

with

AV£(l...Aa\..A')

= -n(f(l)

+ f(l'W(12,l'2')

+

B

oo

fB{EBmP)^VyB0{l2'...B\l'2...B)

+E E E E B=2mP2...B2'...B'

i

x ^ m P ( 2 2 . . . B)BmP{2'2'... B1) L 3 ' • • • 5AA,,

(9.13)

• • B)\2 ,

(9.14)

/ ( l ) = A(1) + E

E

E

fB(EBmp)\4>Bmp(l.

B=2mp2...B oo

A(1) = E W 2 . « ) - E E E £ fB(EBmp) 2 m

B=2mp2...Bl'...B'

x J2 Vg(l ..-B,l'...

B')cj>BrnP{l ... B)4>BmP(l'

...B'),

(9.15)

where fA(E)

= [exp/?(£ - ^/i) - ( - 1 ) * ] _ 1 .

(9.16) VrnJ^;HP

We note that within the HF approximation, the effective potential remains energy independent, i.e. instantaneous. The quantity / ( l ) describes the effective occupation of state 1 due to free and bound states, while exchange is included by the additional terms in A V ^ and A(l), thus accounting for antisymmetrization. Of course, fully self-consistent solution for a cluster embedded in a clustered medium is an extremely demanding objective, which has not yet been achieved. Starting from an uncorrected medium, the first step of iteration has been attempted, including only some bound states, but neglecting all scattering states. The cluster mean field may be viewed as a generalization of the ordinary mean field, where in addition to the mean field produced by the single-particle states, the mean field produced by clusters (bound states) is also taken into account. The modification of bound-state energies as well as wave functions can be evaluated in this approximation. We obtain an optimized set of states which may be of use in evaluating self-energies and spectral functions in a consistent manner, as a prerequisite to evaluating correlation functions and thermodynamic relations, Referring back to the simple approximation sketched above in which the medium is considered as uncorrelated, only the medium terms with / i ( l ) survive. All the higher-cluster distribution functions fs are neglected, but / i ( l ) now denotes the Fermi distribution function for which the effective chemical potential is determined so as to reproduce the total nucleon density. Clearly, this approach is most appropriate in regions of the phase diagram where the contribution of clusters to the total density is small, i.e. at high temperatures and low densities. This approximation is also suitable at densities above the Mott density where the correlations have been destroyed due to the Pauli blocking.

230

G. Ropke and P.

Schuck

A simple special case of the full cluster mean-field approximation is that in which only two-particle bound states are included. The self-consistent system of equations then becomes more tractable, such that solutions can be obtained at least in some limiting situations. As mentioned above, the Bose distribution function of even-A clusters can become singular at low densities, indicating a transition to a quantum condensate. This scenario also holds for the cluster mean-field approximation. However, the inclusion of cluster formation becomes more difficult below the transition temperature. It has been customary to employ a mean-field approximation such as standard BCS theory, and only initial attempts have been made to go beyond a mean-field description by incorporating suitable correlations. 11

9.4. Two-particle condensates at low temperatures Starting from the solution of the in-medium few-body problem, the self-energy of the single-nucleon propagator can be calculated in terms of the T-matrices corresponding to the few-body systems in play. Making use of the expressions for the self-energy, the equation of state in the form (9.2) can be evaluated:

2ImE(l,w-iO)

" ) ,(w . . - pj/2m ^ ^ : . \ T a ~2 + . 7(ImS(l,o; L w , .. - ^iO)) 2 2- DReS(l,w))

(9-17)

For small ImE(l,w), there will be a contribution from the quasiparticle peak at ui—p\/2m — ReE(l,to) = 0. In addition, contributions arise from the bound cluster states. This approach was followed in Ref. 6 under the restriction to two-particle contributions (through the T2-matrix), but otherwise implementing a full treatment including scattering states. In this way a generalized Beth-Uhlenbeck formula was derived, namely p(P,P)

= Pqp(P,P)

+ PCOTI(P,P)

= Pqp(P'P)

+ Abound(P,fl)

+ P s c a t t (P, P)

(9.18)

with 1

Pbound(P,P) =

^2

f2(En,p),

n,P>PMott

PscMP, P) = ^2 / — - ^ ( 4 ^ + E) ~dE 62nP(E^ ~ sm52np(E) npJ

T

m

cosS2np(E)

. (9.19)

_1

The distribution functions appearing here are fi (E) = [exp (3(E — fi) +1] (Fermi) and fo(E) = [exp f3{E — 2^i) — 1 ] _ 1 (Bose). The quasiparticle energy is determined from £ q u a s i ( l ) = p\/2m + ReS(l (i))-

Four-Particle

Condensate

in Nuclear Systems

231

Fig. 9.3. Phase diagram of symmetric nuclear matter showing lines of equal concentration p C orr/p of correlated nucleons, the Mott line, and the critical temperature Tc of the onset of superfluidity.

Evaluation of the Beth-Uhlenbeck formula including two-particle correlations has been carried out in Ref. 6 based on a separable nucleon-nucleon potential. The result 17 for the composition of nuclear matter as function of density and temperature is shown in Fig. 9.3. Two aspects of this study of two-particle condensation deserve special attention. (i) The contribution of the correlated density, which derives both from deuterons as bound states in the isospin-singlet channel and from scattering states, is found to increase with decreasing temperature, in accordance with the law of mass action. This law also predicts the increase of correlated density with increasing nucleon density (as also seen in Fig. 9.3 for the low-density limit). However, under increasing density, the binding energy of the bound state (deuteron) decreases due to Pauli blocking (Mott effect). At the Mott density, introduced above, the bound states with vanishing center-of-mass (c.o.m.) momentum are dissolved in the continuum of scattering states. Bound states with higher c.o.m. momentum merge with the continuum at higher densities. According to Levinson's theorem, if a bound state merges with the continuum, the scattering phase shift in the corresponding channel jumps by an angle ir, such that no discontinuity appears in the equation of state. Accordingly, the contribution of the correlated density will remain finite at the Mott density, but will be strongly reduced at somewhat higher densities.

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G. Ropke and P. Schuck

Thus, one salient result is the disappearance of bound states and correlated density already below the saturation density of nuclear matter. The underlying cause of the Mott effect is Pauli blocking, which prohibits the formation of bound states if the phase space is already occupied by the medium (Fermi sphere), and hence no longer available for the formation of the wave function of the bound state (momentum space). This effect holds also for higher-A bound states such as the triton, helion, and a particle, which disappear at corresponding densities (see Fig. 9.2). (ii) The Bose pole in the correlated density signals the onset of a quantum condensate. As is well known, for the bound-state (deuteron channel) contribution, the T-matrix approach breaks down when the pole corresponding to the bound-state energy coincides with twice the chemical potential. This is the Thouless condition, embodied in T ( l , 2,1", 2"; 2M) = 2^E{l^-E{2)

£

y(12

' i ' 2 ' ) ^ 1 ' ' 2 ' ' l"> 2">2") •

( 9 - 2 °)

The same condition also holds for the contribution of scattering states. Consequently, the transition temperature for the onset of a quantum condensate appears as a smooth function of density, as shown in Fig. 9.4. Below the transition temperature, the T-matrix approach is no longer applicable. However, a mean-field approach becomes possible in this regime after performing a Bogoliubov transformation. Even so, the proper inclusion of correlations below the critical temperature remains a challenging problem. To date, only the first steps have been taken 11 toward solving this problem for general quantum many-particle systems. Another important aspect of the problem of two-particle condensation, indeed one of great current interest, is the interpretation of the critical temperature. At low densities, where the two-body bound states (deuterons) are well-defined composite particles, the mass-action law implies that the deuterons will dominate the composition in the low-temperature region. In this region, the critical temperature for the transition to the quantum condensate coincides with the Bose-Einstein condensation of deuterons as known for ideal Bose systems. At high densities, where bound states are absent, the transition temperature coincides with the solution of the Gor'kov equation describing the formation of Cooper pairs. Thus, BEC and BCS scenarios characterize the low- and high-density regimes, respectively. We observe a smooth crossover transition from BEC to BCS behavior - as predicted generically for fermion systems in Ref. 8 (for example) and now the subject of intense experimental study in cold atomic gases (see Chapters 10 and 11). Already at the mean-field level, the calculation of the transition temperature should include the quasiparticle shifts in the Hartree-Fock approximation. In general, the effective-mass approximation in nuclear matter will reduce the transition temperature. 10 A noteworthy case is the reduction of the transition temperature in the isospin-singlet channel for asymmetric matter. However, we shall not address that example explicitly, nor shall we discuss further effects that can be described in mean-field approximation, such as the shift and/or deformation of the Fermi surfaces18 and the LOFF phases, which are considered in Chapter 6. Going beyond the mean-field approximation, the first remarkable feature 19 emerging at the two-particle level is the formation of a pseudogap in the density of

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233

Fig. 9.4. Onset of quantum condensate in the two-particle channel in symmetric nuclear matter. Crossover from BEC to BCS occurs as the density increases.

states (DOS) above the critical temperature Tc. Compared with the orthodox BCS solution, for which a gap opens in the DOS below Tc, a quite different situation is present in strongly correlated Fermi systems. The full treatment of the (two-body) T-matrix leads to a reduction of the DOS near the Fermi energy already above Tc, within an energy interval of the same order as the BCS gap at zero temperature. This behavior may be traced to fluctuations above Tc that presage the transition to the superfluid state. Similar precursor behavior is known to occur in other systems of strongly correlated fermions. In the Hubbard model, for example, the formation of local magnetic moments already begins above the critical temperature at which long-range order of the moments becomes manifest. The pseudogap phenomenon is of course a widely discussed aspect of compounds exhibiting high Tc superconductivity. 20 In the context of nuclear matter, the occurrence of a pseudogap phase was first considered by Schnell et al. 19 in the quasiparticle approximation, as noted above. They showed that this effect is partially washed out if a self-consistent approximation for the spectral function is implemented, but a full description should take vertex corrections into account. A similar assessment applies to a recent self-

234

G. Ropke and P. Schuck

consistent solution of the Gor'kov equation in terms of the spectral function, 21 which shows a reduction of the transition temperature for quantum condensation (see also Chapter 7). However, vertex corrections should also be included in this case and may partially compensate the self-energy effects.

9.5. Four-particle condensates and quartetting in nuclear matter In general, it is necessary to take account of all bosonic clusters to gain a complete picture of the onset of superfluidity. The picture developed in the preceding section only includes the effects of two-particle correlations leading to two-body deuteron clusters. However, as is well known, the deuteron is weakly bound compared to other nuclei. Higher-A clusters can arise that are more stable. In this section, we will consider the formation of a particles, which are of special importance because of their large binding energy per nucleon (7 MeV). We will not include tritons or helions, which are fermions and not so tightly bound. Moreover, we will not consider nuclei in the iron region, which have even larger binding energy per nucleon than the a and thus comprise the dominant component at low temperatures and densities. The latter are complex structures of many particles and are strongly affected by the medium as the density increases. The in-medium wave equation for the four-nucleon problem has been solved using the Faddeev-Yakubovski technique, with the inclusion of Pauli blocking. The binding energy of an a-like cluster with zero c.o.m. momentum vanishes at around po/10, where po — 0.16 nucleons/fm 3 denotes the saturation density of isospin-symmetric nuclear matter. Thus, the four-body bound states make no significant contribution to the composition of the system above this density. Given the medium-modified bound-state energy .E4,p, the bound-state contribution to the EOS is

n4(/3, M) = J2 [e^'"- 2 ^- 2 ^) - l] _1 .

(9.21)

p

We will not include the contribution of the excited states or that of scattering states. Because of the large specific binding energy of the a particle, low-density nuclear matter is predominantly composed of a particles. This observation underlies the concept of a matter and its relevance to diverse nuclear phenomena. Symmetric nuclear matter is characterized by equality of the proton and neutron chemical potentials, i.e., fip = fin — \x. The four-particle correlations are embodied in the four-particle Green function, which in the ladder approximation is given by G 4 ( 1234,l'2'3'4',n 4 ) = ^ W W W ) / 4 (£ 4 (1234)) +

Yl

* n ^ * S 3 ^ ^4 - £4(1234)

K 4 (1234,l"2"3"4",fi 4 )G 4 (l"2"3"4",l'2'3'4',fi 4 ),

(9.22)

1"2"3"4"

where £ 4 (1234) = E(l)+E(2)+E(3)+E(4) and / 4 is the Bose distribution function. The interaction kernel K is obtained using the technique of the Matsubara Green

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235

functions, which yields K4(1234,l'2'3'4',fi4) = V(12, l ' 2 > 3 3 ^

4

, ^ | ^ p ^

+ ... + ... + ... + ... + ...,

(9.23)

where the terms obtained by relabeling are not shown explicitly. In the zero-density limit p - » 0 where the Fermi distribution function is small compared with unity, this becomes simply ltoK4(1234,m-4-,n4) . ^

^

Y.Vm-m,

(UO

in terms of the bare interaction term of the four-body system. The poles of the analytic continuation of G 4 (1234, 1'2'3'4',-z) into the complex z plane are of special interest. Near the pole at .E4,„, the Green function can be factorized as G 4 (1234,1'2'3'4', z) « V'4,i/(1234)V'I,1,(i'2'3/4')/(;z - £ 4 ,„). In this expression, E^>v and •04,1/(1234) are the eigenvalues and the eigenstates of the fourparticle system, which follow from the solution of the four-particle Schrodinger-like wave equation V>4,i/(1234) =

Y^

K 4 (1234,1'2'3'4',£ , 4 , 1 ,)'!V(1 , 2'3'4').

(9.25)

1'2'3'4'

Importantly, bound states can exist, and we denote the lowest bound state by £4,0, which in the nuclear context is the a particle. Due to the influence of the medium as reflected in the self-energy and the phase-space occupation factors, the bound-state energy depends on the temperature T and chemical potential /x. The four-particle correlations as contained in the four-particle Green function G 4 serve to determine the thermodynamic properties of the system, including the equation of state. For example, the four-particle density is given by I

— / 4 (w)ImG 4 (1234, l'2'3'4', w - iO).

(9.26) / Obviously, this density diverges when, at a given temperature, the chemical potential attains the value fi = £ 4 ,o. At that point, the delta function produced by the pole coincides with the singularity of the Bose distribution function. This singularity is directly related to the onset of superfluidity;22 for example, at low densities it will lead to Bose condensation of a particles. In general, the condition for the onset of superfluidity due to four-particle correlations follows from the equation ^ 4 (1234)=

Yl

K 4 (1234,1'2'3'4',4^)V<4(1'2'3'4'),

(9.27)

1'2'3'4'

which determines the critical temperature Tc(fic) . It should be noted that analogous arguments are used to determine the onset of pairing by considering the behavior of the two-particle propagator. 7 When continued into the complex z plane, the two-particle propagator will feature poles corresponding to two-particle bound states and a cut along the real axis

G. Ropke and P. Schuck

236

T

'

1

'

1

Ttc (deuteron pairing) T4C (a-quartetting)

.

-20

i

i

0

i

i

.

