Functional Renormalization and Ultracold Quantum Gases (Springer Theses)

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Stefan Flörchinger

Functional Renormalization and Ultracold Quantum Gases

Doctoral Thesis accepted by Heidelberg University, Germany

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Author Dr. Stefan Flörchinger Institut für Theoretische Physik Universität Heidelberg Philosophenweg 16 69120 Heidelberg Germany [email protected]

Supervisor Prof. Christof Wetterich Institut für Theoretische Physik Universität Heidelberg Philosophenweg 16 69120 Heidelberg Germany [email protected]

ISSN 2190-5053

e-ISSN 2190-5061

ISBN 978-3-642-14112-6

e-ISBN 978-3-642-14113-3

DOI 10.1007/978-3-642-14113-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010931607  Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Supervisor’s Foreword

The field of ultracold quantum gases has seen an enormous progress in experimental possibilities and methods in the last years. After the first realization of Bose–Einstein condensation in dilute gases of rubidium and sodium, it became possible to investigate many other elements. Particularly interesting is the possibility to tune the interaction strength in a wide range by making use of Feshbach resonances. By using tightly confining traps one can realize situations with three, two or one space dimensions. It is also possible to use optical lattices to simulate elements of solid state physics. Due to this pandora of experimental possibilities, atomic gases became an interesting testing field for many theoretical ideas and methods that were originally developed in high energy quantum field theory or in solid state physics. Quantitative precise comparison between theory and experiments for a number of interesting many-body observables could become possible in the next years. The present thesis develops an approach to many-body physics of cold atomic gases that is based on the functional renormalization group method. The ideas of renormalization are very central in our understanding of quantum field theory and the modern functional formulation provides a valuable tool for theoretical physicists. The method is based on an exact renormalization group flow equation which constitutes an convenient starting point for approximations. In the last years, functional renormalization has been applied to study various topics in quantum and statistical field theory. A few examples are universal critical exponents and amplitude ratios of classical phase transitions, the properties of the chiral phase transition in QCD or condensed matter problems as the Hubbard model or the Kondo effect. Cold atomic quantum gases are another field of application but can also be seen as a particularly interesting testing field. In contrast to other applications one has the big advantage that the microscopic physics is well known and can be controlled experimentally to a large extend. In some cases one expects a far going universality of the macroscopic properties. An example for this is the BCS–BEC crossover close to broad Feshbach resonances. Apart from the temperature, the physics is described by a single parameter, the scattering length a in units of the v

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Supervisor’s Foreword

average inter-particle distance. Observables such as the single particle gap or the critical temperature of the transition to the superfluid state in units of the interparticle distance are predicted to be the same for different materials such as lithium or potassium. It should be possible to use this for a precision comparison between theory and experiment in the quantum many body regime similar to the universal critical exponents that are dominated by thermal fluctuations close to classical second order phase transitions. Heidelberg, May 2010

Christof Wetterich

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Flow Equations to Solve an Integral . . . . . . . . . . . . . . . . . 1.2 Functional Integral Representation of Quantum Field Theory 1.2.1 From the Lattice to Field Theory . . . . . . . . . . . . . . 1.2.2 Expectation Values, Correlation Functions . . . . . . . . 1.2.3 Functional Derivatives, Generating Functionals . . . . 1.2.4 Microscopic Actions in Real Time and Analytic Continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Matsubara Formalism . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Wetterich Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Scale Dependent Schwinger Functional . . . . . . . . . . . . 2.2 The Average Action and its Flow Equation . . . . . . . . . 2.3 Functional Integral Representation and Initial Condition Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalized Flow Equation . . . . . . . . . 3.1 Scale-dependent Bosonization . . . . . 3.2 Flowing Action . . . . . . . . . . . . . . . 3.3 General Coordinate Transformations References . . . . . . . . . . . . . . . . . . . . . .

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Truncations . . . . . . . . . . . . . 4.1 Symmetries as a Guiding 4.2 Separation of Scales. . . . 4.3 Derivative Expansion . . .

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Cutoff Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Investigated Models . . . . . . . . . . . . 6.1 Bose Gas in Three Dimensions . 6.1.1 Lagrangian . . . . . . . . . . 6.2 Bose Gas in Two Dimensions . . 6.3 BCS–BEC Crossover . . . . . . . . 6.3.1 Lagrangian . . . . . . . . . . 6.4 BCS–Trion–BEC Transition . . . 6.4.1 Lagrangian . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Derivative Expansion and Ward Identities 7.1.1 Propagator and Dispersion . . . . . . 7.2 Noethers Theorem . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Truncated Flow Equations . . . . . . . . . . . . . . . . . . . . 8.1 Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Flow Equations for the Effective Potential . 8.1.2 Kinetic Coefficients. . . . . . . . . . . . . . . . . 8.2 BCS–BEC Crossover . . . . . . . . . . . . . . . . . . . . . 8.2.1 Flow of the Effective Potential . . . . . . . . . 8.3 BCS–Trion–BEC Transition . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9

Few-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Repulsive Interacting Bosons . . . . . . . . . . . . . . . 9.1.1 Vacuum Flow Equations and their Solution 9.1.2 Logarithmic Running in Two Dimensions . 9.2 Two Fermion Species: Dimer Formation . . . . . . . 9.2.1 Two-body Problem . . . . . . . . . . . . . . . . . 9.2.2 Renormalization . . . . . . . . . . . . . . . . . . . 9.2.3 Binding Energy. . . . . . . . . . . . . . . . . . . . 9.2.4 Dimer–Dimer Scattering . . . . . . . . . . . . . 9.3 Three Fermion Species: Efimov Effect. . . . . . . . . 9.3.1 SU(3) Symmetric Model . . . . . . . . . . . . . 9.3.2 Experiments with Lithium . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Many-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Bose–Einstein Condensation in Three Dimensions. . 10.1.1 Different Methods to Determine the Density 10.1.2 Quantum Depletion of Condensate . . . . . . . 10.1.3 Quantum Phase Transition . . . . . . . . . . . . .

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Contents

10.1.4 Thermal Depletion of Condensate . . . . . . . . . . . 10.1.5 Critical Temperature . . . . . . . . . . . . . . . . . . . . 10.1.6 Zero Temperature Sound Velocity. . . . . . . . . . . 10.1.7 Thermodynamic Observables . . . . . . . . . . . . . . 10.2 Superfluid Bose Gas in Two Dimensions . . . . . . . . . . . 10.2.1 Flow Equations at Zero Temperature. . . . . . . . . 10.2.2 Quantum Depletion of Condensate . . . . . . . . . . 10.2.3 Dispersion Relation and Sound Velocity . . . . . . 10.2.4 Kosterlitz–Thouless Physics . . . . . . . . . . . . . . . 10.3 Particle–Hole Fluctuations and the BCS–BEC Crossover 10.3.1 Particle–Hole Fluctuations . . . . . . . . . . . . . . . . 10.3.2 Bosonization. . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Critical Temperature . . . . . . . . . . . . . . . . . . . . 10.3.4 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Crossover to Narrow Resonances . . . . . . . . . . . 10.4 BCS–Trion–BEC Transition . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 175

12 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Appendix A: Some Ideas on Functional Integration and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Functional Integral with Probability Measure . . . . . 12.1.2 Correlation Functions from Functional Probabilities 12.1.3 Conservation Laws for On-Shell Excitations . . . . . 12.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Appendix B: Technical Additions . . . . . . . . . . . . . . . . . . 12.2.1 Flow of the Effective Potential for Bose Gas . . . . . 12.2.2 Flow of the Effective Potential for BCS–BEC Crossover. . . . . . . . . . . . . . . . . . . . . . 12.2.3 Hierarchy of Flow Equations in Vacuum . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Functional renormalization in its modern formulation contributes a central part and a valuable tool to our understanding of theoretical physics. It describes how different theories, each of them valid on some momentum scale, are connected to each other. In our modern understanding most theories of physics are ‘‘effective’’ theories. They describe phenomena connected with some typical momentum scale k to a good approximation—often with very high precision. On the other side, they neglect phenomena that are not relevant at this momentum scale. Even the Standard Model of elementary particle physics is of this kind, since it neglects e.g. gravity. Often, the relevant degrees of freedom change with the scale. For example, in quantum chromodynamics (QCD) the high energy theory is described in terms of quarks and gluons, while the low energy limit is governed by mesons and baryons. Functional renormalization describes this transition between different descriptions—from one (effective) theory to another. The functional renormalization group method is also useful for a different task— the statistical description of complex systems with many particles. In atomic systems, the physics at the momentum scale given by the inverse Bohr radius k ¼ ah0 is well known. The solution of Schrödinger’s equation determines the stationary wave functions for electrons, the orbitals. From the structure of the orbitals, the electrostatic, magnetic and spin-related properties, one can calculate predictions for scattering experiments, binding energies and so on. However, if we increase the number of particles, the complexity of the problem rapidly increases as well. It is impossible to find exact solutions to the Schrödinger equation for several—say ten—atoms including all their electrons. To make progress, sensible approximations are needed. Finding a way to describe complex systems in terms of simple, but nevertheless accurate effective theories is not easy. A lot of physical intuition and insight is needed to make the right approximations. Functional renormalization helps us in this important task. An exact renormalization group flow equation connects field theories on different scales. Its close investigation often shows which terms become relevant or irrelevant if the characteristic momentum scale is changed. S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_1, Ó Springer-Verlag Berlin Heidelberg 2010

1

2

1 Introduction

The renormalization group was first developed for the description of critical phenomena close to phase transitions in statistical systems [1–3]. Subsequently, it was realized that the idea of a renormalization group flow as a continuous version of Kadanoff’s block-spin transformation [4] is of great value also for quantum field theory, see e.g. [5]. The development of the renormalization group allowed for a deeper understanding of the formalism including the ‘‘mysterious divergences’’ in perturbation theory and the ‘‘renormalization of coupling constants’’ by introducing counterterms. The modern formulation of functional renormalization uses the concept of the average action (or flowing action) as a modification of the quantum effective action [6–8]. The effective action is the generating functional of the one-particle irreducible correlation functions, see e.g. [9]. A simple, intuitive, but nevertheless exact renormalization group flow equation describes the evolution from microscopic to macroscopic scales [10]. The approach has proven to be successful in many applications ranging from QCD to critical phenomena, for reviews see [11– 17]. However, the method is not yet developed completely. Open issues concern for example the description on non-equilibrium dynamics or the improvement of various approximation schemes. A conceptual point addressed in this thesis is a flow equation for scale-dependent composite operators which allows for a flow from ultraviolet to infrared degrees of freedom [18, 19]. Ultracold quantum gases provide an ideal testing ground for the flow equation method. The ultraviolet physics in the form of atomic physics is well known. In the few-particle sector, exact results are available from quantum mechanical treatments. Many parameters, such as interaction strengths or more obvious temperature and density, can be tuned experimentally over a wide range. Using tightly confining trap potentials, it is even possible to realize different dimensionalities of space. The experimental methods were developed rapidly and improved steadily in the last years and allow for the preparation of very clean samples, nearly without any impurities. It can be expected that future developments lead to more and more accurate measurement techniques which would allow for precision tests of theoretical predictions. At the current stage, many experiments are rather well described by perturbative theories for small coupling constants. Examples are the free theory where interaction effects are neglected completely, or Gaussian approximations such as Bogoliubov theory, Hartree–Fock, or various variants of mean-field theories. In principle, the flow equation method can reproduce the results of all these approaches in the regime where the corresponding approximations are valid. Moreover, it can give corrections and also describe (non-perturbative) features that are not captured in a Gaussian treatment, for example critical phenomena. Experimentally, these corrections should become relevant for strongly interacting systems such as fermions in the BCS-BEC crossover or for lower dimensional systems where fluctuation effects are more important. Also the regions around phase transitions—either quantum or classical—are interesting in this respect. Flow equations have the potential to constitute a systematic extension of perturbative treatments. In contrast to Monte-Carlo methods, the numerical effort is very

1 Introduction

3

small. Physical insight is easier to gain from inspecting flow equations then from complex numerical simulations. Even exact statements can sometimes be made from considering the flow equations in an interesting limit where they can be solved exactly. A large part of the original work presented in this thesis has been published in different articles. For the Bose gas these are ‘‘Functional renormalization for Bose-Einstein condensation’’ [20], ‘‘Superfluid Bose gas in two dimensions’’ [21] and ‘‘Nonperturbative thermodynamics of an interacting Bose gas’’ [22] and for the BSC-BEC crossover ‘‘Particle-hole fluctuations in BCS-BEC crossover’’ [23] as well as ‘‘Functional renormalization group approach for the BCS-BEC crossover’’ [24]. The work on three-component fermions is published in ‘‘Functional renormalization for trion formation in ultracold fermion gases’’ [25], ‘‘Efimov effect from functional renormalization’’ [26] and ‘‘Three-body loss in lithium from functional renormalization’’ [27]. Finally, the new exact flow equation for composite operators is published in ‘‘Exact flow equation for composite operators’’ [28]. In this thesis, the first chapters are devoted to more conceptual issues while concrete applications to ultracold quantum gases are discussed in later chapters. In the remainder of the present chapter, we explain some mathematical ideas underlying the flow equation method at the example of a one-dimensional integral. The functional integral formulation of quantum field theory including the Matsubara formalism is briefly reviewed thereafter. Chapter 2 discusses the flow equation first obtained by Christof Wetterich. We re-derive it starting from the functional integral representation of the partition function. A somewhat generalized form is derived in Chap. 3. This new exact flow equation for scale dependent composite operators allows for example a better treatment of bound states. In Chap. 4 we discuss the idea and use of truncations as an approximate method to solve the flow equation. Chapter 5 is devoted to the choice of the appropriate cutoff function with an emphasis on the particular problems occurring in nonrelativistic field theories. Contact with concrete physics is first made in Chap. 6 where we introduce the microscopic models investigated in this thesis. Besides the repulsive Bose gas in three and two dimensions, this includes a Fermi gas with two (hyperfine) spin components in the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) crossover and fermions with three hyperfine species (‘‘BCS–Trion– BEC transition’’). Chapters 7 and 8 are again a bit more formal and discuss the different symmetries of our models and the used approximation schemes. Results concerning the few-body physics are presented in Chap. 9. We discuss how quantum fluctuations lead to an upper bound on the scattering length of repulsive bosons with contact interaction. For two-component fermions we show that the vacuum model is renormalizable and discuss the properties of the shallow dimer. For the three component fermions, the flow equations in the few-body regime show a limit-cycle scaling and the Efimov tower of three-body bound states. Applied to the case of Lithium they explain recently observed three-body loss features.

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1 Introduction

The main results of this thesis concern the many-body regime and are presented in Chap. 10. We discuss the phase diagram and thermodynamic observables for bosons in three and two spatial dimensions and calculate for example the condensate and superfluid fraction, the critical temperature, the correlation length, the specific heat or properties of sound propagation. The discussion of the Fermi gas in the BCS–BEC crossover concentrates on the effect of particle-hole fluctuations but addresses the complete phase diagram. For three component fermions we extend the vacuum calculations by continuity to non-zero density and find that a new trion phase separates a BCS and a BEC phase close to a common resonance. Finally, we draw some conclusions in Chap. 11. In Appendix A we present some ideas concerning the connection between the functional integral and probability in the foundations of quantum theory. More specific, we discuss a reformulation of the functional integral representation in terms of (quasi-) probabilities. More technical additions such as concrete flow equations for the effective potential and the proof of a theorem concerning the flow equations in vacuum, are given in Appendix B. I would like to thank a number of people who helped and supported me substantially while I was working on this doctoral thesis. In particular, I would like to thank my supervisor, Prof. Dr. Christof Wetterich, for being an excellent mentor and for the outstanding collaboration. I profited enormously from many conversations and good discussions. For great collaborations and many interesting discussions, I would also like to thank very much Michael Scherer, Richard Schmidt, Sergej Moroz, Dr. Sebastian Diehl, Dr. Hans-Christian Krahl, Dr. Philipp Strack, Prof. Dr. Holger Gies, Prof. Dr. Jan M. Pawlowski, Prof. Dr. Markus Oberthaler, Prof. Dr. Selim Jochim, the members of his group, and in particular also all participants of the seminar ‘‘Cold Quantum Coffee’’. I thank Prof. Dr. Holger Gies also for writing the second report on my thesis and the efforts connected with that. Special thanks go to Anne Doster for the great time spent together, welcome and needed distraction as well as a lot of sympathy and support. I am also thankful to my parents, siblings and friends for very many reasons.

1.1 Flow Equations to Solve an Integral In this introductory section we develop a method to calculate simple onedimensional integrals. This may not seem very useful since many integration techniques are known and the method we devise here is not particularly simple. However, it has some advantages, the most important of which is that it can be generalized easily to higher-dimensional or even infinite-dimensional (functional) integrals. No attempt is made to present the following discussion in mathematical rigor. It should be seen as an introductory warm-up to come into contact with some tools and ideas used in later chapters. In particular we will assume that all involved functions have nice enough properties concerning smoothness and convergence.

1.1 Flow Equations to Solve an Integral

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Our goal is to calculate an integral of the form Z1

Z¼

dx f ðxÞ:

ð1:1Þ

1

For simplicity we take the function f to be positive semi-definite, f(x) C 0. One may think of f as describing a probability distribution of the variable x. (For Z = 1 this probability distribution would be normalized.) For convenience we introduce the function S(x) defined by f ðxÞ ¼ eSðxÞ :

ð1:2Þ

The special case where S is quadratic in x 1 SðxÞ ¼ m2 x2 2

ð1:3Þ

with m2 [ 0 can be treated exactly. In that case the integral in (1.1) is of Gaussian form and we obtain Z¼

Z1

1

dx e2m

2 2

x

¼

pffiffiffiffiffiffi 2p=m:

ð1:4Þ

1

It is useful to generalize Z somewhat by introducing the ‘‘source’’ j. We define Z1

ZðjÞ ¼

dx eSðxÞþjx :

ð1:5Þ

1

From Z(j) we can easily derive expectation values, for example 1 hxi ¼ ZðjÞ

Z1

dx xeSðxÞþjx ¼

1 o ZðjÞ: ZðjÞ oj

ð1:6Þ

1

Higher momenta of the probability distribution are obtained as hxn i ¼

1 on ZðjÞ: ZðjÞ ojn

ð1:7Þ

From the ‘‘Schwinger function’’ WðjÞ ¼ ln ZðjÞ we can directly obtain the cumulants of the probability distribution. For example, the variance is given by r ¼ hx2 i  hxi2 ¼

o2 WðjÞ: oj2

ð1:8Þ

6

1 Introduction

More general, the nth cumulant is given by jn ¼

on WðjÞ: ojn

ð1:9Þ

For this reason the functions Z(j) and W(j) are also known as the momentgenerating function and cumulant generating function, respectively. We also introduce the ‘‘effective action’’ as the Legendre transform of the Schwinger function CðuÞ ¼ ju  WðjÞ:

ð1:10Þ

On the right hand side of (1.10) the source j has to be taken such that is fulfills the implicit equation u¼

o WðjÞ: oj

ð1:11Þ

In other words u is the expectation value, u ¼ hxi: It is straightforward to derive some properties of the effective action o CðuÞ ¼ j; ou o2 oj ¼ C¼ ou ou2

 1  2 1 ou o ¼ W : oj oj2

ð1:12Þ

We note that the Legendre transform of the effective action CðuÞ is again the Schwinger function W(j). This shows that if both functions are well defined they carry the same information. After this excursion to probability theory let us now come back to the issue of calculating the integral in (1.1). We show in the following that solving the integral in (1.1) is equivalent to solving a partial differential equation for the flowing action or average action Ck ðuÞ; a generalization of the effective action CðuÞ: We start by generalizing (1.5) Zk ðjÞ ¼ e

Wk ðjÞ

¼

Z1

1 2 2

dx eSðxÞ2k

x þjx

ð1:13Þ

1 1 2 2

where we introduced a ‘‘cutoff’’ term e2k x : For large k2 this factor suppresses the contribution from large values of x to the integral. The k-dependence of Wk(j) is obtained as 1 ok Wk ðjÞ ¼  ðok k2 Þhx2 i 2  2  1 o Wk 2 þ hxi ¼  ðok k2 Þ : 2 oj2

ð1:14Þ

1.1 Flow Equations to Solve an Integral

7

The flowing action is defined by subtracting from the Legendre transform ~ k ðuÞ ¼ ju  Wk ðjÞ; C

ð1:15Þ

~ k ðuÞ  1k2 u2 : Ck ðuÞ ¼ C 2

ð1:16Þ

the cutoff term

As for the effective action the argument of the flowing action is given by u ¼ ojo Wk : We obtain the ‘‘field equation’’ for the average action as oCk ¼ j  k2 u: ou

ð1:17Þ

Similarly as for the effective action one has ~ k o2 W k o2 C ¼1 ou2 oj2

or

o2 W k ¼ oj2



o2 C k þ k2 ou2

1

In order to derive a flow equation for Ck we use (1.15)      oWk   ~ ok Ck ðuÞ u ¼ ok Wk ðjÞ j þ u  ok ju oj   ¼ ok Wk ðjÞ j :

:

ð1:18Þ

ð1:19Þ

Together with (1.14) this leads us to  2 1 1 o Ck 2 2 ok Ck ¼ ðok k Þ þk : 2 ou2

ð1:20Þ

This is the simplest form of the more general flow equation first obtained by Wetterich [10]. For the case of the one-dimensional integral as in (1.1) corresponding to a zero-dimensional field theory (zero time- and zero space-dimensions), this is just a partial differential equation for Ck ðuÞ as a function of k and u: The practical consequences of (1.20) are as follows. Suppose that we know the form of Ck as a function of u for some value of k (say for large k). From solving the flow equation (1.20) we can then obtain Ck ðuÞ for all values of k (at least in principle). The limit k ? 0 is especially interesting since it follows directly from the definitions that the flowing action approaches the effective action lim Ck ðuÞ ¼ CðuÞ:

k!0

ð1:21Þ

Not only can we infer from CðuÞ the value of our integral Z in (1.1), but also all correlation functions or cumulants. To obtain Z we use Z ¼ eWk¼0 ðj¼0Þ

ð1:22Þ

8

1 Introduction

and C ¼ W for j = 0. In other words, we have Z ¼ eCðueq Þ ;

ð1:23Þ

where ueq is determined such that

 o  CðuÞu¼ueq ¼ 0: ou

ð1:24Þ

The cumulants can be obtained from W(j), i.e. the Legendre transform of CðuÞ: This discussion shows already that the flow equation (1.20) is quite powerful and useful if we are interested in the normalization of the probability distribution S(x) as well as in its properties such as cumulants or probabilistic moments. It remains to be shown that the flowing action has a simple form in the limit of large k2 which can serve as an initial condition for the flow equation (1.20). To see that this is indeed the case we consider the integral representation that is easily derived from the definitions Z1 1 2 2 oCk Ck ðuÞ e ¼ dx eSðuþxÞ2k x þ ou x : ð1:25Þ 1

For large k2 we can use k 1 2 2 lim pffiffiffiffiffiffi e2k x ¼ dðxÞ k!1 2p

ð1:26Þ

k lim Ck ðuÞ ¼ SðuÞ þ ln pffiffiffiffiffiffi: 2p

ð1:27Þ

in order to obtain k!1

With this we have a simple initial condition for the function Ck and thus for the flow equation (1.20). The reader may wonder what we have won so far. Solving a partial differential equation is usually at least as complicated as solving an integral. We emphasize again that the big advantage of the flow equation method is that it can be generalized to integrals with many dimensions. In fact we have used besides general arguments only the usual result for Gaussian integration. Higher dimensional Gaussian integrals can be treated very similar. Although it is usually not possible to find exact solutions to the flow equation, it is very valuable as a starting point for different sorts of approximations.

1.2 Functional Integral Representation of Quantum Field Theory In the functional integral formulation of quantum field theory one calculates expectation values and correlation functions on a very large configuration space.

1.2 Functional Integral Representation of Quantum Field Theory

9

For example, for a single complex scalar field one allows the value uðxÞ 2 C to be different for every space–time point x. Different field configurations are weighted by a factor that is determined by the microscopic action S½u: For (stationary) quantum fields in thermal equilibrium this weighting factor eS½u

ð1:28Þ

is real and positive. It has the meaning of a probability for the microscopic field configuration u (‘‘functional probability’’). In the more general case of dynamical (time-dependent) quantum fields, the weighting factor is a complex number. It looses its direct interpretation as a probability. However, on a formal level there is still a close relationship between quantum field theory and statistical theories. For many purposes one can use analytic continuation from real to imaginary time variables to map dynamical quantum field theories with complex weighting factors to statistical theories where the weighting factor is real. Since we are here mainly interested in the properties of thermal and chemical equilibrium, we concentrate on the imaginary time formulation in the following subsections. In Appendix A we present some (preliminary) ideas how it might be possible to reformulate the functional integral description of time-dependent quantum fields in such a way that it deals with real (and positive) probabilities. This shows the probabilistic character of quantum field theory explicitly and might be useful for a more detailed understanding of quantum field theory as well as its philosophical consequences.

1.2.1 From the Lattice to Field Theory One way to approach the infinitely many degrees of freedom of a continuous field theory comes from a discrete lattice of space–time points. Consider a lattice of points 0 1 i ~ xijk ¼ a@ j A; i; j; k 2 Z ð1:29Þ k at times tn ¼ bn;

n 2 Z:

ð1:30Þ

For every set of indices (ijk, n) the field uðxijk ; tn Þ has some value, e.g. uðxijk ; tn Þ 2 C for a complex scalar. The partition sum, i.e. the sum over all possible configurations weighted by the corresponding functional probability is given by 0 1 Y Z ~ijk ; tn ÞAeSðuðx~ijk ;tn ÞÞ : Z¼@ duðx ð1:31Þ ðijk;nÞ2Z4

~ijk ; tn Þ for the different lattice points. The action S depends on the values of uðx Equation (1.31) describes a theory on a discrete space–time lattice. From the

10

1 Introduction

probabilities e-S we can calculate all sorts of expectation values, correlation functions and so on. Our theory becomes a continuum field theory in the limit where a ? 0 and b ? 0. The partition function reads then 0 1 Y Z ~ijk ; tn ÞAeSðuðx~ijk ;tn ÞÞ : Z ¼ lim @ ð1:32Þ duðx a;b!0

ðijk;nÞ2Z4

This can also be written as Z¼

Z

DueS½u :

ð1:33Þ

R The functional integral Du might be defined by the limiting procedure above. ~; tÞ; The microscopic action S is now a functional of the field configuration uðx ~; tÞ 2 R4 : where space and time are now continuous, ðx

1.2.2 Expectation Values, Correlation Functions With our formalism we aim for a statistical description of fields. Important concepts are expectation values of operators and correlation functions. For simplicity, ~ a : The collective index a labels both we denote the field degrees of freedom by U continuous degrees of freedom such as position or momentum and discrete vari~ might consist ables such as spin, flavor or simply ‘‘particle species’’. The field / a of both bosonic and fermionic parts. The fermionic components are described by Grassmann numbers while the bosonic components correspond to ordinary ðCÞ numbers. As an example we consider a theory with a complex scalar field u and a fermionic complex two-component spinor w = (w1, w2). It is useful to decompose the complex scalar into real and imaginary parts 1 u ¼ pffiffiffiðu1 þ iu2 Þ: 2

ð1:34Þ

In momentum space the Nambu spinor of fields reads UðqÞ ¼ ðu1 ðqÞ; u2 ðqÞ; w1 ðqÞ; w2 ðqÞ; w1 ðqÞ; w2 ðqÞÞ

ð1:35Þ

The index a stands in this case for the momentum q and the position in the Nambu-spinor, e.g. Ua ¼ w1 ðqÞ for a = (q, 3). The field expectation value is given by Z ~ ~U ~ a eS½U ~ a i ¼ 1 DU ; ð1:36Þ Ua ¼ hU Z

1.2 Functional Integral Representation of Quantum Field Theory

with Z¼

Z

~

~ S½U : DUe

11

ð1:37Þ

Quite similar one defines correlation functions as Z ~ ~U ~ aU ~ aU ~ bU ~ c . . .i ¼ 1 DU ~ bU ~ c . . .eS½U : cabc... ¼ hU Z

ð1:38Þ

As an example let us consider the two-point function. It is sensible to decompose it into a connected and a disconnected part like ~ aU ~ aU ~ a ihU ~ b i: ~ b i ¼ hU ~ b i þ hU hU c

ð1:39Þ

The connected part is the (full) propagator ~ aU ~ bi : Gab ¼ hU c

ð1:40Þ

Although we discussed here the statistical formulation of the theory (‘‘imaginary time’’) the concepts of expectation values and correlation functions are also useful for the real-time formalism. Formally, the main difference is that the weighting ~ factor eS½U becomes complex after analytic continuation ~

~

eS½U ! eiSt ½U ;

ð1:41Þ

~ is now the real-time action. where St ½U

1.2.3 Functional Derivatives, Generating Functionals To calculate expectation values and correlation functions it is useful to work with sources, functional derivatives and generating functionals. We first explain what a functional derivative is. In some sense it is a natural generalization of the usual derivative to functionals, i.e. to objects that depend on an argument which is itself a function on some space. The basic axiom for the functional derivative is d f ðyÞ ¼ dðx  yÞ df ðxÞ

or

d df ðxÞ

Z

f ðyÞgðyÞ ¼ gðxÞ:

ð1:42Þ

y

R Here we use a notation where the precise meaning of d(x - y) and x depends on the situation. For example when we consider a space with 3 + 1 dimensions we have ~ ~Þ dðx  yÞ ¼ dð4Þ ðx  yÞ ¼ dðx0  y0 Þdð3Þ ðx y

ð1:43Þ

12

1 Introduction

and Z

¼

Z

Z dx0

d3 x:

ð1:44Þ

x

It should always be clear from the context what is meant. Equation (1.42) is the natural extension of the corresponding rule for vectors x; y 2 Rn o xj ¼ dij oxi

or

o X xj yj ¼ yi : oxi j

ð1:45Þ

In addition to (1.42) the functional derivative should obey the usual derivative rules such as product rule, chain rule etc. Using the abstract index notation introduced before (1.36) we write the axiom in (1.42) as d fb ¼ dab dfa

or

d X fb gb ¼ ga : dfa b

ð1:46Þ

With this formalism at hand we can now come back to our task of calculating expectation values and correlation functions. We introduce the source-dependent partition function by the definition Z ~ ~ a Ua ~ S½UþJ Z½J ¼ DUe : ð1:47Þ Expectation values are obtained as functional derivatives ~ ai ¼ U a ¼ hU

1 d Z½J; Z dJa

ð1:48Þ

and similarly correlation functions cabc... ¼

1 d d d    Z½J: Z dJa dJb dJc

ð1:49Þ

The connected part of the correlation functions can be obtained more direct from the Schwinger functional W½J ¼ ln Z½J:

ð1:50Þ

For example the propagator G, the connected two-point function, is given by Gab ¼

d d W½J: dJa dJb

ð1:51Þ

Due to these properties one calls Z[J] (W[J]) the generating functional for the (connected) correlation functions.