20 \i [MeV]

i

40

.

i

60

Fig. 9.5. Transition temperature to quartetting/pairing as a function of chemical potential in symmetric nuclear matter.

corresponding to two-particle scattering states. If, however, the analytic continuation of the four-particle propagator G4 exhibits a separate pole below the cut, its pole dominates the onset of the anomalous phase. An important consequence is that at the lowest temperatures, Bose-Einstein condensation occurs for a particles rather than deuterons. As the density increases within the low-temperature regime, the chemical potential fi first reaches —7 MeV, where the a's Bose-condense. By contrast, Bose condensation of deuterons would not occur until p, rises to —1.1 MeV As quantified by Eq. (9.8), the effect of the medium on the properties of an a particle in mean-field approximation (i.e., for an uncorrelated medium) is produced by the Hartree-Fock self-energy shift and Pauli blocking. The shift of the a-like bound state has been calculated using perturbation theory 9 as well as by solution of the Faddeev-Yakubovski equation. 13 It is found that this bound state merges with the continuum of scattering states at a Mott density p™ott « po/10. The bound states of clusters d, t, and h with A < 4 are already dissolved at the density p^ott. Consequently, if we neglect the contribution of the four-particle scattering phase shifts in the different channels, we can now construct an equation of state p(T, /i) - p free (T, p) + p*™"*.*(T, fj) + p»>ound,a(T) M) such that

Four-Particle

10

Condensate in Nuclear

1

1 —

1

Ttc (deuteron pairing) T4C (a-quartetting)

8 -

>

237

Systems

6

CD

V

0 10"

1

1

10" P [fm"1

10"

\ \ \ \ \ \ \ \ \ \ \ \

>

10"

10u

Fig. 9.6. Transition temperature to quartetting/pairing as a function of nucleon density in symmetric nuclear matter.

a particles determine the behavior of symmetric nuclear matter at densities below pMott a n ( j temperatures below the binding energy per nucleon of the a particle. The formation of deuteron clusters alone, as considered in the Section 9.4, gives an incorrect description because the deuteron binding energy is small, and the abundance of d clusters is small compared with that of a clusters. In this region of the phase diagram, a matter emerges as an adequate model for describing the nuclear-matter equation of state. With increasing density, the medium modifications - especially Pauli blocking - will lead to a deviation of the critical temperature Tc(p) from that of an ideal Bose gas of a particles. (The analogous situation holds for deuteron clusters, i.e., in the isospin-singlet channel.) An extended Thouless condition based on the relation T 4 (1234,l"2"3"4",4/i) =

E 1'2'3'4'

V(12,l'2')[l-/(l)-/(2)] «5(3,3')<5(4,4') 4/i — Ei — E2 — E3 — E4 -l-cycl. f T 4 (l'2'3'4', 1"2"3"4",4/i)

(9.28)

serves to determine the onset of Bose condensation of a-like clusters, noting that existence of a solution of this relation signals a divergence of the four-particle

238

G. Ropke and P. Schuck

correlation function. An approximate solution has been obtained by a variational approach, in which the wave function is taken as Gaussian incorporating the correct solution for the two-particle problem. The results are presented in Figs. 9.5 and 9.6. The calculation reveals that in the low-density region, the critical density tracks that for Bose-Einstein condensation of ideal a particles; hence the Bose condensation of deuterons as considered in the Section 9.4 becomes irrelevant. As expected, with increasing density the transition temperature is depressed from that of the ideal Bose gas of a's due to medium corrections. Moreover, the "quartetting" transition temperature is sharply reduced as the rising density approaches the critical Mott value at which the four-body bound states disappear. At that point, pair formation in the isospin-singlet deuteron-like channel comes into play, and a deuteron condensate will exist below the critical temperature for BCS pairing up to densities above the nuclear-matter saturation density p0, as described in the Section 9.4. The critical density at which the a condensate disappears is estimated to be po/3. However, the variational approach of Ref. 22 on which this estimate is based represents only a first attempt at description of the transition from quartetting to pairing. The detailed nature of this fascinating transition remains to be clarified. Many different questions arise in relation to the possible physical occurrence and experimental manifestations of quartetting: Can we observe the hypothesized "a condensate" in nature? What about thermodynamic stability? What happens with quartetting in asymmetric nuclear matter? Are more complex quantum condensates possible? What is their relevance for finite nuclei? As discussed below, the special type of microscopic quantum correlation associated with quartetting may be important in nuclei, its role in these finite inhomogeneous systems being similar to that of pairing.

9.6. Alpha-particle condensate states in light nuclei What about a-particle condensation in finite nuclei? The only nucleus having a pronounced a-cluster structure in its ground state is 8 Be. In Fig. 9.7(a), we show the result of an exact calculation of the density distribution of 8 Be in the laboratory frame. In Fig. 9.7(b) we show, for comparison, the result of the same calculation in the intrinsic, deformed frame, asking in addition where to find the second a-particle when the first is placed at a given position. We see that the two a's are ~ 4 fm apart, giving rise to a very low average density p ~ /Oo/3 as seen in Fig. 9.7(a). Clearly, 8 Be is a rather unusual nucleus. Nuclear systematics would predict for it an rms radius R = roA1/2 w 2.44 fm, but the experimentally measured value is some 3.7 fm, consistent with the calculated low average density. One may ask the question what happens when one brings a third a-particle alongside the 8 Be nucleus. We know the answer: the 3-a system collapses to the ground state of 12 C, which is much denser than 8 Be. Within its small rms radius of 2.4 fm, the ground state of 12 C cannot accommodate three a-particles in a configuration such that they are barely touching one another, as in the case of 8 Be. One may nevertheless persist and ask the question whether the dilute three-a configuration of 8 Be-a may

Four-Particle

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239

2-

-2

I -

. . . I . . . 4 - 2

I i . . 0 2

I . . . I . . . 4 2

I . .

. -CT> • , i I . • . 0 2 4

r=(x2+y2)W (fm)

Fig. 9.7. Contours of constant density (taken from Ref. 23), plotted in cylindrical coordinates, for 8 B e ( 0 + ) . The left side (a) is in the "laboratory" frame while the right side (b) is in the intrinsic frame.

or may not form an isomeric or excited state of 1 2 C. Answering this question and exploring related issues of quartetting in finite nuclei will consume the rest of the present chapter. In fact, we will be able to offer strong arguments that the Oj state of 12 C at 7.654 MeV is a state of a-condensate nature. First, it should be understood that the O.^ state in 12 C is in fact hadronically unstable (as 8 Be), being situated about 300 keV above the three a-break up threshold. This state is stabilized only by the Coulomb barrier. It has a width of 8.7 eV and a corresponding lifetime of 7.6 x 10~ 17 s. As well known, this state is of paramount astrophysical (and biological!) importance due to its role in the creation of 12 C in stellar nucleosynthesis. Its existence was predicted in 1953 by the astrophysicist Fred Hoyle;24 his prediction was confirmed experimentally a few years later by Willy Fowler and coworkers at Caltech. 25 It is also well known that this Hoyle state, as it is now called, is a notoriously difficult state for any nuclear theory to explain. For example, the most modern no-core shell-model calculations predict the Oj state in 12 C to lie at around 17 MeV above the ground state - more than twice the actual value. 26 This fact alone tells us that the Hoyle state must have a very unusual structure. It is easy to understand that, should it indeed have the proposed loosely bound three a-particle structure, a shell-model type of calculation would have great difficulties in reproducing its properties.

240

G. Ropke and P. Schuck

An important development bearing on this issue took place some thirty years ago. Two Japanese physicists, M. Kamimura 27 and K. Uegaki, 28 along with their collaborators, almost simultaneously reproduced the Hoyle state from a microscopic theory. They employed a twelve-nucleon wave function together with a Hamiltonian containing an effective nucleon-nucleon interaction. At that time, their work did not attract the attention it deserved; the true importance of their achievement has been appreciated only recently. The two groups started from practically the same ansatz for the 12 C wave function, which has the following three a-cluster structure: (ri...f 12 | 12 C) = A [x&Wifafa]

•

(9-29)

In this expression, the operator A imposes antisymmetry in the nucleonic degrees of freedom and i, with i = 1, 2,3 for the three a's, is an intrinsic a-particle wave function of prescribed Gaussian form, <j>(n,?2,r3,r4) = exp { [(r\ - f 2 ) 2 + (fi - r 3 ) 2 + •• •] /b2} ,

(9.30)

where the size parameter b is adjusted to fit the rms of the free a-particle, and x(7?-, s) is a yet-to-be determined three-body wave function for the c.o.m. motion of the three a's, their corresponding Jakobi coordinates being denoted by K and s. The unknown function x w a s determined via the Generator Coordinate Method 28 (GCM) and the Resonating Group Method 27 (RGM) calculations using the Volkov I and Volkov II nucleon-nucleon forces, which fit a-a phase shifts. The precise solution of this complicated three body problem, carried out three decades ago, was truly a pioneering achievement, with results fulfilling expectations. The position of the Hoyle state, as well as other properties including the inelastic form factor and transition probability, successfully reproduced the experimental data. Other states of 12 C below and around the energy of the Hoyle state were also successfully described. Moreover, it was already recognized that the three a's in the Hoyle state form sort of gas-like state. In fact, this feature had previously been noted by H. Horiuchi 29 prior to the appearance of Refs. 27, 28, based on results from the orthogonality condition model (OCM). 30 All three Japanese research groups concluded from their studies that the linear-chain state of three a-particles, postulated by Morinaga many years earlier, 31 had to be rejected. Although the evidence for interpreting the Hoyle state in terms of an a gas was stressed in the cited papers from the late 1970's, two important aspects of the situation were missed at that time. First, because the three a's move in identical Swave orbits, one is dealing with an a-condensate state, albeit not in the macroscopic sense. The second point is that the complicated three-body wave function can be replaced by a structurally and conceptually very simple microscopic three-a wave function of the condensate type, which has practically 100 percent overlap with the previously constructed ones. 32 We now describe this condensate wave function. We start by examining the BCS wave function of ordinary fermion pairing, obtained by projecting the familiar BCS ground-state ansatz onto an A^-particle subspace of Fock space (see Chapter 6). In the position representation, this wave function is (ri...f w |BCS) = AWl,r2)cf>{rz,ri)..4{fN^rN))

,

(9.31)

Four-Particle

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241

where , with one such function for each distinct pair in the reference partition of {1, 2 , . . . , N — 1, N}. Formally, it is now a simple matter to generalize (9.31) to a-particle condensation. We write ( f i , . . . , r/v|$„ Q ) = A [a(fi, r 2 , r 3 , r4)(j>a(rs,...,r8)

• • • a(nv-3, • • •, rN)] , (9.32)

where cf>a is the wave function common to all condensed a-particles. Of course, finding the variational solution for this function is, in general, extraordinarily more complicated than finding the Cooper pair-wave function <> / of Eq. (9.31). Even so, in the present case that the a-particle is the four-body cluster involved, and for applications to relatively light nuclei, the complexity of the problem can be reduced dramatically. This possibility stems from the fact, already known to the authors of Refs. 27, 28, that an excellent variational ansatz for the intrinsic wave function of the a-particle is provided [as in Eq. (9.30)], by a Gaussian form with only the size parameter b to be determined. In addition - and here resides the essential and crucial novelty of our wave function - even the center-of-mass motion of the system of a-particles can be described very well by a Gaussian wave function with, this time, a size parameter B 3> b to account for the motion over the nuclear space. We therefore write 0a(ri,r2,f3,f 4 ) = e-n2'B24>{n

- f2,n

- f3>...),

(9-33)

where R = (fi + r(?i — F2,...) is the same intrinsic a-particle wave function of Gaussian form as already used in Refs. 27,28 and given explicitly in Eq. (9.30). Naturally, in Eq. (9.32) the center of mass Xcm of the three a's, i.e., of the whole nucleus, should be eliminated; this is easily achieved by replacing R by R — Xcm in each of the a wave functions in Eq. (9.32). The a-particle condensate wave function specified by Eqs. (9.32) and (9.33), proposed in Ref. 33 and henceforth called the THSR wave function, now depends on only two parameters, B and b. The expectation value of an assumed microscopic Hamiltonian H, H{B, b)

\vna|Tna/

,

(9.34)

can be evaluated, and the corresponding two-dimensional energy surface can be quantized using the two parameters B and b as Hill-Wheeler coordinates. Before presenting the results, let us discuss the THSR wave function in somewhat more detail. This innocuous-looking variational ansatz, namely Eq. (9.32) together with Eq. (9.33), is actually more subtle than it might at first appear. One should realize that two limits are incorporated exactly. One is obtained by choosing B = b, for which Eq. (9.32) reduces to a standard Slater determinant with harmonicoscillator single-nucleon wave functions, leaving the oscillator length b as the single adjustable parameter. This holds because the right-hand-side of expression (9.33),

242

G. Ropke and P. Schuck

with B = b, becomes a product of four identical Gaussians, and the antisymmetrization creates all the necessary P, D, etc. harmonic oscillator wave functions automatically. 33 On the other hand, when B ~^>b, the density of a-particles is very low, and in the limit B —> oo, the average distance between a's is so large that the antisymmetrisation between them can be neglected, i.e., the operator A in front of Eq. (9.32) becomes irrelevant and can be removed. In this limiting case, our wave function then describes an ideal gas of independent, condensed a-particles - it is a pure product state of a's! Evidently, however, in realistic cases the antisymmetrizer A cannot be neglected, and evaluation of the expectation value (9.34) becomes a nontrivial analytical task. The Hamiltonian in Eq. (9.34) was taken to be the one used in Ref. 34, which features an effective nucleon-nucleon force of the Gogny type, with parameters fitted to a-a scattering phase shifts as available about fifteen years ago. This force also leads to very reasonable properties of ordinary nuclear matter. Our theory is therefore free of any adjustable parameters. The energy landscapes Ti(B, b) for various na nuclei are interesting in themselves, 35 but for the sake of brevity they are not shown here.

9.7. Results for finite nuclei As we have made clear, the variational wave function constructed from the HillWheeler equation based on Eqs. (9.32), (9.33), and (9.34) has practically 100 percent overlap with the wave functions constructed in Refs. 27 and 28, once the same Volkov force is used. 32 It is, therefore, not astonishing that our results are very similar to theirs. For 12 C we obtain two eigenvalues: the ground state and the Hoyle state. Theoretical values for positions, rms values, and transition probabilities are given in Table 9.1 and compared to the data. Inspecting the rms radii, we see that the Hoyle state has a volume 3 to 4 larger than that of the ground state of 1 2 C. This is the primary aspect of the dilute-gas state we highlighted above. Constructing a pure-state a-particle density matrix p(R, R') from our wave function, integrating out of the total density matrix all intrinsic a-particle coordinates, and diagonalizing this reduced density matrix, we find that the corresponding 05 a-particle orbit is occupied to 70 percent by the three a-particles. 36,37 This is a huge percentage, giving vivid support to the view that the Hoyle state is an almost ideal a-particle condensate. As a point of reference, one should remember that in superfluid 4 He close to zero temperature only 8-10 percent of the particles reside in the condensate! Again by way of contrast, we should also mention that in the ground state of 1 2 C, the a-particle occupation is about equally shared between the OS, 0D, and 0G orbits, clearly invalidating a condensate picture of the ground state. (It is important to note that the ground-state energy of 12 C is also reasonably reproduced by our theory.) In our result, the Hoyle-state wave function is obtained by solving the HillWheeler equation based on Eq. (9.32). The experimental values are taken from Ref. 38. Let us now discuss what to our mind is the most convincing evidence that our description of the Hoyle state is the correct one. Like the authors of Ref. 27, we

Four-Particle

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243

Systems

Table 9.1. Comparison of the binding energies, rms radii (iir.m.a.), and monopole matrix elements (M(02~ —> 0+)) for 1 2 C given by solving Hill-Wheeler equation based on Eq. (9.32) and by Ref. 27. T h e effective two-nucleon force Volkov No. 2 was adopted in the two cases for which the 3a threshold energy is calculated to be —82.04 MeV.