1.2 Functional Integral Representation of Quantum Field Theory

13

1.2.4 Microscopic Actions in Real Time and Analytic Continuation In this subsection we discuss the relation between the real-time and the imaginarytime action as well as the analytic continuation in more detail. For concreteness we consider a nonrelativistic repulsive Bose gas in three-dimensional homogeneous space and in vacuum (l = 0). It is straightforward to transfer the discussion also to other cases. In real time the microscopic action is given by   Z Z 1 St ¼  dt d3 x u ðiot  D  iÞu þ kðu uÞ2 : ð1:52Þ 2 The overall minus sign is to match the standard convention. After Fourier transformation the term quadratic in u that determines the propagator reads St;2 ¼

Z

dx 2p

Z

d3 p ð2pÞ

3

 u x  ~ p2 þ i u:

ð1:53Þ

In the basis with the complex fields u; u the inverse microscopic propagator reads GðpÞ1 dðp  p0 Þ ¼

d d St;2 ¼ ðx  ~ p2 þ iÞdðp  p0 Þ: du ðpÞ duðp0 Þ

ð1:54Þ

p2 : From det G1 ðpÞ ¼ 0 we obtain for e ? 0 the dispersion relation x ¼ ~ For the action in (1.52) one can determine the field theoretic expectation values and correlation functions using the formalism described in the previous subsection with the complex weighting function eiSt ½u :

ð1:55Þ

In (1.52), (1.53) and (1.54) the small imaginary term ie is introduced to enforce the correct frequency integration contour (Feynman prescription). In (1.55) it leads to a Gaussian suppression for large values of u u; e

iSt ½u

¼e

iReSt ½u



e

R x

u u

ð1:56Þ

;

which makes the functional integral convergent. Let us now consider the analytic continuation to imaginary time t ! eia s;

0  a\p=2:

ð1:57Þ

For a?p/2 we have t? - is and   o o i  i ! ; ot os

Z

dt ! i

Z ds:

ð1:58Þ

14

1 Introduction

The weighting factor in (1.55) becomes eS½u

ð1:59Þ

with S½u ¼

Z

  1 d3 x u ðos  DÞu þ kðu uÞ2 : 2

Z ds

ð1:60Þ

1.2.5 Matsubara Formalism In statistical physics one is often interested in properties of the thermal (and chemical) equilibrium. For quantum field theories the thermal equilibrium is conveniently described using the Matsubara formalism. In this section we give a short account of the formalism and refer for a more detailed discussion to the literature [29]. The grand canonical partition function is defined as Z ¼ TrebðHlNÞ :

ð1:61Þ

Here we use b ¼ T1 and recall our units for temperature with kB = 1. The trace operation in (1.61) sums over all possible states of the system, including varying particle number. The operator H is the Hamiltonian and N the particle number operator. The factor ebðHlNÞ

ð1:62Þ

is quite similar to an unitary time evolution operator eiDtH evolving the system ~ ¼ H  lN and over some time interval Dt ¼ t2  t1 : Indeed, we can define H evolve the system from time t1 = 0 to the imaginary time t2 = - ib with the operator in (1.62). If we take a (generalized) torus with circumference b in the imaginary time direction as our space–time manifold we can use the functional integral formulation of quantum field theory to write (1.61) as Z Z ¼ D~ ueS½~u ð1:63Þ where S is an action with imaginary and periodic time. From the imaginary time action described in the last subsection it is obtained by replacing also the Hamiltonian H by H - lN. For our Bose gas example this results in S½u ¼

Z

Zb ds

 1 2  d x u ðos  D  lÞu þ kðu uÞ : 2 3





ð1:64Þ

0

Since time is now periodic, the Fourier transform leads to discrete frequencies. The quadratic part of S in (1.64) reads in momentum space

1.2 Functional Integral Representation of Quantum Field Theory

S2 ½u ¼ T

1 Z X

d3 p 3

n¼1

ð2pÞ

u ðixn þ ~ p2  lÞu

15

ð1:65Þ

with the Matsubara frequency xn = 2pT n. In the limit T ? 0 the summation over Matsubara frequencies becomes again an integration Z X dx : ð1:66Þ T ! 2p n For the Fourier decomposition in (1.65) we used the boundary condition uðs ¼ b; ~Þ x ¼ uðs ¼ 0; ~Þ; x   u ðs ¼ b; ~Þ x ¼ u ðs ¼ 0; ~Þ; x

ð1:67Þ

as appropriate for bosonic fields. For fermionic or Grassmann-valued fields w a careful analysis (see e.g. [30]) leads to the boundary conditions wðs ¼ b; ~Þ x ¼ wðs ¼ 0; xÞ w ðs ¼ b; ~Þ x ¼  w ðs ¼ 0; ~Þ: x 

ð1:68Þ

In this case the Matsubara frequencies appearing in (1.65) are of the form   1 xn ¼ 2pT n þ ; n 2 Z: ð1:69Þ 2

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Wilson KG (1971) Phys Rev B 4:3174 Wegner FJ, Houghton A (1973) Phys Rev A 8:401 Wegner FJ (1976) Phase transitions and critical phenomena, vol. 6. Academic Press, London Kadanoff LP (1966) Physica 2:263 Wilson KG, Kogut IG (1974) Phys Rept 12:75 Wetterich C (1991) Nucl Phys B 352:529 Wetterich C (1993) Z Phys C 57:451 Wetterich C (1993) Z Phys C 60:461 Weinberg S (1995) The quantum theory of fields. Cambridge University Press, Cambridge Wetterich C (1993) Phys Lett B 301:90 Berges J, Tetradis N, Wetterich C (2002) Phys Rep 363:223 Bagnuls C, Bervillier C (2001) Phys Rep 348:91 Aoki KI (2000) Int J Mod Phys B 14:1249 Wetterich C (2001) Int J Mod Phys A 16:1951 Salmhofer M, Honerkamp C (2001) Prog Theor Phys 105:1 Metzner W (2005) Prog Theor Phys Suppl 160:58 Pawlowski JM (2007) Ann Phys (NY) 322:2831 Gies H, Wetterich C (2002) Phys Rev D 65:065001 Gies H, Wetterich C (2002) Acta Phys Slov 52:215 Floerchinger S, Wetterich C (2008) Phys Rev A 77:053603

16

1 Introduction

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Floerchinger S, Wetterich C (2009) Phys Rev A 79:013601 Floerchinger S, Wetterich C (2009) Phys Rev A 79:063602 Floerchinger S, Scherer M, Diehl S, Wetterich C (2008) Phys Rev B 78:174528 Diehl S, Floerchinger S, Gies H, Pawlowski JM, Wetterich C (2009). e-print arXiv:0907.2193 Floerchinger S, Schmidt R, Moroz S, Wetterich C (2009) Phys Rev A 79:013603 Moroz S, Floerchinger S, Schmidt R, Wetterich C (2009) Phys Rev A 79:042705 Floerchinger S, Schmidt R, Wetterich C (2009) Phys Rev A 79:053633 Floerchinger S, Wetterich C (2009) Phys Lett B 680:371 Mahan GD (1981) Many-particle physics. Plenum Press, New York Wegner F (1998) Graßmann-variable. Lecture Notes. Universität Heidelberg, Germany

Chapter 2

The Wetterich Equation

In this section we review the properties and derivation of the flow equation first published by Christof Wetterich in 1993 [1]. We will derive this equation from the functional integral representation of quantum field theory. The relation of the flow equation to other methods such as perturbation theory for small interaction strength is then particular clear. In principle one might also consider the flow equation as an own formulation of quantum field theory from which other formulations such as the functional integral representation or the operator formalism can be derived. For practical purposes all this different formulations have their advantages and disadvantages. While some problems are best solved with the flow equation formalism, different methods might be more suitable for other problems. Our physical insight and intuition grows if we look at physics from different perspectives. It is therefore a sensible ambition to further develop the flow equation method and learn how to apply it to various problems in modern physics.

2.1 Scale Dependent Schwinger Functional We start with the Schwinger functional in (1.47), (1.50) which we modify by introducing an cutoff term DSk Z ~ ~ ~ k ½UþJa Ua ~ S½UDS eWk ½J ¼ DUe ð2:1Þ with ~ a ðRk Þ U ~ ¼ 1U ~ DSk ½U ab a : 2

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_2, Ó Springer-Verlag Berlin Heidelberg 2010

ð2:2Þ

17

18

2

The Wetterich Equation

Again we work with an abstract index notation where e.g. a stands for both continuous and discrete variables. For simplicity we will sometimes drop this index when the meaning is clear and write for example ~ k U: ~ ¼ 1UR ~ DSk ½U 2

ð2:3Þ

In praxis one chooses Rk to be an infrared cutoff which is diagonal in momentum ~ we use space. For example, for a single complex scalar field u Z ~  ðqÞRk ðqÞ~ DSk ¼ u uðqÞ ð2:4Þ q

with Rk ðqÞ ¼ Au ðk2  ~ q2 Þhðk2  ~ q2 Þ:

ð2:5Þ

More general the function Rk(q) should have the properties Rk ðqÞ ! 1

ðk ! 1Þ;

Rk ðqÞ ! 0

ðk ! 0Þ;

2

ðq ! 0Þ:

Rk ðqÞ  k

ð2:6Þ

The cutoff term Rk plays a similar role for the functional integral as the parameter k2 in Chap. 1.1 where we developed a flow equation to solve one-dimensional ~ in (2.1) suppresses the contribution of large integrals. For large Rk the term DSk ½U ~ values of U to the functional integral in (2.1). These fluctuations are included as the cutoff Rk is lowered. An additional feature to the one-dimensional case in Chap. 1.1 is the q-dependence of Rk. One can choose the form of Rk(q) such that it vanishes for momenta with q2k2. For a given k only the contribution of modes with q2k2 in (2.1) is then suppressed while the contribution of modes with q2k2 is not modified. We will discuss the choice of an appropriate form of the cutoff function in Chap. 5. Since the only explicitly scale-dependence of the Schwinger functional Wk[J] comes from the cutoff term DSk we can easily calculate its scale-derivative 1 ~ ~ ok Wk jJ ¼  hU a ðok Rk Þab Ub i 2   1 ~ aU ~ b i þ Ua Ub : ¼  ðok Rk Þab hU c 2

ð2:7Þ

~ b i the functional derivative ~ aU We can use for the connected part hU c   ~ b i ¼ W ð2Þ ¼ d d Wk ¼ d Ub ~ aU hU c k ab dJa dJb dJa in order to write (2.7) as

ð2:8Þ

2.1 Scale Dependent Schwinger Functional

19

n o 1 1 ð2Þ ok Wk jJ ¼  STr ðok Rk ÞWk  Uðok Rk ÞU: 2 2

ð2:9Þ

The STr-operation sums over equal indices and includes an extra minus sign for ~ aU ~ b i ¼ hU ~ ai ~ bU fermionic degrees of freedom. This comes from the fact that hU c c ~ ~ if Ua and Ub are fermionic Grassmann-valued fields.

2.2 The Average Action and its Flow Equation From the scale dependent Schwinger functional we can now go to the average action or flowing action. It is defined by subtracting from the Legendre transform ~ k ½U ¼ Ja Ua  Wk ½J C

with Ua ¼

dWk dJa

ð2:10Þ

the cutoff term ~ k ½U  1URk U: Ck ½U ¼ C 2

ð2:11Þ

From the definition it is immediately clear that the average action equals the quantum effective action C½U ¼ Ja Ua  W½J

with Ua ¼

dW dJa

ð2:12Þ

for k ? 0. The quantum effective action is the generating functional of the oneparticle irreducible correlation functions. It is straightforward to show a number of properties of the average action. The field equation follows by taking the functional derivative d ~ Ck ¼ Ja : dUa

ð2:13Þ

The upper (lower) sign is for bosonic (fermionic) field components Ua : The functional derivative in (2.13) is a left derivative for Grassmann valued Ua : For a right-handed derivative we obtain (

~ k d ¼ Ja : C dUa

ð2:14Þ

The second functional derivative is 0 1 (  ð2Þ  d d ~ @C k A ¼ d Jb : C ¼ k ab dUa dUa dUb

ð2:15Þ

20

2

The Wetterich Equation

Comparing this to (2.8) we find the useful relation  ð2Þ    ð2Þ ~ C Wk ¼ dac : k ab

ð2:16Þ

bc

or  1 ð2Þ ð2Þ Wk ¼ Ck þ Rk :

ð2:17Þ

To calculate the flow equation for Ck we use   dWk ~ ¼ ok Wk jJ ok Ck jU ¼ ok Wk jJ þ ok Ja Ua  dJa

ð2:18Þ

and ~ k ½U  1Uðok Rk ÞU: ok Ck ½U ¼ ok C 2

ð2:19Þ

Together with (2.9) and (2.17) we obtain then the central result of this chapter, the Wetterich equation [1]   1 1 ð2Þ ok Ck ¼ STr ðok Rk Þ Ck þ Rk : ð2:20Þ 2

2.3 Functional Integral Representation and Initial Condition From the definition of the average action in (2.10), (2.11) and the scale dependent Schwinger functional in (2.1) we obtain the functional integral representation Z 1 ~ k UþJ ~ UJUþ ~ ~ 1UR ~ S½U 2 2URk U eCk ½U ¼ DUe ¼

Z

~

1~

~

1

~ S½UþU2URk Uþ2 DUe



~ U ~ dCk ðdCk =dUÞUþ dU



ð2:21Þ :

In the last line we performed a change of the integration measure ~ !U ~ þU U

ð2:22Þ

and used 2 ( 3 *

1 dC 1 d d ~ þU ~ k ¼ 4Ck U ~ þU ~ Ck 5 ðdCk =dUÞU 2 2 dU dU dU

ð2:23Þ

2.3 Functional Integral Representation and Initial Condition



1 ~ k U: ~ þU ~ dCk ¼ J U ~  1UR ~  1URk U ðdCk =dUÞU 2 2 2 dU

21

ð2:24Þ

If the cutoff is chosen such that for k?? 1~ ~ 1X rk;a Ua Ua URk U ! 2 2 a

with rk;a ! 1;

ð2:25Þ

we find from (2.21) lim Ck ½U ¼ S½U þ const:

k!1

ð2:26Þ

This is a remarkable and very useful result. In the limit of large k the average action Ck approaches the microscopic action S. Equation (2.26) serves as an initial condition for the flow equation (2.20).

Reference 1. Wetterich C (1993) Phys Lett B 301:90

Chapter 3

Generalized Flow Equation

In this section we derive a generalization of the Wetterich equation discussed in the last chapter. This exact flow equation for composite operators is published in [1]. In contrast to (2.20) we introduce scale-dependent composite operators which describe for example bound states.

3.1 Scale-dependent Bosonization Let us consider a scale-dependent Schwinger functional for a theory formulated in ~ terms of the field w Z w ~ ~ ~ ~ 1w Wk ½g ~ Sw ½w 2 a ðRk Þab wb þga wa : ¼ Dwe ð3:1Þ e Again we use the abstract index notation where e.g. a stands for both continuous variables such as position or momentum and internal degrees of freedom. We now multiply the right hand side of (3.1) by a term that is for Ruk = 0 only a field ~ independent constant. It has the form of the functional integral over the field u with a Gaussian weighting factor Z u 1 D~ ueSpb 2u~  ðRk Þr u~ r þj u~  ; ð3:2Þ where  1 ~  vs Q1 ur  Q1 u s Qr ð~ rq vq Þ: 2  1 u  vQ1 ÞQð~ u  Q1 vÞ; ¼ ð~ 2

Spb ¼

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_3, Ó Springer-Verlag Berlin Heidelberg 2010

ð3:3Þ

23

24

3 Generalized Flow Equation

~ We will often suppress the abstract and v depends on the ‘‘fundamental field’’ w: ~ and the operator v are index as in the last line of (3.3). We assume that the field u bosonic. Without further loss of generality we can then also assume that Q and Ruk are k-dependent symmetric matrices. As an example, we consider an operator v which is quadratic in the original ~ field w; ~ w ~ v ¼ Hab w a b:

ð3:4Þ

The Schwinger functional reads now e

Wk ½g;j

¼

Z

~

~

~ ueSk ½w;~uþgwþj~u DwD~

ð3:5Þ

with   ~ ww ~ þ 1u ~ u ~ þ 1wR ~  ¼ Sw ½w ~ Q þ Ruk u ~ Sk ½w; k 2 2 1 ~ v: þ vQ1 v  u 2

ð3:6Þ

~ ; we can easily shift the variables to obtain In the integration over u eWk ½g;j ¼

Z

~

1~ w ~

~

~ Sw ½w2wRk wþgw Dwe 1

u 1

1

1

 e2ðjþvÞðQþRk Þ ðjþvÞ2vQ Z u 1  D~ ue2u~ ðQþRk Þ~u :

v

ð3:7Þ

~ gives only a (k-dependent) constant. For Ruk = 0 The remaining integral over u and j = 0 we note that Wk[g, j] coincides with Wk[g] in (3.1). We next derive identities for correlation functions of composite operators which follow from the equivalence of (3.5) and (3.7). Taking the derivative with ~ respect to j we can calculate the expectation value for u d Wk ½g; j dj   1  ~ ~ w ¼ Q þ Ruk r jr þ Hrab hw a bi :

u ¼ h~ u i ¼

ð3:8Þ

This can also be written as hvi ¼ Qu  l

ð3:9Þ

3.1 Scale-dependent Bosonization

25

with the modified source l   l ¼ j  Ruk r ur :

ð3:10Þ

For the connected two-point function ðdj dj Wk Þr ¼

d2 ~ r ic Wk ¼ h~ u u dj djr

ð3:11Þ

we obtain from (3.7) ðQ þ Rk Þðdj dj Wk ÞðQ þ Rk Þ   ¼ hðj þ vÞðj þ vÞi  hðj þ vÞihðj þ vÞi þ Q þ Ruk   ¼ hvvi  hvihvi þ Q þ Ruk

ð3:12Þ

or hv vr i ¼



   Q þ Ruk ðdj dj Wk Þ Q þ Ruk r   þ ðQu  lÞ ðQu  lÞr  Q þ Ruk r :

ð3:13Þ

Similarly, the derivative of (3.9) with respect to j yields   ~ s i Q þ Ruk sr u jr  dr h~ u vr i ¼ h~ u u    ¼ u ðQuÞr þ ðdj dj Wk Þ Q þ Ruk r u lr  dr :

ð3:14Þ

We now turn to the scale-dependence of Wk[g, j]. In addition to Rwk and Ruk also Q and H are k-dependent. For H we assume ok Hab ¼ ðok Fq ÞHqab

ð3:15Þ

where we take the dimensionless matrix F to be symmetric for simplicity. For the operator v this gives ~ w ~ ok v ¼ ok Hab w a b ¼ ok Fq vq : From (3.5) and (3.6) we can derive (for fixed g, j)  1 ~  w ~ 1  u ~ o k Rk þ o k Q u ~ ok Rk wi  u ok Wk ¼  hw 2 2 1  hvðok Q1 þ Q1 ðok FÞ þ ðok FÞQ1 Þvi 2 þ h~ uðok FÞvi:

ð3:16Þ

ð3:17Þ

26

3 Generalized Flow Equation

Now we insert (3.13) and (3.15)   1  1  ok Wk ¼  w ok Rwk w  u ok Ruk u 2 2 n  o 1  STr ðdg dg Wk Þ ok Rwk 2   1

 Tr ðdj dj Wk Þ ok Ruk 2   1   Tr Qðok Q1 ÞRuk þ Ruk ok Q1 Q 2 þ Ruk ðok Q1 ÞRuk þ Ruk Q1 ðok FÞðQ þ Rk Þ     þ Q þ Ruk ðok FÞQ1 Ruk ðdj dj Wk Þ  1 þ l ðok Q1 ÞQ þ Q1 ðok FÞQ u 2   þ 12u Qðok Q1 Þ þ Qðok FÞQ1 l  1  l ok Q1 þ Q1 ðok FÞ þ ðok FÞQ1 l 2   1  þ Tr ok Q1 þ Q1 ðok FÞ þ ðok FÞQ1 Ruk 2  1

þ Tr Qok Q1 : 2

ð3:18Þ

The supertrace STr contains the appropriate minus sign in the case that wa are fermionic Grassmann variables. Equation (3.18) can be simplified substantially when we restrict the k-dependence of F and Q such that ok F ¼ Qðok Q1 Þ ¼ ðok Q1 ÞQ:

ð3:19Þ

In fact, one can show that the freedom to choose F and Q independent from each other that is lost by this restriction, is equivalent to the freedom to make a linear change in the source j, or at a later stage of the flow equation in the expectation value u. With the choice in (3.19) we obtain   1  1  ok Wk ¼  w ok Rwk w  u ok Ruk u 2 2 n o  1 w  STr ok Rk ðdg dg Wk Þ 2   1   Tr ok Ruk  Ruk ðok Q1 ÞRuk ðdj dj Wk Þ 2   1 1

þ lðok Q1 Þl þ Tr ok Q1 Q  Ruk : 2 2 The last term is independent of the sources g and j and is therefore irrelevant for many purposes.

3.2 Flowing Action

27

3.2 Flowing Action The average action or flowing action is defined by subtracting from the Legendre transform ~ k ½w; u ¼ gw þ ju  Wk ½g; j C

ð3:20Þ

~ k ½w; u  1wRw w  1uRu u: Ck ½w; u ¼ C 2 k 2 k

ð3:21Þ

the cutoff terms

As usual, the arguments of the effective action are given by wa ¼

d Wk dga

and u ¼

d Wk : dj

ð3:22Þ

By taking the derivative of (3.21) it follows   d Ck ¼ ga  Rwk wb ; ab dwa

ð3:23Þ

where the upper (lower) sign is for a bosonic (fermionic) field w. Similarly, d Ck ¼ j  ðRuk Þr ur ¼ l : du

ð3:24Þ

In the matrix notation ð2Þ Wk

¼

ð2Þ

Ck ¼

dg dg W k ; dg dj W k

; d j dj W k

dw du Ck ; du dw Ck ; du du Ck ! Rwk ; 0 ; 0; Ruk

dj dg W k ; dw dw Ck ;

Rk ¼

ð3:25Þ

it is straight forward to establish ð2Þ ~ ð2Þ Wk C k ¼ 1;

ð2Þ

ð2Þ

Wk ¼ ðCk þ Rk Þ1 :

ð3:26Þ

In order to derive the exact flow equation for the average action we use the identity ~ k j ¼ ok Wk j : ok C w;u g;j

ð3:27Þ

28

3 Generalized Flow Equation

This yields the central result of this chapter   1   1 ð2Þ 1 ok Rk  Rk ðok Q ÞRk ok Ck ¼ STr Ck þ Rk 2  ð1Þ 1 ð1Þ   Ck ok Q1 Ck þ ck 2

ð3:28Þ

with  1

ck ¼  Tr ðok Q1 ÞðQ  Rk Þ : 2

ð3:29Þ

As it should be, this reduces to the standard flow equation for a framework with fixed partial bosonization in the limit qk Q-1 = 0. The additional term is quadratic in the first derivative of Ck with respect to u—we recall that qkQ-1 has non-zero entries only in the u - u block. Furthermore there is a field independent term ck that can be neglected for many purposes. At this point a few remarks are in order. 1. For k?0 the cutoffs Rwk , Ruk should vanish. This ensures that the correlation functions of the partially bosonized theory are simply related to the original correlation functions generated by W0[g], Equation (3.1), namely Z

1 1 ~ ~ Sw ½wþg wþjQ v ~ Dwe W0 ½g; j ¼ ln þ jQ1 j þ const.; 2

ð3:30Þ

W0 ½g; j ¼ 0 ¼ W0 ½g þ const. Knowledge of the dependence on j permits the straightforward computation of correlation functions for composite operators v. 2. For solutions of the flow equation one needs a well known ‘‘initial value’’ which describes the microscopic physics. This can be achieved by letting the cutoffs Rwk , Ruk diverge for k ! K (or k??). In this limit the functional integral in (3.5), (3.6) can be solved exactly and one finds 1 1 CK ½w; u ¼ Sw ½w þ uQK u þ v½wQ1 K v½w  uv½w: 2 2

ð3:31Þ

This equals the ‘‘classical action’’ obtained from a Hubbard–Stratonovich transformation, with v expressed in terms of w. 3. In our derivation we did not use that v is quadratic in w. We may therefore take for v an arbitrary bosonic functional of w. It is straightforward to adapt our formalism such that also fermionic composite operators can be considered. The flow equation (3.28) has a simple structure of a one loop expression with a cutoff insertion—STr contains the appropriate integration over the loop ð1Þ momentum—supplemented by a ‘‘tree-contribution’’  ðCk Þ2 : Nevertheless, it is an exact equation, containing all orders of perturbation theory as well as

3.2 Flowing Action

29

non-perturbative effects. The simple form of the tree contributions will allow for easy implementations of a scale dependent partial bosonization. The details of this can be found in [1].

3.3 General Coordinate Transformations It is sometimes useful to perform a change of coordinates in the space of fields during the renormalization flow. In this section we discuss the transformation behavior of the Wetterich equation (2.20) under such a change of the basis for the fields. We follow here the calculation in [2, 3]. For simplicity we restrict the discussion to bosonic fields. It is straightforward to transfer this to fermions as well [2, 3]. Similarly, one might also consider a general coordinate transformation for the generalized flow equation (3.28). Let us consider a transformation of the form U ! W½U:

ð3:32Þ

Here we denote by U the original fields. The functional W½U is a k-dependent map of the old coordinates to the new ones. We assume that the map in (3.32) is invertible and write the inverse U½W:

ð3:33Þ

In terms of the fields W the definition of the flowing action reads 1 Ck ½W ¼ Ja Ua ½W  Wk ½J  U½WRk U½W 2

ð3:34Þ

where J is determined by Ua ½W ¼

dWk : dJa

ð3:35Þ

In the limit k?0 the flowing action is a Legendre transform with respect to the old fields U but not with respect to the new fields W. This implies for example that C½W is not necessarily convex with respect to the fields W. In addition, only the fields U are expectation values of fields as in (1.48). The relation of the fields W to ~ is more complicated. The field equation reads in terms of the microscopic fields U the new fields dCk dUb d ¼ Jb  DSk : dWa dWa dWa

ð3:36Þ

1 DSk ¼ Ua ½WðRk Þab Ub ½w 2

ð3:37Þ

Note that the cutoff term

30

3 Generalized Flow Equation

is not necessarily quadratic in the fields W: The matrix Rk obtains from d d DSk dUa dUb



dWl dWl d d d2 W m d ¼ DSk þ DSk : dUa dUb dWl dWm dUa dUb dWm

ðRk Þab ¼

ð3:38Þ

ð2Þ

Similarly we obtain for the matrix Ck   ð2Þ Ck

ab

d d Ck dUa dUb



dWl dWl d d d2 W m d ¼ Ck þ Ck : dUa dUb dWl dWm dUa dUb dWm ¼

ð3:39Þ

We now come to the scale dependence of Ck : It is given by ok Ck ½W ¼ ok Ck ½WjU 

dCk ok Wa jU : dWa

ð3:40Þ

For the first term on the right hand side of (3.40) we can use the Wetterich equation (2.20) and obtain 1 1  ð2Þ dCk ok Ck ½W ¼ Tr Ck þ Rk ok R k  ok Wa jU : 2 dWa

ð3:41Þ

ð2Þ

We emphasize that Ck and Rk are now somewhat more complicated objects then usually. They are defined by (3.38) and (3.39). One might also define the transformed matrices

2 d b ð2Þ Þ ¼ d d Ck þ dUa dUb d Wm ðC C k ; k lm dWl dWm dWl dWm dUa dUb dWm ð3:42Þ

2 d d dU dU d W d a b m bkÞ ¼ DSk þ DSk ; ðR lm dWl dWm dWl dWm dUa dUb dWm and similar ð od k Rk Þlm ¼

dUa dUb ðok Rk Þab : dWl dWm

ð3:43Þ

In (3.42) the second functional derivatives are supplemented by connection terms as appropriate for general (non-linear) coordinate systems. With (3.42) and (3.43) the flow equation for Ck reads   1 b dCk 1 d b ok Ck ½W ¼ Trð C k þ R k Þ ok Rk  ok Wa  : ð3:44Þ 2 dWa U

3.3 General Coordinate Transformations

31

Unfortunately this equation has lost its one-loop structure due to the connection terms in (3.42). An important exception is a linear coordinate transformation Wa ½U ¼ Na þ Mab Ub : In that case the terms

d2 W dUdU

vanish and the one-loop structure is preserved.

References 1. Floerchinger S, Wetterich C (2009) Phys Lett B 680:371 2. Gies H, Wetterich C (2002) Phys Rev D 65:065001 3. Wetterich C (1996) Z Phys C 72:139

ð3:45Þ

Chapter 4

Truncations

The exact flow equations derived in the previous sections are powerful and elegant but also complicated. They are functional differential equations i. e. differential equations for the object Ck ½U which depends on the scale parameter k and is a functional of the field configuration U: The mathematics of these kind of differential equations is hardly developed and it is in most cases not possible to find exact solutions in a closed form. However, it is possible to find approximate solutions using truncations in the space of possible functionals. The idea is to take an ansatz for the flowing action of the form Z X N Ck ½U ¼ gi Oi ½U ð4:1Þ x i¼1

where Oi ½U are some operators and the gi are (generalized) coupling constants. In the ideal case the set of operators Oi builds a complete set for N ? ?. Some physical insight into the problem at hand is necessary in order to choose a convenient form of the truncation. In any case it must be possible to write the microscopic action CK ¼ S in the form (4.1). Plugging the ansatz in (4.1) into the flow equation (2.20) or (3.28) for the average action one obtains a set of coupled ordinary differential equations ok gi ¼ bi ðg1 ; . . .; gN ; kÞ

ð4:2Þ

These equations can now be solved either analytically or numerically, depending on the complexity of the problem. For increasing N the approximate solutions found by such a procedure should become better and better. They were even exact if the flow of the couplings gi with i [ N would vanish bi ¼ 0

for i [ N:

ð4:3Þ

These couplings gi would then vanish on all scales and the expansion in (4.1) would give an exact solution to the flow equation. For the example of a free or S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_4, Ó Springer-Verlag Berlin Heidelberg 2010

33

34

4 Truncations

Gaussian theory this is indeed the case. Since all interaction couplings are zero, most of the functions bi also vanish. Exact solutions can also be obtained in another scenario. Suppose that we have a hierarchy in the flow equations of the gi, for example ok g1 ¼ b1 ðg1 ; kÞ ok g2 ¼ b2 ðg1 ; g2 ; kÞ ok g3 ¼ b3 ðg1 ; g2 ; g3 ; kÞ ...

ð4:4Þ

The flow equations for every coupling gi depends only on the couplings gi with i B j. Equation (4.4) can then be solved step by step. First we solve the differential equation for g1(k) [the first line in (4.4)], plug this solution into the second equation for g2, solve it, and so on. In praxis one has to stop at some i = l, of course. The only error in this solution comes from the fact that it may be necessary to solve the (ordinary) differential equations for the gi numerically. What we obtain by this process is not an exact solution for the complete functional Ck ½U but for the coefficients gi with i B l. However, all observables that depend on these couplings only, can be determined exactly. As we will see later, an hierarchy of flow equations similar to (4.4) is indeed found for the nonrelativistic few-body problem. From a flow equation point of view this hierarchy is the reason why no ‘‘renormalization of coupling constants’’ is needed in quantum mechanics.

4.1 Symmetries as a Guiding Principles How should one choose a truncation? The choice of the appropriate ansatz for the flowing action is certainly one of the most important points for someone who wants to work with the flow equation method in praxis. Besides the necessary physical insight there is one major guiding principle: symmetries. As will be discussed in Chap. 7 the flowing action Ck respects the same symmetries as the microscopic action S if no anomalies of the functional integral measure are present and if the cutoff term DSk is also invariant. In the notation of (4.1) this implies that the coefficient gi of an operator Oi ½U that is not invariant under all symmetries will not be generated by the flow equation such that gi = 0 for all k. As an example we consider the microscopic action of a Bose gas Z 1 S ¼ u ðos  D  lÞu þ ku ðu uÞ2 : ð4:5Þ 2 It is invariant under the global U(1) symmetry u ! eia u u ! eia u :

ð4:6Þ

4.1 Symmetries as a Guiding Principles

35

This implies that only operators that are invariant under this transformation may appear in the flowing action Ck ½U: For example, the part that describes homogeneous fields, the effective potential is of the form Z Ck ¼    þ Uðu uÞ ð4:7Þ x

where U(q) is a function of the U(1)-invariant combination q = u*u, only. The action in (4.5) has more symmetries such as translation, rotation or, at zero temperature, Galilean invariance. A useful strategy to find a sensible truncation is to start from the microscopic action S or an effective action C calculated in some (perturbative) approximation scheme such as for example mean-field theory. One now renders the appearing coefficients to become k-dependent ‘‘running couplings’’ and adds also additional terms after checking that they are allowed by the symmetries of the microscopic action.

4.2 Separation of Scales Some symmetries are realized only in some range of the renormalization group flow. For example, Galilean symmetry is broken explicitly by the thermal heat bath for T [ 0. Nevertheless, for k2  T the flowing action Ck (or its real-time version obtained from analytic continuation) will still be invariant under Galilean boost transformations. This comes since the scale parameter k sets the infrared scale on the right hand side of the flow equation. As long as k2/T is large, the flow equations are essentially the same as for T ? 0. In other words, the flow only ‘‘feels’’ the temperature once the scale k is of the same order of magnitude k2 & T. On the other side, for k2  T the flow equations may simplify again. Now they are equivalent to those obtained in the large temperature limit T ? ?. Different symmetries may apply to the action in this limit. This separation of scales is often very useful for practical purposes. In different regimes of the flow different terms are important, while others might be neglected. For example, the universal critical properties such as the critical exponents or amplitude ratios can be calculated in the framework of the classical theory, i.e. in the large temperature limit T ? ?. The flow equations in this limit are much simpler then the ones obtained for arbitrary temperature T. The scale-separation is also useful for the fixing of the initial coupling constants at the initial scale k ¼ K: If this scale is much larger then the temperature K2  T and the relevant momentum scale for the density, the inverse interparticle distance K  n1=3 ; the initial flow is the same as in vacuum where T = n = 0. One can then also use the same initial values for most of the couplings and only change the temperature and the chemical potential appropriately to describe points in the phase diagram that correspond to T [ 0 and n [ 0.

36

4 Truncations

4.3 Derivative Expansion A central part of a truncation is the form of the propagator. It follows from the second functional derivative of the flowing action. For the example of a Bose gas one has in the normal phase 0 G1 k ðpÞdðp  p Þ ¼

d d Ck ½U: du ðpÞ duðp0 Þ

ð4:8Þ

The inverse propagator G-1 may be a quite complicated function of the k momentum p which consists of the spatial momentum and the (Matsubara-) frequency, p ¼ ðp0 ; ~Þ: p From rotational invariance it follows that G-1 k depends on 2 the spatial momentum only in the invariant combination ~ p : At zero temperature it follows from Galilean invariance that G-1 is a (analytic) function of the k combination ip0 þ ~ p2 ; provided that Galilean invariance is not broken by the cutoff. Using a derivative expansion, one truncates the flowing action in the form Z ~ 2  Vo2 þ   Þu þ Uðu uÞ: ð4:9Þ Ck ¼ u ðZos  Ar s x

The ‘‘kinetic coefficients’’ Z, A, V, etc. depend on the scale parameter k and for more advanced approximations also on the U(1) invariant combination q = u*u. One can improve the expansion in (4.8) by promoting the coefficients Z, A, V, etc. to functions of p0 and ~ p2 : In praxis one usually neglects terms higher then quadratic in the momenta. Nevertheless, derivative expansion often leads to quite good results. The reason is the following. On the right hand side of the flow equation the cutoff insertion Rk in the propagator ðCð2Þ þ Rk Þ1 suppresses the contribution of the modes with small momenta. On the other side, the cutoff derivative qkRk suppresses the contribution of very large momenta provided that Rk(q) falls of sufficiently fast for large q. Effectively mainly modes with momenta of the order k2 contribute. It would therefore be sensible to use on the right hand side of the flow equations the coefficients Zðp0 ¼ k2 ; ~ p2 ¼ k2 Þ;

Aðp0 ¼ k2 ; ~ p2 ¼ k2 Þ; . . .

ð4:10Þ

One main effect of the external frequencies and momenta in Zðp0 ; ~ p2 Þ; etc. is to 2 provide an infrared cutoff scale of order Maxðp0 ; ~ p Þ: Such an infrared cutoff scale is of course also provided by Rk itself and one might therefore also work with the k-dependent couplings Zðp0 ¼ 0; ~ p2 ¼ 0Þ;

Aðp0 ¼ 0; ~ p2 ¼ 0Þ; . . .

ð4:11Þ

We emphasize that it is important that the cutoff Rk(q) falls off sufficiently fast for large q. If this is not the case, the derivative expansion might lead to erroneous

4.3 Derivative Expansion

37

results since the kinetic coefficients as appropriate for small momenta and frequencies are then also used for large momenta and frequencies. Only when the scale derivative qkRk provides for a sufficient ultraviolet cutoff does the derivative expansion work properly.

Chapter 5

Cutoff Choices

In this section we discuss the choice of the cutoff function Rk(q). In principle, this function can be chosen quite arbitrary as long as the general requirements in (2.6) are fulfilled. In praxis, the choice of Rk is more involved than it seems on first sight. There is a number of points that have to be taken into account for nonrelativistic systems. (i) Hierarchy of flow equations. In nonrelativistic few-body physics there exists an interesting and important hierarchy: The n-body problem partly decouples from the (n + 1)-body problem. More precisely, one can solve the n-body (scattering)problem without any information about the additional couplings of the (n + 1)body problem. In other words, there are no quantum corrections to a n-point function involving couplings that describe interactions between more then n particles. We will describe this in more detail in Chap. 9. Formally, this hierarchy is closely connected with the frequency pole structure of the correlation functions. For example, the microscopic propagator for a Bose gas 1 iq0 þ ~ q2  l

ð5:1Þ

~2  lÞ: In vacuum, i.e. for l  0 this pole is always in the has a pole for q0 ¼ iðq upper half of the complex plane. For loop expressions where all particle-number arrows point is the same direction (therefore constituting a closed tour of particles) this is similar: all frequency poles are in the same half-plane. If it is possible to close the frequency integration contour in the other half-plane these expressions vanish. In choosing a regulator function Rk ðqÞ (with q ¼ ðq0 ; ~Þ) q one has to be careful to maintain this feature. The cutoff may shift the frequency pole or give rise to additional poles, but (at least for l  0) these poles must remain in the original half plane. If the cutoff depends on the frequency in such a way that

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_5, Ó Springer-Verlag Berlin Heidelberg 2010

39

40

5 Cutoff Choices

1 iq0 þ ~ q  l þ Rk ðq0 ; ~Þ q 2

ð5:2Þ

is a non-analytic function in q0 (in the sense of complex analysis), one has to check directly that the hierarchy feature is not violated. This is very important for practical purposes. (ii) Matsubara summation. At nonzero temperature the Matsubara frequency is discrete, q0 ¼ 2pTn (for bosons). Loop expressions involve a summation over the integer number n from 1 to ?. In praxis one has to find a way to perform these summations—either analytically or numerically. An analytic treatment has the advantage that it is also well controlled in the limit of zero temperature T ! 1 or in the high temperature limit T ! 1. In addition, it usually allows for simpler expressions and lower numerical effort. Methods to perform the Matsubara summation analytically are explained in the literature, e.g. [1]. Usually they require that the integrand in the loop expressions depend on q0 in an analytic way (again in the sense of complex analysis). It is then important to choose the cutoff Rk appropriately. (iii) Ultraviolet regulator property. The cutoff function Rk(q) does not only provide an infrared cutoff but may also provide for an ultraviolet regulator for large momenta q within loop expressions. When Rk(q) falls off sufficiently fast for large q, the cutoff derivative ok Rk on the right hand side of the flow equation suppresses the contribution of large loop-momenta q. This feature is especially important if a derivative expansion is used to treat the momentum dependence of the propagator (and higher correlation functions). As discussed at the end of Chap. 4 derivative expansion might lead to fake results if loop expressions are not sufficiently regularized in the ultraviolet. (iv) Symmetry. As emphasized already in Chap. 4, the symmetries of a problem play a very important role for the flow equation method. If no anomalies of the functional integral measure are present, the effective action C ¼ Ck¼0 shows the same invariances as the microscopic action S. This is also the case for the flowing action Ck for intermediate scales 0\k\K provided that the cutoff term DSk is also invariant under the corresponding symmetry. This is a very useful feature since it strongly restricts the space of possible functionals Ck and helps in finding an effective truncation. Under any circumstances should one choose DSk to be invariant under global ‘‘internal’’ symmetries such as U(1) or SU(N). A cutoff that is invariant under Galilean symmetry (at zero temperature and after analytic continuation to real time) would also be of advantage. (v) Optimization ideas. In a given truncation one observes that different choices of the cutoff lead to different results for physical observables. The range of observed outcomes should become smaller as the truncation becomes better. It may therefore be taken as a rough error estimation. One might ask which cutoff (or which class of cutoffs) leads to the ‘‘best’’ result. It would be sensible to work with such an optimized cutoff function. These ideas are discussed in more detail for systems with relativistic invariance in the literature [2–5] and apply in slightly modified form also to the nonrelativistic case.

5 Cutoff Choices

41

(vi) Simplicity. The cutoff function should be chosen such that the loop expressions become as simple as possible. Not only does this simplify life but often also makes analytic investigations possible that would be very cumbersome otherwise. If it is possible to find analytic expressions for the flow equations (no numerical summation or integration left), the numerical effort to solve them is strongly reduced. Often, this makes larger truncations possible. This in turn should lead to more precise results (which again makes life simpler). (vii) Fermi surface. For fermionic systems at nonvanishing density one has to deal with additional complications from the fact that the actual pole or infrared singularity of the (not regularized) propagator at zero frequency does not occur at ~ q2 ¼ 0 but at the Fermi surface ~ q2 ¼ f ðlÞ: (In the simplest approximation and for homogeneous systems one has f ðlÞ ¼ l:) The cutoff function Rk has to be constructed such that it regularizes this singularity at the Fermi surface. After this discussion of the general requirements we now turn to the regulator functions used in this thesis. To cope with points (i) (hierarchy), (ii) (Matsubara summation) and (vi) (simplicity) of the above listing the cutoff is chosen to be independent of the frequency q0, i.e. ~Þ: Rk ¼ Rk ðq

ð5:3Þ

This is certainly not optimal with respect to the points (iii) (ultraviolet regulator property) and (iv) (symmetry). More explicit, we use for the Bose gas Z  2 ~   Aðk DSk ¼ u q2  m2 Þhðk2  ~ q2  m2 Þ u q

¼

Z

ð5:4Þ u ðk2  ~ q2  m2 Þhðk2  ~ q2  m2 Þu;

q

with m2 ¼ U 0 ðqÞjq0 and m2 = 0 in the regime with spontaneous U(1) symmetry breaking. The cutoff in (5.4) is similar to the optimized cutoff proposed by Litim for relativistic systems [4]. It respects U(1)-symmetry but breaks Galilean invariance. The loop expressions obtained with the cutoff (5.4) are very simple since all Matsubara summations and momentum integrations can be performed analytically for the truncations investigated in this thesis. A drawback is that ok Rk does not serve as an ultraviolet cutoff for the Matsubara summation. In the flow equation for the pressure this leads to convergence problems for a truncation with linear and quadratic frequency terms, indeed. For the BCS-BEC crossover we use a slide modification of (5.4), Z ~2  lÞk2  ðq ~2  lÞÞhðk2  jq ~2  ljÞw DSk ¼ wy ðsignðq q

þ

Z q

ð5:5Þ u ðk2  ~ q2 =2Þhðk2  ~ q2 =2Þu:

42

5 Cutoff Choices

Again, this is an optimized choice in the sense of [2, 4] but now properly regularizes around the Fermi-surface for the fermions w. In ‘‘pure’’ diagrams involving only fermionic or only bosonic lines, it is still possible to perform frequency summations and momentum integrations in closed form. For mixed diagrams, one has to perform the momentum integration numerically. The question arises whether one can find a cutoff which fulfills all the requirements (i)–(vii). Already the combination of points (i), (ii) and (iii) restrict the space of possible functions Rk(q) quite a bit. A cutoff that respects Galilean invariance must be a function of the combination iq0 þ ~ q2 : A possible choice for bosons would be of the form Z DSk ¼ u Ak2 rk ðzÞu ð5:6Þ q

with z ¼ ðiq0 þ nq Þ=k2 ; nq ¼ ~ q2 þ m2 and the dimensionless function rk ðzÞ ¼

1 : 1 þ c 1 z þ c 2 z2 þ    þ c n zn

ð5:7Þ

The coefficients ci might be chosen for convenience, a simple choice is ci = 1. The ultraviolet properties become better if n is large. On the other hand, expressions are expected to be simpler for small n. The cutoff in (5.7) is analytic in the sense of complex analysis and fulfills therefore criterion (ii). It respects the symmetries of the microscopic propagator (iv) and at least for small n it also fulfills the requirement of simplicity (vi). For large n, real and positive argument z and ci = 1 we find that the regulator in (5.7) approaches the optimized regulator in (5.4). This can be seen from summing the geometric series in the denominator of (5.7). In order to fulfill point (i) (hierarchy of flow equations) one has to choose the coefficients ci in (5.7) conveniently.

References 1. 2. 3. 4. 5.

Mahan GD (1981) Many-particle physics. Plenum Press, New York Pawlowski JM (2007) Ann Phys (NY) 322:2831 Litim DF (2000) Phys Lett B 486:92 Litim DF (2001) Phys Rev D 64:105007 Litim DF (2001) Int J Mod Phys A 16:2081

Chapter 6

Investigated Models

In this chapter we explain the physical systems investigated in this thesis. We introduce the microscopic models in the form of Lagrange densities, discuss their scope and comment on experimental realizations. After a discussion of the symmetries of the microscopic models in the next chapter and a presentation of the approximation scheme thereafter, we discuss our results for few-body physics in Chap. 9 and for many-body physics in Chap. 10.

6.1 Bose Gas in Three Dimensions A gas of non-relativistic bosons with a repulsive pointlike interaction is one of the simplest interacting statistical systems. Since the first experimental realization [1–3] of Bose–Einstein condensation (BEC) [4–6] with ultracold gases of bosonic atoms, important experimental advances have been achieved, for reviews see [7–12]. Thermodynamic observables like the specific heat [13] or properties of the phase transition like the critical exponent m [14] have been measured in harmonic traps. Still, the theoretical description of these apparently simple systems is far from being complete. For ultracold dilute non-relativistic bosons in three dimensions, Bogoliubov theory gives a successful description of most quantities of interest [15]. This approximation breaks down, however, near the critical temperature for the phase transition, as well as for the low temperature phase in lower dimensional systems, due to the importance of fluctuations. One would therefore like to have a systematic extension beyond the Bogoliubov theory, which includes the fluctuation effects beyond the lowest order in a perturbative expansion in the scattering length. Such extensions have encountered obstacles in the form of infrared divergences in various expansions [16–18]. Only recently, a satisfactory framework has been found to cure these problems [19–21]. In this thesis, we extend this formalism to a nonvanishing temperature. We present a quantitative rather accurate picture of Bose–Einstein S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_6, Ó Springer-Verlag Berlin Heidelberg 2010

43

44

6 Investigated Models

condensation in three dimensions and find that the Bogoliubov approximation is indeed valid for many quantities. The same method is also applied for two spatial dimensions (see Sect. 6.2) and can also be applied for one dimension. For dilute non-relativistic bosons in three dimensions with repulsive interaction we find an upper bound on the scattering length a. This is similar to the ‘‘triviality bound’’ for the Higgs scalar in the standard model of elementary particle physics. As a consequence, the scattering length is at most of the order of the inverse effective ultraviolet cutoff K1 ; which indicates the breakdown of the pointlike approximation for the interaction at short distances. Typically, K1 is of the order of the range of the Van der Waals interaction. For dilute gases, where the interparticle distance n-1/3 is much larger than K1 ; we therefore always find a small concentration c = a n1/3. This provides for a small dimensionless parameter, and perturbation theory in c becomes rather accurate for most quantities. For typical experiments with ultracold bosonic alkali atoms one has K1  107 cm; n1=3  104 cm1 ; such that c . 103 is really quite small. Bosons with pointlike interactions can also be employed for an effective description of many quantum phase transitions at zero temperature, or phase transitions at low temperature T. In this case, they correspond to quasi-particles, and their dispersion relation may differ from the one of non-relativistic bosons, 2

~ p x ¼ 2M : We describe the quantum phase transitions for a general microscopic dispersion relation, where the inverse classical propagator in momentum and 2 p2 (in units where the frequency space takes the form G1 0 ¼ Sx  Vx þ ~ particle mass M is set to 1/2). We present the quantum phase diagram at T = 0 in dependence on the scattering length a and a dimensionless parameter ~v  V=S2 ; which measures the relative strength of the term quadratic in x in G-1 0 . In the limit S ! 0 ð~v ! 1Þ our model describes relativistic bosons.

6.1.1 Lagrangian Our microscopic action describes nonrelativistic bosons, with an effective interaction between two particles given by a contact potential. It is assumed to be valid on length scales where the microscopic details of the interaction are irrelevant and the scattering length is sufficient to characterize the interaction. The microscopic action reads  Z  1 2 2   S½u ¼ u ðSos  Vos  D  lÞu þ kðu uÞ ; ð6:1Þ 2 x

with x ¼ ðs; ~Þ; x

Z x

1

¼

ZT

Z ds

0

d3 x:

ð6:2Þ

6.1 Bose Gas in Three Dimensions

45

The integration goes over the whole space as well as over the imaginary time s, which at finite temperature is integrated on a circle of circumference b = 1/T according to the Matsubara formalism. We use natural units h ¼ kB ¼ 1: We also scale time and energy units with appropriate powers of 2M, with M the particle mass. In other words, our time units are set such that effectively 2M = 1. In these units time has the dimension of length squared. For standard non-relativistic bosons one has V = 0 and S = 1, but we also consider quasiparticles with a more general dispersion relation described by nonzero V. After Fourier transformation, the kinetic term reads Z   u ðqÞ iSq0 þ Vq20 þ ~ q2 uðqÞ; ð6:3Þ q

with q ¼ ðq0 ; ~Þ; q

Z

¼

q

Z Z

Z ;

q0

~ q

¼

~ q

1 ð2pÞ3

Z

d3 q:

At nonzero temperature, the frequency q0 = 2pT n is discrete, with Z 1 X ¼T ;

ð6:4Þ

ð6:5Þ

n¼1

q0

while at zero temperature this becomes Z q0

1 ¼ 2p

Z1 dq0 :

ð6:6Þ

1

The dispersion relation encoded in (6.3) obtains by analytic continuation Sx þ Vx2 ¼ ~ q2 =2M:

ð6:7Þ

In this thesis, we consider homogeneous situations, i.e. an infinitely large volume without a trapping potential. Many of our results can be translated to the inhomogeneous case in the framework of the local density approximation. One assumes that the length scale relevant for the quantum and statistical fluctuations is much smaller than the characteristic length scale of the trap. In this case, our results can be transferred by taking the chemical potential position dependent in ~ÞÞ; where Vt ðx ~Þ is the trapping potential. the form lð~Þ x ¼ 2Mðl  Vt ðx The microscopic action (6.1) is invariant under the global U(1) symmetry which is associated to the conserved particle number, u ! eia u:

ð6:8Þ

On the classical level, this symmetry is broken spontaneously when the chemical potential l is positive. In this case, the minimum of lu u þ 12kðu uÞ2 is situated

46

6 Investigated Models

at u u ¼ lk: The ground state of the system is then characterized by a macroscopic field u0, with u0 u0 ¼ q0 ¼ lk: It singles out a direction in the complex plane and thus breaks the U(1) symmetry. Nevertheless, the action itself and all modifications due to quantum and statistical fluctuations respect the symmetry. For V = 0 and S = 1, the situation is similar for Galilean invariance. At zero temperature, we can perform an analytic continuation to real time and the microscopic action (6.1) is then invariant under transformations that correspond to a change of the reference frame in the sense of a Galilean boost. It is easy to see that in the phase with spontaneous U(1) symmetry breaking also the Galilean symmetry is broken spontaneously: a condensate wave function, that is homogeneous in space and time, would be represented in momentum space by ~ÞdðxÞ: uðx; ~Þ p ¼ u0 ð2pÞ4 dð3Þ ðp

ð6:9Þ

~; this would Under a Galilean boost transformation with a boost velocity 2q transform according to uðx; ~Þ p !uðx  ~ q2 ; ~ p  ~Þ q ~  ~Þdðx q ~ q2 Þ: ¼ u0 ð2pÞ4 dð3Þ ðp

ð6:10Þ

This shows that the ground state is not invariant under such a change of reference frame. This situation is in contrast to the case of a relativistic Bose–Einstein condensate, like the Higgs boson field after electroweak symmetry breaking. A relativistic scalar transforms under Lorentz boost transformations according to uðpl Þ ! uððK1 Þlm pm Þ;

ð6:11Þ

such that a condensate wave function u0 ð2pÞ4 dð4Þ ðpl Þ !u0 ð2pÞ4 dð4Þ ððK1 Þlm pm Þ ¼ u0 ð2pÞ4 dð4Þ ðpl Þ

ð6:12Þ

transforms into itself. We will investigate the implications of Galilean symmetry for the form of the effective action in Chap. 7. An analysis of general coordinate invariance in nonrelativistic field theory can be found in [22].

6.2 Bose Gas in Two Dimensions Bose–Einstein condensation and superfluidity for cold nonrelativistic atoms can be experimentally investigated in systems of various dimensions [9–11]. Two dimensional systems can be achieved by building asymmetric traps, resulting in different characteristic sizes for one ‘‘transverse extension’’ lT and two ‘‘longitudinal extensions’’ l of the atom cloud [23–31]. For l  lT the system behaves effectively two-dimensional for all modes with momenta ~ q2 . l2 T : From the

6.2 Bose Gas in Two Dimensions

47

two-dimensional point of view, lT sets the length scale for microphysics—it may be as small as a characteristic molecular scale. On the other hand, the effective size of the probe l sets the scale for macrophysics, in particular for the thermodynamic observables. Two-dimensional superfluidity shows particular features. In the vacuum, the interaction strength k is dimensionless such that the scale dependence of k is logarithmic [32]. The Bogoliubov theory with a fixed small k predicts at zero ~j; temperature a divergence of the occupation numbers for small q ¼ jq ð2Þ ~Þ  nC d ðq ~Þ [15]. In the infinite volume limit, a nonvanishing condensate nðq 0 is allowed only for T = 0, while it must vanish for T [ 0 due to the nc ¼ q Mermin–Wagner theorem [33, 34]. On the other hand, one expects a critical temperature Tc where the superfluid density q0 jumps by a finite amount according to the behavior for a Kosterlitz–Thouless phase transition [35–38]. We will see that Tc/n (with n the atom-density) vanishes in the infinite volume limit l ! 1: Experimentally, however, a Bose–Einstein condensate can be observed for temperatures below a nonvanishing critical temperature Tc—at first sight in contradiction to the theoretical predictions for the infinite volume limit. A resolution of these puzzles is related to the simple observation that for all practical purposes the macroscopic size l remains finite. Typically, there will be a 0 =n; Tc/n or k on the dependence of the characteristic dimensionless quantities as q scale l. This dependence is only logarithmic. While kðn ¼ T ¼ 0; l ! 1Þ ¼ 0; ð q0 =nÞðT 6¼ 0; l ! 1Þ ¼ 0; ðTc =nÞðl ! 0Þ ¼ 0; in accordance with general theorems, even a large finite l still leads to nonzero values of these quantities, as observed in experiment. The description within a two-dimensional renormalization group context starts with a given microphysical or classical action at the ultraviolet momentum scale 1 KUV  l1 T : When the scale parameter k reaches the scale kph  l ; all fluctuations are included since no larger wavelength are present in a finite size system. The experimentally relevant quantities and the dependence on l can be obtained from Ckph : For a system with finite size l we are interested in Ckph ; kph ¼ l1 : If statistical quantities for finite size systems depend only weakly on l, they can be evaluated from Ckph in the same way as their thermodynamic infinite volume limit follows from C: Details of the geometry, etc. essentially concern the appropriate factor between kph and l-1. The microscopic model we use for the two-dimensional Bose gas is basically the one for the three-dimensional case in (6.1). The difference is that now ~ x and the space-integral are two-dimensional x ¼ ðs; ~Þ; x

Z x

1

¼

ZT

Z ds

d2 x;

ð6:13Þ

0

and similarly in momentum space. The dimensionless interaction parameter k in (6.1) describes now a reduced two-dimensional interaction strength and is directly related to the scattering length in units of the transverse extension a/lT.

48

6 Investigated Models

The few-body physics and the logarithmic scale-dependence of k is discussed in Sect. 9.1 (Chap. 9).

6.3 BCS–BEC Crossover Besides the bosons we also investigate systems with ultracold fermions. A qualitative new feature for fermions in comparison to bosons is the antisymmetry of the wavefunction and the tightly connected ‘‘Pauli blocking’’. Due to the antisymmetry of the wavefunction it is not possible to have two identical fermions in the same state. This feature has many interesting consequences. For example, a s-wave interaction between two identical fermions is not possible. This in turn implies that a gas of fermions in the same spin- (and hyperfine-) state has many properties of a free Fermi gas provided the p-wave and higher interactions are suppressed. The situation changes for a Fermi gas with two spin or hyperfine states. S-wave interactions and pairing are now possible. In the simplest case the densities of the two components are equal. Depending on the microscopic interaction the system has different properties. For a repulsive interaction one expects Landau Fermi liquid behavior (for not too small temperature) where many qualitative properties are as for the free Fermi gas [39]. For weak attractive interaction the theory of Baarden, Cooper and Schrieffer (BCS) [40, 41] is valid. Cooper-pairs are expected to form at small temperatures and the system is then superfluid. On the other hand, for strong attractive interaction one expects the formation of bound states of two fermions. These bound states are then bosons and undergo Bose–Einstein condensation (BEC) at small temperatures. Again, the system shows superfluidity. As first pointed out by Eagles [42] and Leggett [43] there is a smooth and continuous crossover (BCS–BEC crossover) between the two limits described above. Experimentally, this crossover can be realized using Feshbach resonances. The detailed mechanism how these resonances work can be found in the literature, e.g. [11, 12]. It is important that the scattering length a which serves as a measure for the s-wave interaction can be tuned to arbitrary values. As an example we consider the case of 6Li where the resonance was investigated in Refs. [44, 45] and is shown in Fig. 6.1. For magnetic fields in the range around B = 1,200 G the scattering length is relatively small and negative. In this regime the many-body ground state is of the BCS-type. Fermions with different spin and with momenta on opposite points on the Fermi surface form pairs. These Cooper pairs are (hyperfine-) spin singlets and have small or vanishing momentum. They are condensed in a Bose– Einstein condensate (BEC). The system is superfluid and the U(1) symmetry connected with particle number conservation is spontaneously broken. The macroscopic wavefunction of the BEC can be seen as an order parameter which is quadratic in the fermion field u0  hw1 w2 i: Increasing the temperature, the system will at some point undergo a second order phase transition to a normal state where the order parameter vanishes, u0 ¼ 0:

6.3 BCS–BEC Crossover

49

Fig. 6.1 Scattering length a in units of the Bohr radius a0 as a function of the magnetic field B for the lowest hyperfine states of 6Li [44, 45]

In the magnetic field range around B = 600 G in Fig. 6.1 the scattering length a is small and positive. There is now a bound state of two fermions in the spectrum and the ground state of the many-body system is BEC-like. Pairs of fermions with different spin constitute bound states (dimers) which are pairs in position space. The interaction between these dimers is repulsive and proportional to the scattering length between fermions. When this repulsive interaction is weak the dimers are completely condensed in a BEC at zero temperature (no quantum depletion of the condensate). Again the order parameter is the macroscopic wavefunction of this condensate which is quadratic in the fermion fields u0  hw1 w2 i: The phase transition between the superfluid state at small temperatures and the normal state is of second order, again. Now we come to the magnetic field in the intermediate crossover regime 700 G . B . 1; 000 G: The scattering length is now large and positive or large and negative with a divergence at B  834 G [45]. Since the two-body scattering properties are solely governed by the requirement of unitarity of the scattering matrix for a ! 1; the point B = 834 G is also called the ‘‘unitarity point’’. Due to the divergent scattering length one speaks of strongly interacting fermions. Perturbative methods for small coupling constants fail in the crossover regime. Non-perturbative methods show that the ground state is superfluid and governed by a order parameter u0  hw1 w2 i as before. The crossover from the BCS- to the BEC-like ground state is conveniently parameterized by the inverse scattering length in units of the Fermi momentum c1 ¼ ðakF Þ1 where the Fermi momentum is determined by the density n ¼ 3p1 2 kF3 (in units with  h ¼ kB ¼ 2M ¼ 1). The dimensionless parameter c-1 varies from large negative values on the BCS side to large positive values on the BEC side of the crossover. It crosses zero at the unitarity point. We will also use the Fermi energy which equals the Fermi temperature in our units EF ¼ TF ¼ kF2 : The quantitatively precise understanding of BCS–BEC crossover physics is a challenge for theory. Experimental breakthroughs as the realization of molecule condensates and the subsequent crossover to a BCS-like state of weakly attractively interacting fermions have been achieved [46–51]. Future experimental precision measurements could provide a testing ground for non-perturbative

50

6 Investigated Models

methods. An attempt in this direction are the recently published measurements of the critical temperature [52] and collective dynamics [53, 54]. A wide range of qualitative features of the BCS–BEC crossover is already well described by extended mean-field theories which account for the contribution of both fermionic and bosonic degrees of freedom [55, 56]. In the limit of narrow Feshbach resonances mean-field theory becomes exact [57, 58]. Around this limit perturbative methods for small Yukawa couplings [57] can be applied. Using -expansion [59–64] or 1/N-expansion [65] techniques one can go beyond the case of small Yukawa couplings. Quantitative understanding of the crossover at and near the resonance has been developed through numerical calculations using various quantum Monte–Carlo (QMC) methods [66–71]. Computations of the complete phase diagram have been performed from functional field-theoretical techniques, in particular from t-matrix approaches [72–76], Dyson–Schwinger equations [57, 77], 2-partice irreducible (2-PI) methods [78], and renormalization-group flow equations [79–82]. These unified pictures of the whole phase diagram [57, 65, 72–78, 80–82], however, do not yet reach a similar quantitative precision as the QMC calculations. In this thesis we discuss mainly the limit of broad Feshbach resonances for which all thermodynamic quantities can be expressed in terms of two dimensionless parameters, namely the temperature in units of the Fermi temperature T/TF and the concentration c = akF. In the broad resonance regime, macroscopic observables are to a large extent independent of the concrete microscopic physical realization, a property referred to as universality [57, 65, 80]. This universality includes the unitarity regime where the scattering length diverges, a-1 = 0 [83], however it is not restricted to that region. Macroscopic quantities are independent of the microscopic details and can be expressed in terms of only a few parameters. In our case this is the two-body scattering length a or, at finite density, the concentration c = akF. At nonzero temperature, an additional parameter is given by T/TF. For small and negative scattering length c1 \0; jcj 1 (BCS side), the system can be treated with perturbative methods. However, there is a significant decrease in the critical temperature as compared to the original BCS result. This was first recognized by Gorkov and Melik-Barkhudarov [84]. The reason for this correction is a screening effect of particle–hole fluctuations in the medium [85]. There has been no systematic analysis of this effect in approaches encompassing the full BCS–BEC crossover so far. In Sect. 10.3 (Chap. 10), we present an approach using the flow equation described in Chap. 2. We include the effect of particle–hole fluctuations and recover the Gorkov correction on the BCS side. We calculate the critical temperature for the second-order phase transition between the normal and the superfluid phase throughout the whole crossover. We also calculate the critical temperature at the point a-1 = 0 for different resonance widths DB: As a function of the microscopic Yukawa coupling hK ; we find a smooth crossover between the exact narrow resonance limit and the broad

6.3 BCS–BEC Crossover

51

resonance result. The resonance width is connected to the Yukawa coupling via DB ¼ h2K =ð8plM ab Þ where lM is the magnetic moment of the bosonic bound state and ab is the background scattering length.