E(MeV)

o2+

fir.m.s.(fm)

M(0+ — 0+)(fm 2 )

condensate w.f. (Hill-Wheeler)

RGM27

Exp.

-89.52

-89.4

-92.2

-81.79

-81.7

-84.6 2.44

2.40

2.40

3.83

3.47

6.45

6.7

5.4

IO -2 IO -3

io- 4 <2_10~5

^io-6

io-7 IO -8

io-9 0

5 10 qi [fm-2]

15

Fig. 9.8. Experimental values of inelastic form factor in 1 2 C to the Hoyle state are compared with our values and those given by Kamimura et al. in Ref. 27 (RGM).

reproduce very accurately the inelastic form factor 0+ —> Oj of 1 2 C, as shown in Fig. 9.8. As such, the agreement with experiment is already quite impressive. Additionally, however, the following study was made, results from which are presented in Fig. 9.9. We artificially varied the extension of the Hoyle state and examined the influence on the form factor. It was found that the overall shape of the form factor shows little variation, for example in the position of the minimum. On the other hand, we found a strong dependence of the absolute magnitude of the form factor;

244

G. Ropke and P. Schuck

0 I

1

1

-0.2

0

0.2

1-

0.4

8 Fig. 9.9. The ratio of the value of the maximum height, theory versus experiment, of the inelastic form factor, i.e. m a x | F ( 9 ) | 2 / m a x | F ( q ) | 2 x p , is plotted as a function of <5 = (RT.m.s. — Ro)/Ro- Here •Rr.m.s. and Ro are the rms radii corresponding, respectively, to the wave function of Eq. (9.32) and that obtained by solving the Hill-Wheeler equation based on Eq. (9.32).

Fig. 9.9 illustrates this behavior with a plot the variation of the height of the first maximum of the inelastic form factor as a function of the percentage change of the rms radius of the Hoyle state. 39 It can be seen that a 20 percent increase of the rms radius produces a remarkable decrease of the maximum - by a factor of two! This strong sensitivity of the magnitude of the form factor to the size of the Hoyle state enhances our firm belief that the agreement with the actual measurement is tantamount to a proof that the calculated wide extension of the Hoyle state corresponds to reality. We thus advocate and support the view that the Hoyle state can be regarded as the ground state of an a-particle condensate. Exciting one a-particle out of the condensate and putting it into the QD orbit reproduces the experimentally measured position of the 2 j state in 1 2 C. Without going into details, we also affirm that the width of this state is correctly reproduced. 40 It is tempting to imagine that the O3 state which - experimentally - is almost degenerate with the 2% state, is obtained by lifting one a-particle into the IS orbit. Initial theoretical studies 41 indicate that this scenario might indeed apply. However, the width of the 0^" state (~ 3 MeV) is very broad, rendering a theoretical treatment rather delicate. Further investigations are necessary to validate or reject this picture. At any rate, it would be quite satisfying if the triplet of states (02 + , 22 + , 0 3 + ) could all be explained from the a-particle perspective, since those three states are precisely the ones which cannot be explained within a (no core) shell-model approach. 26 Summarizing our inquiry into the possible role of a clustering in 1 2 C, we have accumulated enough facts to be convinced that the Hoyle state is, indeed, what one

Four-Particle

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245

5 0

Fig. 9.10. Alpha-particle mean-field potential for three a ' s in lower Coulomb barrier for 2 4 M g (from Ref. 36).

12

C and six a ' s in

24

Mg. Note the

may call an a-particle condensate state. At the same time, we acknowledge that referring to only three particles as a "condensate" constitutes a certain abuse of the word. However, in this regard it should be remembered that also in the case of nuclear Cooper pairing, only a few pairs are sufficient to obtain clear signatures of superfluidity in nuclei! What about a-particle condensation in heavier nuclei? Once one accepts the idea that the Hoyle state is essentially a state of three free a-particles held together only by the Coulomb barrier, it is hard to see why analogous states would not also exist in heavier n a nuclei like 1 6 0 , 20 Ne, 24 Mg, etc. In fact, our calculations on such nuclei systematically yield a 0 + -state close to the a-particle disintegration threshold. For example in 1 6 0 we obtain three 0 + -states: the ground state at £0 = —124.8 MeV (experimental value: 127.62 MeV), a second state at E0i+ = 8.8 MeV, and a third one at 0 3 + = 14.1 MeV. The threshold in 1 6 0 is at 14.4 MeV. Unfortunately, the relevant experimental information in 1 6 0 is not nearly so complete as in 1 2 C. In particular, no measurements are available for transition probabilities of 0 + -states near the threshold or for inelastic form factors. Recently Wakasa 43 identified a new 0 + -state at 13.5 MeV in 1 6 0 , which is the 5th measured 0 + -state below threshold. Actually there exists another one at 14.03 MeV, so that at present six 0 + -states up to threshold have been identified in 1 6 0 . In contrast to the situation for 1 2 C, the THSR wave function is certainly not able to describe the structure of all 0 + -states in 1 6 0 lying below the disintegration threshold. A case in point is the first excited state in 1 6 0 , i.e., the 02 + -state at 6.06 MeV, which is believed to have a structure corresponding to an a-particle orbiting in an S wave around a 1 2 C core in its ground state. Such a configuration is clearly missing from our wave function (9.32). On the other hand, if one supposes that the 12 C core in this model is excited to the Hoyle state, then one arrives at a total excitation energy 6.06 MeV + 7.65 MeV = 13.71 MeV, which is surprisingly close to the aforementioned 0s + -state at 13.5 MeV. The latter state has been excited by inelastic a-particle scattering, and first attempts to reproduce the magnitude of the cross section using our a-particle wave

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function indicate that the cross section is indeed sensitive to the greatly extended size of this state. 44 However, more studies, both experimental and theoretical, need to be carried out before one or another of the 0 + -states close to the threshold in 16 0 can be unambiguously identified with an a-particle condensed state. We also did the THSR cluster calculation for 20 Ne. In this case the threshold is at 17.37 MeV and we obtained a condensate state at 17.91 MeV, but it must be added that the experimental situation in 20 Ne is even less clear than in 1 6 0 . One interesting question that can be asked at this point is: How many a's can maximally exist in a self-bound a-gas state? Seeking an answer, we performed a schematic investigation using an effective a-a interaction within an a-gas meanfield calculation of the Gross-Pitaevskii type. 45 The parameters of the force were adjusted to reproduce our microscopic results for 1 2 C. The corresponding a mean-field potential is shown in Fig. 9.10. One sees the OS-state lying slightly above threshold but below the Coulomb barrier. As more a-particles are added, the Coulomb repulsion drives the loosely bound system of a-particles farther and farther apart, so that the Coulomb barrier fades away. According to our estimate, 42 a maximum of ten a-particles can be held together in a condensate. However, there may be ways to lend additional stability to such systems. We know that in the case of 8 Be, adding one or two neutrons produces extra binding without seriously disturbing the pronounced a-cluster structure. Therefore, one has reason to speculate that adding a few of neutrons to a many-a state may stabilize the condensate. But again, stateof-the-art microscopic investigations are necessary before anything definite can be said about how extra neutrons will influence an a-particle condensate. Another interesting idea concerning a-particle condensates was put forward by von Oertzen and collaborators. 46,47 Adding more and more a-particles to the 40 Ca core (for example), one will arrive sooner or later at the point of a-particle drip. Therefore minimal further excitation may be sufficient to shake loose some a-particles, so that an n a-condensate could be created on top of an inert 4 0 Ca core. Similar ideas also have been advanced by Ogloblin,48 who envisions a threea-particle condensate on top of 100 Sn, and earlier by Brenner and Gridnev, who have presented evidence of experimental detection of gaseous a-particles in 28 Si and 32 S on top of an inert 1 6 0 core. 49 In conclusion, we see that the idea of a-particle condensation in nuclei has already triggered many new ideas and calculations, in spite of the fact that, so far, a compelling case for such a state has only been made in 1 2 C. Even so, the possible existence of a completely new nuclear phase in which a-particles play the role of quasi-elementary constituents is surely fascinating. Hopefully, many more a-particle states of nuclei will be detected in the near future, bringing deeper insights into the role of clustering and quantum condensates in systems of strongly interacting fermions.

9.8. Reduction of the a condensate with increasing density The properties of a matter can be used to frame the discussion of the structure of n a nuclei. As described in the preceding section, computational studies of these

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nuclei based on THSR cluster states have demonstrated that an a condensate is established at low nucleon density. More specifically, states lying below but near the threshold for decomposition into a particles, notably the ground state of 8 Be, 12 C in the 0^ Hoyle state, and the corresponding states in 1 6 0 and other na nuclei are dilute, being of low mean density and unusually extended for their mass numbers. We have shown quantitatively within a variational approach that a-like clusters are well formed, with the pair correlation function of a-like clusters predicting relatively large mean distances. For example, in determining the sizes of the 12 C nucleus in its Of (ground) state and in its Oj excited state, we obtained rms radii of 2.44 fm and 3.83 fm, respectively. The corresponding mean nucleon densities estimated from 36/47rrfms are close to the nuclear-matter saturation density po = 0.16 nucleon/fm 3 in the former state and 0.03 nucleon/fm 3 in the latter. The expected low densities of putative alpha-condensate states are confirmed by experimental measurements of form factors. 39 All of our considerations indicate that quartetting is possible in the low-density regime of nucleonic matter, and that a condensates can survive until densities of about 0.03 nucleons/fm 3 are reached. Here, we are in the region where the concept of a matter can reasonably be applied. 50 ' 51 It is then clearly of interest to use this model to gain further insights into the formation of the condensate, and especially the reduction or suppression of the condensate due to repulsive interactions. We will show explicitly that in the model of a matter, as in our studies of finite nuclei, condensate formation is diminished with increasing density. Already within an amatter model based on a simple a — a interaction, we can demonstrate that the condensate fraction - the fraction of particles in the condensate - is significantly reduced from unity at a density of 0.03 nucleon/fm 3 and essentially disappears below the nuclear matter-saturation density. The quantum condensate formed by a homogeneous interacting boson system at zero temperature has been investigated in the classic 1956 paper of Penrose and Onsager, 52 who characterize the phenomenon in terms of off-diagonal long-range order of the density matrix. Here we recall some of their results that are most relevant to our problem. Asymptotically, i.e., for |r — r'| ~ oo, the nondiagonal density matrix in coordinate representation can be decomposed as p{?,7)~r0{f)M?)+!{?-?)• (9-35) In the limit, the second contribution on the right vanishes, and the first approaches the condensate fraction, formally defined by P0

_ QElaSooItt) ~
(

}

Penrose and Onsager showed that in the case of a hard-core repulsion, the condensate fraction is determined by a "filling factor." They applied the theory to liquid 4 He, and found that for a hard-sphere model of the atom-atom interaction yielding a filling factor of about 28%, the condensate fraction at zero temperature is reduced from unity (its value for the noninteracting system) to around 8%. (Remarkably, but to some extent fortuitously, this estimate is in rather good agreement with current experimental and theoretical values for the condensate fraction in liquid 4 He.)

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To make a similar estimate of the condensate fraction for a matter, we follow Ref. 53 and assume an "excluded volume" for a particles of 20 fm 3 . At a nucleonic density of po/3, this corresponds to a filling factor of about 28%, the same as for liquid 4 He. Thus, a substantial reduction of the condensate fraction from unity (for a noninteracting a-particle gas at zero temperature) is also expected in low-density a matter. Turning to a more systematic treatment, we proceed in much the same way as Clark and coworkers,50 referring especially to the most recent study with M. T. Johnson. Adopting the a — a interaction potential Va{r) = 475 e - ( 0 7 r / f m ) 2 M e V - 130 e - ( 0 - 4 7 5 r / f m ) 2 MeV

(9.37)

54

introduced by Ali and Bodmer, we calculate the reduction of the condensate fraction as function of density within what is now a rather standard variational approach. Alpha matter is described as an extended, uniform Bose system of interacting a particles, disregarding any change of the internal structure of the a clusters with increasing density. In particular, the dissolution of bound states associated with Pauli blocking (Mott effect) is not taken into account in the present description. The simplest form of trial wave function incorporating the strong spatial correlations implied by the interaction potential (9.37) is the familiar Jastrow choice,

(9-38)

*(ri, • • •, ?A) = J ] /(I* " fiI) • i<j

The normalization condition /•OO

47rpa / [f(r) - 1] r2dr = - 1 , (9.39) Jo in which p is the number density of a-particles, is imposed as a constraint on the variational wave function, in order to promote the convergence of the cluster expansion used to calculate the energy expectation value. 55 In the low-density limit, the energy functional [binding energy per a cluster as a functional of the correlation factor /(r)] is given by

m

~ *"- f {£ W

+/V)V W r2

" } *•

where ma is the a-particle mass, while the condensate fraction is given by po = exp I-4xPa

J°°[f(r)

- l ] 2 r2dr\

.

(9.41)

The variational two-body correlation factor / was taken as one of the forms employed by Clark and coworkers,50 namely f{r) = (1 - e~ar)(l + be~ar + ce~2ar).

(9.42)

At given density p, the expression for the energy expectation value is minimized with respect to the parameters a, b, and c, subject to the constraint (9.39). It is important to note that these approximations, based on truncated cluster expansions,

(9

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are reliable only at densities low enough that the length scale associated with decay of / 2 (1) — 1 is sufficiently small compared to the average particle separation, which is inversely proportional to the cube root of the density. 50 ' 51 ' 55 ' 56 To give an example, for the nucleon density ipa = 0.06 fm - 3 , a minimum of the energy expectation value (9.40) was found at a — 0.616 f m - 1 , b = 1.221, and c = —5.306, with a corresponding energy per a cluster of —9.763 MeV and a condensate fraction of 0.750. The dependence of the condensate fraction on the nucleon density p = Apa as determined in this exploratory calculation is displayed in Fig. 9.11. The reduction of the condensate fraction of a matter to roughly 0.8

£0.8 O O 03

£0.6

cd

C0.4 C O O0.2 i

0

l

i

0.05

l

0.1

i

i

i

0.15

0.2

I

.3

nucleonic density [fm ] Fig. 9.11. Reduction of condensate fraction in a matter with increasing nucleon density. Exploratory calculations (full line) are compared with HNC calculations of Johnson and Clark 5 0 (crosses). For comparison, we show estimates of the condensate fraction in t h e 0% (Hoyle) state of 1 2 C , according to Refs. 36,57 (stars).

as given by our calculation at nucleonic density 0.03 f m - 3 agrees well with results of Suzuki 57 and Yamada 42 for 12 C in the Hoyle Oj state. Using many-particle approaches to the ground-state wave function and to the THSR (Oj) state of 1 2 C, the occupation of the inferred natural a orbitals is found to be quite different in the two cases. Roughly 1/3 shares (approaching equipartition) are found for the S, D, and, G orbits in the ground (0^) state, with a-cluster occupations of 1.07, 1.07, and 0.82, respectively. On the other hand, in the Hoyle (Oj) state, one sees enhanced occupation (2.38) of the S orbit and reduced occupation (0.29, 0.16, respectively) of the D and G orbits. This corresponds to an enhancement of about 70% compared with equipartition. A more accurate and reliable variational description of a matter can be realized within the hypernetted-chain (HNC) approach to evaluation of correlated integrals; this approach 50 ' 55 largely overcomes the limitations of the cluster-expansion treatment, including the need for an explicit normalization constraint. Such an improved

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approach is certainly required near the saturation density of nuclear matter, where it predicts only a small condensate fraction.50 Of course, at high densities the simple Ali-Bodmer interaction 54 ceases to be valid, and it becomes crucial to include the effects of Pauli blocking. Once again, this conclusion reinforces the point that we can expect signatures of an a condensate only for dilute nuclei near the threshold of na decay, but no signatures from configurations with saturated density.