6.3.1 Lagrangian We start with a microscopic action including a two-component Grassmann field w = (w1, w2), describing fermions in two hyperfine states. Additionally, we introduce a complex scalar field u as the bosonic degrees of freedom. In different regimes of the crossover, it can be seen as a field describing molecules, Cooper pairs or simply an auxiliary field. Using the resulting two-channel model we can describe both narrow and broad Feshbach resonances in a unified setting. Explicitly, the microscopic action at the ultraviolet scale K reads

S½w; u ¼

Z1=T

Z ds



d x wy ðos  D  lÞw 3

0

  1 þ u os  D  2l þ mK u 2   hK ðu w1 w2 þ h:c:Þ ;

ð6:14Þ

where we choose nonrelativistic natural units with h ¼ kB ¼ 2M ¼ 1; with M the mass of the atoms. The system is assumed to be in thermal equilibrium, which we describe using the Matsubara formalism. In addition to the position variable ~; x the fields depend on the imaginary time variable s which parameterizes a torus with circumference 1/T. The variable l is the chemical potential. The Yukawa coupling h couples the fermionic and bosonic fields. It is directly related to the width of the Feshbach resonance. The parameter m depends on the magnetic field and determines the detuning from the Feshbach resonance. Both h and m get renormalized by fluctuations, and the microscopic values hK ; and mK have to be determined by the properties of two body scattering in vacuum. For details, we refer to [80] and Sect. 9.2 in Chap. 9. More formally, the bosonic field u appears quadratically in the microscopic action in (6.14). The functional integral over u can be carried out. This shows that our model is equivalent to a purely fermionic theory with an interaction term   Z h2 Sint ¼  w ðp0 Þw ðp1 Þ Pu ðp1 þ p2 Þ 1 1 1 ð6:15Þ p1 ;p2 ;p01 ;p02

w2 ðp02 Þw2 ðp2 Þdðp1 þ p2  p01  p02 Þ;

52

6 Investigated Models

where p ¼ ðp0 ; ~Þ p and the classical inverse boson propagator is given by Pu ðqÞ ¼ iq0 þ

~ q2 þ mK  2l: 2

ð6:16Þ

On the microscopic level the interaction between the fermions is described by the tree level expression kw;eff ¼ 

h2 : x þ 12~ q2 þ mK  2l

ð6:17Þ

Here, x is the real-time frequency of the exchanged boson u: It is connected to the Matsubara frequency q0 via analytic continuation x = -iq0. Similarly, ~ q ¼~ p1 þ ~ p2 is the center of mass momentum of the scattering fermions w1 and w2 with momenta ~ p1 and ~ p2 ; respectively. The limit of broad Feshbach resonances, which is realized in current experiments, e.g. with 6Li and 40K corresponds to the limit h ! 1; for which the microscopic interaction becomes pointlike, with strength h2 =mK :

6.4 BCS–Trion–BEC Transition In the last section we discussed the interesting BCS–BEC crossover that is realized in a system consisting of two fermion species. We restricted ourselves to the case where the density for both components is equal. Interesting physics is also found if this constraint is released. The phase diagram of the imbalanced Fermi gas shows also first order phase transitions and phase separation, see [86, 87] and references therein. Another interesting generalization is to take a third fermion species into account. A very rich phase diagram can be expected for the general case where the total density is arbitrarily distributed to the different components. Even the simpler case where the densities for all three components are equal is far less understood as the analogous two-component case. For simplicity we restrict much of the discussion to the case where all properties of the three components apart from the hyperfine-spin are the same. In particular, we assume that they have equal mass, chemical potential and scattering properties. We label the different hyperfine states by 1, 2 and 3. The s-wave scattering length a12 for scattering between fermions of components 1 and 2 is the same as for scattering between fermions of species 2 and 3 or 3 and 1, a12 = a23 = a31 = a. Close to a common resonance where a ! 1 one expects the three-body problem to be dominated by the Efimov effect [88, 89]. This implies the formation of a three-body bound state (the ‘‘trion’’). Directly at the resonance an infinite tower of three-body bound states, the Efimov-trimers, exists. We refer to the Efimov trimer with the lowest lying energy as trion. The few-body physics is discussed in more detail in Sect. 9.3 (Chap. 9).

6.4 BCS–Trion–BEC Transition

53

The many-body phase diagram is far less understood. Not too close to the resonance one expects a superfluid ground state which is similar to the BCS ground state for a \ 0 or a BEC-like ground state for a [ 0. However, there are also some important differences. While in the two-component case the order parameter is a singlet under the corresponding SU(2) spin symmetry, the order parameter for the three component case with SU(3) spin symmetry is a (conjugate) triplet. In the superfluid phase the spin symmetry is therefore broken spontaneously. Due to some similarities with QCD this was called color superfluidity [90–95]. Between the extended BCS and BEC phase one can expect the ground state to be dominated by trions. Since trions are SU(3) singlets, the spin symmetry is unbroken in this regime such that there have to be true quantum phase transitions at the border to the BCS and BEC regimes. Such a trion phase has first been proposed for fermions in an optical lattice by Rapp et al. [96, 97], see also [98]. We will further discuss the many-body physics in Sect. 10.4 (Chap. 10). To the knowledge of the author, there have been no experiments addressing the many-body issues so far. Only recently, experiments with 6Li probing the few-body physics found interesting phenomena [99, 100]. For the case of 6Li the assumption of equal scattering properties for the three different species are not fulfilled. We will present a more general model where SU(3) symmetry is broken explicitly and where the parameters can be chosen to describe 6Li in Sect. 9.3 (Chap. 9). We also discuss the experiments and show how their results can be explained in our framework. The remainder of this section is devoted to the discussion of the microscopic model in the SU(3) symmetric case.

6.4.1 Lagrangian As our microscopic model we use an action similar to the one for the BCS–BEC crossover in (6.14)   Z  1 S¼ wy ðos  D  lÞw þ uy os  D  2l þ mu u 2 x   1 þ v os  D  3l þ mv v 3  1 þ hijk ui wj wk  ui wj wk 2       þ g ui wi v  ui wi v : The (Grassmann valued) fermion field has now three components w = (w1, w2, w3) and similar the boson field u ¼ ðu1 ; u2 ; u3 Þ¼ðw ^ 1 w2 ; w2 w3 ; w3 w1 Þ: In addition we also include a single component fermion field v. This trion field represents the totally antisymmetric combination w1w2w3. One choose the parameters such that g = 0 and

54

6 Investigated Models

mv ! 1 at the microscopic scale. The trion field v is then only an non-dynamical auxiliary field. We assumed in (6.18) that the fermions w1, w2, and w3 have equal mass M and chemical potential l. We also assume that the interactions are independent of the spin (or hyperspin) so that our microscopic model is invariant under a global SU(3) symmetry transforming the fermion species into each other. While the fermion field w = (w1, w2, w3) transforms as a triplet 3, the boson field u ¼ 3: The trion field v is a singlet under ðu1 ; u2 ; u3 Þ transforms as a conjugate triplet  SU(3). In concrete experiments, for example with 6Li [99], the SU(3) symmetry may be broken explicitly since the Feshbach resonances of the different channels occur for different magnetic field values and have different widths. In addition to the SU(3) spin symmetry our model is also invariant under a global U(1) symmetry w !eia w; u ! e2ia u; and v !e3ia v: The conserved charge related to this symmetry is the total particle number. Since we do not expect any anomalies the quantum effective action C ¼ Ck¼0 will also be invariant under SU(3) 9 U(1). Apart from the terms quadratic in the fields that determine the propagators, (6.18) contains the Yukawa-type interactions  h and  g: The energy gap parameters mu for the bosons and mv for the trions are sometimes written as m2u ¼ mu  2l; m2v = mv - 3l, absorbing an explicit dependence on the chemical potential l. In (6.18), the fermion field v can be ‘‘integrated out’’ by inserting the ðw; uÞ-dependent solution of its field equation into Ck : For m2v ! 1 this results in a contribution to a local three-body interaction, kuw ¼ g2 =m2v : Furthermore one may integrate out the boson field u; such that (for large m2u ) one replaces the parts containing u and v in Ck by an effective pointlike fermionic interaction Z 3 1 1  kw ðwy wÞ2 þ k3 wy w ; ð6:18Þ Ck;int ¼ 2 3! x

with kw ¼ 

h2 ; m2u

k3 ¼ 

h2 g2 : m4u m2v

ð6:19Þ

We note that the contribution of trion exchange to kuw or k3 depends only on the combination g2/m2v. The sign of g can be changed by v ! v; and the sign of g2 can be reversed by a sign flip of the term quadratic in v. Keeping the possible reinterpretation by this mapping in mind, we will formally also admit negative g2 (imaginary g).

References 1. Anderson MH, Ensher JR, Matthews MR, Wieman CE, Cornell EA (1995) Science 269:198 2. Bradley CC, Sackett CA, Tollett JJ, Hulet RG (1995) Phys Rev Lett 75:1687

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3. Davis KB, Mewes MO, Andrews MR, van Druten NJ, Durfee DS, Kurn DM, Ketterle W (1995) Phys Rev Lett 75:3969 4. Einstein A (1924) Sitzungsber Preuss Akad Wiss 261 5. Einstein A (1925) Sitzungsber Preuss Akad Wiss 3 6. Bose SN (1924) Z Phys 26:178 7. Dalfovo F, Giorgini S, Pitaevskii LP, Stringari S (1999) Rev Mod Phys 71:463 8. Leggett AJ (2001) Rev Mod Phys 73:307 9. Morsch O, Oberthaler M (2006) Rev Mod Phys 78:179 10. Bloch I, Dalibard J, Zwerger W (2008) Rev Mod Phys 80:885 11. Pethick CJ, Smith H (2002) Bose–Einstein condensation in dilute gases. Cambridge University Press, Cambridge 12. Pitaevskii L, Stringari S (2003) Bose–Einstein condensation. Oxford University Press, Oxford 13. Gati R, Esteve J, Hemmerling B, Ottenstein TB, Appmeier J, Weller A, Oberthaler MK (2006) New J Phys 8:189 14. Donner T, Ritter S, Bourdel T, Öttl A, Köhl M, Esslinger T (2007) Science 315:1556 15. Bogoliubov NN (1947) J Phys (Moscow) 11:23 16. Beliaev ST (1958) Sov Phys JETP 7:289–299 17. Gavoret J, Nozières P (1964) Ann Phys (N. Y.) 28:349 18. Nepomnyashchii AA, Nepomnyashchii YA (1975) JETP Lett 21:1 19. Castellani C, Di Castro C, Pistolesi F, Strinati GC (1997) Phys Rev Lett 78:1612 20. Pistolesi F, Castellani C, Di Castro C, Strinati GC (2004) Phys Rev B 69:024513 21. Wetterich C (2008) Phys Rev B 77:064504 22. Son DT, Wingate M (2006) Ann Phys (N. Y.) 321:197 23. Görlitz A, Vogels JM, Leanhardt AE, Raman C, Gustavson TL, Abo-Shaeer JR, Chikkatur AP, Gupta S, Inouye S, Rosenband T, Ketterle W (2001) Phys Rev Lett 87:130402 24. Rychtarik D, Engeser B, Nägerl H-C, Grimm R (2004) Phys Rev Lett 92:173003 25. Smith NL, Heathcote WH, Hechenblaikner G, Nugent E, Foot CJ (2005) J Phys B 38:223 26. Burger S, Cataliotti FS, Fort C, Maddaloni P, Minardi F, Inguscio M (2002) EPL 57:1 27. Köhl M, Moritz H, Stöferle T, Schori C, Esslinger T (2005) J Low Temp Phys 138:635 28. Orzel C, Tuchman AK, Fenselau ML, Yasuda M, Kasevich MA (2001) Science 291:2386 29. Spielman IB, Phillips WD, Porto JV (2007) Phys Rev Lett 98:080404 30. Hadzibabic Z, Stock S, Battelier B, Bretin V, Dalibard J (2004) Phys Rev Lett 93:180403 31. Hadzibabic Z, Kruger P, Cheneau M, Battelier B, Dalibard J (2006) Nature 441:1118 32. Lapidus IR (1986) Am J Phys 54:459 33. Mermin ND, Wagner H (1966) Phys Rev Lett 17:1133 34. Hohenberg PC (1967) Phys Rev 158:383 35. Berezinskii VL (1971) Sov Phys JETP 32:493 36. Berezinskii VL (1972) Sov Phys JETP 34:610 37. Kosterlitz JM, Thouless DJ (1973) J Phys C 6:1181 38. Nelson DR, Kosterlitz JM (1977) Phys Rev Lett 39:1201 39. Landau LD (1957) Sov Phys JETP 3:920 40. Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 106:162 41. Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 108:1175 42. Eagles DM (1969) Phys Rev 186:456 43. Leggett AJ (1980) Modern trends in the theory of condensed matter. Springer, Berlin 44. O’Hara KM, Hemmer SL, Granade SR, Gehm ME, Thomas JE, Venturi V, Tiesinga E, Williams CJ (2002) Phys Rev A 66:041401 45. Bartenstein M, Altmeyer A, Riedl S, Geursen R, Jochim S, Chin C, Denschlag JH, Grimm R, Simoni A, Tiesinga E, Williams CJ, Julienne PS (2005) Phys Rev Lett 94:103201 46. Regal CA, Greiner M, Jin DS (2004) Phys Rev Lett 92:040403 47. Zwierlein MW, Stan CA, Schunck CH, Raupach SMF, Kerman AJ, Ketterle W (2004) Phys Rev Lett 92:120403 48. Kinast J, Hemmer SL, Gehm ME, Turlapov A, Thomas JE (2004) Phys Rev Lett 92:150402

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49. Bourdel T, Khaykovich L, Cubizolles J, Zhang J, Chevy F, Teichmann M, Tarruell L, Kokkelmans SJJMF, Salomon C (2004) Phys Rev Lett 93:050401 50. Chin C, Bartenstein M, Altmeyer A, Riedl S, Jochim S, Hecker Denschlag J, Grimm R (2004) Science 305:1128 51. Partridge GB, Strecker KE, Kamar RI, Jack MW, Hulet RG (2005) Phys Rev Lett 95:020404 52. Luo L, Clancy B, Joseph J, Kinast J, Thomas JE (2007) Phys Rev Lett 98:080402 53. Altmeyer A, Riedl S, Kohstall C, Wright MJ, Geursen R, Bartenstein M, Chin C, Hecker Denschlag J, Grimm R (2007) Phys Rev Lett 98:040401 54. Wright MJ, Riedl S, Altmeyer A, Kohstall C, Guajardo ERS, Denschlag JH, Grimm R (2007) Phys Rev Lett 99:150403 55. Nozières P, Schmitt-Rink S (1985) J Low Temp Phys 59:195 56. Sá de Melo CAR, Randeria M, Engelbrecht JR (1993) Phys Rev Lett 71:3202 57. Diehl S, Wetterich C (2006) Phys Rev A 73:033615 58. Gurarie V, Radzihovsky L (2007) Ann Phys (N. Y.) 322:2 (January special issue 2007) 59. Nussinov Z, Nussinov S (2006) Phys Rev A 74:053622 60. Nishida Y, Son DT (2006) Phys Rev Lett 97:050403 61. Nishida Y, Son DT (2007) Phys Rev A 75:063617 62. Nishida Y (2007) Phys Rev A 75:063618 63. Arnold P, Drut JE, Son DT (2007) Phys Rev A 75:043605 64. Chen J-W, Nakano E (2007) Phys Rev A 75:043620 65. Nikolic´ P, Sachdev S (2007) Phys Rev A 75:033608 66. Carlson J, Chang S-Y, Pandharipande VR, Schmidt KE (2003) Phys Rev Lett 91:050401 67. Astrakharchik GE, Boronat J, Casulleras J, Giorgini S (2004) Phys Rev Lett 93:200404 68. Bulgac A, Drut JE, Magierski P (2006) Phys Rev Lett 96:090404 69. Bulgac A, Drut JE, Magierski P (2008) Phys Rev A 78:023625 70. Burovski E, Prokof’ev N, Svistunov B, Troyer M (2006) Phys Rev Lett 96:160402 71. Akkineni VK, Ceperley DM, Trivedi N (2007) Phys Rev B 76:165116 72. Haussmann R (1993) Z Phys B 91:291 73. Chen Q, Kosztin I, Levin K (2000) Phys Rev Lett 85:2801 74. Pieri P, Strinati GC (2000) Phys Rev B 61:15370 75. Perali A, Pieri P, Pisani L, Strinati GC (2004) Phys Rev Lett 92:220404 76. Pieri P, Pisani L, Strinati GC (2004) Phys Rev B 70:094508 77. Diehl S, Wetterich C (2007) Nucl Phys B 770:206 78. Haussmann R, Rantner W, Cerrito S, Zwerger W (2007) Phys Rev A 75:023610 79. Birse MC, Krippa B, McGovern JA, Walet NR (2005) Phys Lett B 605:287 80. Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:021602 81. Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:053627 82. Gubbels KB, Stoof HTC (2008) Phys Rev Lett 100:140407 83. Ho T-L (2004) Phys Rev Lett 92:090402 84. Gorkov LP, Melik-Barkhudarov TK (1961) Sov Phys JETP 13:1018 85. Heiselberg H, Pethick CJ, Smith H, Viverit L (2000) Phys Rev Lett 85:2418 86. Ketterle W, Zwierlein MW (2007) In: Inguscio M, Ketterle W, Salomon C (eds.), Ultra-cold Fermi Gases (Proceedings of the International School of Physics ‘‘Enrico Fermi’’) 87. Chevy F (2007) In: Inguscio M, Ketterle W, Salomon C (eds) Ultra-cold Fermi gases. (Proceedings of the International School of Physics ’’Enrico Fermi’’) 88. Efimov V (1970) Phys Lett B 33:563 89. Efimov V (1973) Nucl Phys A 210:157 90. Honerkamp C, Hofstetter W (2004) Phys Rev B 70:094521 91. Paananen T, Martikainen J-P, Törmä P (2006) Phys Rev A 73:053606 92. Paananen T, Törmä P, Martikainen J-P (2007) Phys Rev A 75:023622 93. Cherng RW, Refael G, Demler E (2007) Phys Rev Lett 99:130406 94. Zhai H (2007) Phys Rev A 75:031603 95. Bedaque PF, D’Incao JP (2006) e-print arXiv:cond-mat/0602525

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Chapter 7

Symmetries

The symmetries of the microscopic action play an important role for the flow equation method. Provided the measure of the functional integral shows no anomalies, the effective action C is invariant under the same symmetry transformations as the microscopic action S. This also holds for the flowing action Ck if the cutoff term DSk is invariant. If this is not the case, the constraints from symmetries on Ck (Ward identities) are modified [1]. In devising truncations, the symmetries are a very useful guiding principle as was already emphasized in Chap. 4. In the following section we will discuss the implication of Ward identities on the form of the effective action C (and the flowing action Ck ) in more detail. In Sect. 7.2 we discuss the second important implication of symmetries, the conservation of Noether currents. We concentrate the discussion to the Bose gas but it applies with minor modifications also for the systems with fermions. A discussion of the symmetries for the BCS–BEC crossover model (6.14) can be found in [2].

7.1 Derivative Expansion and Ward Identities Let us consider the microscopic model in (6.1) using the flow equation (2.20). We use a derivative expansion for the truncation of the effective average action with derivative operators up to four momentum dimensions Z  1 Ck ¼ Uðq; lÞ þ Z1 ðq;lÞðu os u  uos u Þ 2 x

  1 1 þ Z2 ðq;lÞðu ðDÞu þ uðDÞu Þ þ V1 ðq;lÞ u ðo2s Þu þ uðo2s Þu 2 2     1 2 2    þ V2 ðq;lÞðu ðos DÞu  uðos DÞu Þþ V3 ðq;lÞ u ðD Þu þ uðD Þu 2 ð7:1Þ S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_7, Ó Springer-Verlag Berlin Heidelberg 2010

59

60

7 Symmetries

Here, we employ the renormalized fields  1=2 u ; u¼A  q ¼ A  u u  q ¼ u u ¼ A

ð7:2Þ

and coupling functions U, Zi, Vi. We fix the wave function renormalization factor  such that Z1(q0, l0) = 1. Terms of the form qðDÞq or q(- q2s )q are not A included here, since they are expected to play a sub-leading role. For a systematic derivative expansion they have to be added—the terms with up to two derivatives can be found in [3]. In terms of dimensions, the operator qs counts as two space derivatives for the nonrelativistic model with V = 0, while it counts as one space dimension for the relativistic model with S = 0. We expand the k-dependent functions U(q, l), Z1(q, l), Z2(q, l), V1(q, l), V2(q, l) and V3(q, l) around the k-dependent minimum q0(k) of the effective potential and the k-independent value of the chemical potential l0 that corresponds to the physical particle number density n. For example, with Z1 = Z1(q0, l0), one has Z1 ðq; lÞ ¼ Z1 þ Z10 ðq0 ; l0 Þðq  q0 Þ ðlÞ

þ Z1 ðq0 ; l0 Þðl  l0 Þ þ   

ð7:3Þ

Let us concentrate on the non-relativistic model where S = 1, V = 0 in the microscopic action. At zero temperature, we can perform an analytic continuation to real time s = it. The microscopic action (6.1) is then S½u ¼  

Z1

Z dt

d3 x

1

 1   u ðiot  l  DÞu þ kðu uÞ : 2

ð7:4Þ

In addition to the global U(1) symmetry u ! eia u; u ! eia u ; space translations, rotations, time translations and the discrete symmetries parity and time reflection, two further symmetries constrain the form of the effective action C. In order to derive these constraints, we extend (7.4) to a t-dependent source l(t). First, there is a semi-local U(1) symmetry of the form uðt; ~Þ x ! eiaðtÞ uðt; ~Þ x u ðt; ~Þ x ! eiaðtÞ u ðt; ~Þ x l ! l þ ot a:

ð7:5Þ

This holds since the combination (- iqt - l) acts as a covariant derivative. In addition, we have the invariance under Galilean boost transformations of the fields 2

~tÞ uðt; ~Þ x ! u0 ðt; ~Þ x ¼ eiðq~ tq~x~Þ uðt; ~ x 2q 2

~tÞ: u ðt; ~Þ x ! u0 ðt; ~Þ x ¼ eiðq~ tq~x~Þ u ðt; ~ x 2q

ð7:6Þ

7.1 Derivative Expansion and Ward Identities

61

While the invariance of the interaction term under this symmetry is obvious, its realization for the kinetic term is more involved. Performing the transformation explicitly, one finds ~u ~u r u Du ! u Du  ~ q2 u u þ 2iq ð7:7Þ ~ u; ~u r u iot u ! u iot u þ ~ q2 u u  2iq such that indeed the combination iot þ D

ð7:8Þ

leads to this invariance. On the other hand, the validity of the Galilean symmetry for an effective action guarantees that only the combination (7.8) or powers of this operator act on u. An operator of the form ðiot þ CDÞ with C 6¼ 1 would break the symmetry. (Note, that Dq is also invariant.) Both the semi-local U(1) symmetry and the Galilean symmetry are helpful only at zero temperature. At nonzero temperature, the analytic continuation to real time is no longer useful. An analog version of the semi-local U(1) transformation for Euclidean time s would involve the imaginary part of the chemical potential l, which has no physical meaning. The dependence of physical quantities on l + l* is not restricted. In addition, the Galilean symmetry is broken explicitly by the thermal heat bath. Combining semi-local U(1) symmetry and Galilean symmetry at T = 0, we find that the derivative operators iqt, D and the chemical potential term (l - l0) are combined to powers of the operator D ¼ ðiot  ðl  l0 Þ  DÞ:

ð7:9Þ

In addition to powers of that operator acting on u, only spatial derivatives of terms, that are invariant under U(1) transformations, like qDq, may appear. Since the symmetry transformations act linearly on the fields, the full effective action C½u is also invariant. This also holds for the average action Ck ½u, provided that the cutoff term DSk ½u is invariant. We can write the effective action as an expansion in the operator D Z  1~ C½u ¼ U0 ðqÞ þ ZðqÞ ðu ðiot  ðl  l0 Þ  DÞu þ c:cÞ 2 x ð7:10Þ   1~   þ VðqÞ u ðiot  ðl  l0 Þ  DÞ2 u þ c:c þ    2 Performing the Wick rotation back to Euclidean time, we can compare this to (7.1), and find for T = 0 the relations

62

7 Symmetries

~ Z1 ðq; l0 Þ ¼ Z2 ðq; l0 Þ ¼ ZðqÞ; ~ V1 ðq; l0 Þ ¼ V2 ðq; l0 Þ ¼ V3 ðq; l0 Þ ¼ VðqÞ; ðlÞ

~ 0 ðq0 ÞÞ; ~ 0 Þ þ q0 V Z1 ðq0 ; l0 Þ ¼ 2ðVðq

ð7:11Þ

ðlÞ ~ 0 Þ; Z2 ðq0 ; l0 Þ ¼ 2Vðq

and therefore ~ 0 Þ þ q0 Z~ 0 ðq0 ÞÞ; a ¼ ðZðq ~ 0 Þq0 ; nk ¼ Zðq ~0

ð7:12Þ

~ 00

b ¼ ð2Z ðq0 Þ þ q0 Z ðq0 ÞÞ: We next compute the inverse propagator in a constant background field by expanding Ck to second order in the fluctuations around this background. For this purpose, it is convenient to decompose 1 uðs; ~Þ x ¼ u0 þ pffiffiffiðu1 ðs; ~Þ x þ iu2 ðs; ~Þ x Þ: 2

ð7:13Þ

The constant condensate field u0 can be chosen to be real without loss of generality. The fluctuating real fields are the radial mode u1 and the Goldstone mode u2, pffiffiffi and q ¼ q0 þ 2u0 u1 þ 12u21 þ 12u22 : The truncation of the effective average action (7.1) reads in that basis Z  pffiffiffi 1 Ck ½u ¼ Uðq; lÞ þ Z1 ðq; lÞði 2u0 os u2 þ iu1 os u2  iu2 os u1 Þ 2 x

pffiffiffi 1 þ Z2 ðq; lÞð 2u0 ðDÞu1 þ u1 ðDÞu1 þ u2 ðDÞu2 Þ 2 pffiffiffi 1 þ V1 ðq; lÞð 2u0 ðo2s Þu1 þ u1 ðo2s Þu1 þ u2 ðo2s Þu2 Þ 2 pffiffiffi þ V2 ðq; lÞði 2u0 ðos DÞu2 þ iu1 ðos DÞu2  iu2 ðos DÞu1 Þ  pffiffiffi 1 2 2 2 þ V3 ðq; lÞð 2u0 ðD Þu1 þ u1 ðD Þu1 þ u2 ðD Þu2 Þ : 2 ð2Þ

ð7:14Þ

The inverse propagator Ck can be inferred from an expansion to second order in u1 and u2. We keep the linear order in l - l0, which will be needed for the flow equation for the density. This yields Z  1 1 Uðq0 ; l0 Þ þ U ðlÞ ðl  l0 Þ þ ðU 0 þ 2q0 U 00 Þu21 þ U 0 u22 Ck ½u ¼ 2 2 x

þ

 1 ðlÞ Z1 þ Z10 q0 þ Z1 ðl  l0 Þ ðiu1 os u2  iu2 os u1 Þ 2

7.1 Derivative Expansion and Ward Identities

 1 ðlÞ 1 þ 2Z20 q0 þ Z2 ðl  l0 Þ ðu1 ðDÞu1 Þ 2  1 ðlÞ þ 1 þ Z2 ðl  l0 Þ ðu2 ðDÞu2 Þ 2   1 ðlÞ þ V1 þ 2V10 q0 þ V1 ðl  l0 Þ u1 ðo2s Þu1 2   1 ðlÞ þ V1 þ V1 ðl  l0 Þ u2 ðo2s Þu2 2 

63

þ

ð7:15Þ

ðlÞ

þ V2 þ V20 q0 þ V2 ðl  l0 Þ ðiu1 ðos DÞu2  iu2 ðos DÞu1 Þ   1 ðlÞ þ V3 þ 2V30 q0 þ V3 ðl  l0 Þ u1 ðD2 Þu1 2    1 ðlÞ þ V3 þ V3 ðl  l0 Þ u2 ðD2 Þu2 ; 2 where we dropped the argument (q0, l0) at several places and used the implicit rescaling condition Z2(q0, l0) = 1. In a simple truncation, we take at l = l0 only S ¼ Z1 ðq0 ; l0 Þ þ Z10 ðq0 ; l0 Þq0 ; V ¼ V1 ðq0 ; l0 Þ

ð7:16Þ

into account. We neglect the contribution of the other couplings, i.e. set Z20 ¼ V2 ¼ V3 ¼ V10 ¼ V20 ¼ V30 ¼ 0:. As shown above, it follows from the ~ Z1ðlÞ ¼ 2ðV ~þ symmetry requirements at zero temperature, that V1 ¼ V2 ¼ V3 ¼ V; ~ 0 q0 Þ and Z ðlÞ ¼ 2V: ~ The truncation V2 = V3 = 0 therefore violates the Galilean V 2 symmetry, as does our choice of the cutoff term *Rk. Within our approximation, (l) it is consistent to set Z(l) 1 = Z2 = 2V at zero temperature. Also the deviations from this relation at finite temperature are neglected for simplicity in this thesis. This yields the truncation used in order to obtain the numerical results which will be discussed in Chap. 10.

7.1.1 Propagator and Dispersion The inverse propagator is given by the second functional derivative of the effective action 0* 1   d ( u ðqÞ ACk ( Cð2Þ ¼ @ * 1 d u1 ðpÞ ; d u2 ðpÞ ð7:17Þ d u2 ðqÞ ¼ G1 dðp  qÞ; and we find from the truncation (7.1)

64

7 Symmetries

G1 ¼



pffiffiffiffiffiffi

þ 2qV10 Þq20 ; q0 2K H þ 2J þ ðV p1ffiffiffiffiffiffi : q0 2K ; H þ V1 q20

ð7:18Þ

Here we use the abbreviations H ¼ Z2~ p2  V3~ p4 þ U 0 J ¼ qZ20 ~ p2  qV30~ p4 þ qU 00  2 ~2 : 2K ¼ Z1 þ qZ10  2ðV2 þ qV20 Þp

ð7:19Þ

At zero temperature, we can analytically continue to real time q0 ! ix; and find pffiffiffiffiffiffi

þ 2qV10 Þx2 ; ix 2K H þ 2J  ðV p1ffiffiffiffiffiffi G1 ¼ : ð7:20Þ ix 2K ; H  V1 x2 The dispersion relation is found from the on shell condition det G1 ¼ 0

ð7:21Þ

which yields   H 2 þ 2HJ  2 HðV1 þ qV10 Þ þ JV1 þ KÞ x2 þ V1 ðV1 þ 2qV10 Þx4 ¼ 0:

ð7:22Þ

The solutions for x define the dispersion relation. We find two branches, according to   1 ðx2 Þ ¼ H V1 þ qV10 þ JV1 þ K 0 V1 ðV1 þ 2qV1 Þ

      2 1=2  ðK þ JV1 Þ2 þ 2H K V1 þ qV10  JVqV10 þ H 2 qV10 : ð7:23Þ In the phase with spontaneous symmetry breaking, the (+) branch of this solution is an ‘‘optical mode’’, while the (-) branch is a sound mode. The microscopic 0 00 sound velocity is cS ¼ ox op jp¼0 : Using q = q0, U = 0, U = k and Z2 = 1, we find c2S ¼

1 ðZ1 þq0 Z10 Þ2 2kq0

¼ þ V1

S2

2kq0 : þ 2kq0 V

ð7:24Þ

The ‘‘optical mode’’ has at vanishing spatial momentum the frequency ~2 ¼ 0Þ ¼ x2þ ðq

2kq0 ðZ1 þ q0 Z10 Þ þ 0 V1 þ 2q0 V1 V1 ðV1 þ 2q0 V10 Þ

which diverges x2þ ! 1 in the limit V1 ! 0:

ð7:25Þ

7.2 Noethers Theorem

65

7.2 Noethers Theorem In the following we further discuss the role of continuous symmetries of the microscopic action S[u]. Since all these symmetries are linear in the fields, the full effective action C½u is also symmetric. From Noether’s theorem it follows that there exists a conserved current jl ¼ ðj0 ;~ j Þ connected with every such symmetry. If the action is formulated as an integral over the imaginary time s the conservation equation implies for the current ~~ os jðsÞ þ r j ¼ 0:

ð7:26Þ

At zero temperature, we can perform a Wick rotation to real time, s ! it; and (7.26) takes the usual form ~~ ot jðtÞ þ r j ¼ 0:

ð7:27Þ

The Noether charge C = $d3 x j(t) is conserved in time, i.e. dtd C ¼ 0: This holds if ~ j falls off sufficiently fast at spatial infinity. At finite temperature however, the situation is different. A simple analytic continuation to real time is no longer possible, since the configuration space is now a torus with periodicity 1/T in the s-direction. Instead, we can integrate (7.26) over complex time s, giving ~~ ~ r J ¼r

Z1=T

dsj~ ¼ jðsÞ ð0Þ  jðsÞ ð1=TÞ ¼ 0:

ð7:28Þ

0

From the symmetry, it now follows that there exists a solenoidal vector field or R three component current ~ J ¼ s~ j: A global symmetry of an action C½u (where C could be replaced by S or Ck if appropriate) can be formulated in its infinitesimal form as C½u þ su ¼ C½u;

ð7:29Þ

with  independent of x. Here s is the infinitesimal generator of the symmetry transformation. For a local transformation, where  depends on x,  ¼ ðxÞ; we can expand Z  C½u þ su ¼ C½u þ ðol ÞJ l þ ðol om ÞKlm þ    : ð7:30Þ x

The global symmetry implies that the expansion on the r.h.s. of (7.30) starts with ql. Here and in the following it is implied that  as well as its derivatives are infinitesimal, i.e. we keep only terms that are linear in . The index l goes over (0, 1, 2, 3), representing (t, x1, x2, x3) in the real time case and (s, x1, x2, x3) for imaginary time. Equation (7.30) implies for arbitrary u(x)

66

7 Symmetries

Z 

dC½u su  ðol ÞJ l  ðol om ÞKlm þ    du

 ¼ 0:

ð7:31Þ

x

Our notation is for real fields and implies a summation over components, if dC dC  appropriate. In a complex basis one replaces dC dusu by dusu þ du su : Equation (7.31) is valid for all field configurations u and not only for those that fulfill the field equation dC½u du ¼ 0: In consequence, the integrand is a total derivative dC½u su  ðol ÞJ l  ðol om ÞKlm þ    ¼ ol ðjl  þ jlm om  þ   Þ: du

ð7:32Þ

We can now specialize to ol  ¼ ol om  ¼    ¼ 0 and find dC½u suðxÞ ¼ ol jl : du This defines the Noether current jl. For solutions of the field equation, the current jl is conserved, ql jl = 0. For a given x we can also specialize to ðxÞ ¼ 0;

ol ðxÞ 6¼ 0;

ol om ðxÞ ¼ 0; . . .;

ð7:33Þ dC½u du

¼ 0;

ð7:34Þ

which leads to jl ¼ J l  om jml :

ð7:35Þ

This process can be continued, leading us to a whole tower of identities for the conserved current jl. If the action C½u includes derivatives only up to a finite order n, i.e. can be written in the form Z C½u ¼ Lðu; ou; oou; . . .; oðnÞ uÞ; ð7:36Þ x

the expansion on the right hand side of (7.30) only contains terms up to order q(n) such that the tower of equations for jl can be solved. Moreover, for homogeneous situations where dC½u du is solved by a constant u, the second term on the r.h.s. of (7.35) vanishes since it includes a derivative. We have then jl ¼ J l : A convenient way to find the local currents employs parameters (x) that decay sufficiently fast at infinity such that we can partially integrate (7.31)   Z dC½u su þ ol J l  ol om Klm þ    ¼ 0: ðxÞ ð7:37Þ du x

7.2 Noethers Theorem

67

This yields the local identity dC½u su ¼ ol J l þ ol om Klm     du

ð7:38Þ

An expansion of the l.h.s. in derivatives often yields substantial information on J l ; etc. by inspection. Our construction yields a unique conserved local current jl for every generator of a continuous symmetry. We note, however, that ajl + bl is also a conserved local current if a and bl are independent of x. This remark is important if we want to associate jl with the current for a physical quantity. A rotation invariant setting implies bi = 0, but b0 and a may differ from zero. After these general considerations we now specialize to nonrelativistic real time actions of the form C½u ¼

Z1

Z dt

d 3 xLðu; ðiot þ DÞu; ðiot þ DÞ2 u; . . .Þ:

ð7:39Þ

1

We assume, that C invariant under the same symmetries as the action (7.4). From the symmetry under time translations u ! u þ ðst Þu ¼ u þ ot u L ! L þ ot L ¼ L þ ol ðDl0 LÞ;

ð7:40Þ

we find a conserved current (jE)l. Up to a possible additive constant its t-component is the energy density, while the spatial components describe energy flux density. The multiplicative constant a gets fixed if we choose the units to measure energy. The choice h ¼ 1 corresponds to a = 1. Similarly, the invariance under spatial translations u ! u þ i ðsM Þi u ¼ u  i oi u L ! L  i oi L ¼ L  i ol ðdli LÞ

ð7:41Þ

implies a conserved current (jM)li for each spatial direction i = 1, 2, 3. Up to an additive constant (bM)0i the t-component is the conserved momentum density, pi = (jM)0i + (bM)0i , while the spatial components can be interpreted as a momentum flux density, with the diagonal components (jM)ii describing pressure. From the global U(1) symmetry u ! u þ ðsC Þu ¼ u  iu u ! u þ ðsC Þu ¼ u þ iu L!L

ð7:42Þ

we can infer the conservation of the current (jC)l associated to the conserved particle number. In order to identify the total particle number with the charge of

68

7 Symmetries

this current, $d3x (jC)0, we need to fix a possible multiplicative constant a. For this purpose, we use the Galilean boost invariance, described already in (7.6). It reads in its infinitesimal form u ! u þ i ðsG Þi u ¼ u þ 2i toi u  ii xi u u ! u þ i ðsG Þi u ¼ u þ 2i toi u þ ii xi u i

L!Lþ

ð7:43Þ

ol ð2dli tLÞ;

and the conserved charge of (sG) is the center of mass, again up to an additive constant. The generator (sG) can be decomposed as ðsG Þi ¼ xi ðsC Þ  2tðsM Þi :

ð7:44Þ

ðjG Þli ¼ xi ðjC Þl  2tðjM Þli :

ð7:45Þ

This implies for the current

Specializing to the t-component, identifying the momentum density pi = (jM)0i + (bM)0i and reintroducing the particle mass 2M = 1 we find ðjG Þ0i ¼ xi ðjC Þ0  t

pi  ðbM Þ0i : M

ð7:46Þ

From this we can conclude that up to an additive constant (jC)0 is the particle density n = (jC)0 + (bC)0. pffiffiffiffiffi For the effective action (7.10) we find for l = l0 and constant uðxÞ ¼ q0 the current ~ 0 Þq0 : ðjC Þ0 ¼ Zðq

ð7:47Þ

~ 0 Þ ¼ 1; this gives (jC)0 = q0. At zero Using the normalization condition Zðq temperature, this is the particle density and the additive constant (bC)0 vanishes. At nonzero temperature we can compare to (10.14) and find (bC)0 = nT. For completeness we also mention the symmetry under spatial rotations uðt; ~Þ x ! uðt; R1~Þ x x Lðt; ~Þ x ! Lðt; R1~Þ;

ð7:48Þ

~

with orthogonal matrix Rij ¼ ðeig~J Þij ; generators (Ji)jk = ieijk, and eijk the antisymmetric tensor in three dimensions. The infinitesimal transformation reads uðt; ~Þ x ! uðt; ~Þ x þ gi eijk xk oj uðt; ~Þ x Lðt; ~Þ x ! Lðt; ~Þ x þ gi ol ðeijk xk Dlj LÞ:

ð7:49Þ

The time component of the conserved current (jR)li is, of course, the angular momentum density.

References

69

References 1. Ellwanger U (1994) Phys Lett B 335:364 2. Diehl S, Floerchinger S, Gies H, Pawlowski JM, Wetterich C (2009). e-print arXiv:0907.2193 3. Wetterich C (2008) Phys Rev B 77:064504

Chapter 8

Truncated Flow Equations

8.1 Bose Gas We start with the nonrelativistic Bose gas. The microscopic model for this system is shown in (6.1). The infrared cutoff function we use is shown in (5.4). For the approximate solution of the flow equation (2.20) we use a truncation with up to two derivatives Z      Vo  2s u    Ck ¼ Sos  AD u ð8:1Þ x   q; lÞg;   l0 Þ u ðos  DÞ u þ Uð þ2Vðl ¼u  u  . This particular form is motivated by a more systematic derivative with q expansion and an analysis of symmetry constraints (Ward identities) in Sect. 7.1  q, the renor 1=2 u  ; q ¼ A (Chap. 7). We introduce the renormalized fields u ¼ A S  V malized kinetic coefficients S ¼ A , V ¼ A and we express the effective potential in terms of the renormalized invariant q, with  q; lÞ: Uðq; lÞ ¼ Uð

ð8:2Þ

This yields Ck ¼

Z



  u Sos  D  Vo2s u

x

ð8:3Þ



þ2Vðl  l0 Þu ðos  DÞu þ Uðq; lÞg: For the effective potential, we use an expansion around the k-dependent minimum q0(k) of the effective potential and the k-independent value of the chemical

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_8, Ó Springer-Verlag Berlin Heidelberg 2010

71

72

8 Truncated Flow Equations

potential l0 that corresponds to the physical particle number density n. We determine q0(k) and l0 by the requirements ðoq UÞðq0 ðkÞ; l0 Þ ¼ 0 for all k  ðol UÞðq0 ; l0 Þ ¼ n at k ¼ 0:

ð8:4Þ

More explicitly we take a truncation for U(q, l) of the form Uðq; lÞ ¼ Uðq0 ; l0 Þ  nk ðl  l0 Þ   þ m2 þ aðl  l0 Þ ðq  q0 Þ 1 þ ðk þ bðl  l0 ÞÞðq  q0 Þ2 : 2

ð8:5Þ

In the symmetric phase we have q0 = 0, while in the phase with spontaneous symmetry breaking, we have m2 = 0. In summary, the flow of Ck for fixed l = l0 is described by four running renormalized couplings q0, k, S and V. In addition, we  A computation of n requires a flow need the anomalous dimension g ¼ kok lnA. equation of nk, which involves the couplings linear in l - l0, namely a and b. The pressure is calculated by following the k-dependence of the height of the  nk, pk, a, b depend on k minimum pk = -U(q0, l0). All couplings q0, k, S, V, A, and T. The physical renormalized couplings obtain for k ? 0. They specify the thermodynamic potential U(q0, l0) as well as suitable derivatives of the potential and the correlation function. The ‘‘initial values’’ at the scale k ¼ K are determined by the requirement CK ½u ¼ S½u;

ð8:6Þ

using the microscopic action S½u in (6.1). This implies the initial values q0;K ¼ nK ¼ hðl0 Þl0 =kK ; m2K ¼ hðl0 Þl0 ;  K ¼ 1; aK ¼ 1; bK ¼ 0: A

ð8:7Þ

K . The We remain with the free microscopic couplings kK , SK ¼ SK ; VK ¼ V coupling kK will be replaced by the scattering length a in Sect. 9.1 (Chap. 9). We further choose units for s where SK ¼ 1: Then our second free coupling is ~v ¼

VK K2 ¼ VK K2 : S2K

ð8:8Þ

In consequence, besides the thermodynamic variables T and l0 our model is characterized by two free parameters, a and ~v. Often, we will concentrate on ‘‘standard’’ non-relativistic bosons with a linear s derivative, such that ~v ¼ 0. The scattering length a remains then the only free parameter. In the vacuum, where T = n = 0, this sets the relevant unit of length. It is convenient to work with real fields u1;2 ðxÞ; uðxÞ ¼ p1ffiffi2ðu1 ðxÞ þ iu2 ðxÞÞ, with Fourier components

8.1 Bose Gas

73

uj ðs; ~Þ x ¼

Z

Z Z

eiqx uj ðqÞ ¼

q

q0

eiðq0 sþq~x~Þ uj ðq0 ; ~Þ: q

ð8:9Þ

~ q

 becomes a 2 9 2 matrix in the space of u 1 The inverse propagator for the fields u  2 , given by the second functional derivative of Ck . For a real constant and u pffiffiffiffiffiffi  2 ðxÞ ¼ 0 the latter becomes diagonal in  1 ðxÞ ¼ 2 q; u background field u momentum space ð2Þ

0 Ck ðq; q0 Þ ¼ G1 k ðqÞdðq  q Þ:

ð8:10Þ

For our truncation one has at l = l0 1

G

 2  ~ Sq0 q þ Vq20 þ U 0 þ 2qU 00 ;  ¼A : ~ Sq0 ; q2 þ Vq20 þ U 0

ð8:11Þ

). In the phase with Here, primes denote derivatives with respect to q (not q spontaneous symmetry breaking, the infrared cutoff in (5.4) adds to the diagonal term in (8.11) a piece  2 ~  k ðq ~Þ ¼ Aðk q2 Þhðk2  ~ q2 Þ: Ar

ð8:12Þ

q2 \k2 , thus providing for an This effectively replaces ~ q2 ! k2 in (2.20) whenever ~ efficient infrared regularization.

8.1.1 Flow Equations for the Effective Potential We project the flow equation of the effective average action onto equations for the coupling constants by using appropriate background fields and taking functional derivatives. The flow equation for the effective potential obtains by using a spaceand time-independent background field in (2.20), with t ¼ lnðk=KÞ   1 ot U q ¼ kok U q ¼ f ¼ 2

Z

 k Þ: tr Gk ot ðAr

ð8:13Þ

  ~ ~  P11 ; P12 ¼A ~ 22 ; ~ 21 ; P P

ð8:14Þ

q

The propagator Gk is here determined from  1  rk G1 ¼ G þ A k 0 with

0 rk



74

8 Truncated Flow Equations

~ 11 ¼ k2 þ Vq20 þ U 0 þ 2qU 00 þ 2Vðl  l0 Þq ~2 ; P ~ 21 ¼ P ~ 12 ¼ Sq0 þ 2Vðl  l0 Þq0 ; P

ð8:15Þ

~ 22 ¼ k2 þ Vq20 þ U 0 þ 2Vðl  l0 Þq ~2 : P Again primes denote a differentiation with respect to q. We switch to renormalized fields by making a change of variables in the differential equation (8.13)  ot U q ¼ f þ gqU 0 :

ð8:16Þ

We can now derive the flow equations for the couplings q0(k) and k(k) by appropriate differentiation of (8.16) with respect to q. The flow equation for U is given more explicitly in Appendix B.1. Differentiation with respect to l yields the flow of nk, a, b. We use in detail d d 2

k¼ o U ðq0 ; l0 Þ dt dt q



¼ o2q ot U ðq0 ; l0 Þ þ o3q U ðq0 ; l0 Þot q0 ;  ¼ o2q fq0 ;l0 þ 2gk;

ð8:17Þ

where we recall, that o3q U ¼ 0 in our truncation. To determine the flow equation of q0, we use the condition that U0 (q0) = 0 for all k, and therefore d ðoq UÞðq0 ; l0 Þ ¼ 0; dt

ðoq ot UÞðq0 ; l0 Þ þ o2q U ðq0 ; l0 Þot q0 ¼ 0;

ð8:18Þ

1 1  ot q0 ¼  ðoq ot UÞðq0 ; l0 Þ ¼ gq0  oq fq0 ;l0 : k k We show the flow of k and q0 in Figs. 8.1 and 8.2 for n = 1, T = 0 and different values of kK (with ~v ¼ 0). The change in q0 is rather modest. This will be different for nonzero temperature. The flow of nk is given by d d nk ¼ ðol UÞðq0 ; l0 Þ dt dt ¼ ðol ot UÞðq0 ; l0 Þ  ðoq ol UÞðq0 ; l0 Þot q0  ¼ ol fq0 ;l0  aot q0 ; and similar for the flow of a and b,

ð8:19Þ

8.1 Bose Gas

75

Fig. 8.1 Flow of the interaction strength k with the scale parameter k for different start values, corresponding to kK ¼ 103 (dotted), kK ¼ 0:044 (solid) and kK ¼ 0:0026 (dashed). The first case is plotted for zero density (n = 0) only, while the last two cases are plotted also for unit density (n = 1). The curves n = 0 and n = 1 are identical within the plot resolution. The solid and the dashed curve correspond to a = 10-3 and a = 10-4, respectively

d d a ¼ ðoq ol UÞðq0 ; l0 Þ dt dt



¼ ðoq ol ot UÞðq0 ; l0 Þ þ o2q ol U ðq0 ; l0 Þot q0  ¼ oq ol fq0 ;l0 þ ga þ bðgq0 þ ot q0 Þ;

d d 2 b¼ oq ol U ðq0 ; l0 Þ dt dt



¼ o2q ol ot U ðq0 ; l0 Þ þ o3q ol U ðq0 ; l0 Þot q0 ;  ¼ o2q ol fq0 ;l0 þ 2gb;

where the last equation holds since o3q ol U ¼ 0 in our truncation.

Fig. 8.2 Flow of the minimum of the effective potential for n = 1. The parameters for the solid and the dashed curves are the same as in Fig. 8.1

ð8:20Þ

76

8 Truncated Flow Equations

8.1.2 Kinetic Coefficients  A  and the flow equations for S and V, we have For a derivation of g ¼ ðot AÞ= to evaluate the flow equation (2.20) for a background field depending on q0 and ~. q We use an analytic continuation q0 = ix and obtain the flow equation for S from d d  ¼ iX1 o ot Ck jx¼0 ; ot ðSAÞ ox d u2 ðx; 0Þ d u1 ðx; 0Þ with four-volume X ¼ T1

R

~ x.

ð8:21Þ

The projection prescription for V is

d d  ¼ X1 o ot ðV AÞ ot Ck jx¼0 ; 2 ox d u2 ðx; 0Þ d u2 ðx; 0Þ

ð8:22Þ

 and similar for A d d  ¼ X1 o ot A ot Ck j~p2 ¼0 : 2 d ~Þ d u2 ð0; ~Þ p ~ u2 ð0; p op

ð8:23Þ

After the functional differentiation, we evaluate the expressions in (8.21), (8.22), and (8.23) at homogeneous background fields. These calculations are a little intricate, but standard and straightforward in principle. A more detailed description of the calculation can be found in [1]. More explicit flow equations are given below. Eventually, it is always possible to perform the Matsubara sums and also the spatial momentum integration analytically. In Fig. 8.3 we show the  S and V at zero temperature and for density n = 1. The kinetic flow of A,  starts on the large scale with A  ¼ 1, increases a little around coefficient A k = n1/3 and saturates to a constant. In contrast, the coefficient S starts to  For very tiny scales k, S would decrease after a short period of increase with A. finally go to zero. The frequency dependence of the propagator is then governed by the quadratic frequency coefficient V. In three spatial dimensions, however, this decrease of S is so slow that it is not relevant on the length scales of experiments. This is one of the reasons why Bogoliubov theory, which neglects the appearance of V, describes experiments with ultracold bosonic quantum gases in three dimensions with so much success. The coefficient V is always generated in the phase with spontaneous symmetry breaking [1]. Its k-dependence is also shown in Fig. 8.3.  and V. We show now our flow equation obtained for the kinetic coefficients S, A We neglect all contributions from momentum dependent vertices. In other words,  ¼ Z2 and V = V1. In our we use q-independent constants S ¼ Z1 þ q0 Z10 ; A 0 truncation with Z 2 = V2 = V3 = 0, and with the cutoff (5.4), we can perform all momentum integrations analytically, leading us to

8.1 Bose Gas

77

Fig. 8.3 Flow of the kinetic  (solid), coefficients A S (dashed) and V for a scattering length a = 10-3, temperature T = 0 and density n = 1



 X g ot V ¼ gV  1  32vd k2þd k2 q0 T dþ2 n  2 2     2  2  k S þ k V þ S þ 2k2 V kq0  2V S2 þ k2 V þ Vkq0 xn 2  3V 3 xn 4      3  d k4 þ 2k2 kq0 þ S2 þ 2k2 V þ 2Vkq0 xn 2 þ V 2 xn 4 ;   X g 32vd k2þd Sk2 q0 ot S ¼ gS  1  T dþ2 n       k4  2kq0 k2 þ kq0 þ S2 xn 2 þ 2V k2  kq0 xn 2 þ V 2 xn 4      3  d k4 þ 2k2 kq0 þ S2 þ 2k2 V þ 2Vkq0 xn 2 þ V 2 xn 4 ; X  ot A 16vd k2þd k2 q0 g¼  ¼T A n   4    2  d k þ 2k2 kq0 þ S2 þ 2k2 V þ 2Vkq0 xn 2 þ V 2 xn 4 : ð8:24Þ Here, d is the number of spatial dimensions, vd ¼ ð2dþ1 pd=2 Cðd=2ÞÞ1 and xn = 2pTn. The Matsubara sums over n can be performed analytically by using (B.3) and derivatives thereof. In the limit T ? 0, the Matsubara frequencies are R P 1 1 dq . continuous xn ? q0 and the sum becomes an integral T n ! 2p 0 1

78

8 Truncated Flow Equations

8.2 BCS–BEC Crossover Let us now come to the approximation used to investigate the BCS–BEC crossover model. Our truncation of the flowing action reads

Ck ½v ¼

Z1=T

Z

d x wy ðos  D  lÞw 3

ds 0

  1  k ð   Zu os  A  þU þu D u q; lÞ: u 2   w2 w1 Þ :  hð u w1 w2 þ u

ð8:25Þ

 q; lÞ contains no derivatives and is a function of Here the effective potential Uð  (see below) our trunca¼u  u  and l. Besides the couplings parameterizing U q  u , Zu and h. The tion contains three further k-dependent (‘‘running’’) couplings A truncation in (8.25) can be motivated by a systematic derivative expansion and analysis of symmetry constraints (Ward identities), see [2, 3] and Sect. 7.1 (in Chap. 7). The truncation in (8.25) does not yet incorporate the effects of particle–hole fluctuations and we will come back to this issue in Sect. 10.3  1=2 uq ; q ¼ A , renormalized (Chap. 10). In terms of renormalized fields u ¼ A u u pffiffiffiffiffiffi      q; lÞ, couplings Zu ¼ Zu =Au ; h ¼ h= Au and effective potential Uðq; lÞ ¼ Uð (8.25) reads

Ck ½v ¼

Z1=T

Z ds

d3 x wy ðos  D  lÞw

0

 1 þ u Zu os  D u þ Uk ðq; lÞ: 2      ðu w1 w2 þ uw2 w1 Þ : 



ð8:26Þ

For the effective potential, we use an expansion around the k-dependent location of the minimum q0(k) and the k-independent value of the chemical potential l0 that corresponds to the physical particle number density n. We determine q0(k) and l0 by the requirements ðoq UÞðq0 ðkÞ; l0 Þ ¼ 0 for all k  ðol UÞðq0 ; l0 Þ ¼ n

at k ¼ 0:

ð8:27Þ

8.2 BCS–BEC Crossover

79

More explicitly, we employ a truncation for U(q, l) of the form Uðq; lÞ ¼ Uðq0 ; l0 Þ  nk ðl  l0 Þ þ ðm2 þ aðl  l0 ÞÞðq  q0 Þ 1 þ kðq  q0 Þ2 : 2

ð8:28Þ

In the symmetric or normal gas phase, we have q0 = 0, while in the regime with spontaneous symmetry breaking, we have m2 = 0. The atom density n ¼ oU=ol corresponds to nk in the limit k ? 0. In total, we have the running couplings m2(k), k(k), a(k), nk, Zu(k) and h(k). (In the phase with spontaneous symmetry breaking m2 is replaced by q0.)  u . We project the flow In addition, we need the anomalous dimension g ¼ kok lnA of the average action Ck on the flow of these couplings by taking appropriate (functional) derivatives on both sides of (2.20). We thereby obtain a set of coupled nonlinear differential equations which can be solved numerically. At the microscopic scale k ¼ K the initial values of our couplings are determined from CK ¼ S with the microscopic action in (6.14). This gives m2 ðKÞ ¼ mK  2l0 , q0 = 0, kðKÞ ¼ 0, Zu ðKÞ ¼ 1, hðKÞ ¼ hK , aðKÞ ¼ 2 and nK ¼ 3p2 l0 hðl0 Þ. The initial values mK and hK can be connected to the two particle scattering in vacuum close to a Feshbach resonance. For this purpose one follows the flow of m2(k) and h(k) in vacuum, i.e. l0 = T = n = 0 and extracts the renormalized parameters m2 = m2(k = 0), h = h(k = 0). The scattering length a obeys a = -h2/(8pm2) and the renormalized Yukawa coupling h determines the width of the resonance. Broad Feshbach resonances with large h become independent of h.

8.2.1 Flow of the Effective Potential For our choice of Rk in (5.5) and with the approximation (8.26), we can perform the momentum integration and the Matsubara sums explicitly pffiffiffi 5

2k ð0Þ kok Uk ¼ gAu qUk0 þ 2 1  2gAu =5 sB 3p Zu k5 ð0Þ  2 lð~ lÞsF ; 3p

l  1Þð~ l  1Þ3=2 : lð~ lÞ ¼ hð~ l þ 1Þð~ l þ 1Þ3=2  hð~

ð8:29Þ

~ ¼ l=k2 and the anomalous Here we use the dimensionless chemical potential l  u =olnk. The threshold functions sB and sF depend on dimension gA u ¼ olnA 0 2 0 w1 ¼ Uk =k ; w2 ¼ ðUk þ 2qUk00 Þ=k2 ; w3 ¼ h2u q=k4 , as well as on and the

80

8 Truncated Flow Equations

dimensionless temperature T~ ¼ T=k2 . They describe the decoupling of modes if the effective masses wj get large. They are normalized to unity for vanishing arguments and T~ ! 0 and read ð0Þ sB

ð0Þ

sF

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ w1 1 þ w2 ¼ þ 1 þ w2 1 þ w1   pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  þ NB ð 1 þ w1 1 þ w2 =Su Þ 2   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  NF ð 1 þ w3 Þ 1 þ w3 2

ð8:30Þ

(For s(0) B , only all its q derivatives vanish for w1  w2 ! 1. The remaining constant part is a shortcoming of the particular choice of the cutoff acting only on spacelike momenta.) In (8.30) the temperature dependence arises through the Bose and Fermi functions NB=F ðÞ ¼

1 : 1

ð8:31Þ

e=T~

For T~ ! 0 the ‘‘thermal parts’’  NB;F vanish, whereas for large T~ one has ð0Þ sF ! T~ 1 ;

ð0Þ ~ u ð1 þ w1 Þ1 ð1 þ w2 Þ1 : sB ! 2TZ

ð8:32Þ

In this high-temperature limit the fermionic fluctuations become unimportant. For the boson fluctuations only the n = 0 Matsubara frequency contributes substantially. Inserting (8.32) into (8.29) yields the well known flow equations for the classical three-dimensional scalar theory with U(1) symmetry [4, 5]. In Appendix B.2 we derive the flow equations (8.29) and discuss the threshold functions sB and sF more explicitly. We recall that for k ! 0 pk ¼ Uk ðq0 ; l0 Þ is the pressure. The flow equations for pk, m2 or q0(k), and k are given by   ok pk ¼ ok Uk q0  Uk0 q0 ok q0 ;  ok m2 ¼ ok Uk0 q¼0 for q0 ¼ 0;    1 ok q0 ¼  Uk00 q0 ok Uk0 q0 for  ok k ¼ ok Uk00 q0 :

ð8:33Þ q0 [ 0;

Taking a derivative of (8.29) with respect to q one obtains for T~ ¼ 0

8.2 BCS–BEC Crossover

kok Uk0

81

pffiffiffi   2k 2 g ¼  þ 2 1 3p Zu d þ 2 Au h

i ð1;0Þ ð0;1Þ ð3Þ ð0;1Þ  2qðUk00 Þ2 sB;Q þ 3sB;Q þ 4q2 Uk00 Uk sB;Q gAu ðUk0

þ

qUk00 Þ

ð8:34Þ

k 2 ð1Þ h lð~ lÞsF;Q : 3p2 u

(1,0) (1) The threshold functions s(0,1) B,Q , sB,Q , and sF,Q are defined in Appendix B.2 and describe again the decoupling of the heavy modes. They can be obtained from q (0) ~ derivatives of s(0) B and sF . Setting q = 0 and T ! 0, we can immediately infer from (8.34) the running of m2 in the symmetric regime.

kok m2 ¼ kok Uk0 ¼ gAu m2 þ

k 2 ð1Þ h lð~ lÞsF;Q ðw3 ¼ 0Þ: 3p2

ð8:35Þ

One can see from (8.35) that fermionic fluctuations lead to a strong renormalization of the bosonic ‘‘mass term’’ m2. In the course of the renormalization group flow from large scale parameters k (ultraviolet) to small k (infrared) the parameter m2 decreases strongly. When it becomes zero at some scale k [ 0 the flow enters the regime where the minimum of the effective potential Uk is at some nonzero value q0. This is directly related to spontaneous breaking of the U(1) symmetry and to local order. If q0 = 0 persists for k ? 0 this indicates superfluidity.  u ; Zu ; hu , (8.29) is a nonlinear differential equation for Uk, which For given A depends on two variables k and q. It has to be supplemented by flow equations for  u ; Zu ; h. The flow equations for the wave function renormalization Zu and the A  u cannot be extracted from the effective potential, but are gradient coefficient A obtained from the following projection prescriptions,  o   ot Zu ¼ ot ðP u Þ12 ðq0 ; 0Þ q0 ¼0 ; oq0   u ¼ ot 2 o ð P  Þ ð0; ~Þ q ~¼0 ot A q ; 2 u 22 ~ oq

ð8:36Þ

where the momentum dependent part of the propagator is defined by  d2 Ck  pffiffiffiffiffi  u Þ ðqÞdðq þ q0 Þ: ¼ ðP  ab d ua ðqÞd ub ðq0 Þ u1 ¼ 2q0 ;u2 ¼0

ð8:37Þ

The computation of the flow of the gradient coefficient is rather involved, since the ~  ~Þ q2 , where ~ p is the loop loop depends on terms of different type,  ðq p 2; ~ momentum. An outline of the calculation and explicit expressions can be found in [2].

82

8 Truncated Flow Equations

8.3 BCS–Trion–BEC Transition Now we turn to the truncation used to investigate the model with three fermion species in (6.18). For this model the focus will be on the few-body problem where the approximation scheme can be simpler in some respects. We use the following truncation for the average action Z

Ck ¼ wy ðos  D  lÞw þ uy os  D=2 þ m2u u x



 2 þ hijk ui wj wk  ui wj wk =2 þ ku uy u =2

  þ v os  D=3 þ m2v v þ g ui wi v  ui wi v      þ kuw ui wi uj wj :

ð8:38Þ

Here we use as always natural nonrelativistic units with h ¼ kB ¼ 2M ¼ 1, where M is the mass of the original fermions. The integral in (8.38) goes over homogeneous space and over imaginary time as appropriate for the Matsubara forR R R 1=T malism x ¼ d3 x 0 ds. On the level of the three-body sector, the symmetry of the problem would allow also for a term  wy wuy u in (8.38). This term plays a similar role as for the case of two fermion species, where it was investigated in [6]. The qualitative features of the three-body scattering are dominated by the term  kuw in (8.38). The quantitative influence of a term  wy wuy u on the flow equations was also investigated in [7]. At the microscopic scale k ¼ K, we use the initial values of the couplings in (8.38) g ¼ ku ¼ kuw ¼ 0 and m2v ? ?. Then the fermionic field v decouples from the other fields and is only an auxiliary field which is not propagating. However, depending on the parameters of our model we will find that v, which describes a composite bound state of three original fermions v = w1w2w3, becomes a propagating degree of freedom in the infrared. The initial values of the boson energy gap mu and the Yukawa coupling h will determine the scattering length a between fermions and the width of the resonance, see below. The pointlike limit (broad resonance) corresponds to mu ! 1, h2 ? ? where the limits are taken such that the effective renormalized four fermion interaction remains fixed. In (8.38) we use  v¼A  1=2  1=2 ðkÞw;  1=2 renormalized fields u¼A u; w ¼ A v; with u ðkÞ v ðkÞ w 2 2      =A u ; h ¼ Au ðKÞ ¼ Aw ðKÞ ¼ Av ðKÞ ¼ 1, and renormalized couplings m ¼ m u

u

  2 ; m2 ¼ m  v ; g ¼ g=  1=2   1=2  1=2  1=2  2v =A ku ¼  ku =A ðA and h=ðA u Aw Þ; v Au Av Þ, u v    kuw ¼ kuw =ðAu Aw Þ. To derive the flow equations for the couplings in (8.38) we have to specify an infrared regulator function Rk. Here we use the particularly simple function Rk ¼ rðk2  ~ p2 Þhðk2  ~ p2 Þ;

ð8:39Þ

8.3 BCS–Trion–BEC Transition

83

where r = 1 for the fermions w, r = 1/2 for the bosons u, and r = 1/3 for the composite fermionic field v. This choice has the advantage that we can derive analytic expressions for the flow equations and that it is optimized in the sense of [8].

References 1. 2. 3. 4. 5. 6. 7. 8.