9.9. Conclusions In this chapter we have investigated the roles that pairing and multiparticle correlations may play in the diverse forms of nuclear matter existing in dense astrophysical objects and in finite nuclei. A complete and quantitative description of nuclear matter must allow for the presence of clusters of nucleons, bound or metastable, possibly forming a quantum condensate. In particular, quartetting correlations, responsible for the emergence of o-like clusters, are identified as uniquely important in determining the behavior of nuclear matter in the limiting regime of low density and low temperature. We have calculated the transition temperature for the onset of quantum condensates made up of a-like and deuteron-like bosonic clusters, and considered in considerable detail the intriguing example of Bose-Einstein condensation of a particles. It is inevitable that under increasing density or pressure, the bound a, d, or other nuclidic clusters present at low density experience significant modification due to the background medium (and eventually merge with it). We have shown how self-energy corrections and Pauli blocking alter the properties of single-particle and cluster states, and we have formulated a cluster mean-field approximation to provide an initial description of this process. One result of special interest is the suppression of the a-like condensate, which is dominant at lower densities, as the density reaches and exceeds the Mott value, allowing the pairing transition to occur. A truly remarkable manifestation of a-particle condensation seems to be present in finite nuclei. Indeed, the so-called Hoyle state (0 2 + ) in 12 C at 7.654 MeV is very likely a dilute gas of three a-particles, held together only by the Coulomb barrier. This view is encouraged by the fact that we can explain all the experimental data in terms of a conceptually simple wave function of the quartet-condensate type. Within the same model, we also systematically predict such states in heavier na nuclei, and the search is on for their experimental identification. It is quite natural that such states should exist up to some maximum number of a particles. We estimate that the phenomenon will terminate at about ten a's as the confining Coulomb barrier fades away. However, there is the possibility that larger condensates could be stabilized by addition of a few neutrons. There is also the exciting prospect that macroscopic a condensates may exist during the collapse of massive stars.

Acknowledgments This work is part of a collaboration with Y. Funaki, H. Horiuchi, A. Tohsaki, and T. Yamada. We thank P. Nozieres for his interest in this work, and we are greatful to A. Sedrakian for discussions and contributions.

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30. S. Saito, Prog. Theor. Phys. 40 (1968); 4 1 , 705 (1969); Prog. Theor. Phys. Suppl. 62, 11 (1977). 31. H. Morinaga, Phys. Rev. 101, 254 (1956); Phys. Lett. 2 1 , 78 (1966). 32. Y. Funaki, A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Phys. Rev. C 67, 051306(R) (2003). 33. A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Phys. Rev. Lett. 87, 192501 (2001). 34. A. Tohsaki, Phys. Rev. C 49, 1814 (1994). 35. A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Proc. of the 8th Int. Conf. on Clustering Aspects of Nuclear Structure and Dynamics, Nara, Japan, 2003, ed. K. Ikeda, I. Tanihata and H. Horiuchi {Nucl. Phys. A 738, 259 (2004)). 36. T. Yamada and P. Schuck, Eur. Phys. J. A 26, 185 (2005). 37. H. Matsumura and Y. Suzuki, Nucl. Phys. A 739, 238 (2004). 38. I. Sick and J. S. McCarthy, Nucl. Phys. A 150, 631 (1970); A. Nakada, Y. Torizuka and Y. Horikawa, Phys. Rev. Lett. 27, 745 (1971); and 1102 (Erratum); P. Strehl and Th. H. Schucan, Phys. Lett. 27 B , 641 (1968). 39. Y. Funaki, A. Tohsaki, H. Horiuchi, P. Schuck, and G. Roepke, Eur. Phys. J. A 28, 259 (2006). 40. Y. Funaki, H. Horiuchi, A. Tohsaki, P. Schuck and G. Ropke, Eur. Phys. J. A 24, 321 (2005). 41. C. Kurakowa and K. Kato, Phys. Rev. C 71, 021301 (2005). 42. T. Yamada and P. Schuck, Phys. Rev. C 69, 024309 ( 2004). 43. T. Wakasa, private communication. See also http://www.rcnp.osaka-u.ac.jp/ annurep/2002 /seel/wakasa2.pdf 44. Y. Funaki et al., RCNP workshop, April, 2006, Osaka, to appear in Mod. Phys. Lett. A, World Scientific. 45. L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys. JETP13, 451 (1961)]; E. P. Gross, Nuovo Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). 46. Tz. Kokalova, N. Itagaki, W. von Oertzen, and C. Wheldon, Phys. Rev. Lett. 96, 192502 (2006). 47. W. von Oertzen et al., to appear in Eur. Phys. J. A. 48. A. A. Ogloblin et al., Proceedings of the International Nuclear Physics Conference, Peterhof, Russia, June 28-July 2, 2005. 49. M. W. Brenner et al., Proceedings of the International Conference " Clustering Phenomena in Nuclear Physics", St. Petersburg, published in 'Physics of Atomic Nuclei (Yadernaya Fizika), 2000. 50. M. T. Johnson and J. W. Clark, Kinam 2, 3 (1980) (PDF available at Faculty web page of J. W. Clark at http://wuphys.wustl.edu); see also J. W. Clark and T. P. Wang, Ann. Phys. (NY.) 40, 127 (1966) and G. P. Mueller and J. W. Clark, Nucl. Phys. A155, 561 (1970). 51. A. Sedrakian, H. Muther, and P. Schuck, Nucl. Phys. A 766, 97 (2006). 52. O. Penrose and L. Onsager, Phys. Rev. 140, 576 (1956). 53. H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Progr. Theor. Phys. 100, 1013 (1998). 54. S. AM and A. R. Bodmer, Nucl. Phys. A 80, 99 (1966). 55. J. W. Clark, Prog. Nucl. Part. Phys. 2, 89 (1979). 56. R. Pantforder, T. Lindenau, and M. L. Ristig, J. Low Temp. Phys. 108, 245 (1997). 57. Y. Suzuki and M. Takahashi, Phys. Rev. C 65, 064318 (2002).

C h a p t e r 10 Realization, Characterization, and Detection of Novel

Superfluid

Phases with Pairing between Unbalanced Fermion Species

K u n Yang National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306, USA In this chapter we review recent experimental and theoretical work on various novel superfluid phases in fermion systems, that result from pairing fermions of different species with unequal densities. After briefly reviewing existing experimental work in superconductors subject to a strong magnetic field and trapped cold fermionic atom systems, we discuss how to characterize the possible pairing phases based on their symmetry properties, and the structure/topology of the Fermi surface(s) formed by the unpaired fermions due to the density imbalance. We also discuss possible experimental probes that can be used to directly detect the structure of the superfluid order parameter in superconductors and trapped cold atom systems, which may establish the presence of some of these phases unambiguously.

Contents 10.1. Introduction and brief review of experimental work 10.2. Characterization of phases based on symmetry and topology of Fermi surface(s) . . . 10.3. Detection of phases based on "phase sensitive" experimental probes 10.3.1. Detecting spatial structure of pairing order parameter in superconductors using phase sensitive experimental probes 10.3.2. Detection of novel pairing phases in cold atom systems 10.4. Summary Bibliography

253 255 259 259 264 265 266

1 0 . 1 . I n t r o d u c t i o n a n d b r i e f r e v i e w of e x p e r i m e n t a l w o r k It is well known t h a t superfluidity and superconductivity in fermionic systems result from pairing of fermions, and the Bose condensations of these so-called Cooper pairs. In a specific fermion system, Cooper pairs are often made of fermions of different species; for example in superconductors they are electrons of opposite spins. T h u s t h e most favorable situation for pairing is when the two species of fermions have the same density, so t h a t there is no unpaired fermion in the ground state. T h e physics of pairing and resultant superfluidity under such condition is well described by the highly successful Bardeen-Cooper-Schrieffer (BCS) theory. It has been a longstanding fundamental question as to what kind of pairing states fermions can form 253

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K. Yang

when the two fermion species have different densities. A closely related issue is that in any paired or superfluid state formed under such situation, some of the majority fermions will necessarily be unpaired; thus a related question is how the system accommodates these unpaired fermions. An early suggestion was due to Fulde and Ferrell,1 and Larkin and Ovchinnikov,2 who argued that the Cooper pairs may condense into either a single finite momentum state, or a state that is a superposition of finite-momentum states. Such states are known collectively as the Fulde-FerrellLarkin-Ovchinnikov (FFLO) state; they break translation and rotation symmetries. More recently other suggestions have been put forward, including deformed Fermi surface pairing (DFSP) 3 ' 4 and breached pairing (BP) 5 ~ 7 states, each with their distinct symmetry properties. Experimentally, the issue of unbalanced pairing arises in several different contexts. Historically, it first arose in the context of a singlet superconductor subject to a large Zeeman splitting. The Zeeman splitting could be due to either a strong external magnetic field, or an internal exchange field (in the case of a ferromagnetic metal/superconductor). Under such a strong magnetic or exchange field, there is a splitting between the Fermi surfaces of spin-up and -down electrons. The original FFLO proposal was advanced in this context. However the FFLO state has not been observed in conventional low-Tc superconductors. The reason for that, we believe, is because these superconductors are mostly three-dimensional, and the magnetic field that gives rise to Zeeman splitting also has a very strong orbital effect, which suppresses the superconductivity before the Zeeman effect becomes significant. The situation has changed recently, as experimental results suggestive of the FFLO state in heavy-fermion, organic, and high-T c superconductors have been found.8~15 These compounds are quasi-one or quasi-two-dimensional, so the orbital effect is weak when the magnetic field is aligned in the conducting plane or along the chain; as a consequence the upper critical field of the superconductor is comparable to or exceeds the so-called Pauli paramagnetic limit, 16 a field at which the Zeeman splitting becomes comparable to the superconducting gap. The the FFLO state becomes possible in such cases. Recent experimental results in CeCoIns, a quasi-2D
Phases with Pairing between Unbalanced Fermion

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255

field; the field, however, also brings the orbital effect that complicates the situation considerably. Indeed, in the very first such experiments unequal numbers of two hyperfine states of fermionic 6 Li atoms were mixed and scanned across a Feshbach resonance. 24 ' 25 In these experiments it was found that paired and unpaired fermions phase separate 26 when the imbalance is large. In one of the experiments 25 it was found that the fermions do not phase separate when the imbalance is sufficiently small; further experiments are needed to clarify the nature of the state in such a situation. Another place where pairing between unbalanced fermion species arises is quark and nucleon pairing in high density quark or nuclear matter, such as in the core of a neutron star. There the origin of density imbalance is due to the difference in the rest mass of quarks or nucleons that form the pairs; when the different pairing species are in chemical equilibrium (meaning they have the same chemical potential), their Fermi momenta and therefore densities are different. The physics of quark and nucleon pairing have been previously reviewed in Ref. 27, and is also covered with great detail in other chapters of this volume.

10.2. Characterization of phases based on symmetry and topology of Fermi surface(s) As discussed in Section 10.1, a number of possible phases have been proposed theoretically in fermionic supernuids with unbalanced pairing species. The purpose of this Section is to classify these phases based on their symmetry and other properties, and discuss the relations between these phases based on such classification. This also gives us insight into the nature of the phase transitions between various phases. We note that classification (or characterization) of classical phases and phase transitions are based on Landau theories, whose forms are completely determined by the symmetry properties of the phases involved; thus our classification based on symmetry considerations are complete at finite temperature (T). It has been realized recently, however, that such classification may not be complete for quantum phases and phase transitions; 28 additional classification schemes may be necessary at T = 0. Here we propose that in these pairing phases the structure and in particular, topology of the Fermi surfaces formed by unpaired fermions can be used as additional classification scheme to characterize phases and phase transitions. Most of the ideas behind such considerations were originally presented in Refs. 29-31, which we review below. We begin with symmetry considerations, and start our discussion from the FFLO state, which has the longest history of studies. Following the superconductivity terminology, throughout the rest of this chapter we will use "spin" indices a =T, I to label the two different species of fermions that form Cooper pairs, "Zeeman splitting" A/z = /xj — /x^ to represent the chemical potential difference between the two fermion species that form Cooper pairs, and "magnetization" m to represent their density difference. When A/i ^ 0, up- and down-spin electrons form Fermi seas with different Fermi momenta pjrf and ppi in the normal state; it was thus suggested 1 ' 2 that when pairing interaction is turned on, the initial pairing instability

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is for fermions with opposite spins on their respective Fermi surfaces to pair up and form a Cooper pair with a net momentum p « ppi — PFI • This results in a pairing order parameter A(r) that is oscillatory in real space, with period 2nh/p. In general the structure of A(r) is characterized not just by a single momentum p, but also by its higher harmonic components. More detailed mean-field study 32 suggested the following real space picture for FFLO state: it is a state with a finite density of uniformly spaced domain walls; across each domain wall the order parameter A (which is real in the mean-field theory) changes sign, and the excess magnetization due to spin imbalance are localized along the domain walls, where A (which is also the gap for unpaired fermions) vanishes; see Fig. l a of Ref. 29 for an illustration. 33 Thus the total magnetization is proportional to the domain wall density. This picture was made more precise by an exact solution in one-dimension (ID) based on bosonized description of spin-gapped Luttinger liquids 34 (also refered to as the Luther-Emery liquid in condensed matter literature), where the domain walls are solitons of the sine-Gorden model that describes the spin sector; each soliton carries one half-spin. While quantum and thermal fluctuations do not allow true long-range order in ID, such order can be stabilized by weak interchain couplings. 34 Coming back to isotropic high D cases, it is clear that the presence and ordering of these domain walls break rotation symmetry, and translation symmetry in the direction perpendicular to the walls, although translation symmetry along the wall remains intact. Thus the symmetry properties of the FFLO state is identical to that of the smectic phase of liquid crystals (smectic-A phase to be more precise). 35 Once the mean-field FFLO state is identified with the smectic phase of liquid crystals based on symmetry considerations, one can borrow insights as well as known results on the thermodynamic phases of liquid crystals to the present problem. 29 In classical liquid crystals it is known that as one increases thermal fluctuations, the broken symmetries of the smectic phase are restored in the following sequence: 35 the translation symmetry is restored first when the smectic melts into a nematic that breaks the rotation symmetry only, and then the nematic melts into an isotropic liquid that has no broken spatial symmetry. We thus expect the same sequence of phases and phase transitions occur in superfluids with unbalanced fermion pairing, as we increase the strength of either thermal or quantum fluctuations. Very interestingly, the nematic and isotropic phases have precisely the same symmetry properties as the deformed Fermi sea pairing (DFSP) 3 ' 4 and breached pairing (BP) 5 - 7 states, which were proposed as alternative phases for unbalanced pairing that compete with the FFLO phase. In the DFSP phase, the (originally mismatched) Fermi surfaces of the majority and minority fermion species deform spontaneously, so that they match in certain regions in momentum space to facilitate pairing; the rotation symmetry is broken by the Fermi surface distortion, but the translation symmetry remains intact (see Fig. 2 of Ref. 3 for an illustration). In the BP phase, an isotropic shell in momentum space is used to accommodate the excess magnetization, while pairing occurs in the rest of the momentum space; both rotation and translation symmetries are intact in this phase. It should be noted that the variational states studied in Refs. 3-7 are quite simple and essentially of mean-field type; they look quite different from the real space picture developed in Ref. 29 (see its Fig. 1) based on considerations of fluctuation effects. We would like to emphasize, however, that