Wetterich C (2008) Phys Rev B 77:064504 Diehl S, Floerchinger S, Gies H, Pawlowski JM, Wetterich C (2009) e-print arXiv:0907.2193 Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:021602 Wetterich C (1993) Phys Lett B 301:90 Berges J, Tetradis N, Wetterich C (2002) Phys Rept 363:223 Diehl S, Krahl HC, Scherer M (2008) Phys Rev C 78:034001 Moroz S, Floerchinger S, Schmidt R, Wetterich C (2009) Phys Rev A 79:042705 Litim DF (2000) Phys Lett B 486:92

Chapter 9

Few-Body Physics

In this chapter we investigate the flow equation applied to nonrelativistic systems in the limit of vanishing temperature and density (‘‘vacuum limit’’). The effective action as the generating functional of the one-particle irreducible correlation functions contains then directly the information about the few-body physics such as as scattering properties, binding energies etc. Usually, the flow equations simplify substantially in the vacuum limit. Whenever they can be solved exactly, this leads to the same results as an quantum mechanical treatment. This is of course expected, since the flow equation itself is exact. Since the flow equation and the quantum mechanical treatment are equivalent, one might conjecture that many (all?) problems that can be solved in the quantum mechanical formalism find an equivalent solution in the flow equation context. (It is another question whether the flow equation solution is particular simple or gives any new insights.) This is especially interesting since the flow equation is more general. As briefly discussed at the beginning of Chap. 2, one might consider functional renormalization as a formulation of quantum field theory. In the flow equation method both quantum mechanics and quantum field theory find a unified framework. An important feature of nonrelativistic field theory in the vacuum limit is a hierarchy between n-point correlation functions. This implies that the n-particle problem can be solved independent of the n + 1-particle problem. More formally, the flow equation of the n-point function depends only on the correlation functions of lower order and the n-point function itself but not on correlation functions of higher order. To see this, let us consider a nonrelativistic field theory for the field u. This may be a spinor containing both fermionic and bosonic degrees of freedom. We only require that all components have particle number one. In other words u must not contain any composite degrees of freedom such as bound states and no exchange particles such as photons or phonons. Technically, the requirement is that the microscopic model is invariant under the global U(1) transformation

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_9, Ó Springer-Verlag Berlin Heidelberg 2010

85

86

9 Few-Body Physics

u ! eia u

ð9:1Þ

where the charge a is the same for all components of u. To fulfill the condition above it will sometimes be necessary to ‘‘integrate out’’ composite fields. The second premise is that the microscopic propagator for the field u is of the nonrelativistic form iq0 þ ~ q2 =ð2MÞ þ m  l: Here we work with a imaginary-time (or Matsubara) frequency q0. The mass M and the gap parameter m might be different for the different components of u. One can then proof the following theorem: The flow equation for the n-point function Gn is independent of correðnÞ lation functions Gm with order m [ n. Here we use a notation where Gn ¼ Ck for ð2Þ

n [ 2 and G2 ¼ ðCk þ Rk Þ1 is the regularized propagator. The proof will be given in appendix B.3. We now come to the discussion of the flow equations in the vacuum limit for the different models introduced in Chap. 6.

9.1 Repulsive Interacting Bosons In this section we will investigate the flow equations for the Bose gas in the limit of vanishing temperature and density. The correlation functions then directly contain the information about few-body scattering. For a repulsive Bose gas with pointlike interaction in three or two dimensions, the main result is that the interaction strength is bounded from above. This is a vacuum screening effect and formally very similar to the triviality in the Higgs sector of the standard model of particle physics. In three dimensions the interaction strength has the canonical dimension of a length and the bound is of the order of the microscopical scale k . K1 : In two dimensions k is dimensionless and the running is only logarithmically. But let us first concentrate on the three-dimensional case and discuss the case of d = 2 thereafter.

9.1.1 Vacuum Flow Equations and their Solution for d 5 3 The vacuum is defined to have zero temperature T = 0 and vanishing density n = 0, which also implies q0 = 0. The interaction strength k at the scale k = 0 determines the four point vertex at zero momentum. It is directly related to the scattering length a for the scattering of two particles in vacuum, which is experimentally observable. We therefore want to replace the microscopic coupling kK by the renormalized coupling a. In our units (2M = 1), one has the relation a¼

1 kðk ¼ 0; T ¼ 0; n ¼ 0Þ: 8p

ð9:2Þ

9.1 Repulsive Interacting Bosons

87

The vacuum properties can be computed by taking for T = 0 the limit n ? 0. We may also perform an equivalent and technically simple computation in the symmetric phase by choosing m2(k = K) such, that m2(k ? 0) = 0. This guarantees that the boson field u is a gap-less propagating degree of freedom. We first investigate the model with a linear s-derivative, SK ¼ 1; VK ¼ 0: Projecting the flow equation (2.20 in Chap. 2) to the truncation in (8.3), (8.5) in Chap. 8 in Chap. 8, we find the following equations: ot m2 ¼ 0  2 2 3=2 k ðk  m2 Þ Hðk2  m2 Þ: ot k ¼ 6 k 2 p2 S

ð9:3Þ

 ¼ 0, g = 0, ot a ¼ 0, and The propagator is not renormalized, ot S ¼ ot V ¼ ot A one finds ot nk ¼ 0: The coupling b is running according to  ot b ¼

 2 3=2 1 2 1 2 ðk  m2 Þ ak  k bk Hðk2  m2 Þ: 3 3 k 4 p2 S

ð9:4Þ

Since b appears only in its own flow equation, it is of no further relevance in the vacuum. Also, no coupling V is generated by the flow and we have therefore set V = 0 on the r.h.s. of (9.3) and (9.4). Inserting in (9.3) the vacuum values m2 = 0 and S = 1, we find ot k ¼

k 2 k : 6p2

ð9:5Þ

The solution kðkÞ ¼

1 1 kK

þ

1 6p2 ðK

 kÞ

ð9:6Þ

tends to a constant for k?0, k0 = k(k = 0). The dimensionless variable k~ ¼ kkS goes to zero, when k goes to zero. This shows the infrared freedom of the theory. For fixed ultraviolet cutoff, the scattering length a¼

k0 1 ¼ ; 4 8p k8pK þ 3p K

ð9:7Þ

as a function of the initial value kK, has an asymptotic maximum amax ¼

3p : 4K

The relation between a and kK is shown in Fig. 9.1.

ð9:8Þ

88

9 Few-Body Physics

Fig. 9.1 Scattering length a in dependence on the microscopic interaction strength kK (solid line). The asymptotic 3p maximum amax ¼ 4K is also shown (dashed line)

As a consequence of (9.8), the nonrelativistic bosons in d = 3 are a ‘‘trivial theory’’ in the sense that the bosons become noninteracting in the limit K ? ?, where a ? 0. The upper bound (9.8) has important practical consequences. It tells us, that whenever the ‘‘macrophysical length scales’’ are substantially larger than the microscopic length K-1, we deal with a weakly interacting theory. As an example, consider a boson gas with a typical inter-particle distance substantially larger than K-1. (For atom gases K-1may be associated with the range of the Van der Waals force.) We may set the units in terms of the particle density n, n = 1. In these units K is large, say K = 103. This implies a very weak interaction, a.2:5  103 : In other words, the scattering length cannot be much larger than the microscopic length K-1. For such systems, perturbation theory will be valid in many circumstances. We will find that the Bogoliubov theory indeed gives a reliable account of many properties. Even for an arbitrary large microphysical coupling (kK ? ?), the renormalized physical scattering length a remains finite. Let us mention, however, that the weak interaction strength does not guarantee the validity of perturbation theory in all circumstances. For example, near the critical temperature of the phase transition between the superfluid and the normal state, the running of k(k) will be different from the vacuum. As a consequence, the coupling will vanish proportional to the inverse correlation length n-1 as T approaches Tc ; k  T 2 n1 : Indeed, the phase transition will be characterized by the non-perturbative critical exponents of the Wilson–Fisher fixed point. Also for lower dimensional systems, the upper bound (9.8) for k0 is no longer valid—for example the running of k is logarithmic for d = 2. For our models with VK = 0, the upper bound becomes dependent on VK. It increases for VK [ 0. In the limit SK ? 0, it is replaced by the well known ‘‘triviality bound’’ of the four dimensional relativistic model, which depends only logarithmically on K. Finally, for superfluid liquids, as 4 He; one has n * K3, such that for a * K-1 one finds a large concentration c. The situation for dilute bosons seems to contrast with ultracold fermion gases in the unitary limit of a Feshbach resonance, where a diverges. One may also think about a Feshbach resonance for bosonic atoms, where one would expect a large

9.1 Repulsive Interacting Bosons

89

scattering length for a tuning close to resonance. In this case, however, the effective action does not remain local. It is best described by the exchange of molecules. The scale of nonlocality is then given by the gap for the molecules, mM. Only for momenta ~ q2 \ m2M the effective action becomes approximately local, such that K = mM for our approximation. Close to resonance, the effective cutoff is low and again in the vicinity of a-1.

9.1.2 Logarithmic Running in Two Dimensions It is well known that in two dimensions the scattering properties cannot be determined by a scattering length as it is the case in three-dimensions. In experiments where a tightly confining harmonic potential restricts the dynamics of a Bose gas to two dimensions, the interaction strength has a logarithmic energy dependence in the two-dimensional regime. For low energies and in the limit of vanishing momentum the scattering amplitude vanishes. In our formalism this is reflected by the logarithmic running of the interaction strength k(k) in the vacuum, where both temperature and density vanish. In general, the flow equations in vacuum describe the physics of few particles like for example the scattering properties or binding energies. Following the above calculation for the three dimensional case, we find the flow equations for the interaction strength ðt ¼ lnðk=KÞÞ ot k ¼

k2 ð k 2  m 2 Þ 2 hðk  m2 Þ: 4k2 pS

ð9:9Þ

 ¼ ot m2 ¼ 0; Since in vacuum the propagator is not renormalized, ot S ¼ ot V ¼ ot A we set S = 1 and V = 0 on the right-hand side of (9.9). The vacuum corresponds to m2 = 0 and we obtain the flow equation ot k ¼

k2 : 4p

ð9:10Þ

The vacuum flow is purely driven by quantum fluctuations. It will be modified by the thermal fluctuations for T = 0 and for nonzero density n. The solution of (9.10) kðkÞ ¼

1 1 kK

þ

1 4plnðK=kÞ

ð9:11Þ

goes to zero logarithmically for k ? 0, k(k = 0) = 0. In contrast to the threedimensional system, the flow of the interaction strength k does not stop in two dimensions. To relate the microscopic parameter kK to experiments exploring the

90

9 Few-Body Physics

scattering properties, we have to choose a momentum scale qexp, where experiments are performed. To a good approximation the relevant interaction strength can be computed from (9.11) by setting k = qexp. If not specified otherwise, we will use a renormalized coupling k ¼ kðkph Þ: For our calculation we also have to use a microscopic scale K below which our approximation of an effectively two-dimensional theory with pointlike interaction becomes valid. Our two-dimensional computation only includes the effect of fluctuations with momenta smaller then K. In experiments K-1 is usually given either by the range of the van der Waals interaction or by the length scale of the potential that confines the system to two dimensions. We choose in the following K ¼ 10;

kph ¼ 102 :

ð9:12Þ

At this stage the momentum or length units are arbitrary, but we will later often choose the density to be n = 1, so that we measure length effectively in units of the interparticle spacing n-1/2. For typical experiments with ultracold bosonic alkali atoms one has n1=2  104 cm: The flow of k(k) for different initial values kK is shown in Fig. 9.2. Following the flow from K to kph yields the dependence of k ¼ kðkph Þ on kK as displayed in Fig. 9.3. It follows from (9.11) that for positive initial values kK the interaction strength k is bounded by k\

4p  1:82: lnðK=kph Þ

ð9:13Þ

The last equation holds for our choice of K and kph : We emphasize that our bound holds only if the interactions are approximately pointlike for all momenta below K. Close to a Feshbach resonance this may not be true and our

Fig. 9.2 Flow of the interaction strength k(k) at zero temperature and density for different initial values kK = 100, kK = 10, kK = 4, kK = 2, kK = 1, and kK = 0.4 (from top to bottom)

4

3

2

1

4

3

2

1

0

1

2

9.1 Repulsive Interacting Bosons Fig. 9.3 Interaction strength k at the macroscopic scale kph ¼ 102 in dependence on the microscopic interaction strength kK at K = 10 (solid lines). The upper bound 4p kmax ¼ lnðK=k is also shown Þ ph

91

1.5

1.0

(dashed line) 0.5

5

10

15

20

25

30

formalism would need to be extended by considering nonlocal interactions or introducing an additional field for the exchanged two-atom state in the ‘‘closed channel’’. In other words, close to a Feshbach resonance the effective cutoff K for the validity of a two-dimensional model with pointlike interactions may be substantially lower, and the upper bound on k correspondingly higher.

9.2 Two Fermion Species: Dimer Formation In this section, we discuss the few-body properties of the BCS–BEC crossover model in (6.14) in Chap. 6. The treatment here is condensed to the basics, a more extensive discussion can be found in [1]. The vacuum problem can be structured in the following way: The two-body sector describes the pointlike fermionic two-body interactions. It involves couplings up to fourth order in the fermion field in a purely fermionic setting. In the language with a composite boson field u  ww we have in addition terms quadratic in u or *wwu. The two-body sector decouples from the sectors involving a higher number of particles, described by higher-order interactions parameters such as the dimer–dimer scattering (which corresponds to interactions of eighth order in w in a fermionic language). This decoupling reflects the situation in quantum mechanics, where a two-body calculation (in vacuum) never needs input from states with more than two particles. For two fermion species and not too far from unitarity the solution of the twobody problem fixes the independent renormalized couplings completely. The couplings in the sectors with higher particle number then are derived quantities which can be computed as functions of the parameters of the two-body problem. Here, we will study the dimer–dimer or molecular scattering length aM as an important example. On the technical side, the vacuum limit leads to a massive simplification of the diagrammatic structure as compared to the nonzero density and temperature system. This is discussed in [1–3].

92

9 Few-Body Physics

9.2.1 Two-body Problem The two-body problem is best solved in terms of the bare couplings. Their flow equations read  h2u hðk2 þ lÞðk2 þ lÞ3=2 6p2 k3  h2u ¼  2 5 hðk2 þ lÞðk2 þ lÞ3=2 6p k  h2u ¼  2 5 hðk2 þ lÞðk2 þ lÞ3=2 6p k

 2u ¼ ok m ok Zu u ok A

ð9:15Þ

ok  hu ¼ 0: The flow in the two-body sector is driven by fermionic diagrams only. There is no renormalization of the Yukawa coupling  h: The (9.15) are solved by direct integration with the result  2u ðKÞ  2u ðkÞ ¼ m m

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#    h2u qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 þ l l 3pffiffiffiffiffiffiffi 2 K þl 1 2   hðK þ lÞ 2 l arctan pffiffiffiffiffiffiffi 2 6p l 2K " pffiffiffiffiffiffiffiffiffiffiffiffiffi!#  3pffiffiffiffiffiffiffi  h2u pffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ l l 2 k2 þ l 1  2  þ hðk þ lÞ 2 l arctan pffiffiffiffiffiffiffi 2k 2 6p l 2

Zu ðkÞ ¼ Zu ðKÞ

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#  2   h2u qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5K þ 2l K2 þ l 3 2 K  hðK þ lÞ þ l  arctan p ffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffi 48p2 l l K4 " !# ffiffiffiffiffiffiffiffiffiffiffiffiffi p  h2u pffiffiffiffiffiffiffiffiffiffiffiffiffi ð5k2 þ 2lÞ k2 þ l 3 2 2 þ hðk þ lÞ  pffiffiffiffiffiffiffi arctan k þl pffiffiffiffiffiffiffi k4 48p2 l l 2

 u ðkÞ ¼ A  u ðKÞ A

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#  2   h2u qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5K þ 2l K2 þ l 3 2 K þl  hðK þ lÞ  pffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffi 4 2 48p l l K " !# ffiffiffiffiffiffiffiffiffiffiffiffiffi p  2þl h2u pffiffiffiffiffiffiffiffiffiffiffiffiffi ð5k2 þ 2lÞ k 3 2 þ hðk þ lÞ  pffiffiffiffiffiffiffi arctan k2 þ l : pffiffiffiffiffiffiffi k4 48p2 l l 2

ð9:16Þ Here, K is the initial ultraviolet scale. Let us discuss the initial value for the boson mass. It is given by

9.2 Two Fermion Species: Dimer Formation

93

 2u ðKÞ ¼ mðBÞ  2l þ dmðKÞ: m

ð9:17Þ

The detuning mðBÞ ¼ lM ðB  B0 Þ describes the energy level of the microscopic state represented by the field u with respect to the fermionic state w. At a Feshbach resonance, this energy shift can be tuned by the magnetic field B, lM denotes the magnetic moment of the field u, and B0 is the resonance position. Physical observables such as the scattering length and the binding energy are obtained from the effective action and are therefore related to the coupling constants at the infrared scale k = 0. The quantity dm(K) denotes a renormalization counter term that has to be adjusted conveniently, see below.

9.2.2 Renormalization We next show that close to a Feshbach resonance the microscopic parameters 2 ðk ¼ KÞ are related to B - B0 and a by two  2u ðk ¼ KÞ and   u;K  m h2u;K  h m u simple relations  2u;K ¼ lM ðB  B0 Þ  2l þ m

h2 u;K 6p2

K

ð9:18Þ

and  h2u;K : a¼ 8plM ðB  B0 Þ

ð9:19Þ

Away from the Feshbach resonance the Yukawa coupling may depend on B,  h2u ðBÞ ¼  h2u þ c1 ðB  B0 Þ þ    : Also the microscopic difference of energy levels between the open and closed channel may show corrections to the linear B-dependence, mðBÞ ¼ lM ðB  B0 Þ þ c2 ðB  B0 Þ2 þ    or lM ! lM þ h2u ðBÞ and lM ðBÞ our formalism can easily be adapted to c2 ðB  B0 Þ þ . . .: Using  a more general experimental situation away from the Feshbach resonance. The relations in (9.18) and (9.19) hold for all chemical potentials l and temperatures T. For a different choice of the cutoff function the coefficient dm(K) being the term linear in K in (9.18) might be modified.  2u;K and h2u;K with the magnetic field We want to connect the bare parameters m B and the scattering length a for fermionic atoms as renormalized parameters. In our units, a is related to the effective interaction kw;eff by a¼

kw;eff : 8p

ð9:20Þ

The fermion interaction kw;eff is determined by the molecule exchange process in the limit of vanishing spatial momentum

94

9 Few-Body Physics

kw;eff ¼ 

 h2u;K  u ðx; ~ P p2 ¼ 0; lÞ

:

ð9:21Þ

Even though (9.21) is a tree-level process, it is not an approximation, since  u jk!0 denotes the full bosonic propagator which includes all fluctuation u  P P effects. The frequency in (9.21) is the sum of the frequency of the incoming fermions which in turn is determined from the on-shell condition x ¼ 2xw ¼ 2l:

ð9:22Þ

On the BCS side we have l = 0 and find with  u ðx ¼ 0; ~  2u ðk ¼ 0Þ  m  2u;0 q ¼ 0Þ ¼ m P

ð9:23Þ

 h2u;K kw;eff ¼  2 ;  u;0 m

ð9:24Þ

the relation

 2u;0 ¼ m  2u ðk ¼ 0Þ: For the bosonic mass terms at l = 0, we can read off where m from (9.16) and (9.17) that  2u;0 ¼ m  2u;K  m

 h2u;K

h2 u;K K ¼ l ðB  B Þ þ dmðKÞ  K: 0 M 2 6p 6p2

ð9:25Þ

To fulfill the resonance condition a ? ± ? for B = B0, l = 0, we choose dmðKÞ ¼

 h2u;K 6p2

K:

ð9:26Þ

 2u as a relevant The shift dm(K) provides for the additive UV renormalization of m coupling. It is exactly canceled by the fluctuation contributions to the flow of the mass. This yields the general relation (9.18) (valid for all l) between the bare mass  2u;K and the magnetic field. On the BCS side we find the simple vacuum term m relation  2u;0 ¼ lM ðB  B0 Þ: m

ð9:27Þ

Furthermore, we obtain for the fermionic scattering length  h2u;K : a¼ 8plM ðB  B0 Þ

ð9:28Þ

This equation establishes (9.19) and shows that h2u;K determines the width of the resonance. We have thereby fixed all parameters of our model and can express  2u;K and  h2u;K by B - B0 and a. The relations (9.18) and (9.19) remain valid also m at nonzero density and temperature. They fix the ‘‘initial values’’ of the flow h2u;K Þ at the microscopic scale K in terms of experimentally accessible ð h2u !  quantities, namely B - B0 and a.

9.2 Two Fermion Species: Dimer Formation

95

On the BEC side, we encounter l \ 0 and thus x [ 0. We therefore need the bosonic propagator for x = 0. Even though we have computed directly only  u at x = 0 and derivatives with respect to x (Zu), we can quantities related to P obtain information about the boson propagator for nonvanishing frequency by using the semilocal U(1) invariance described in Sect. 7.1 in Chap. 7. In momentum space, this symmetry transformation results in a shift of energy levels wðx; ~Þ p ! wðx  d; ~Þ p uðx; ~Þ p ! uðx  2d; ~Þ p l ! l þ d:

ð9:29Þ

Since the effective action is invariant under this symmetry, it follows for the bosonic propagator that  u ðx; ~;  u ðx  2d; ~; P p lÞ ¼ P p l þ dÞ:

ð9:30Þ

To obtain the propagator needed in (9.21), we can use d = -l and find as in (9.28) kw;eff ¼ 

 h2u;K  u ðx ¼ 0; ~ P p2 ¼ 0; l ¼ 0Þ

¼

h2 u;K

: lM ðB  B0 Þ

ð9:31Þ

 u;K and h2u;K in terms of Thus the relations (9.18) and (9.19) for the initial values m B - B0 and a hold for both the BEC and the BCS side of the crossover.

9.2.3 Binding Energy We next establish the relation between the molecular binding energy M ; the scattering length a, and the Yukawa coupling  h2u;K : From (9.16), we obtain for k = 0 and l B 0  2u;0 ¼ lM ðB  B0 Þ  2l m "  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  h2u;K l þ 2 K  K2 þ l 1  2 6p 2K pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# K2 þ l 3pffiffiffiffiffiffiffi l arctan þ : pffiffiffiffiffiffiffi 2 l

ð9:32Þ

pffiffiffiffiffiffiffi In the limit K= l ! 1 this yields  2u;0 ¼ lM ðB  B0 Þ  2l þ m

ffi h2 pffiffiffiffiffiffi u;K l 8p

:

ð9:33Þ

96

9 Few-Body Physics

Together with (9.28), we can deduce a¼

 h2u;K ffi;  h2 pffiffiffiffi l  2u;0 þ 2l  u;K8p 8p m

ð9:34Þ

 2u;0 ¼ 0 this yields which holds in the vacuum for all l. On the BEC side where m a¼

1  : pffiffiffiffiffiffiffi 16p pffiffiffiffiffiffiffi l 1 þ h2 l

ð9:35Þ

u;K

The binding energy of the bosons is given by the difference between the energy for  2u =Zu and the energy for two fermions -2l. On the BEC side, we can a boson m 2  u;0 ¼ 0 and obtain use m  2u m M ¼  þ 2ljk!0 ¼ 2l: Zu

ð9:36Þ

From (9.35) and (9.36) we find a relation between the scattering length a and the binding energy M pffiffiffi 2 1 M 3=2 4 2p 2 ð8pÞ þ ð ¼ Þ þ ð Þ ð9:37Þ M M h4 :  a2 2 h2u;K u;K In the broad resonance limit  h2u;K ! 1; this is just the well-known relation between the scattering length a and the binding energy M of a dimer (see for example [4]) M ¼ 

2 1 ¼  2: 2 a Ma

ð9:38Þ

The last two terms in (9.37) give corrections to (9.38) for more narrow resonances. The solution of the two-body problem turns out to be exact as expected. In our formalism, this is reflected by the fact that the two-body sector decouples from the flow equations of the higher-order vertices: no higher-order couplings such as ku enter the set of (9.15). Extending the truncation to even higher order vertices or by including a boson–fermion vertex wy wu u does not change the situation.

9.2.4 Dimer–Dimer Scattering So far we have considered the sector of the theory up to order u ww; which is equivalent to the fermionic two-body problem with pointlike interaction in the limit of broad resonances. Higher-order couplings, in particular the four-boson coupling ku ðu uÞ2 ; do not couple to the two-body sector. Nevertheless, a four-boson

9.2 Two Fermion Species: Dimer Formation

97

coupling emerges dynamically from the renormalization group flow. In vacuum we have q0 = 0 and ku is defined as ku ¼ Uk00 ð0Þ: The flow equation for ku can be found by taking the q-derivative of (8.34) in Chap. 8 pffiffiffi 3   2k 2 gA u 1 kok ku ¼ 2gAu Uk00  2 dþ2 3p Su   h4u ð2Þ ð1;0Þ ð0;1Þ  2ðUk00 Þ2 sB;Q þ 3sB;Q þ 2 3 sF;Q 3p k pffiffiffi 5 2 ð9:39Þ 2k k u ¼ 2gAu ku þ ð1  2gAu =5Þ 3p2 Su ðm2u þ k2 Þ2 

h4u hðl þ k2 Þðl þ k2 Þ3=2 : 4p2 k6

There are contributions from fermionic and bosonic vacuum fluctuations, but no contribution from higher q derivatives of U. The fermionic diagram generates a four-boson coupling even for zero initial value. This coupling then feeds back into the flow equation via the bosonic diagram. The scattering lengths are related to the corresponding couplings by the relation (cf. [5]) aM ku ¼2 ; a kw;eff

kw;eff ¼ 8pa:

ð9:40Þ

Omitting the bosonic fluctuations, a direct integration yields the mean field result aM =a ¼ 2: This value is lowered when the bosonic fluctuations are taken into account. With our truncation and choice of cutoff one finds aM =a ¼ 0:718: The calculation can be improved by extending the truncation to include a boson– fermion vertex kuw which describes the scattering of a dimer with a fermion [6]. Inspection of the diagrammatic structure shows that this vertex indeed couples into the flow equation for ku. The ratio aM/a has been computed by other methods. Diagrammatic approaches give aM =a ¼ 0:75ð4Þ [7], whereas the solution of the 4-body Schrödinger equation yields aM =a ¼ 0:6 [8], confirmed in QMC simulations [9] and with diagrammatic techniques [10].

9.3 Three Fermion Species: Efimov Effect In the following we discuss the few-body properties of the model for three fermion species in (6.18) in Chap. 6. We first concentrate on the SU(3) symmetric case where the mass and the scattering properties are equal for all three species. The three-body problem is governed by the Efimov effect. We later generalize the model in (6.18) Chap. 6 to cover also the case where SU(3) symmetry is broken

98

9 Few-Body Physics

explicitly. Subsequently we apply this to the case of 6Li and discuss the relation to some recent experiments.

9.3.1 SU(3) Symmetric Model We use now the truncation in (8.38) in Chap. 8 to derive flow equations for the model with global SU(3) symmetry in (6.18) in Chap. 6 that describes three fermion species at a common two-body resonance. 9.3.1.1 Flow Equations for Two-body Sector To obtain the flow equations that govern the two-body sector, we insert our ansatz equation (8.38 in Chap. 8) into the Wetterich equation (2.20 in Chap. 2). Functional derivatives with respect to the fields for zero temperature T = 0 and density n = 0 lead us to a system of ordinary coupled nonlinear differential equations for w; A u; A v; m  2u ; h; A  2v ;  the couplings m g and  kuw : One finds that the propagator of the  w ðkÞ ¼ 1; hðkÞ ¼ h: The flow equations original fermions w is not renormalized, A for the couplings determining the two body sector, namely the boson gap parameter (with t = ln(k/K))  2u ¼ ot m

 k5 h2 ; 6p2 ðk2  lÞ2

ð9:41Þ

and the boson wave function renormalization 2 k5 u ¼  h ot A 6p2 ðk2  lÞ3

ð9:42Þ

decouple from the other flow equations. These flow equations are very similar to the ones for the two-body sector of the BCS–BEC crossover model discussed in Sect. 9.2. The difference comes from the fact that we use a slightly different cutoff function ((8.39) in Chap. 8 to be compared with (5.5) in Chap. 5 ). They can be solved analytically,    l K k h2  2u ðkÞ ¼ m  2u ðKÞ  2 ðK  kÞ  m  2 K2  l k 2  l 6p  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi

l l 3pffiffiffiffiffiffiffi þ l arctan  arctan ; K k 2 " ð9:43Þ  h2 Kð5K2  3lÞ kð5k2  3lÞ   Au ðkÞ ¼ 1  2 6p 8ðK2  lÞ2 8ðk2  lÞ2  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi# l l 3 þ pffiffiffiffiffiffiffi arctan  arctan : 8 l K k

9.3 Three Fermion Species: Efimov Effect

99

Using this explicit solution (9.43) we can relate the initial value of the boson gap  2u as well as the Yukawa coupling h to physical observables. The parameter m interaction between fermions w is mediated by the exchange of the bound state u. Again in terms of bare quantities, the scattering length between fermions is given  2u Þ where the couplings   2u are evaluated at the macroby a ¼  h2 =ð8pm h and m scopic scale k = 0 and for vanishing chemical potential l = 0. This fixes the initial value  2u ðKÞ ¼  m

  h2 1 h2 a þ 2 K  2l: 8p 6p

ð9:44Þ

In addition to the l-independent part we have added here the chemical potential term -2l where the factor 2 accounts for the bosons consisting of two fermions.  2u is proportional to the detuning of In vacuum, the gap parameter of the bosons m 2  u ðk ¼ 0; l ¼ 0Þ ¼ lM ðB  B0 Þ: Here lM is the magnetic the magnetic field m moment of the bosonic dimer and B0 is the magnetic field at the resonance. From  1 h2 a¼ 8p lM ðB  B0 Þ

ð9:45Þ

one can read off that  h2 is proportional to the width of the resonance. We have now fixed all initial values of the couplings at the scale k = K or, in other words, the parameters of our microscopic model. At vanishing density n = 0, the chemical potential is negative or zero, l B 0, and will be adjusted such that the lowest excitation of the vacuum is a gapless propagating degree of freedom in the infrared, i.e. at k = 0. Depending on the value of a-1 this lowest energy level may be the original fermion w, the boson u, or the composite fermion v.