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it is the common symmetry properties that allowed us to identify the DFSP state as the nematic phase, and the BP state as the istropic phase of the liquid crystal. The symmetry considerations also suggests a unified understanding of all three of these states as different phases of a liquid crystal. We note in passing that very similar considerations have also led to deeper understanding of different phases in cuprate superconductors 36 and quantum Hall liquids. 37 ' 38 Coming back to the smectic state, one should in principle also consider the possibility that further symmetry breaking occurs along directions parallel to the domain walls; this will result in breaking of translation symmetries in all directions, and result in a crystal version of the FFLO state. 3 3 As usual the strength of thermal fluctuation is controlled by temperature (T). The quantum fluctuations (QF), on the other hand, are controlled by the strength of pairing interactions; QF is weaker for weak pairing interactions (the BCS regime, where the superfluid state is well described by the mean-field theory), while stronger for strong pairing interaction (the BEC regime). In Ref. 29 a phase diagram has been proposed based on such considerations, in which an infinitesimal density imbalance is present (assuming there is no phase separation in this case, and the smectic phase is the least symmetric phase that is realized), while both temperature and pairing interaction strength are varied. As discussed earlier, the pairing interaction strength can be controlled by manipulating the Feshbach resonance in trapped cold atom systems; thus one may be able to explore the entire phase diagram in such systems. At finite temperature, all phases and phase transitions are classical, and are fully characterized by symmetries. We thus conclude that our classification of the possible pairing phases based on symmetry is complete, and the crystal (FFLO), smectic (FFLO), nematic (DFSP) and isotropic (BP) phases discussed above exhaust all possible phases in this case. Furthermore, symmetry also dictates the nature of the phase transitions. Again, based on known results from studies of classical liquid crystals, 35 we expect the transition between nematic (DFSP) and isotropic (BP) phases to be generically first order, while the transition between nematic (DFSP) and smectic (FFLO) phases is most likely 2nd order. A direct transition between FFLO (either crystal or smectic) and isotropic (BP) phases is unlikely; should such a transition occur, it will be first order. At zero temperature, the possible phases are characterized by the ground state of the system, and the low-lying excitations above it, which are intrinsically quantum mechanical. It has become increasingly clear in recent years that characterization of quantum phases based on symmetry alone is often insufficient, and additional characterization schemes are needed to classify such "quantum" or "topological" order. 28 At present we do not yet have a complete classification scheme for quantum order. 28 In the following we will argue that in the problem of unbalanced pairing discussed here, one may use the properties of the Fermi surface(s) formed by unpaired fermions, especially their topology, to characterize all the possible phases; combined with symmetry properties discussed above, they most likely provide a complete classification scheme. We note that Fermi surfaces are sharp and welldefined objects only at T = 0; finite T smears Fermi surfaces, and as a consequence they are no longer well-defined.

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It is intuitively clear that in the presence of imbalance, some of the majority fermions will be unpaired; these unpaired fermions will form a Fermi sea of its own with at least one, but possibly more Fermi surfaces. Recently this intuitive picture has been made quantitatively precise in the form of a mathematically rigorous theorem, 31 which is a generalization of the Luttinger's theorem 39 for normal metals to the case with pairing interaction and superfluidity. The theorem makes distinction between two different cases: 31 (i) In the absence of superfluidity, there are two Fermi surfaces for spin-up and -down fermions, whose volumes are individually conserved:

"T = ^ n

T >

"1 = ^

,

(lo.D

where d is the dimensioanlity, A is the (real space) volume of the system, and fi is the (momentum space) volume enclosed by the Fermi surface. We emphasize that this is an exact result that applies even when the pairing interaction is so strong that some of the fermions may form very closely bound pairs or "molecules"; in this case one might intuitively expect that these fermions in closely bound states would not contribute to the Fermi surface volume. Our result indicate that as long as there is no superfluidity (or the pairs do not condense), it is the total numbers of fermions that dictate the volumes of Fermi surfaces. (ii) In the presence of superfluidity, or when Cooper pairs Bose condense and the U(l) symmetry associated with charge conservation is spontaneously broken, the spin-up and -down Fermi surface volumes are no longer individually conserved. However their difference remains to be conserved, and is dictated by the imbalance: AN = Ni-Ni = —^nt-nj(2TT)°

(io-2)

In this case we can have either one or two Fermi surfaces; when there is only one Fermi surface we simply have fi^ = 0. In our discussion so far we have assumed the system to be uniform or translationally invariant. These results, however, can be generalized to cases with spontaneously broken translational symmetry, which is the case for the FFLO state. In such cases, the Fermi surface volumes are well-defined modulo the Brillouin zone volume QB': a s a consequence all of our statements on the constraints on Fermi surface volumes are modulo fig. The situation is identical to electrons moving in a periodic potential considered by Luttinger originally.39 The theorem discussed above, in particular the constraint of Eq. (10.2), dictates that the ground states of the systems considered here can be characterized by their Fermi surfaces, and there must be gapless quasiparticle excitations near these Fermi surfaces. We can thus use the Fermi surfaces as an additional classification scheme for the unbalanced pairing phases at T = 0, and expect the following generic cases: (i) One Fermi surface for spin-up fermions. In this case its volume is fixed to be AN = N^-Nl

A_ = 7—^T-

(10-3)

(2TT)«

(ii) Two Fermi surfaces, whose volumes are not fixed individually, but their difference are fixed by Eq. (10.2).

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(iii) No Fermi surface. In this case Eq. (10.2) indicates AN = 0, so there is no imbalance. We believe combining the symmetry property (crystal, smectic, nematic or isotropic) with the number of Fermi surfaces, we have an essentially complete characterization of all the possible quantum pairing phases. As an example, we expect two possible isotropic phases. The breached pair (BP) phase (also known as Sarma phase 40 ), in its original form,5 has two Fermi surfaces, which may be stable at weak coupling. On the other hand we expect an isotropic phase with a single Fermi surface at strong coupling. 29,41 ' 42 For the nematic case, we can in principle again have one or two Fermi surfaces; in this case the pairing order parameter is uniform and does not break any spatial symmetry, while the Fermi surface(s) should be anisotropic and break rotation symmetry spontaneously. As already mentioned, in the FFLO phase, due to the broken translation symmetry, Brillouin zones form and the Fermi surface(s) are folded into a Brillouin zone. The transitions between different phases with different numbers of Fermi surfaces have been discussed in Refs. 30, 31.

10.3. Detection of phases based on "phase sensitive" experimental probes As discussed in the previous Section, the possible phases for systems with pairing between unbalanced fermion species can be characterized by (i) their symmetry properties, especially those associated with the spatial structure of the pairing order parameter; and (ii) in the case T = 0, the structure and in particular, topology of the Fermi surfaces formed by unpaired fermions. Thus to experimentally identify a phase unambiguously, one needs to have experimental methods that probe (i) and/or (ii) directly. While there have been quite a few experiments that study possible FFLO phases in various systems, and the studies of CeCoIns are getting more and more detailed, none of the existing experiments probes (i) or (ii) directly. In the following we will discuss a few possible experiments that probe either (i) or (ii), in either electronic superconductors or trapped cold atom systems.

10.3.1. Detecting spatial structure of pairing order parameter in superconductors using phase sensitive experimental probes In this Subsection we will briefly discuss three possible experimental methods that directly probe the spatial structure of the pairing order parameter, in the FFLO state of electronic superconductors, which we have considered recently. 43-45 We also discuss a proposed experiment 46 that probes physics similar to that of Ref. 43, as well as the possibility of using neutron or muon scattering to detect the spin structure of the FFLO state.

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10.3.1.1. Josephson effect between a BCS and an FFLO superconductor In Ref. 43 we demonstrated that one can use the Josephson effect between an FFLO superconductor and a BCS superconductor to measure the momenta of (in principle) all the Fourier components of the pairing order parameter of the FFLO superconductor. The idea behind this proposal is quite simple. Consider a two-dimensional BCS superconductor, described by a spatially dependent superconducting order parameter ^ B C S W I which is coupled to a two-dimensional FFLO superconductor, described by an order parameter ^FFLO{Y)We consider the two Josephson junction geometries shown in Fig. 1 of Ref. 43. Since the physics for the two geometries are similar we focus our discussion on geometry of Fig. l a of Ref. 43, in which the two superconductors are stacked on top of each other. In the Ginzburg-Landau description, the Josephson coupling term in the free energy takes the form (in the absence of any magnetic flux going through the junction, or in between the two superconductors) Hj = -t J d2v[^FFLO(r)^BCS(r)

+ cc],

(10.4)

where t is the Josephson coupling strength. In the ground state of a BCS superconductor, ^ B c s ( r ) = i>o is a constant. However, in an FFLO superconductor the order parameter is a superposition of components carrying finite momenta:

*™?(r) = £>meik-r,

(10.5)

m

and is oscillatory in space. In the absence of magnetic flux inside the junction, the total Josephson current is

Im tfd2r^*BCS(r)^FFLo(r)

=]>IIm W m

2 ik J i re "

(10.6)

Clearly, due to the oscillatory nature of the integrand, the Josephson current is suppressed in such a junction. Mathematically, the reason that the Josephson current is suppressed here is similar to the suppression of Josephson current by an applied magnetic field in an ordinary Josephson junction between two BCS superconductors. However, the physics is very different: here the suppression is due to the spatial oscillation of the order parameter in the FFLO state, while in the case of ordinary Josephson junction in a magnetic field, the phase of the Josephson tunneling matrix element is oscillatory (in a proper gauge choice). Nevertheless, the mathematical similarity allows these two effects to cancel each other and restore the Josephson current, by applying an appropriate amount of magnetic flux through the junction, and the amount of flux that restores the Josephson effect is a direct measure of the momentum of one of the Fourier components of the pairing order parameter of the FFLO superconductor. This was demonstrated in Ref. 43, and we refer the reader to this paper for detailed analyzes using both the effective Ginzburg-Landau description and microscopic theory, as well as for an alternative geometry. This idea has some similarity to the so called "phase sensitive" experiments that unambiguously determined the d-wave nature of the pairing order parameter of high Tc cuprate

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superconductors. 47 ' 48 However unlike the cuprate experiments that attempt to determine the internal structure of the Cooper pairs (or their anugular momentum), here 43 we use the sensitivity of the Josephson coupling to the phase of the pairing order parameter to determine its spatial structure, or the momentum of the Cooper pairs. In the following we discuss a few practical issues that may arise when trying to implement this proposal experimentally. (i) We want to use the Josephson effect to probe the spatial structure of the pairing order parameter of the FFLO superconductor, using the BCS superconductor (whose pairing order parameter is uniform in space) as a reference point. In order for this idea to work however, the BCS and FFLO superconductors should have the same internal structure for their pairing order parameter, i.e., the two superconductors should be both s-wave or both d-wave etc, otherwise the Josephson current will vanish simply due to the mismatch in internal symmetry. As noted earlier, the most promising candidate for FFLO state thus far is CeCoIns, which is a d-wave superconductor. Thus to implement this idea on CeCoIns one needs to use another d-wave superconductor for the reference BCS state. A natural choice is thus a cuprate superconductor, which has the additional advantage that it has a much bigger gap and higher Pauli limit than CeCoIns; thus when placed in a strong magnetic field (about 10T, necessary to drive CeCoIns into the FFLO state), it is still in the BCS phase. (ii) The key ingredient that makes the Josephson effect useful in the determination of the structure of the FFLO pairing order parameter is that one needs to adjust the magnetic flux in the junction to have the Josephson effect; the order parameter momentum can be determined from the magnetic flux. On the other hand we also need to put the superconductors in a strong magnetic field to drive one of them into the FFLO phase, unless it is a ferromagnetic superconductor that has a spontaneous magnetization. Thus the magnetic field that stabilizes the FFLO state may interfere with the flux through the junction. The configuration that avoids this complication is the one depicted in Fig. l b of Ref. 43, in which the BCS and FFLO superconductors, both assumed to be (quasi) two-dimensional, are placed side-byside. The advantage of this configuration is that the magnetic field that stabilizes the FFLO state is an in-plane magnetic field, which does not contribute to the flux through the junction that controls the Josephson effect. As a result the in-plane field and (out of plane) flux through the junction can be tuned independently. (iii) In an infinite system, which was analyzed in Ref. 43, the Josephson current is exactly zero unless the relative phase oscillation between the BCS and FFLO superconductors are canceled exactly by the phase oscillation in Josephson coupling induced by the flux through the junction. In real systems the junction has a finite size; we thus expect a Fraunhofer pattern in the flux-dependence of the Josephson current, which is peaked at a finite flux strength determined by the momentum of pairing order parameter of the FFLO superconductor.