9.3.1.2 Three-body Problem Now that we have solved the equations for the two-body problem within our approximation, we can address the three-body sector. It is described by the flow equations for the trion gap parameter, ot m2v ¼

6g2 k5 þ gv m2v ; p2 ð3k2  2l þ 2m2u Þ2

ð9:46Þ

and the Yukawa-type coupling, g ot g ¼ ðgu þ gv Þ þ m2v ot a 2 6k2  5l þ 2m2u 2gh2 k5 :  3p2 ðk2  lÞ2 ð3k2  2l þ 2m2u Þ2

ð9:47Þ

100

9 Few-Body Physics

These equations are supplemented by the anomalous dimension  v 24g2 ot A k5 gv ¼   ¼ 2 p ð3k2  2l þ 2m2u Þ3 Av

ð9:48Þ

and the variable ot a ¼ 

9k2  7l þ 4m2u h4 k5 : 12p2 g ðk2  lÞ3 ð3k2  2l þ 2m2u Þ2

ð9:49Þ

Here a determines the scale dependence of the trion field v ot v ¼ ðot aÞui wi  gv v=2

ð9:50Þ

in the sense of the general coordinate transformation in (3.41) in Chap. 3. We neglect here and in the following correction terms coming from the connection in the space of fields since they are expected to be small. Instead of working with the coordinate transformation one might also work with the generalized flow equation (3.28) in Chap. 3. Since this is an exact equation and the structure is simpler compared to (3.41) in Chap. 3 we propose to employ (3.28) in Chap. 3 in future work. While the second term in (9.50) is the usual wave-function renormalization, the first term describes a nonlinear change of variables. It is chosen such that the flow of kuw vanishes on all scales, kuw(k) = 0. The scattering between bosons and fermions is then described by the exchange of the trion bound state v. This reparametrization, which is analogous to rebosonization [11, 12], is crucial for our description of the system in terms of the composite trion field v. Note that the flow equations (9.46), (9.47), (9.48), (9.49), that describe the threebody sector are independent from the flow of the boson–boson interaction ku that belongs to the four-body sector. Since the three-body sector is driven by fermionic and bosonic fluctuations, it is not possible to find simple analytic solutions to the flow equations in the general case. However, it is no problem to solve them numerically. This is most conveniently done using again ‘‘bare’’ couplings. As a general feature we note that a negative chemical potential l acts as an infrared cutoff for the fermionic fluctuations while a positive value of the bosonic gap m2u suppresses bosonic fluctuations in the infrared. We can find numerical solutions for different scattering length between the fermions a and varying chemical potential l. We use an iteration process to determine the chemical potential l B 0 where the lowest excitation of the vacuum is gapless. On the far BCS side for small and negative scattering length a ? 0-, where this lowest excitation is the fermion w, this implies simply l = 0. On the far BEC side for small and positive scattering length  2u ¼ 0: We can then use a ? 0+ the lowest excitation is the boson u, implying m our analytic solution of the two-body sector (9.43) to obtain the chemical poten 2u ¼ 0 at the macroscopic scale k = 0. In the limit K/ tial l from the condition m |l| ? ? we find

9.3 Three Fermion Species: Efimov Effect

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2  h2 a1  h4 h2 l¼ þ :  16p 32p ð32pÞ2

101

ð9:51Þ

As it should be, this is equivalent to the corresponding relation found for the two-fermion case (9.35). In the broad resonance limit h2 ! 1 this reduces to the well-known result l = -1/a2. For scattering lengths close to the Feshbach 1 1 resonance a1 c1 \a \ac2 we find that the lowest vacuum excitation is the trion v. Numerically, we determine l from the implicit equation m2v ¼ 0 at k = 0. Our result for l obtained for  h2 ¼ 100K as a function of the inverse scattering length a-1 is shown in Fig. 9.4. With the choice K = 1/a0, this value of h2 corresponds to the width of the Feshbach resonance of 6 Li atoms in the ðmF ¼ 1=2; mF ¼ 1=2Þchannel [13]. For small Yukawa couplings  h2 =K 1; or narrow resonances we find that the range of scattering length where the trimer is the lowest excitation of the vacuum 2 1 increases linear with  h2 [14, 15]. More explicit, we find a1 c1 ¼ 0:0015h , ac2 ¼ 2 2 0:0079 h : However, for very broad Feshbach resonances h =K 1 the range 1 depends on the ultraviolet scale a1 c1 ; ac2  K: We show this behavior in Fig. 9.5 The chemical potential lU at the unitarity point a-1 = 0 is plotted in the inset of Fig. 9.5. It increases with the width of the resonance similar to lU ¼ 3:5  106 h4 for small  h2 =K: It also approaches a constant value which depends on the cutoff h2 ! 1: scale lU*K2 in the broad resonance limit  The dispersion relation for the atoms, dimers and trimers can be computed by analytical continuation, s = it, of the inverse propagators to ‘‘real frequencies’’ x. In our truncation, they read (in terms of renormalized fields) ~ p2  xw  l; 2M ~ p2  xu  2l þ mu ðl þ xu =2  ~ Pu ¼ p2 =ð8MÞÞ; 4M ~ p2  xv  3l þ mv ðl þ xv =3  ~ p2 =ð18MÞÞ: Pv ¼ 6M Pw ¼

ð9:52Þ

We note that the functions mu and mv depend only on the particular combinations of l, x, and ~ p2 given above. This is a result of the symmetries of our problem. The real-time microscopic action S = Ck=K with t = -is is invariant under the time-dependent U(1) symmetry transformation of the fields w ? eiEtw, u ? ei2Etu, and v ? ei3Etv if we also change the chemical potential according to ~ ¼ l þ E: Since the microscopic action has this symmetry and since we do l!l not expect any anomalies, the quantum effective action C = Ck=0 is also invariant under these transformations. In consequence, any dependence on l must be accompanied by a corresponding frequency dependence, such that only the invariant combinations xw ? l, xu ? 2l, xv ? 3l appear in the effective action. The relation between the dependence on x and on ~ p2 is a consequence of Galilean

102

9 Few-Body Physics

0.0

0.1

0.2

0.3

0.4

0.5

0.1

0.0

0.1

0.2

0.3

0.4

0.5

pffiffiffiffiffiffi Fig. 9.4 Dimensionless chemical potential  jlj=K in vacuum as a function of the dimen-1 sionless scattering length a /K. For comparison, we also plot the result for two fermion species where the original fermions are the propagating particles for a-1 \ 0 (dotted line) and composite bosons for a-1 [ 0 (dashed line). The inset is a magnification of the little box and shows the energy of the first excited Efimov state

symmetry. Even though we compute directly with the flow equations only mu ðl; xu ¼ 0; ~ p2 ¼ 0Þ and mv ðl; xv ¼ 0; ~ p2 ¼ 0Þ; we can use the symmetry information (‘‘Ward identities’’) for an extrapolation to arbitrary x and ~ p2 : The 2 ~ Þ follows from (9.52) by solving Pu ðxu ¼ dispersion relation for the bosons E u ðp ~2 Þ and the trions E u Þ ¼ 0 and similarly the dispersion relation of the fermions E w ðp 2 ~ Þ: E v ðp

0.4 0.3 0.2 0.1 0.0 0.1 0

20

40

60

80

100

120

140

1 1 Fig. 9.5 Interval of scattering length a1 c1 \a \ac2 where the lowest vacuum excitation is the 2 trimer fermion v (solid lines). For comparison we also plot the linear fits a1 c1 ¼ 0:0015h and h2 (dotted lines). The inset shows the chemical potential lU/K2 at the resonance ¼ 0:0079 a1 c2 a-1 = 0 as a function of  h2 =K in the same range of  h2 =K: Here we also plot the curve lU ¼ 6 4 3:5  10 h for comparison (dotted lines)

9.3 Three Fermion Species: Efimov Effect

103

We are interested in the dimer and trimer energy levels relative to the energy of ~2 ¼ 0Þ the atoms. For this purpose we consider their dispersion relations at rest ðp and substract twice or three times the atom energy E w ¼ l; ~2 ¼ 0Þ þ 2l ¼ mu ðEu =2Þ; Eu ¼ E u ðp ~2 ¼ 0Þ þ 3l ¼ mv ðEv =3Þ: Ev ¼ E v ðp

ð9:53Þ

When the dimers are the lowest states one has m2u ¼ 0 and therefore mu ¼ 2l ¼ Eu ; while for lowest trimers m2v ¼ 0 implies mv ¼ 3l ¼ Ev : We have shown Eu and Ev in Fig. 10.44 in Chap. 10. A typical value for K is of the order of the inverse Bohr radius a1 0 : We also mention that the energy levels have been computed in the absence of a microscopic three-body interaction, k3(K) = 0. The same computation may be repeated with nonzero k3(K), and thus microscopic parameters may be fixed by a comparison to the experimentally measured spectrum. So far we were only concerned with the lowest energy excitation of the vacuum. We found that near a Feshbach resonance this lowest excitation is given by a trimer. However, it is known from the work of Efimov [16, 17] that one can expect not only one trimer state close to resonance but a whole spectrum. In fact, the solution mv = 3l may not be the only solution for the equation fixing the trimer energy levels, i.e.   1 E v ¼ mv l þ E v  3l: ð9:54Þ 3 For an investigation of the energy dependence of mv we use the symmetry trans~ ¼ l þ E w ¼ 0: We can formation and shift the chemical potential by E w , l ~ and (9.54) turns into a simple implicit therefore follow the flow at vanishing l equation for the trion energy levels     Ev Ev Ev ¼ m v ¼ m2v ð9:55Þ 3 3 ~ ¼ 0 the flow equations simplify considerably. For example, (9.43) For l becomes  h2 k h2  2u ðKÞ  2 ðK  kÞ ¼ lM ðB  B0 Þ þ 2 ;  2u ðkÞ ¼ m m 6p 6p 2 1 1  h  u ðkÞ ¼ 1 þ  A : 6p2 k K

ð9:56Þ

~ u ¼ xu þ 2l ¼ 0: The We observe that the two-body sector is evaluated here for x physical chemical potential l remains, of course, negative in the vicinity of the Feshbach resonance, and our evaluation therefore corresponds to a positive energy xu as compared to the lowest energy level of the trimer for which xv = 0.

104

9 Few-Body Physics

An especially interesting point in the spectrum is the unitarity limit, B = B0, where the scattering length diverges, a-1 = 0. At that point all length scales drop out of the problem and we expect a sort of scaling solution for the flow equations.  u ðkÞ is dominated completely by the term with In the limit k? 0 the solution for A 2 2    u ¼ k2 : For Ev ? 0 the flow  2u =A 1=k; Au ðkÞ ¼ h =ð6p kÞ; and we find m2u ¼ m equations for the three-body sector simplify and we find 36 2 k2  g 2 25  h 64 2 13 2 h2 2  ¼   : ot g g  m 25 25 v k2

 2v ¼ ot m

ð9:57Þ

In contrast, for Ev = 0, as needed for (9.55), we will have to solve the flow with a ^ ¼ Ev =3; which will cause a nonvanishing ‘‘effective chemical potential’’ l departure from the scaling flow.

9.3.1.3 Limit Cycle Scaling First, let us consider the scaling solution obeying (9.57). It is convenient to rescale ~2 ¼ m  2v ðk= the variables according to m hÞh , ~ g2 ¼  g2 ðk=hÞ2þh which gives the linear differential equation  2   2  36 h; ~ ~ m m 25 ot : ð9:58Þ ¼ 64 13 ~ ~g2 g2 25; 2 þ h  25 The matrix on the right hand side of (9.58) has the eigenvalues b1=2 ¼ h  ð7=25Þ  pffiffiffiffiffiffiffiffi i 419=25: The flow equation (9.58) leads therefore to an oscillating behavior. It is straightforward to solve (9.58) explicitly. Restricting to real solutions and using initially  g2 ðKÞ ¼ 0 we find the following for the trimer gap parameter and coupling  257   k k 2  v ðKÞ cos s0 ln ¼ m K K  

7 k þpffiffiffiffiffiffiffiffi sin s0 ln ; K 419  257   13 h2 k k 2 2 p ffiffiffiffiffiffiffi ffi g ðkÞ ¼   v ðKÞ sin s0 ln m : K 419k2 K

 2v ðkÞ m

ð9:59Þ

 2v and we find for  2v ðKÞ drops out of the ratio g2 =m As it should be, the initial value m the three-body coupling

9.3 Three Fermion Species: Efimov Effect

105

  sin s0 ln Kk 468p4     k4 : p ffiffiffiffiffiffiffi ffi k3 ðkÞ ¼ k 7 ffi k ffiffiffiffiffi p þ cos s ln sin s ln 419 0 0 K K 419

ð9:60Þ

pffiffiffiffiffiffiffiffi We obtain for the ‘‘frequency’’ s0 ¼ 419=25  0:82: Since we use a truncation in the space of functionals Ck this result is only a rough estimate. It has to be compared with the result of other methods which find s0&1.00624 [4, 16–20]. Considered the simplicity of our approximation, the agreement is quite reasonable. A more elaborate truncation that includes the full momentum dependence of vertices, confirms Efimovs value, indeed [21]. For a determination of the trion energy levels we have to solve the flow with an ~ ¼ Ev =3: This acts as an infrared cutoff, effective negative chemical potential l such that the flow deviates from the limit cycle once k2  ~ l: Qualitatively, pffiffiffiffiffiffiffi the flow eventually stops once k becomes smaller than ~ l: For an evaluation of (9.55) we may therefore use (9.59) with a specific value for k, namely k2 ¼ Ev =3: The possible energy levels therefore obey  1 ðkÞm  2v ðkÞ ¼ 0: 3k2 þ A v

ð9:61Þ

With  v ðkÞ ¼ ok A

24  g2 ðkÞ 125p2 k2

ð9:62Þ 32

 v  k25 : We can write (9.61) one finds, up to oscillatory behavior an increase of A in the form     k 7 k FðkÞ ¼ cos s0 ln þ pffiffiffiffiffiffiffiffi sin s0 ln K K 419 ð9:63Þ 7   k2 k 25  ¼ 3 2 Av ðkÞ:  v ðKÞ K m For small k K we infer that F(k) has to vanish   k=h2 : Since F(k) is periodic, solutions will occur for roughly equidistant values in ln Kk : For k  h2 the possible solutions simply correspond to F(k) = 0. The first solution with the largest k corresponds to the ground state level with E0 \ 0. The subsequent solutions obey   knþ1 kn  ln s0 ln ¼ p ð9:64Þ K K or   Enþ1 2p ¼ exp  ; s0 En



 2pn En ¼ exp  E0 : s0

ð9:65Þ

106

9 Few-Body Physics

This corresponds to the tower of trimer bound states at the unitarity limit, with En approaching zero exponentially for n ? ?. For (9.65) we have actually taken into account all zeros of F(k). Since  g2 ðkÞ oscillates periodically, only half of these 2 zeros correspond to  g ðkÞ [ 0; while the other half has formally g2 ðkÞ\0: We may use the mapping discussed after (6.19) in Chap. 6 to obtain an equivalent picture with positive  g2 : We may understand the repetition of states by the following qualitative picture. ~ 2 ; that is proportional to the energy gap of the trimer, starts on the The coupling m ultraviolet scale with some positive value. The precise initial value is not important. The Yukawa-type coupling  g vanishes initially so that the trion field v is simply an auxiliary field which decouples from the other fields and is not propagating. However, quantum fluctuations lead to the emergence of a scattering amplitude between the original fermions w and the bosons u. We describe this by the exchange of a composite fermion v. This leads to an increase of the coupling ~g2 ~ 2 : At some scale t1 ¼ lnðk1 =KÞ with k12  l the and a decrease of the trion gap m ~ 2 crosses zero which indicates that a trion state v becomes the lowest coupling m energy excitation of the vacuum. Indeed, would we consider the flow without modifying the chemical potential, this would set an infrared cutoff that stops the pffiffiffiffiffiffi flow at the scale k1  jlj and the trion v would be the gapless propagating particle while the original fermions w and the bosons u are gapped since they have higher energy. Following the flow further to the infrared, we find that the Yukawa coupling ~g2 decreases again until it reaches the point ~ g2 ¼ 0 at t ¼ t10 (see Fig. 9.6). Naive continuation of the flow below that scale would lead to ~g2 \0 and therefore imaginary Yukawa coupling ~ g: However, since the trion field v decouples from the other fields for ~ g ¼ 0; we are not forced to use the same field v as before. We can  v to describe  2v2 =A simply use another auxiliary field v2 with very large gap m2v2 ¼ m 2 the scattering between fermions and bosons on scales t \ t10 : We are then in the same position as on the scale k = K and the process repeats. Starting from a ~ 2v2 decreases as the infrared cutoff k is positive value, the rescaled gap parameter m lowered. At the scale k2 it crosses zero which indicates that there is a second trimer bound state in the spectrum with energy per original fermion E2 ¼ l  k22 : ~ ¼ l þ E2 ¼ k22 ; the flow would Would we use the modified chemical potential l be stopped at the scale k2 and the second trimer v2 would be the propagating degree of freedom. This cycle repeats and corresponds precisely to the limit cycle scaling of (9.58). At the unitarity point with a-1 = 0 and at the threshold energy E = -l with ~ ¼ l þ E ¼ 0; this limit cycle scaling is not stopped and leads to an infinite tower l of trimer bound states. The energy of this states comes closer and closer to the fermion–boson threshold energy E = -l. In our language we recover the effect first predicted by Efimov [16]. A similar limit cycle description of the Efimov effect for identical bosons was given in the context of effective field theory in [18–20].

9.3 Three Fermion Species: Efimov Effect Fig. 9.6 Limit cycle in the renormalization group flow at the unitarity point a-1 = 0, and for energy at the fermion ~ ¼ l þ E ¼ 0: We threshold l plot the rescaled gap param~ 2 ðtÞ (solid eter of the trimer m lines) and the rescaled Yukawa coupling ~g2 ðtÞ (dashed lines). The dotted curves would be obtained from naive continuation of the flow after the point where ~g2 ¼ 0

107

1.0

0.5

0.0

0.5

1.0 10

8

6

4

2

0

The infinite limit cycle scaling occurs only directly at the Feshbach resonance ~; also a nonzero with a-1 = 0. Similar to a nonzero (modified) chemical potential l inverse scattering length a-1 provides an infrared cutoff that stops the flow. For ~ ¼ l þ E ¼ 0; the example for negative scattering length a and energy E with l solution of the two body sector (9.43) implies for the boson gap in the infrared m2u ¼ 3pa1 k=4 þ k2 so that bosonic fluctuations are suppressed in comparison to the unitarity point with a1 ¼ 0: This leads to a stop of the limit cycle scaling at the scale k&3p|a-1|/4 so that the spectrum consists only of a finite number of trimer states. In the inset of Fig. 9.4 we show the numerical result for the modified ~ of the first excited Efimov trimer. chemical potential l

9.3.2 Experiments with Lithium From the work of Efimov it is known that the trion bound state (lowest energy Efimov state) is also expected to persist if the SU(3) symmetry is violated by a different location and strength for the Feshbach resonances between different pairs of atomic components [17, 4]. In this subsection we generalize the SU(3) symmetric model discussed above to the case where the three resonances are at different positions. Recent measurements of the threebody loss coefficient in a three-component system of 6Li [22, 23] may find an interpretation in this way. We investigate here a simple setting, where the loss arises from the formation of an intermediate trion bound state. The trion is not stable and subsequently decays into unspecified degrees of freedom—possibly the ‘‘molecule type’’ dimers associated to the nearby Feshbach resonances. In turn, the trion formation from three atoms proceeds by the exchange of an effective bosonic field, as shown in Fig. 9.7. Evaluating the matrix elements corresponding to the diagrams shown in Fig. 9.7, we estimate the loss coefficient K3 as being proportional to

108

9 Few-Body Physics

2    3   X hi gi 1  :   p ¼   2 C  i¼1 mui m2v  i 2v 

ð9:66Þ

Here m2v and Cv are the trion gap parameter and decay width, while m2ui describes a type of gap parameter for the effective boson, such that its propagator can be approximated by m2 ui : The Yukawa couplings hi couple the fermionic atoms to the effective boson, and the trion coupling gi accounts for the coupling between trion, atom and effective boson. We sum over the ‘‘flavor’’ indices i = 1, 2, 3. We will estimate m2v ; m2ui ; hi and gi using the flow equation method. In the previous subsection we studied the SU(3) invariant Fermi gas close to a common Feshbach resonance and explored the manifestation of Efimov’s effect. In contrast to this theoretical model, the system consisting of three-component 6Li atoms, which is of current experimental interest [22, 23], does not possess this SU(3) symmetry. The main difference is that the resonances do not occur at the same magnetic field, and thus, for a given magnetic field B, the scattering lengths of different pairs of atoms, (1, 2), (2, 3), and (3, 1) differ from each other. In this section we adapt the model in (6.18) in Chap. 6 to cope with this more general situation. Our truncation of the (euclidean) average action reads then Z Ck ¼ wi ðos  D  lÞwi x h i þ ui Aui ðos  D=2Þ þ m2ui ui ð9:67Þ h i þ v os  D=3 þ m2v v

      þ hi ijk ðui wj wk  ui wj wk Þ þ gi ðui wi v  ui wi v Þ ; where we choose natural units  h ¼ 2M ¼ 1; with the atom mass M. We sum over the indices i, j, k wherever they appear. Here w ¼ ðw1 ; w2 ; w3 ÞT denotes the fermionic atoms, u ¼ ðu1 ; u2 ; u3 ÞT a bosonic auxiliary field which mediates the four-fermion interaction and v is a fermionic field representing the bound state of three atoms. Formally, this trion field is introduced via a generalized Hubbard– Stratonovich transformation on the microscopic scale and mediates the interaction between atoms w and bosons u. We show this schematically in Fig. 9.8. In the Fig. 9.7 Three-body loss process involving the trion

9.3 Three Fermion Species: Efimov Effect

109

limit m2ui ! 1, h2i =m2ui ! jki j, m2v ! 1, g2i =m2v ! jkð3Þ j the action describes pointlike two-body interactions between the atoms with strength ki as well as a pointlike three-body interaction between atoms and dimers with strength k(3). This corresponds to a zero-range approximation and describes the universal limit of dilute gases with large s-wave interactions. No detailed knowledge of the microscopic physics in necessary and physics depends only on the scattering length and a three-body parameter fixing the Efimov trimer energy spectrum [4]. We consider the ‘‘vacuum limit’’ where temperature and atom density go to zero. Then the chemical potential l in (9.67) satisfies l B 0. A negative chemical potential l has the meaning of an energy gap for the fermions when some other particle (boson or trion) has a lower energy. The dominant difference to the SU(3) symmetric model arises from the different propagators of the bosonic fields u1 ¼w b 2 w3 , u2 ¼w b 3 w1 , and u3 ¼w b 1 w2 : In addition, we allow in general for different Yukawa couplings hi corresponding to different widths of the three resonances. Also the Yukawa-like coupling gi that couples the different combinations of fermions wi and bosons ui to the trion field v ¼w b 1 w2 w3 is permitted to vary with the species involved. Although the SU(3) symmetry is explicitly broken, the system exhibits three global U(1) symmetries corresponding to the three conserved numbers of species of atoms. The renormalization flow of the various couplings from the microscopic (UV), k = K, to the physical, macroscopic (IR) scale, k = 0, is obtained by inserting the ‘‘truncation’’ (9.67) into the exact flow equation (2.20 in Chap. 2). The flow equations for the two-body sector, i.e. for the boson propagator parameterized by Aui and m2ui ; are very similar to the SU(3) symmetric case (t = ln(k/K)) ot Aui ¼  ot m2ui

¼

h2i k5 6p2 ðk2  lÞ2 h2i k5

6p2 ðk2  lÞ3

; ð9:68Þ

:

Since the Yukawa couplings hi are not renormalized, ot hi ¼ 0;

Fig. 9.8 Interaction between atoms w and effective bosons u as mediated by the trion field v

ð9:69Þ

110

9 Few-Body Physics

we can immediately integrate the (9.68). The solutions can be found in (9.43). The microscopic values m2ui ðKÞ (bare couplings) have to be chosen such that the physical scattering lengths (at k = 0) between two fermions (renormalized couplings) are reproduced correctly. They are given by the exchange of the boson field u. For example, the scattering length between the fermions 1 and 2 obeys a12 ¼ 

h23 ; 8pm2u3

ð9:70Þ

where all ‘‘flowing parameters’’ are evaluated at the macroscopic scale k = 0 and for atoms at threshold energy and therefore l = 0. We use this description for the scattering between fermions w in terms of a composite boson field u also away from the resonance. We emphasize that the field u is not related to the closed channel Feshbach molecules of the nearby resonance. It rather describes an additional ‘‘effective boson’’, which may be seen as an auxiliary or Hubbard– Stratonovich field, allowing for a simple but effective description of the interaction between two fermions. For the numeric calculations in this note we will use large values of h2i on the initial scale K. This corresponds to pointlike atom–atom interactions in the microscopic regime. Quite similar to the scattering between fermions w in terms of the bosonic composite state u we use a description of the scattering between fermions w and bosons u in terms of the trion field v. As an example, a process where the fermion wi and the boson ui scatter to a fermion wj and a boson uj is given by a tree level diagram as in Fig. 9.8. For vanishing center-of-mass momentum the effective atom–boson coupling reads ð3Þ

ki;j ¼ 

gi gj : m2v

ð9:71Þ

The flow equations for the three-body sector within our approximation are given by the flow of the ‘‘mass term’’ for the trion field ot m2v ¼

3 X

2g2i k5

i¼1

p2 Aui ð3k2  2l þ 2m2ui =Aui Þ2

and the Yukawa-like coupling g1 with flow equation   2m2 g2 h2 h1 k5 6k2  5l þ Au2u2 ot g1 ¼    2m2 2 3p2 Au2 ðk2  lÞ2 3k2  2l þ Au2u2   2m2 g3 h3 h1 k5 6k2  5l þ Au3u3    : 2m2u3 2 2 2 2 2 3p Au3 ðk  lÞ 3k  2l þ Au3

ð9:72Þ

ð9:73Þ

9.3 Three Fermion Species: Efimov Effect

111

The flow equations for g2 and g3 can be obtained from (9.73) by permuting the indices 1, 2, 3. For simplicity, we neglected in the flow equations (9.72) and (9.73) a contribution that arises from box-diagrams contributing to the atom–boson interaction. As described in the previous subsection this term can be incorporated into our formalism using scale-dependent fields, a procedure referred to as refermionization. Also terms of the form wi wi uj uj with i = j, that are in principle allowed by the symmetries are neglected by our approximation in (9.67). We expect that their quantitative influence is sub dominant as it is the case for the SU(3) symmetric case [21]. We apply our formalism to 6Li by choosing the initial values of m2ui at the scale K such that the experimentally measured scattering lengths (see Fig. 9.9) are reproduced. For Aui ðKÞ ¼ 1; the value of hi parameterizes the momentum dependence of the interaction between atoms on the microscopic scale. Close to the Feshbach resonance it is also connected to the width of the resonance h2i  DB: We choose here equal and large values for all three species h1 ¼ h2 ¼ h3 ¼ h: This corresponds to pointlike interactions at the microscopic scale K. The zero-range limit is valid if the absolute values of the scattering lengths are much larger than the range of the interaction potentials which is typically given by the van der Waals length lvdW : For 6Li one has lvdW  62:5a0 and the loss resonances appear for values aij [ 2lvdW : Therefore the zero-range approximation might be questionable. Since the precise value of h is not known, we use the dependence of our results on h as an estimate of their uncertainty within our truncation (9.67). The initial values of the couplings m2v and gi are parameters in addition to the scattering lengths which have to be fixed from experimental observation. For equal interaction between atoms w and bosons u in the UV, the parameter to be fixed is kð3Þ ¼ 

g2 ðKÞ m2v ðKÞ

ð9:74Þ

with g ¼ g1 ¼ g2 ¼ g3 : Pointlike interactions at the microscopic scale may be realized by m2v ðKÞ ! 1: We solve the flow equations (9.68), (9.69), (9.72) and (9.73) numerically. We find m2v ¼ 0 at k = 0 for some range of k(3) and l B 0 for large enough values of the scattering lengths a12 ; a23 and a31. This indicates the presence of a bound state of three atoms v ¼w b 1 w2 w3 : The binding energy ET of this bound state is given by the chemical potential, ET = 3|l| with l fixed such that m2v ¼ 0: To compare with the recently performed experimental investigations of 6Li [22, 23], we adapt the initial value k(3) such that the appearance of this bound state corresponds to a magnetic field B ¼ 125 G; the point where strong three-body losses have been observed. Using the same initial value of k(3) also for other values of the magnetic field, all microscopic parameters are fixed. We can now proceed to the predictions of our model. First we find that the bound state of three atoms exists in the magnetic field region from B = 125G to B ¼ 498 G: The binding energy ET is plotted as the

112

9 Few-Body Physics 400 200 0 200 400 600 800 1000 0

10 20 30 40

0

100

200

300

400

500

600

Fig. 9.9 Upper panel scattering length a12 (solid line), a23 (dashed line) and a31 (dotted line) as a function of the magnetic field B for 6Li. These curves were calculated by P. S. Julienne and b w1 w2 w3 : The taken from Ref. Lower panel binding energy ET of the three-body bound state v ¼ solid line shows the result without the inclusion of box diagrams contributing to the atom–boson interaction and it corresponds to the initial value h2 ¼ 100a1 0 ; while the shaded region gives the 2 1 result in the range h2 ¼ 20a1 0 (upper border) to h ¼ 300a0 (lower border). The dashed line corresponds to the calculated binding energy ET when the refermionization of the atom–boson interaction is taken into account

solid line in the lower panel of Fig. 9.9. We choose here h2 ¼ 100a1 0 ; as appropriate for 6Li in the (1, 2)-channel close to the resonance, while the shaded 1 region corresponds to h2 2 ð20a1 0 ; 300a0 Þ: If one includes the contribution to the flow of the atom–dimer interaction arising from box-diagrams by means of a refermionization procedure (see Sect. 9.3.1), the flow equations for the Yukawa couplings gi, as in (9.73), receive an additional contribution ! ð3Þ ð3Þ ð3Þ g o k o k o k i t t t jl ij  il þ m2v  : ð9:75Þ 2gj 2gl 2gj gl Here we define (i, j, l) = (1, 2, 3) and permutations thereof and we find   4m2 k5 h1 h2 h3 hl 9k2  7l þ Aulul ð3Þ ð9:76Þ ot kij ¼   : 2m2 2 6p2 Aul ðk2  lÞ3 3k2  2l þ Aulul This leads to a reduction of the trion binding energy ET and the result is shown as the dashed line in Fig. 9.9. As a second prediction, we present an estimate of the three-body loss coefficient K3 that has been measured in the experiments by Jochim et al. [22] and O’Hara et al. [23]. For this purpose it is important to note that the fermionic bound state particle v might decay into states with lower energies. These may be some deeply bound molecules not included in our calculation here. In order to include these decay processes we introduce a decay width Cv of the trion. We first assume

9.3 Three Fermion Species: Efimov Effect

113

that such a loss process does not depend strongly on the magnetic field B and therefore work with a constant decay width Cv. The decay width Cv appears as an imaginary part of the trion propagator when continued to real time G1 v ¼ x

~ p2 Cv  m2v þ i : 3 2

ð9:77Þ

We now perform the calculation of the loss for the fermionic energy gap l = 0, which corresponds to the open channel energy level. In the region from B ¼ 125G to B ¼ 498G the energy gap of the trion is then negative m2v \0: The three-body loss coefficient K3 for arbitrary Cv is obtained as follows. The amplitude to form a trion out of three fermions with vanishing momentum and energy is given by P3 2 i¼1 hi gi =mui : The amplitude for the transition from an initial state of three atoms to a final state of the trion decay products (cf. Fig 9.7) further involves the trion P propagator that we evaluate in the limit of small momentum ~ p2 ¼ ð i ~ pi Þ2 ! 0; P p2i , x ¼ i xi ! 0: A thermal distribution and small on-shell atom energies xi ¼ ~ of the initial momenta will induce some corrections. Finally, the loss coefficient involves the unknown vertices and phase space factors of the trion decay—for this reason our computation contains an unknown multiplicative factor cK. In terms of p given by (9.66) we obtain the three-body loss coefficient K3 ¼ cK p:

ð9:78Þ

Our result as well as the experimental data points [22] are shown in Fig. 9.10. The agreement between the form of the two curves is already quite remarkable. We have used three parameters, the location of the resonance at B0 ¼ 125 G which we translate into k(3), the overall amplitude cK and the decay width Cv. They are essentially fixed by the peak at B0 ¼ 125 G: The extension of the loss rate away from the peak involves then no further parameter. Our estimated decay width Cv corresponds to a short lifetime of the trion of the order of 108 s: Our simple prediction involves a rather narrow second peak around B1  500 G; where the trion energy becomes again degenerate with the open channel, cf. Fig. 9.9. The width of this peak is fixed so far by the assumption that the decay width Cv is independent of the magnetic field. This may be questionable in view of the close-by Feshbach resonance and the fact that the trion may actually decay into the associated molecule-like bound states which have lower energy. We have tested several reasonable approximations, which indeed lead to a broadening or even disappearance of the second peak, without much effect on the intermediate range of fields 150 G\B\400 G; see [24]. Shortly before finishing the calculations discussed here [24a] similar work by Braaten et al. [25] as well as Naidon and Ueda [26] was published. In Ref. [25] the scattering amplitudes of the loss process are calculated in the zero-range approximation with the help of generalized STM equations and a two parameter fit is given for the measured loss coefficient. Naidon and Ueda [26] perform a quantum mechanical calculation using the hyperspherical formalism as in [16].

114

9 Few-Body Physics

Fig. 9.10 Loss coefficient K3 in dependence on the magnetic field B as measured in [22] (dots). The solid line is the fit of our model to the experimental curve. We use here a decay width Cv that is independent of the magnetic field B

In addition to a three parameter fit for the loss coefficient the authors calculate the Efimov trimer binding energy. The complex three-body parameters introduced in [25, 26] correspond to the parameters k(3) and Cv in our setting. The authors of both papers conclude that the three-body loss is due to an Efimov state near the three-atom threshold which is consistent with the scenario depicted here. In the few-body regime our predictions should be equivalent to those obtained in Refs [25, 26] and, indeed, the results for the loss coefficient are consistent with our calculation shown in Fig. 9.10. Without refermionization we find a difference in the prediction of the trion binding energy compared to Ref. [26]. However, with the improved truncation (dashed line in Fig. 9.9) the agreement becomes better and the inclusion of an interaction of the type wi wi uj uj with i = j will induce further quantitative corrections. In conclusion, a rather simple bound state exchange picture describes rather well the observed enhancement of the three-body loss coefficient in a range of magnetic fields between 100 and 520 G. A similar trion dominated three-body loss is possible for large BðBJ850 GÞ; where also a trion bound state with energy below the open channel exists. However, the dimer bound states are now above the open channel level, such that the trion decay may be strongly altered. The role of trion bound states in the resonance region is an interesting subject by its own, that can be explored by our functional renormalization group methods with an extended truncation. The calculated energy spectrum may be experimentally tested by radio frequency spectroscopy although this might be difficult due to the short lifetimes of the trion. An advantage of the method we use is that the flow equations can also be extended to the case of nonvanishing temperature and density similar as for the BCS–BEC crossover in the two-component Fermi gas, see Sect. 10.3 in Chap. 10. The few-body calculation we presented here is the necessary premise for finite temperature and density calculations and fixes the microscopic parameters for the three-component 6Li Fermi gas. Therefore this work provides a good starting point for the investigation of the interesting phase diagram of this system, a challenge for both theory and experiment.