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K. Yang

Exotic vortex structure of FFLO superconductors

The FFLO state is stabilized by the Zeeman effect of an external magnetic field. On the other hand the field can also generate an orbital effect; for example in a purely 2D system, the Zeeman effect is determined by the total magnetic field, while the orbital effect is generated by the out-of-plane component of the magnetic field, when it is nonzero. Thus the relative importance between the Zeeman and orbital effect can be controlled by the angle between the magnetic field and the 2D plane. The orbital effect generates vortex states, which can be used to detect FFLO physics. The idea here goes back to an early observation by Bulaevskii, who pointed out that 4 9 depending on the interplay between the orbital and Zeeman effects of the magnetic field, the order parameter of a FFLO state near its upper critical field can correspond to a high Landau level (LL) index Cooper pair wave function. Recent work 50-54 on the FFLO vortex lattice structure (VLS) in specific situations has demonstrated that these high LL index VLS's can be very different from the triangular lattice Abrikosov VLS favored by lowest Landau level (LL) Cooper pairs. However it remained a challenging task to determine the vortex lattice structure for FFLO superconductors under general conditions. The difficulty has its origin in the complicated Ginzburg-Landau theory appropriate for FFLO superconductors: 55 F oc | ( - V 2 - q2)iP\2 + a\i>\2 + b\ip\4 + clVflVVf + + V 2 (VV'*) 2 ]+e|Vf + --- ,

d[^*)2(ViP)2 (10.7)

where a, b, c, d, e and q are parameters that depend on both temperature and Zeeman splitting. The fundamental difference between FFLO and BCS superconductors is expressed by the first term in F which describes the kinetic energy of the order parameter; in an FFLO superconductor this term is minimized when the order parameter carries a finite wave vector (or momentum) q. Thus far we have only taken into account the Zeeman effect of the external magnetic field; for a 2D superconductor with the field B tilted out of system plane, orbital coupling must be accounted for by performing a minimal substitution V ^ —» ~Dip = (V — 2ieA/c)ip with V x A = B±z. This leads to Landau quantization of the kinetic energy term, namely the eigenvalues of D 2 = (V—2ieA/c) 2 are — (2n+l)/£2, where I = y/hc/2eB is the Cooper pair magnetic length and n = 0,1,2, • • • is the LL index. There are two specific sources of difficulty, compared to the vortex states of a BCS superconductor, that results in the Abrikosov lattice, (i) For a BCS superconductor the kinetic energy is minimized by n = 0, i.e., tp is a lowest LL wave function. For an FFLO superconductor, however, the kinetic energy is minimized by the index n that minimizes |(2rc + l)/i2 — q2\. This can lead to high Landau level wave functions which are much more complicated, (ii) For the BCS case, one only needs to minimize the \ip\4 term in Eq. (10.7). For FFLO superconductors however, due to the fact that the order parameter carries a finite momentum, there are additional quartic terms (which involve spatial gradients) that make substantial contribution to the free energy, and higher order (|V'|6 a n < i beyond) terms need to be kept because very often the quartic terms make negative contributions. Fortunately, the complicated high LL wave functions have been studied in great detail in the context of quantum Hall effect.56 In Ref. 44 we have used techniques developed in

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the studies of quantum Hall effect to advance a very efficient method to evaluate the free energy (10.7) for the high LL wave function I/J, and minimize it to determine the optimal VLS. The method is somewhat technical and we refer the readers to Ref. 44 for details. More importantly, from the details of the VLS one can extract the LL index n, from which we can get an estimate of the order parameter momentum:

q « V2n + 1/e. 10.3.1.3.

Spectra of Andreev surface bound state of d-wave FFLO superconductors probed by tunneling

The idea here is specific to d-wave superconductors, which CeCoL-15 is believed to be. In a d-wave superconductor, the sign of the pairing order parameter depends on the direction. As a consequence of this there exist low-energy quasiparticle states that are bound to the surface of a superconductor. 57 These so-called Andreev surface bound states (ASBS) result from the change of sign of pairing order parameter when a quasiparticle bounces off the surface; they give rise to a zero bias conductance peak (ZBCP) 57,58 in the tunneling spectrum between a normal metal and the dwave superconductor (with proper orientation), separated by a potential barrier. The ZBCP was recently observed in CeCoIns. 59 In Ref. 45, we find the spectrum of ASBS changes when the d-wave superconductor is driven into the FFLO state, and depends on the momentum of the pairing order parameter. In particular, this leads to a shift and split of the ZBCP in the tunneling spectrum, with the split proportional to the order parameter momentum. This provides yet another way to measure the order parameter momentum using tunneling. 10.3.1.4.

Other possible experiments

In Ref. 46, Bulaevskii and coworkers proposed using interlayer transport in quasi-2D superconductors in the presence of an in-plane magnetic field to detect the FFLO state. The idea bears some similarity to that of Ref. 43: in the superconducting phase, interlayer transport is dominated by Josephson tunneling; for the FFLO state the order parameter has spatial modulation, and the Josephson effect is enhanced when the order parameter modulation is commensurate with the phase modulation of the interlayer Josephson coupling due to the in-plane field. The authors of Ref. 46 have worked out the commensuration condition based on certain assumption on the spatial structure of the order parameter, under which the interlayer transport is enhanced (i.e., enhanced critical current or conductance). Experimentally one can tune the in-plane magnetic field to look for such enhancement associated with the commensuration, from which the wave vector of the order parameter may be extracted. This experiment is also "phase-sensitive". In addition to probing the spatial structure of the superconducting order parameter directly using the experiments discussed above, one can also try to probe the spatial distribution of the unpaired spin-up electrons in the FFLO state, which is closely related to the order parameter structure. For example, for the onedimensional, Larkin-Ovchinnikov type order parameter structure (or the smectic phase), one expect the unpaired spin-up electrons to localize along the domain

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K. Yang

walls where the order parameter changes sign; thus the periodicity (or wave length) of the spin modulation should be one-half of that of the superconducting order parameter. The spatial structure of the spins can be detected from elastic neutron or muon scattering experiments. While such experiments do not directly probe the order parameter structure and are thus not "phase-sensitive", being able to detecting the spin structure should also provide convincing evidence for the FFLO state, and allow us to extract the order parameter structure from it.

10.3.2. Detection

of novel pairing phases

in cold atom

systems

As discussed in Section 10.1, recent experiments 24 ' 25 have started to explore trapped cold atom systems with pairing between atoms of different species (or hyperfine quantum number) and unequal densities. Such systems have also generated strong theoretical interest recently. 41 ' 42 ' 60-65 In particular, the possibility of realizing the FFLO state has been discussed, and in Ref. 61 it was suggested that it can be detected by imaging the density profiles of each of the pairing species, which should be oscillatory in real space for the FFLO state. In another paper, 65 it was suggested that radio-frequency spectroscopy can be used to detect both phase separation and the FFLO state. In Ref. 62 we proposed two alternative methods to detect the FFLO state, which directly probes the momenta of the Cooper pairs, using the methods advanced in Refs. 66-68. In Ref. 66 one projects the Cooper pairs of a BCS state onto molecules by sweeping the tuning field through the Feshbach resonance, and then removes the trap and uses time-of-flight (TOF) measurement to determine the molecular velocity distribution and the condensate fraction. One can do exactly the same experiment on the FFLO state; the fundamental difference here is that in this case because the Cooper pairs carry intrinsic (non-zero) momenta, the condensate will show up as peaks corresponding to a set of finite velocities in the distribution. Another method to detect the Cooper pairs is to study the correlation in the shot noise of the fermion absorption images in TOF, 6 8 first proposed in Ref. 67. In Ref. 68 the shot noise correlation clearly demonstrates correlation in the occupation of k and — k states in momentum space when weakly bound diatom molecules are dissociated and the trap is removed. In principle the same measurement can be performed on fermionic supernuid states, and for an FFLO state, it would reveal correlation in the occupation of k and —k + q states, where q is one of the momenta of the pairing order parameter. 62 Both methods allow one to directly measure q, which defines the FFLO state. These methods are unique to the cold atom systems; very similar ideas have also been discussed in Ref. 42. As discussed in Section 10.2, in addition to the spatial structure of the pairing order parameter, we also need to detect the structure and in particular the topology of the Fermi surface(s) of the unpaired fermions. As discussed in Ref. 31, this can be detected from the momentum distributions of the atoms, using TOF after removing the trap. Such a measurement was recently performed in a gas of 4 0 K across a Feshbach resonance, 69 and hopefully will be performed in systems with unbalanced pairing in the future. In the experiment of Ref. 69 the effect of the trap on the momentum distribution appears to be quite strong, such that the discontinuity in momentum distribution gets wiped out even for non-interacting fermions. We hope that by manipulating the form of the trap potential, its effect can be minimized

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so that discontinuities in momentum distribution associated with Fermi surfaces can be detected in future experiments; this would probably require a trap potential that is flat inside the trap and rises very fast near the boundary. It has also been pointed out 6 3 that in the deformed Fermi surface pairing state, the TOF experiment will find anisotropy in the distribution of the fermion velocity. Again one needs to carefully analyze the effect of the trapping potential in this case. We note in passing that the possibility of detecting some of the novel unbalanced phases in nuclear matter has been discussed recently, 70 as well as the possibility that at large imbalance the system may switch from s-wave to p-wave pairing, 71 including its detection.

10.4. Summary In this chapter we have discussed how to characterize and detect various possible phases that may result from pairing fermions with different species and density imbalance. We argue in Section 10.2 that all the possible phases may be completely characterized by (i) the spatial structure of the pairing order parameter; and (ii) the structure and in particular, topology of the Fermi surfaces formed by unpaired fermions. These novel pairing phases may be realized in spin-singlet superconductors subject to a Zeeman splitting between electron spin states (either due to an external magnetic field or spontaneous magnetization), trapped cold atom systems, and high density quark/nuclear matter. For superconductors, the best case so far is a quasitwo-dimensional heavy fermion superconductor CeCoIns, where evidence for the realization of FFLO state has been found when it is subject to a large in-plane magnetic field. While the existing evidence from various experiments are quite strong, they are all circumstantial in the sense that they do not directly probe the spatial structure of the pairing order parameter. We hope some of the "phase sensitive" experiments we discussed in Subsection 10.3.1 will lead to definitive proof of the FFLO state in this or other superconductors. Experimental work on unbalanced pairing in trapped cold atom systems have just started. Thus far clear evidence of phase separation 26 between paired and unpaired fermions have been found 24 ' 25 when the imbalance is large. Further work is needed to clarify whether some of the novel pairing phases discussed in this and other chapters in this book are realized at small imbalance, and the methods discussed in Subsection 10.3.2 will hopefully be useful in that task.

Acknowledgments Over a period of nearly ten years, the author has benefitted greatly from collaborations with Dan Agterberg, Qinghong Cui, Denis Dalidovich, Chia-Ren Hu, Allan MacDonald, Subir Sachdev, Shivaji Sondhi, and John Wei on the problem of pairing between unbalanced fermion species. His work on this subject has been supported by National Science Foundation grants DMR-9971541 and DMR-0225698, as well

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as the Alfred P. Sloan Foundation and the Research Corporation. This chapter was written while the author was on sabbatical leave and visiting Harvard University and University of California at Los Angeles; he t h a n k s Professors Subir Sachdev and Sudip Chakravarty for their warm hospitality, and acknowledges partial supp o r t from a Florida State University Research Foundation Cornerstone grant during his sabbatical leave.

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11. 12.

13. 14. 15. 16. 17. 18. 19.

20. 21.

P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 20, 762 (1965). H. Muther and A. Sedrakian, Phys. Rev. Lett. 88, 252503 (2002); H. Muther and A. Sedrakian, Phys. Rev. C 67, 015802 (2003); Phys. Rev. D 67, 085024 (2003). W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002 (2003). E. Gubankova, W. V. Liu, and F. Wilczek, Phys. Rev. Lett. 9 1 , 032001 (2003). M. M. Forbes, E. Gubankova, W. V. Liu, and F. Wilczek, Phys. Rev. Lett. 94, 017001 (2005). K. Gloos, R. Modler, H. Schimanski, C. D. Bredl, C. Geibel, F. Steglich, A. I. Buzdin, N. Sato, and T. Komatsubara, Phys. Rev. Lett. 70, 501 (1993). M. Tachiki, S. Takahashi, P. Gegenwart, M. Weiden, M. Lang, C. Geibel, F. Steglich, R. Modler, C. Paulsen, and Y. Onuki, Z. Phys. B: Condens. Matter 100, 369 (1996). R. Modler, P. Gegenwart, M. Lang, M. Deppe, M. Weiden, T. Lhmann, C. Geibel, F. Steglich, C. Paulsen, J. L. Tholence, N. Sato, T. Komatsubara, Y. Onuki, M. Tachiki, and S. Takahashi, Phys. Rev. Lett. 76, 1292 (1996). P. Gegebwart, M. Deppe, M. Koppen, F. Kromer, M. Lang, R. Modler, M. Weiden, C. Geibel, F. Steglich, T. Fukase, and N. Toyota, Ann. Phys. (Leipzig) 5, 307 (1996). J. L. O'Brien, H. Nakagawa, A. S. Dzurak, R. G. Clark, B. E. Kane, N. E. Lumpkin, R. P. Starrett, N. Muira, E. E. Mitchell, J. D. Goettee, D. G. Rickel, and J. S. Brooks, Phys. Rev. B 6 1 , 1584 (2000). J. Singleton, J. A. Symington, M.-S. Nam, A. Ardavan, M. Kurmoo, and P. Days, J. Phys.: Condens. Matter 12, L641 (2000). L. Balicas, J. S. Brooks, K. Storr, S. Uji, M. Tokumoto, H. Tanaka, H. Kobayashi, A. Kobayashi, V. Barzykin, and L. P. Gor'kov, Phys. Rev. Lett. 87, 067002 (2001) M. A. Tanatar, T. Ishiguro, H. Tanaka, and H. Kobayashi, Phys. Rev. B 66, 134503 (2002). A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962); B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962). H. A. Radovan, N. A. Fortune, T. P. Murphy, S. T. Hannahs, E. C. Palm, S. W. Tozer, and D. Hall, Nature (London) 425, 51 (2003). A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 9 1 , 187004 (2003). C. Capan, A. Bianchi, R. Movshovich, A. D. Christianson, A. Malinowski, M. F. Hundley, A. Lacerda, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. B 70, 134513 (2004). T. Watanabe, Y. Kasahara, K. Izawa, T. Sakakibara, Y. Matsuda, C. J. van der Beek, T. Hanaguri, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. B 70, 020506 (2004). C. Martin, C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C. Palm, T. P. Murphy, and J. L. Sarrao, Phys. Rev. B 7 1 , 020503 (2005).

Phases with Pairing between Unbalanced Fermion

Species

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22. K. Kakuyanagi, M. Saitoh, K. Kumagai, S. Takashima, M. Nohara, H. Takagi, and Y. Matsuda, Phys. Rev. Lett. 94, 047602 (2005). 23. R. Movshovich, A. Bianchi, C. Capan, P.G. Pagliuso and J.L. Sarrao, Physica B 359, 416 (2005). 24. M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311, 492 (2006). 25. G. B. Partridge, W. Li, R. I. Kamar, Y. Liao, and R. G. Hulet, Science 311, 503 (2006). 26. P. F. Bedaque, H. Caldas, and G. Rupak, Phys. Rev. Lett. 91, 247002 (2003); Heron Caldas, Phys. Rev. A 69, 063602 (2004); H. Caldas, C. W. Morais, and A. L. Mota, Phys. Rev. D 72, 045008 (2005). 27. R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004). 28. Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems, (Oxford Graduate Texts, Oxford, 2004). 29. K. Yang, cond-mat/0508484. 30. K. Yang and S. Sachdev, Phys. Rev. Lett. 96, 187001 (2006). 31. S. Sachdev and K. Yang, Phys. Rev. B 73, 174504 (2006). 32. H. Burkhardt and D. Rainer, Ann. PhysikS, 181 (1994). 33. More recent studies have suggested that the lowest energy mean-field solutions may have complicated crystalline structures; see, e.g., J. A. Bowers and K. Rajagopal, Phys. Rev. D 66, 065002 (2002). At this point some of the approximations (such as keeping only a finite number of plane wave states for the order parameter) used in these studies are not yet under control and lead to divergence in energy. We will briefly discuss this possibility below. 34. K. Yang, Phys. Rev. B 63, 140511 (2001). 35. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, )Oxford University Press, New York, 1993). 36. S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998). 37. E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065 (1999). 38. Leo Radzihovsky and A. T. Dorsey, Phys. Rev. Lett. 88, 216802 (2002). 39. J. M. Luttinger, Phys. Rev. 119, 1153 (1960). 40. G. Sarma, J. Phys. Chem. Solids 24, 1029 (1963). 41. D. T. Son and M. A. Stephanov, cond-mat/0507586. 42. D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 96, 060401 (2006). 43. K. Yang and D. F. Agterberg, Phys. Rev. Lett. 84, 4970 (2000). 44. K. Yang and A. H. MacDonald, Phys. Rev. B 70, 094512 (2004). 45. Qinghong Cui, Chia-Ren Hu, J.Y.T. Wei, and Kun Yang, Phys. Rev. B 73, 214514 (2006). 46. L. Bulaevskii, A. Buzdin, and M. Maley, Phys. Rev. Lett. 90, 067003 (2003). 47. D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995). 48. C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000). 49. L. N. Bulaevskii, Sov. Phys. JETP 38, 634 (1974). 50. H. Shimahara and D. Rainer, J. Phys. Soc. Jpn. 66, 3591 (1997). 51. U. Klein, D. Rainer, and H. Shimahara, J. Low. Temp. Phys. 118, 91 (2000). 52. M. Houzet and A. Buzdin, Europhys. Lett. 50, 375 (2000). 53. M. Houzet, A. Buzdin, L. Bulaevskii, and M. Maley, Phys. Rev. Lett. 88, 227001 (2002). 54. U. Klein, Phys. Rev. B 69, 134518 (2004). 55. Early discussions of the Ginzburg-Landau theory appropriate for FFLO superconductors can be found in: A. I. Buzdin and M. L. Kulic, J. Low Temp. Phys. 54, 203

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(1984); M. L. Kulic and U. Hofmann, Solid State Commun. 77, 717 (1991). 56. The Quantum Hall Effect, 2nd Ed., edited by R. E. Prange and S. M. Girvin (Springer, New York, 1990). 57. C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994). 58. Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995). 59. P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, 107005 (2005). 60. R. Combescot, Europhys. Lett. 55, 150 (2001). 61. T. Mizushima, K. Machida, and M. Ichioka, Phys. Rev. Lett. 94, 117003 (2005). 62. K. Yang, Phys. Rev. Lett. 95, 218903 (2005). 63. A. Sedrakian, J. Mur-Petit, A. Polls, and H. Muther, Phys. Rev. A 72, 013613 (2005). 64. J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 (2005); T. D. Cohen, Phys. Rev. Lett. 95, 120403 (2005); C.-H. Pao, Shin-Tza Wu, and S.-K. Yip, condmat/0506437; P. Pieri, G.C. Strinati and cond-mat/0512354; W. Yi and L.-M. Duan, cond-mat/0601006; T. N. De Silva and E. J. Mueller, cond-mat/0601314; M. Haque and H.T.C. Stoof, cond-mat/0601321; Tin-Lun Ho and Hui Zhai, cond-mat/0602568; Zheng-Cheng Gu, Geoff Warner, and Fei Zhou, cond-mat/0603091. 65. J. Kinnunen, L. M. Jensen, and P. Torma, Phys. Rev. Lett. 96, 110403 (2006). 66. C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004). 67. E. Altman, E. Dernier, and M. D. Lukin, Phys. Rev. A 70, 013603 (2004). 68. M. Greiner, C. A. Regal, J. T. Stewart, and D. S. Jin, Phys. Rev. Lett. 94, 110401 (2005). 69. C. A. Regal, M. Greiner, S. Giorgini, and D. S. Jin, Phys. Rev. Lett. 95, 250404 (2005). 70. A. A. Isayev, JETP Letters 82, 551 (2005). 71. A. Bulgac, M. McNeil Forbes, and A. Schwenk, cond-mat/0602274.

Chapter 11 Phase Transition in Unbalanced Fermion Superfluids

Heron Caldas Universidade Federal de Sao Joao del Rey, Sao Joao del Rei, 36300-000, MG, Brazil hcaldas@ufsj. edu. br In this chapter recent theoretical work on phase transition in unbalanced fermion superfluids is reviewed. Unbalanced systems are those in which the two fermionic species that form pairs have different Fermi surfaces or densities. We consider systems subjected to weak interactions. In this scenario two distinct phase transitions are predicted to occur, namely a thermodynamic phase transition, induced by the temperature (T), and a quantum phase transition as a function of the increasing chemical potential asymmetry, taking place at zero temperature. We also briefly discuss some recent experimental work at non-zero T with unbalanced Fermi gases in cold atomic traps.

Contents 11.1. Introduction 11.2. The model 11.3. Phase transitions 11.3.1. Phase transitions in fully gapped systems 11.3.2. Phase transitions in gapless systems 11.3.3. The quantum phase transition 11.4. Recent experimental work on trapped Fermi gases at finite temperature 11.5. Summary Bibliography

11.1.

269 270 273 273 274 276 278 279 280

Introduction

It is well known t h a t fermions with opposite spins near a common Fermi surface suffer the Cooper pairing instability at arbitrarily weak coupling, below a certain critical t e m p e r a t u r e , which leads to the phenomenon of superfluidity. W h e n the two fermion species have the same density, the ground s t a t e is described successfully by the Bardeen-Cooper-Shrieffer ( B C S ) 1 theory of superconductivity. A novelty in the pairing mechanism is brought about by an unbalance in the number densities of the two species: certainly there will be fermions without a partner. [These different species could be two fermionic atoms ( 4 0 K or 6 Li) or hyperfine states of 269

270

H. Caldas

the same atom. 2 - 4 The cores of neutron stars could also have the basic ingredients for asymmetric pairing: quarks with different flavors.5-7] For this asymmetrical scenario various exotic phases have been suggested, such as the first proposal by Sarma, 8 who considered a model exhibiting asymmetry between the (fixed) chemical potentials of two particle species at zero and finite temperature, the Larkin, Ovchinnikov, Fulde and Ferrell (LOFF)-phase 9 (where the gap parameter breaks translational invariance), deformed Fermi surfaces 10 ' 11 (in which the Fermi-surfaces of the two species are deformed into ellipsoidal form at zero total momentum of Cooper pairs), the breached pair superfluid phase (BP) 1 2 ' 1 3 (composed of a homogeneous mixture of normal and superfluid phases), and phase separation in real space 15 ' 16 (defined as an inhomogeneous mixed phase formed by normal and superfluid components). Alternatives beyond mean field have also been presented, such as (induced) P-wave superfluidity.17 The very recent experimental demonstration of superfluidity through the observation of vortices 18 and phase separation with controlled population imbalance in atomic gases 19 ' 20 have promoted a great interest in the field, including prospects from the theoretical point of view. 21-28 In these experiments, the strength of the interactions can be tuned by the use of the Feshbach resonance, by varying the external magnetic field. In this way, the crossover from BCS superfluidity to the BoseEinstein condensation (BEC) can be accessed. For a more complete description of recent experimental work on various novel superfluid phases in fermion systems, see Chapter 10 or Ref. 29. In passing we note that the counterpart for ordering in the quantum chromodynamics (QCD) context (for the case where the quarks have approximately the same Fermi surface) at high baryon density, is color superconductivity. 5-7 ' 30 For high density asymmetric quark matter the analog of the LOFF a state is crystalline color superconductivity. 33 In QCD with two light flavors, the ground state is the twoflavor color superconductor (2SC). 34 The pairing between u and d quarks under the condition of charge neutrality and /3-equilibrium at intermediate baryon densities, termed gapless color superconductivity, has been studied in Refs. 35,36. Quark and nuclear pairing is discussed in other chapters of this volume.

11.2. The model To begin with, let us consider a nonrelativistic dilute cold fermionic system. We assume that the particles interact through a short-range attractive interaction. The system is described by the following Hamiltonian H = # - ] P / x Q 7 i Q = ^2ela\ak k,a

k

+ ebkb\bk - 0^a£,&t f c ,6_ f e a f c ,

(H-l)

fc,fc'

where ak, ak are the creation and annihilation operators for the a particles (and the same for the b particles) and e£ are their dispersion relation, defined by ek = £k—fJ.a, a

A review of the theoretical approach and phenomenological applications of the L O F F state in Condensed Matter and QCD, can be found in Ref. 31. An analysis of the competition between the BP and LOFF pairing mechanisms in asymmetric fermion superfluids is shown in Ref. 32.

Phase Transition in Unbalanced Fermion

Superfluids

271

with £jj* = k2/2ma and [ia is the chemical potential of the (non-interacting) aspecies, a = a,b. To reflect an attractive (s-wave) interaction between particles a and b we take — g < 0. 11.2.0.1. The thermodynamic potential We now derive the thermodynamic potential at fixed chemical potentials and the finite temperature gap equation for an asymmetrical fermion system, in order to determine the critical temperature. (The situation where the chemical potentials are kept fixed arises, for instance, in a gas of fermionic atoms connected to reservoirs of species a and b, so that the number densities are allowed to change in the system). We follow the usual derivation in the textbooks; 37 however we extend the analysis to the case of the unbalanced systems we wish to investigate. The thermodynamic potential is given by n = E-TS,

(11.2)

where E is the internal energy and S is the entropy. Let us define fk as the probability of an a particle being excited with momentum k, and similarly gk as the probability of a b particle excited with momentum — k. One can write the following possible probabilities for the particles states: The probability that a given pair of k is unexcited is Pk(0) = (1 - fk)(l

- gk).

(11.3)

The probability that one of the states (k, for instance) is excited, and the other is not, is Pk(2) = fk(l - 9k).

(11.4)

Now the probability that the state —k is excited, and the other is not, is Pk(3) = (I - fk)gk.

(11.5)

And finally, the probability that both states are excited is Pfc(l) = fk9k-

(11.6)

The entropy is defined as S = — 2_] probability of state i x In [probability of state i] = — ^~]Pi ln(Pj), k

k

(11.7) where we have set the Boltzmann constant equal to one. It is left as an exercise for the reader to show that S for an asymmetrical fermion gas is found to be S

= ~ J2 {fk Hfk)

+ (1 - fk) ln(l - fk) + 9k m(sfc) + (1 - gk) ln(l - gk)} .

k

(11.8) The BCS ground state, which describes a superposition of empty and occupied (paired) states, is given by 1 \BCS) = H[uk

+ vkalblk]\0),

(11.9)

272

H. Caldas

where the arbitrary complex (a priori) coefficients Uk and Vk are to be determined by a variational calculation. They are required to be normalized, |ufc|2 + \vk\2 = 1, and to be a spin singlet, uk = w_fc, Vk = v_k- At zero temperature, the internal or ground state energy E is simply the expectation value of the Hamiltonian, E = ( BCS|7i| BCS). At finite temperature the energy has to take into account the excitation probabilities, E = J2k EiPi.37 Then we find

E = J2 K K l -fk- 9k)u\ + fk] + ebk[(l -fk-

9k)ul + gk}}

(11.10)

k

-g^Uk'Vk,UkVk{\

- fk -5fc)(l - fk'

~9k')-

k,k'

Plugging Eqs. (11.8) and (11.10) in Eq. (11.2), and using the minimum conditions 89 n — = 0, ofk

69 „ 59 n j - = 0 , F-=0, ogk 5uk

, 11.11

we find, respectively, fk = l / ( e ^ + 1),

(11.12)

gk = l / ( e ^ + 1),

(11.13)

*-\{1 + k)>

(1L14)

where Sf

= ±e^ + Ekt

(11.15)

are the quasiparticle excitations, with Ek = ye£ + A 2 (T) and e^ = (e£ ± ebk)/2In terms of fk, gk and Uk the thermodynamic potential becomes 9 = —+ £

[e+ - Ek - T l n ( e - ^ + 1) -T\n(e~^

+ 1)] .

(11.16)

In the definition of ££' we have also defined A(T)=gY,ukVk(l-fk-gk).

(11.17)

k

Since v\ = 1 — u\, then UkVk = A/2Ek, and the gap equation can be written as

1=9

^2k ( 1 ~ / f c _ f f f e ) -

(U 18)

-

k

We note that the equation above can also be obtained from the condition 89/dA 0.

=

Phase Transition in Unbalanced Fermion

11.3.

Superfluids

273

Phase transitions

11.3.1. Phase transitions

in fully gapped

systems

The critical temperature Tc is, by definition, the temperature at which A = 0. Then Eq. (11.18) becomes 1

= 9 Y, - r j - T [1 - exp(-/?c^) - exp(-/? c 4)] . k

€

k

+

(11.19)

€

k

After the momentum integration, Eq. (11.19) can be written as 3 8

- ^ - t a ( A , S E ) = -!*(.),

(n.20,

where p(0) = Mkpj-K1 is the density of states at the Fermi level, with kp = yJ2Mp, being the "average" Fermi surface having fi = // a +W» and M = mamb/ (ma+m,b) is the reduced mass. We also introduced a = M/y/mamb as a dimensionless parameter reflecting the mass or chemical potential asymmetry, and

^(a) = * ( - + - . j + * ( - - with \t being the digamma function, defined as $(z) = T'(z)/T(z), where z is a complex number with a positive real component, T is the gamma function, and V is the derivative of the gamma function and /? /? mbfib - mafJ.a a = — r] = — . 2 2 ma+mb Eq. (11.20) can be put in the form Tc = — e x p

•^{ac)

(11.21)

where ac = (3cr]/2 and Ao = 2u£> exp[— l/p(0)g] is the BCS gap parameter in the weak coupling limit, p(0)g
(11.22)

7T

where we have used that ^(0) = —2j — 41n(2), where 7 is Euler's constant. An important feature of Eq. (11.22) is that, although Eq. (11.19) requires regularization, the regulator dependence cancels from the result (11.21). 38 In the symmetric limit, namely ma = mb (or \ia = /j,b) the standard BCS relation T c /Ao = e7/7r 1,4 ° is recovered. We can observe from Eq. (11.22) that the critical temperature for the

H. Caldas

274

0.60

"lM Fig. 11.1. Tc/Ao as a function of m^/ma

for a system constrained to ma^a

= mbVb-

e7/7rAo for system constrained to Pp = PF (or ma^a = ^fcMfc) g ° e s with 2^ mfc greater than ma and approaches zero for m j 3> m a . This shows that pair formation is disfavored for very large mass asymmetry, even in systems were the Fermi surfaces match. In Fig. 11.1 we show the ratio T c /Ao as a function of m t / m a . As one can see, T c /Ao is a smooth function of the mass asymmetry, and goes to zero for rrib/ma —» oo. 11.3.2. Phase

transitions

in gapless

systems

Depending on the relative magnitudes of the particle's Fermi surfaces and masses, the quasiparticle excitations ££' can be negative. If we choose PF > PF and mb ~> ma, only S\ will cross zero. From the equation e£e£ = —A2 we determine the roots of ££14"i6,39 k2 "-1,2

\&Pf

~ 16m 0 m 6 A 2 ] 1 / 2 ,

=F \[{&Pf?

(11.23)

where SPp± = PF ± PF2. The negativity of £\ between k\ and hi means that the corresponding states (the b particles) are singly occupied. In Fig. 11.2 we show the quasiparticle excitations (QPE) as a function of the momentum k. For some values of A, £% may be negative for momenta fci < k < faWe can observe from Eq. (11.23) that A has a critical value _2.

\SP,

, _

4y/mamb'

(11.24)

Phase Transition in Unbalanced Fermion

Superfluids

275

For A > A c , ki£ are not real and ££' never crosses zero. This corresponds to standard BCS with pairing for all k. The situation where A < A c , called the "Sarma phase", was first pointed out in Ref. 8. The thermodynamic potential in the Sarma phase is obtained when we find a state |\I>) which minimizes the internal energy. The smallest energy is reached when the modes with negative £%' are filled and the remaining modes are left empty. More precisely, the ground state \^>) satisfies 15 ' 16 ak,bk\$)

= 0 if £%>0,

4 , 6 t | t f ) = 0 if 4 < 0 .

(11.25)

This state can be written in terms of the a\ and b'k operators and the vacuum state |0) as

i*) = k
vka b

fc2

i - k]fcin bt i°) •

(n-26)

k>k2

The state above corresponds to having BCS pairing in the modes k where £ £ > 0 and a state filled with particles b in the modes where £\ < 0. Using this state in the computation of the entropy and the internal energy, the thermodynamic potential of the Sarma phase turns out to be

fi(5,T,A
=

fis(T)

= —+J2[4-Eks

r i n ( e - ^ ° + 1) - T l n f e " ^ + 1)'

fck2

+ J2 k-Tln(e+^+l) .

(11.27)

Since there are gapless states in Eq. (11.27), one cannot define a critical temperature as performed in the previous situation, i.e., in the fully gapped system. To find the critical temperature (T*) in this case, it is necessary to perform a comparison between QS(T) and the normal free energy, fis(A = 0,T), at a given and fixed asymmetry 6fi < S/j,c for increasing temperature. 41 [The prediction for the break down of the fermionic superfluidity is S/j,c = (fib — /x a )/2 = A 0 . 1 5 ' 1 6 ' 5 7 This picture has been confirmed qualitatively experimentally. 19 ] We then define T* for gapless systems as the temperature at which fi(<5/i,T = T*,A — Ao) = ft(6fi,T — T*, A = 0). Solving this equality for T* we obtain the transition temperature. We point out here that the effect of the temperature in the thermodynamic potential of Eq. (11.27) is to induce a transition from a stable phase at Cl(5/j,,T < T*, A = A 0 ) to an also stable phase Q(Sfi, T > T*, A = 0). 41 - 54 The nature of this transition, however, needs further investigation and will be presented elsewhere. 41

276

H. Caldas

Fig. 11.2. Dispersion relations for the quasi-particles a and /3 showing a region where E^k is negative for mi, = 7ma, PF = 1.45P F , and A = 0.29Ao, obtained from Eq. (11.32). Solid curve corresponds to £% and dashed curve corresponds to £g.

11.3.3. The quantum phase

transition

The zero temperature limit of Eq. (11.27) yields

fi(ff, T = 0, A < A c ) = ns(T = 0) = — + J^ [ 4 - Ek] + J2 ek- ( 1 L 2 8 ) ^

fc
fci

fc>fc2

With d£ls(T = 0)/<9A = 0 (remembering that the partial derivative also hits k\ and &2 in the limits of the integrals) we obtain the gap equation dk d k

_£ f

"

L' HJt ' \Je+ - A 2

(11.29)

~~ 2./fc
k>k2

We can find the gap in the Sarma phase through the identity 8 ' 15,16 ' 39 jlf 2TT

al ~ /

t

i

A^U 3

(27r) ;

r

^31

AT2 , . 2 ~ /fc
Vefc

+ A

°

k>k2

V£t

+A2

'

(11.30)

For small values of the gaps (Ao, A < < /J.a,Hb) the integrals can be approximated and it is found that (11.31)

A2 which has the solution -2,

Ao~

\

Ao

\SP,

y2y/mamb

Ao .

(11.32)

Phase Transition in Unbalanced Fermion

277

Superfluids

It is easy to verify that the Fermi surfaces asymmetry and the gap in the Sarma phase are restricted to: 2 v ^ ^ 5 A o < \SPp2\ < 4 y ^ 7 ^ A 0 0

< As <

(11.33)

A0.

In Fig. 11.3 we show the thermodynamic potential as a function of A for different

Q.

Fig. 11.3. Thermodynamic potential for different values of Pp and Pp (constant kp). The top curve corresponds to Pp = Pp and the lower curves correspond to increasing values of \Pp — PF\.

values of PF and Pp, keeping the combination kp/M = PF2/ma + Pp jrm, fixed, computed from Eq. (11.28) by a numerical evaluation. As we can see, there exist a special combination of Pp and PF for which QS(T = 0) has double minima. Then, given a Pp, we want to know the corresponding value of PF that satisfies this requirement. We note from Eq. (11.23) that when A = 0, fci = Pp and fc2 = PF, k2 = ^{Pp-2 + PP)/2. and Pp is

whereas when A = AQ, ki the relation between PF

Thus, the condition to find

ns(T = o, A = o) = ns(T = o, A = A0) f

3z

dd kk ,, +.

ppb

., ^...

Z3

[r rF dd kk

+

(11.34) 3

b

f dk , ,

^x

Ag

U p ^ « - WD 4 (Soft - J (5^(4 - B>

(11.35)

Note that the left side is just the free energy of the unpaired a and b particles and the right side is the standard BCS free energy. Solving both sides we get x

F

ma

F

Kp

mb

~M

r

+

+

15MkFAl,

(11.36)

where kF = [M{Pfi2/ma + P^/mt)]1/2, and A 0 = (4/i/e 2 )e 3 , ^ r T i s the standard 16 BCS gap parameter. The solution of the equation above we define as P^F,DM> where DM stands for double minima. Given these fixed values of PF and PF (or

278

H. Caldas

lib and Ha) found above, the system could either be found in the normal, BCS or in a mixed phase, since those states would have the same minimal energy. When the free energy F(PFiDM) has its minimum at A 0 , after an increase in PF the gap jumps from A 0 to 0, characterizing a first order quantum phase transition from the superfluid to the normal phase, as has been found theoretically in several physical situations, see, for instance, Refs. 42,43,46. Summarizing, we find that at fixed chemical potentials the Sarma gap corresponds to an extremum of the free energy, it never represents its minimum. For any curve having momenta PF > PF (which are fixed in each curve of Fig. 11.3) both particle species have the same density

na = nb =

'- = —2h- +

^

d(j,aib

67r

n oi

2ir2kF

° 5+ \

\a\k

up to the momentum PFDM. For PF > PF r>M, there is absorption of particles from the reservoirs so the asymmetry in the number densities gets its maximum value, and the system goes to the normal state with number densities pa3

pfc 3

F

— — F U ° - 67r2 ' » - 6?r 2 • The mixed phase (MP) or heterogeneous composition is formed by a normal and a superfluid components, and (differently from the two homogeneous ground states found above) accommodates na and rib particle densities in a trap. 1 5 ' 1 6 The MP has been found to be stable, with no surface energy cost at weak coupling. 44 The issue of the phase transitions in the MP is under investigation. 45 U

11.4. Recent experimental work on trapped Fermi gases at finite temperature For unitary Fermi gases, current experiments 47-50 produce temperatures down to about 0.05TF, where TF is what the Fermi temperature would be for a noninteracting gas with the same number of atoms and in the same trap conditions. Typically TF is of order /j,K. However, a weakly interacting Fermi gas requires much lower T to achieve superfluidity. For the conditions of these experiments, the mean field approximation with an interaction energy proportional to the scattering length is not valid. However, the mean field approximation with a unitary limit appears approximately valid, furnishing a good agreement with predictions of the collective frequencies, and a very good agreement on the transition temperature. Recently, measurements of the T-dependent momentum distribution of a trapped Fermi gas consisting of an equal mixture of the two lowest spin states of 4 0 K in the BCS-BEC crossover regime have been presented. 52 The results show the existence of a competition between the T dependence of the fermionic excitation gap and thermal broadening, leading to non-monotonic behavior in the T dependence of the momentum profiles. Semi-quantitative agreement between theory and experiment using a simple mean-field theory has been found.

Phase Transition in Unbalanced Fermion

Superfluids

279

Thus, even when the measurements are done in strongly interacting Fermi gases, mean field theory qualitatively explains the behavior of these systems, and we expect that our weak coupling mean field BCS results should be also valid, at least, qualitatively. More recently, direct normal-to-superfluid phase transition has been observed in a strongly interacting Fermi gas with unequal mixtures of the two spin components. 53 Both the thermodynamical and quantum phase transitions have been detected. A quantum phase transition at S « 70% has been observed, where S =z (JVj — Ni)/(Nj + N±) is the imbalance parameter, with iVj(iVj.) being the number of majority(minority) atoms. The authors of Ref. 53 found that the critical temperature will in general depend on the population imbalance. This is an (expected) guide for theoretical investigations. 45

11.5. Summary In this chapter we have discussed phase transitions in cold fermionic gases composed of two particle species whose Fermi surfaces or densities do not match. We have seen that depending on the relative difference between the Fermi surfaces, two distinct phase transitions can occur. Fully gapped asymmetrical systems undergo a second order phase transition, driven by the temperature, given by

*T ° = -e-TL -176— 2 c r ~ ' 2CT' c

where a = ^mamb/(ma + mj,) = y / /i a /i(,/(/i a + /n&) appears to be a universal con55 stant. Since experiments are being set up to study pairing between fermions of unequal mass (for example 40 K and 6 Li), we expect that this expression can be verified soon. 56 A system with different Fermi surfaces is found to be in the BCS state while the asymmetry between the chemical potentials is smaller than the critical difference 6[ic. A quantum first order phase transition from superfluid to normal phase, at zero temperature, happens as a function of the increasing chemical potentials asymmetry. As systematic studies at non-zero temperature in unbalanced ultracold systems are just starting, we hope that the work presented in this chapter could stimulate new investigations in this interesting field.

Acknowledgments The author thanks the Nuclear Science Division of LBL for hospitality and support, and the Institute for Nuclear Theory at the University of Washington for its hospitality, where part of this work was done. He also thanks Paulo Bedaque for enlightening discussions and Martin Zwierlein for helpful comments and for a critical reading of the manuscript. This work was partially supported by the Brazilian agencies CNPq, FAPEMIG and CAPES.

280

H. Caldas

Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

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Subject Index

't Hooft interaction, 66, 67, 72 alpha cluster, 224 condensate, 242, 246 Hoyle state, 239, 244 rms radius, 244 Bardeen-Cooper-Schrieffer (BCS) limit, 129, 159, 222 BCS weak-coupling formula, 142 beta function, 127 Beth-Uhlenbeck formula, 230 Bose-Einstein condensation alpha particles, 223, 250 deuterons, 160, 232 eta meson, 121 kaons, 5, 119 limit, 129 pions, 121, 147, 207 chiral perturbation theory, 118 chiral symmetry, 4, 69, 110, 117 broken phase, 75, 79, 81 closed-shell nuclei, 178 coherence length, 162 color chemical potentials, 38, 66, 94 color gauge invariance, 110 color neutrality, 5, 40, 64, 129 color superconductivity, 2, 37, 64, 113 2SC phase, 38, 64, 92 chromomagnetic instabilities, 95 color-flavor-locked (CFL) phase, 2, 3, 72, 110 crystalline phase, 3, 10, 17, 37, 51, 64

gapless phase, 2, 5, 38, 64, 92 mixed phases, 11, 64 single-flavor pairing, 10 color symmetry, 3 Cooper instability, 72, 92, 181, 269 correlated basis functions (CBF), 145, 149 Coulomb barrier, 239 Deformed Fermi surface pairing, 38, 157, 254, 256, 270 deuteron, 138, 193 Dyson-Schwinger equations, 110, 139, 147, 181 effective field theory, 110 high density, 39, 110 electric charge neutrality, 5, 14, 59, 64, 92, 110, 117, 125 electrostatic chemical potential, 38, 66,94 eta meson condensation, 121 Fermi liquid, 140, 146, 207 parameters, 111, 114, 144, 146 Fermi surface deformation, 157 topology, 255 fermion number symmetry, 3, 4 Feshbach resonance, 161, 254, 270 flavor symmetry, 110 free energy, 43, 58, 66, 224, 271 Ginzburg-Landau theory, 12, 17, 19, 48, 53, 202, 218, 260, 262

284

Subject

gluon propagator, 4, 113 Goldstone boson, 4, 5, 117 Goldstone modes, 4, 118 hard dense loop, 112 high Tc superconductors, 260 Hill-Wheeler method, 241 hypernetted chain methods, 151, 153, 249 Jastrow correlation factor, 150 kaon condensation, 5, 15, 119, 153 Landau damping, 113 Landau levels, 262 Larkin, Ovchinnikov, Fulde, Ferrell (LOFF) phase, 7, 45, 92, 100, 130, 156, 254, 270 lepton fraction, 84 Levinson theorem, 231 Luttinger liquids, 256 Luttinger's theorem, 258 mixed phases, 270, 278 Monte-Carlo methods, 153 Mott effect, 223 Nambu-Gorkov propagators, 39, 67, 94, 139 Nambu-Jona-Lasinio model, 10, 12, 40, 65, 75, 93 neutrino emission, 30 neutrino trapping, 75 neutron stars, 30, 68, 76, 136 non-adiabatic superconductivity, 146 nuclear matter asymmetric, 153 dressed propagators, 181, 229 short range correlations, 180, 222 tensor correlations, 180, 202, 222 Pauli paramagnetic limit, 254 penetration depth, 163 phase shifts, 138, 183, 207 phase transition

Index

chiral, 69, 70 first order, 23, 69, 72 second order, 22, 50 pion condensation, 121, 147, 207 polarization effects, 143, 152, 176,197 polarization potential theory, 145 polarization tensor, 144 proton-neutron stars, 65, 85 pseudo-gap, 187 pulsars, 136 glitches, 30, 136 post-glich relaxation, 165 precession, 136, 165 no-go theorems, 166 QCD Higgs phase, 3 perturbative, 118 phase diagram, 29, 61, 68, 79, 110 quark gluon vertex, 113 quark mass, 70, 115, 123 matrix, 38, 67 strange, 4, 59, 64, 116, 120 quark propagator, 58, 113 quasiparticle approximation, 140, 181, 189, 230, 233 random-phase approximation, 148, 176, 228 rotational vortices, 31, 162, 270 scattering matrix, 115, 182 secondary pairing, 9 separation method, 204, 209 shear modulus, 31 single-particle strength, 177 specific heat, 30 spectral function, 185, 225, 234 pions, 148 spin-orbit force, 207 superconductor type-I, 167 type-II, 163 variational theories, 151, 272 vortex clusters, 164

Subject

vortex states, 162, 262 wave function, 225, 238, 262 bound state, 229 condensate, 149, 162, 240 trial, 149, 248 wave-function renormalization, 140, 192

Index

weak equilibrium, 4, 37, 64, 92 weak interactions, 14, 117 Weyl spinors, 39 Zeeman effect, 262

285

Series on Advances in Quantum Many-Body Theory - Vol. 8

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