References

115

References 1. Diehl S, Floerchinger S, Gies H, Pawlowski JM, Wetterich C (2009) arXiv:0907.2193 (e-print) 2. Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:053627 3. Diehl S (2006) The BCS–BEC crossover in ultracold fermion gases. Ph.D. thesis, Universität Heidelberg, Germany 4. Braaten E, Hammer HW (2006) Phys Rep 428:259 5. Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:021602 6. Diehl S, Krahl HC, Scherer M (2008) Phys Rev C 78:034001 7. Pieri P, Strinati GC (2000) Phys Rev B 61:15370 8. Petrov DS, Salomon C, Shlyapnikov GV (2004) Phys Rev Lett 93:090404 9. Astrakharchik GE, Boronat J, Casulleras J, Giorgini, S. (2004) Phys Rev Lett 93:200404 10. Brodsky IV, Klaptsov AV, Kagan MY, Combescot R, Leyronas X (2005) JETP Lett 82:273 11. Gies H, Wetterich C (2002) Phys Rev D 65:065001 12. Pawlowski JM (2007) Ann Phys (N. Y.) 322:2831 13. Bartenstein M, Altmeyer A, Riedl S, Geursen R, Jochim S, Chin C, Denschlag JH, Grimm R, Simoni A, Tiesinga E, Williams CJ, Julienne PS (2005) Phys Rev Lett 94:103201 14. Petrov DS (2004) Phys Rev Lett 93:143201 15. Gogolin AO, Mora C, Egger R (2008) Phys Rev Lett 100:140404 16. Efimov V (1970) Phys Lett B 33:563 17. Efimov V (1973) Nucl Phys A 210:157 18. Bedaque PF, Hammer H-W, van Kolck U (1999) Phys Rev Lett 82:463 19. Bedaque PF, Hammer HW, van Kolck U (1999) Nucl Phys A 646:444 20. Bedaque PF, Braaten E, Hammer H-W (2000) Phys Rev Lett 85:908 21. Moroz S, Floerchinger S, Schmidt R, Wetterich C (2009) Phys Rev A 79:042705 22. Ottenstein TB, Lompe T, Kohnen M, Wenz AN, Jochim S (2008) Phys Rev Lett 101:203202 23. Huckans JH, Williams JR, Hazlett EL, Stites RW, O’Hara KM (2009) Phys Rev Lett 102:165302 24. Schmidt R (2009) Trion formation in ultracold quantum gases. Diploma thesis, Universität Heidelberg, Germany 24a. Floerchinger S, Schmidt R, Wetterich C (2009) Phys Rev A 79:053633 25. Braaten E, Hammer HW, Kang D, Platter L (2009) Phys Rev Lett 103:073202 26. Naidon P, Ueda M (2009) Phys Rev Lett 103:073203

Chapter 10

Many-Body Physics

10.1 Bose–Einstein Condensation in Three Dimensions In this section we discuss the many-body results obtained for the Bose gas model in (6.1) using the approximation scheme described in Sect. 8.1. We start with the phase diagram at zero temperature (quantum phase diagram) and move then to nonzero temperature. After describing the general features of the thermal phase transition we discuss the critical temperature for an interacting Bose gas and several thermodynamic observables ranging from pressure and energy density to different sound velocities.

10.1.1 Different Methods to Determine the Density The density sets a crucial scale for our problem. Its precise determination is mandatory for quantitative precision. We will discuss two different methods for its determination and show that the results agree within our precision. For T = 0, we also find agreement with the Ward identity n = q0. The first method is to derive flow equations for the density. This has the advantage that the occupation numbers for a given momentum ~ p are mainly sensitive to running couplings with k2 ¼ ~ p2 : In the grand canonical formalism, the density is defined by n¼

 o 1  C½u ol X u¼u0 ;l¼l0

We can formally define a k-dependent density nk by  o 1  Ck ½u nk ¼  ¼ ðol UÞðq0 ; l0 Þ: ol X u¼u0 ;l¼l0 S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_10, Ó Springer-Verlag Berlin Heidelberg 2010

ð10:1Þ

ð10:2Þ

117

118

10

Many-Body Physics

The flow equation for nk is given in (8.19) and the physical density obtains for k = 0. The term ol fjq0 ;l0 that enters (8.19) is the derivative of the flow equation (8.13) for U with respect to l. To compute it, we need the l-dependence of the propagator Gk in the vicinity of l0. Within a systematic derivative expansion, we use the expansion of U(q, l) and the kinetic coefficients Z1 and Z2 to linear order in (l - l0), as described in Sect. 7.1. Here, Z1(q, l) and Z2(q, l) are the coefficient functions of the terms linear in the s-derivative and linear in D, respectively. No reasonable qualitative behavior is found, if the linear dependence of Z1 and Z2 on (l - l0) is neglected. Also, the scale dependence of a and b are quite important. The flow equations for a and b can be obtained directly by differentiating the flow equation of the effective potential with respect to l and q, ðlÞ cf. (8.20). The situation is more complicated for the kinetic coefficients Z1 ¼ ðlÞ

ol Z1 ðq0 ; l0 Þ and Z2 ¼ ol Z2 ðq0 ; l0 Þ: Their flow equations have to be determined by taking the l-derivative of the flow equation for Z1(q, l) and Z2(q, l). ðlÞ ðlÞ As discussed in Sect. 7.1, we use the approximation Z1 ¼ Z2 ¼ 2V ¼ 2V1 ðq0 ; l0 Þ: As a check of both our method and our numerics, we also use another way to determine the particle density. This second method is more robust with respect to shortcomings of the truncation, but less adequate for high precision calculations as needed, e.g. to determine the condensate depletion. The second method determines the pressure p = -U(q0, l0) as a function of the chemical potential l0. Here, the effective potential is normalized by U(q0 = 0, l0) = 0 at T = 0, n = 0. The flow of the pressure can be read of directly from the flow equation of the effective potential and is independent of the couplings a and b. We calculate the pressure as a function o of l and determine the density n ¼ ol p by taking the l-derivative numerically. It turns out that p is in very good approximation given by p = cl2, where the constant c can be determined from a numerical fit. The density is thus linear in l. At zero temperature and for ~v ¼ 0; we can additionally use the Ward identities connected to Galilean symmetry, which yield n = q0. We compare our methods in Fig. 10.1 and find that they give numerically the same result. We stress again the importance of a reliable method to determine the density, since we often rescale variables by powers of the density to obtain dimensionless variables.

10.1.2 Quantum Depletion of Condensate We want to split the density into a condensate part nc and a density for uncondensed particles or ‘‘depletion density’’ nd = n - nc. For our model the condensate density is given by the ‘‘bare’’ order parameter 0 ¼ q 0 ðk ¼ 0Þ: nc ¼ q

ð10:3Þ

~Þ for the modes with In order to show this, we introduce occupation numbers nðp momentum ~ p with normalization

10.1

Bose–Einstein Condensation in Three Dimensions

119

Fig. 10.1 Pressure and density as a function of the chemical potential at T = 0. We use three different methods: n ¼ ol Umin from the flow equation (triangles), n = q0 as implied by Galilean symmetry (stars) and n ¼ ol p, where the pressure p = -U (boxes) was obtained from the flow equation and phenomenologically fitted by p = 56.5 l2 (solid lines). Units are arbitrary and we use a = 3.4 9 10-4, K = 103

Z

~Þ ¼ n: nðp

ð10:4Þ

~ p

~Þ in the grand One formally introduces a ~ p dependent chemical potential lðp canonical partition function eCmin ½l ¼ Tre with three-dimensional volume X3 ¼ numbers by ~Þ ¼  nðp

bðHX3

R

~ x:

R ~ p

lðp ~Þnðp ~ÞÞ

;

ð10:5Þ

Then one can define the occupation

 d 1  ~Þ C½u; lðp : ~Þ bX3 dlðp u¼u0 ;lðp ~Þ¼l0

ð10:6Þ

This construction allows us to use k-dependent occupation numbers by the definition ~Þ ¼  nk ðp

 d 1  Ck ½u; l : ~Þ bX3 ~Þ¼l0 dlðp u¼u0 ;lðp

ð10:7Þ

~Þ [1]: One can derive a flow equation for this occupation number nk ðp n o 1 d 1 Tr ðCð2Þ þ Rk Þ1 ok Rk ~Þ bX3 2 dlðp o d 1 0 Þ: þ C½u; lðok q ~Þ bX3 o q dlðp

~Þ ¼  ok nk ðp

ð10:8Þ

120

10

Many-Body Physics

We split the density occupation number into a d-distribution like part and a depletion part, which is regular in the limit ~ p!0 ~Þ ¼ nc;k dðp ~Þ þ nd;k ðp ~Þ: nk ðp

ð10:9Þ

~Þ that the only contribution to ok nc;k One can see from the flow equation for nk ðp comes from the second term in equation (10.8). Within a more detailed analysis [1] one finds 0;k : ok nc;k ¼ ok q

ð10:10Þ

We, therefore, identify the condensate density with the bare order parameter q 0 ¼ 0 ¼ u  20 : nc ¼ q A

ð10:11Þ

Correspondingly, we define the k-dependent quantities 0;k ; nc;k ¼ q

nk ¼ nc;k þ nd;k

ð10:12Þ

and compute nd = nd(k = 0) by a solution of its flow equation. Even at zero temperature, the repulsive interaction connected with a positive scattering length a causes a portion of the particle density to be outside the condensate. From dimensional reasons, it is clear, that nd/n = (n - nc)/n should be a function of an1/3. The prediction of Bogoliubov theory or, equivalently, mean field theory, is nd =n ¼ 3p8 ffiffip ðan1=3 Þ3=2 : We may determine the condensate depletion from the solution to the flow equation for the particle density, n = nk=0, and 0 ¼ q 0 ðk ¼ 0Þ: nc ¼ q From Galilean invariance for T = 0 and ~v ¼ 0; it follows that 0 nd q0  q 1 ¼ ¼1; n q0 A

ð10:13Þ

 ¼ Aðk  ¼ 0Þ: This gives an independent determination of nc. In Fig. 10.2 we with A 0 over several orders of plot the depletion density obtained from the flow of n and q magnitude. Apart from some numerical fluctuations for small an1/3, we find that our result is in full agreement with the Bogoliubov prediction.

10.1.3 Quantum Phase Transition For T = 0 a quantum phase transition separates the phases with q0 = 0 and q0 [ 0. In this section, we investigate the phase diagram at zero temperature in the ~ ¼ Kl2 ; ~a ¼ aK and ~v ¼ VS2K K2 : cube spanned by the dimensionless parameters l K

This goes beyond the usual phase transition for nonrelativistic bosons, since we also include a microscopic second s-derivative  ~v; and, therefore, models with a

10.1

Bose–Einstein Condensation in Three Dimensions

121

Fig. 10.2 Condensate depletion (n - nc)/n as a function of the dimensionless scattering length an1/3. For the solid curve, we vary a with fixed n = 1, for the dashed curve we vary the density at fixed a = 10-4. The dotted line is the Bogoliubov-Result ðn  nc Þ=n ¼ 3p8 ffiffip ðan1=3 Þ3=2 for reference. We find perfect agreement of the three determinations. The fluctuations in the solid curve for small an1/3 are due to numerical uncertainties. Their size demonstrates our numerical precision

generalized microscopic dispersion relation. For non-vanishing ~v (i.e. for a nonzero initial value of V1 with V2 = V3 = 0 in Sect. 7.1), the Galilean invariance at zero temperature is broken explicitly. For large ~v; we expect a crossover to the ‘‘relativistic’’ O(2) model. If we send the initial value of the coefficient of the linear s-derivative SK to zero, we obtain the limiting case ~v ! 1: The symmetries of the model are now the same as those of the relativistic O(2) model in fourdimensions. The space-time-rotations or Lorentz symmetry replace Galilean symmetry. It is interesting to study the crossover between the two cases. Since our cutoff explicitly breaks Lorentz symmetry, we investigate in this paper only the regime ~v . 1: Detailed investigations of the flow equations for ~v ! 1 can be found in the ~-~v plane with ~a ¼ 1 is shown in literature [2–7]. The phase diagram in the l Fig. 10.3. The critical chemical potential first increases linearly with ~v and then saturates to a constant. The slope in the linear regime as well as the saturation value depend linearly on ~ a for ~ a\1: Fig. 10.3 Quantum phase ~-~v plane for diagram in the l ~a ¼ 1

122

10

Many-Body Physics

At T = 0, the critical exponents are everywhere the mean field ones (g = 0, m = 1/2). This is expected: It is the case for ~v ¼ 0 [8–10], and for ~v ¼ 1 the theory is equivalent to a relativistic O(2) model in d = 3 + 1-dimensions. This is just the upper critical dimension of the Wilson–Fisher fixed point [11]. From Sect. 3 we know that for ~v ¼ 0 the parameter ~a is limited to ~a\ 3p 4  ~ 2:356: For v ¼ 0 and a small scattering length a?0, a second order quantum phase transition divides the phases without spontaneous symmetry breaking for l \ 0 from the phase with a finite order parameter q0 [ 0 for l [ 0. It is an interesting question, whether this quantum phase transition at l ¼ 0; ~v ¼ 0 also occurs for larger scattering length a. We find in our truncation that this is indeed the case for a large range of a, but not for ~ a [ 1:55: Here, the critical chemical potential suddenly increases to large positive values as shown in Fig. 10.4. For ~v [ 0 this increase happens even earlier. (For a truncation with V1:0, the phase transition ~-~ would always occur at l = 0.) We plot the l a plane of the phase diagram for different values of ~v in Fig. 10.4. The form of the critical line can be understood by considering the limits ~v ! 0 as well as ~ a ! 0: For a fixed chemical potential, the order parameter q0 as a function of a goes to zero at a critical value ac as shown in Fig. 10.5. This happens in a continuous way and the phase transition is, therefore, of second order. For l?0, we find ac = 1.55 K-1. We emphasize, however, that ac is of the order of the microscopic Fig. 10.4 Quantum phase ~-~a plane for diagram in the l ~v ¼ 1 (dotted), ~v ¼ 0:01 (dashed) and ~v ¼ 0 (solid)

Fig. 10.5 Quantum phase transition for fixed chemical potential l = 1, with K = 103. The density q0 = n as a function of the scattering length a goes to zero at a critical ac K = 1.55, indicating a second order quantum phase transition at that point

10.1

Bose–Einstein Condensation in Three Dimensions

123

distance K-1. Universality may not be realized for such values, and the true phase transition may depend on the microphysics. For example, beyond a critical value for the repulsive interaction, the system may form a solid. Ultracold atom gases correspond to metastable states which may lose their relevance for a?K-1. For ~v [ 0 and l  K2 the phase transition occurs for acK  1 such that universal behavior is expected.

10.1.4 Thermal Depletion of Condensate So far, we have only discussed the vacuum and the dense system at zero temperature. A nonvanishing temperature T will introduce an additional scale in our problem. For small T  n2/3 we expect only small corrections. However, as T increases the superfluid order will be destroyed. Near the phase transition for T  Tc and for the disordered phase for T [ Tc, the characteristic behavior of the boson gas will be very different from the T?0 limit. For T [ 0 the particle density n receives a contribution from a thermal gas of bosonic (quasi-) particles. It is no longer uniquely determined by the superfluid density q0. We may write n ¼ q 0 þ nT

ð10:14Þ

and observe, that nT = 0 is enforced by Galilean symmetry only for T = 0, VK = 0. The heat bath singles out a reference frame, such that for T [ 0 Galilean symmetry no longer holds. In our formalism, the thermal contribution nT appears due to modifications of the flow equations for T = 0. We start for high k with the same initial values as for T = 0. As long as k  p T the flow equations receive only minor modifications. For k  pT or smaller, however, the discreteness of the Matsubara sum has important effects. We plot in Fig. 10.6 the density as a function of T for fixed l = 1. In Fig. 10.7 we show n(l), similar to Fig. 10.1, but now for different a and T. For T = 0 the scattering length sets the only scale besides n and l, such that by dimensional arguments a2l = f(a3 n). Bogoliubov theory predicts Fig. 10.6 Density n/l2/3 (solid) and order parameter q0/l2/3 (dashed) as a function of the temperature T/l. The units are arbitrary with a = 2 9 10-4 and K = 103. The plot covers only the superfluid phase. For higher temperatures, the density is given by the thermal contribution n = nT only

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Fig. 10.7 Density n for different temperatures and scattering length. We plot n(l) in arbitrary units, with K = 103, and for a scattering length a = 2 9 10-4 (solid and dotted), a = 10-4 (dashed and dashed-dotted). The temperature is T = 0 (solid and dashed) and T = 1 (dotted and dashed-dotted)

  32 1=2 f ðxÞ ¼ 8px 1 þ pffiffiffi x : 3 p

ð10:15Þ

The first term on the r.h.s. gives the contribution of the ground state, while the second term is added by fluctuation effects. For small scattering length a, the ground state contribution dominates. We have then l*a for n = 1 and l/n can be treated as a small quantity. For T = 0 and small a one expects l = g(T/n2/3)an. The curves in Fig. 10.7 for T = 1 show that the density, as a function of l, is below the curve obtained at T = 0. This is reasonable, since the statistical fluctuations now drive the order parameter q0 to zero. At very small l, the flow enters the symmetric phase. The density is always positive, but for simplicity, we show the density as a function of l in Fig. 10.7 only in those cases, where the flow remains in the phase with spontaneous U(1) symmetry breaking. For temperatures above the critical temperature, the order parameter q0 vanishes at the macroscopic scale and so does the condensate density nc ¼ q0 ¼ 1  q0 : The density is now given by a thermal distribution of particles with nonzero A momenta. Up to small corrections from the interaction *aT, it is described by a free Bose gas, n¼

T 3=2 ð4pÞ

3=2

g3=2 ðebl Þ;

ð10:16Þ

with the ‘‘Bose function’’ 1 gp ðzÞ ¼ CðpÞ

Z1

dx xp1

1 : z1 ex  1

ð10:17Þ

0

In Fig. 10.8 we show the dimensionless order parameter q0/n as a function of the dimensionless temperature T/n2/3. This plot shows the second order phase transition from the phase with spontaneous U(1) symmetry breaking at small temperatures to the symmetric phase at higher temperatures. The critical

10.1

Bose–Einstein Condensation in Three Dimensions

125

Fig. 10.8 Order parameter q0/n as a function of the dimensionless temperature T/(n2/3) for scattering length a = 10-4. Here, we varied T keeping l fixed. Numerically, this is equivalent to varying l with fixed T

temperature Tc is determined as the temperature, where the order parameter just   1Þ  1; the vanishes—it is investigated in the next section. Since we find ðA 2/3  as a function of T/n resembles the condensate fraction nc =n ¼ q0 =n ¼ q0 =ðAnÞ  as a function of T/n2/3 in Fig. 10.9. Except for a order parameter q0/n. We plot A narrow region around Tc, the deviations from one remain indeed small. Near Tc the  diverges according to the anomalous dimension, A   ng ; gradient coefficient A with g the anomalous dimension. The correlation length n diverges with the critical exponent m, n*|T - Tc|-m, such that   jT  Tc jgm : A

ð10:18Þ

Here, g and m are the critical exponents for the Wilson Fisher fixed point of the classical three-dimensional O(2) model, g = 0.0380(4), m = 0.67155(27) [2, 4–7, 12].

 ¼ q0 =nc ; as a function of the Fig. 10.9 Order parameter divided by the condensate density A dimensionless temperature T/(n2/3), and for scattering length a = 10-4. Here, we varied T keeping l fixed. Numerically, this is equivalent to varying l with fixed T. The plot covers only the phase with spontaneous symmetry breaking. For higher temperatures, the symmetric phase  for T?Tc reflects the anomalous dimension g of the has q0 = nc = 0. The divergence of A Wilson-Fisher fixed point

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Fig. 10.10 Order parameter divided by the density, q0/n, as a function of the chemical potential. We use arbitrary units with K = 103. The curves are given for a scattering length a = 2 9 10-4 (solid and dotted), a = 10-4 (dashed and dashed-dotted) and temperature T = 0.1 (solid and dashed) and T = 1 (dotted and dashed-dotted). At zero temperature, Galilean invariance implies q0 = n, which we find within our numerical resolution

In Fig. 10.10 we plot q0/n as a function of the chemical potential l for different temperatures and scattering lengths. We find, that q0/n = 1 is indeed approached in the limit T?0, as required by Galilean invariance. All figures of this section are for ~v ¼ 0: The modifications for ~v 6¼ 0 are mainly quantitative, not qualitative.

10.1.5 Critical Temperature The critical temperature Tc for the phase transition between the superfluid phase at low temperature and the disordered or symmetric phase at high temperature depends on the scattering length a. By dimensional reasoning, the temperature shift DTc = Tc(a) - Tc(a = 0) obeys DTc/Tc*an1/3. The proportionality coefficient cannot be computed in perturbation theory [13]. It depends on ~v and we concentrate here on ~v ¼ 0: Monte-Carlo simulations in the high temperature limit, where only the lowest Matsubara frequency is included, yield DTc/Tc = 1.3 an1/3 [14, 15]. Within the same setting, renormalization group studies [16–20] yield a similar result, for composite bosons see [21]. Near Tc, the long wavelength modes with momenta ~ p2  ðpTÞ2 dominate the ‘‘long distance quantities’’. Then a description in terms of a classical three-dimensional system becomes valid. This ‘‘dimensional reduction’’ is achieved by ‘‘integrating out’’ the nonzero Matsubara frequencies. However, both DTc/Tc and n are dominated by modes with momenta ~ p2  ðpTc Þ2 such that corrections to the classical result may be expected. We have computed Tc numerically by monitoring the zero of q0, as shown in Fig. 10.8, q0(T? Tc)?0. Our result is plotted in Fig. 10.11. In the limit a?0 we find for the dimensionless critical temperature Tc/(n2/3) = 6.6248, which is in good

10.1

Bose–Einstein Condensation in Three Dimensions

127

Fig. 10.11 Dimensionless critical temperature Tc/(n2/3) as a function of the dimensionless scattering length an1/3 (points). We also plot the linear fit DTc/Tc = 2.1an1/3 (solid line)

agreement with the expected result for the free theory Tc =ðn2=3 Þ ¼

4p fð3=2Þ2=3

¼

6:6250: For the shift in Tc due to the finite interaction strength, we obtain DTc ¼ j an1=3 ; Tc

j ¼ 2:1:

ð10:19Þ

We expect that the result for j depends on the truncation and may change somewhat if additional higher order couplings are included.

10.1.6 Zero Temperature Sound Velocity The macroscopic sound velocity vS is a crucial quantity for the hydrodynamics of the gas or liquid. It is accessible to experiment. As a thermodynamic observable, the adiabatic sound velocity is defined as v2S ¼

1 op j M on s

ð10:20Þ

where M is the particle mass (in our units 1/M = 2), p is the pressure, n is the particle density, and s is the entropy per particle. It is related to the isothermal sound velocity vT by v2S

      1 op op  oT  op  oT  2 ¼ þ ¼ vT þ 2  M onT oT n on s oT n on s

ð10:21Þ

where we use our units 2M = 1. One needs the ‘‘equation of state’’ p(T, n) and  S 1 op  sðT; nÞ ¼ ¼ : ð10:22Þ N n oT l

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By dimensional analysis, one has p ¼ n5=3 F ðt; cÞ;

t¼

T ; n2=3

c ¼ an1=3 ;

with F ð0; cÞ ¼ 4pc (in Bogoliubov theory), and F ðt; 0Þ ¼

fð5=2Þ 5=2 t ð4pÞ3=2

ð10:23Þ (free theory),

such that for small c F ¼

fð5=2Þ ð4pÞ3=2

t5=2 þ gðtÞc:

ð10:24Þ

At zero temperature the second term in (10.21) vanishes, such that vS = vT. For the isothermal sound velocity one has     1 op  op  on  ¼2  ¼2  : on T ol T ol T

ð10:25Þ

 op dUmin ¼ ol Uðq0 Þ ¼ n ¼  ol T dl

ð10:26Þ

v2T We can now use

and infer v2T

  o ln n 1 ¼2 : ol

ð10:27Þ

One may also define a microscopic sound velocity cS, which characterizes the propagation of (quasi-) particles. At zero temperature, where we can perform the analytic continuation to real time, we can calculate the microscopic sound velocity ~j). In turn, the dispersion relation is from the dispersion relation x(p) (with p ¼ jp obtained from the effective action by setting det(G-1) = 0, where G-1 is the full inverse propagator. We perform the calculation explicitly at the end of Sect. 7.1 and find c2 S ¼

S2 þ V: 2kq0

ð10:28Þ

The Bogoliubov result for the sound velocity is in our units c2S ¼ 2kq0 ¼ 16pan:

ð10:29Þ

In three-dimensions, the decrease of S is very slow and the coupling V remains comparatively small even on macroscopic scales, cf. Fig. 10.28. We thus do not expect measurable deviations from the Bogoliubov result for the sound velocity at

10.1

Bose–Einstein Condensation in Three Dimensions

129

Fig. 10.12 Dimensionless sound velocity cs/(n1/3) at zero temperature, as a function of the scattering length an1/3. Within the plot resolution the curves obtained by varying a with fixed n, by varying n with fixed a, and the Bogoliubov result, cs ¼ pffiffiffiffiffiffiffiffi 16pðanÞ1=2 ; coincide

T = 0. In Fig. 10.12, we plot our result over several orders of magnitude of the dimensionless scattering length and, indeed, find no deviations from Bogoliubov’s result. We finally show that for T = 0 the macroscopic and microscopic sound velocities are equal, vS = vT = cS. For this purpose, we use   on d  dq ol Uðq0 Þ ¼ o2l Uðq0 Þ  oq ol Uðq0 Þ 0 : ¼ olT dl dl

ð10:30Þ

From the minimum condition oq U ¼ 0; it follows dq0 oq ol U a ¼ 2 ¼ : k dl oq U

ð10:31Þ

Combining this with the Ward identities from Sect. 7.1, namely o2l U ¼ 2Vq0 and a ¼ oq ol U ¼ S; valid at T = 0, it follows that the macroscopic sound velocity equals the microscopic sound velocity v2S ðT ¼ 0Þ ¼ c2S :

ð10:32Þ

10.1.7 Thermodynamic Observables We now come to the discussion of some thermodynamic properties at nonzero temperature. With our method we can determine many thermodynamic observables from the effective potential U. It is related to the pressure by  pðT; lÞ ¼ Umin ðT; lÞk¼kph

ð10:33Þ

dp ¼ s dT þ n dl:

ð10:34Þ

which has the differential

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Here we use s = S/V for the entropy density and n = N/V for the particle density. The formal infinite volume limit corresponds to kph = 0. Derivatives of U with respect to T and and l are taken numerically by solving the flow equation for close enough values of T and l. The numerical effort is reduced and the accuracy increased by using an additional flow equation for nk ¼ 

  o Uk  ; ol ¼ q q0 ðkÞ

ð10:35Þ

with n ¼ nkph : The approximation scheme we use in the following is basically the same as the one described in Sect. 8.1. Since we use an infrared cutoff only for momenta but not for frequencies, the correct ultraviolet convergence for the sum of the Matsubara frequencies is not automatically obeyed for the flow equations. We have checked that all thermodynamic quantities discussed in the following show a satisfactory convergence of the Matsubara sum, except for the pressure. In the flow equation for pk we set the frequency coefficients to their microscopic  ¼ 0 for very large Matsubara frequencies |q0| [ K2UV. values  S ¼ 1; V For bosons with a pointlike repulsive interaction we found in Sect. 9.1 that the scattering length is bounded by the ultraviolet scale a \ 3p/(4K). This is an effect due to quantum fluctuations similar to the ‘‘triviality bound’’ for the Higgs scalar in the standard model of elementary particle physics. For a given value of the dimensionless combination an1/3 we cannot choose K/n1/3 larger than 3p/(4an1/3). For our numerical calculations we use K=n1=3  10: Other momentum scales are set by the temperature and the chemical potential. The lowest nonzero Matsubara frequency gives the momentum scale K2T ¼ 2pT: For a Bose gas with a = 0 one has Tc =n2=3  6:625 such that KTc =n1=3  6:45: The momentum scale associated to the chemical potential is K2l = l. For small temperatures and scattering length pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi one finds l 8pan and thus Kl =n1=3  8pan1=3 : We finally note that the thermodynamic relations for intensive quantities can only involve dimensionless ratios. We may set the unit of momentum by n1/3. The thermodynamic variables are then T/n2/3 and l/n2/3. The thermodynamic relations will depend on the strength of the repulsive interaction k or the scattering length a, and, therefore, on a ‘‘concentration’’ type parameter an1/3.

10.1.7.1 Density, Superfluid Density, Condensate and Correlation Length Let us start our discussion with the density. In the grand canonical formalism it is obtained by taking the derivative of the thermodynamic potential with respect to l n¼

 1 o op  XG ¼ : V ol ol T

ð10:36Þ

10.1

Bose–Einstein Condensation in Three Dimensions

Fig. 10.13 Density in units of the scattering length na3 as a function of the (rescaled) chemical potential la2. We choose for the temperatures Ta2 = 2 9 10-4 (solid curve), Ta2 = 4 9 10-4 (dashed-dotted curve) and Ta2 = 6 9 10-4 (dashed curve). For all three curves we use aK = 0.1

1.4 10

6

1.2 10

6

1. 10

6

8. 10

7

6. 10

7

4. 10

7

2. 10

7

0.00001

131

0.00002

0.00003

0.00004

We could compute the l-derivative of p numerically by solving the flow equation for U with neighboring values of l. As desribed above, we use also another method which employs a flow equation directly for n. Since we often express dimensionful quantities in units of the interparticle distance n-1/3, it is crucial to have an accurate value for the density n. Comparison of the numerical evaluation and the solution of a separate flow equation for n shows higher precision for the latter method and we will, therefore, employ the flow equation. We plot in Fig. 10.13 the density in units of the scattering length, na3, as a function of the dimensionless combination la2. For a comparison with experimentally accessible quantities we have to replace the interaction parameter k in the microscopic action (6.1) by a scattering length a which is a macroscopic quantity. For this purpose we start the flow at the UV-scale KUV with a given k, and then compute the scattering length in vacuum (T = n = 0) by following the flow to k = 0, see Sect. 9.1. This is a standard procedure in quantum field theory, where a ‘‘bare coupling’’ (k) is replaced by a renormalized coupling (a). For an investigation of the role of the strength of the interaction we may consider different values of the ‘‘concentration’’ c = an1/3 or of the product la2. While the concentration is easier to access for observation, it is also numerically more demanding since for every value of the parameters one has to tune l in order to obtain the appropriate density. For this reason we rather present results for three values of la2, i.e. la2 = 2.6 9 10-5 (case I), la2 = 0.0040 (case II) and la2 = 0.044 (case III). The prize for the numerical simplicity is a week temperature dependence of the concentration c = an1/3 for the three different cases, as shown in Fig. 10.14. Here and in the following figures case I, which corresponds to an1=3  0:01, is represented by the little crosses, case II with an1=3  0:05 by the dots and case III with an1=3  0:1 by the stars. It is well known that the critical temperature depends on the concentration c = an1/3. From our calculation we find Tc/(n2/3) = 6.74 with c = 0.0083 at T = Tc in case I, Tc/(n2/3) = 7.16 with c = 0.044 at T = Tc in case II and finally Tc/(n2/3) = 7.75 with c = 0.088 at T = Tc in case III. This values are obtained by following the superfluid fraction of the density nS/n, or equivalently the condensate part of the density nC/n as a function of

132 Fig. 10.14 Concentration c = an1/3 as a function of temperature T/(n2/3) for the three cases investigated in this paper. Case I corresponds to an1=3  0:01 (crosses), case II to an1=3  0:05 (dots) and case III has an1=3  0:01 (stars)

10

Many-Body Physics

0.10

0.08

0.06

0.04

0.02

2

4

6

8

temperature. For small temperatures T?0 all of the density is superfluid, which is a consequence of Galilean symmetry. However, in contrast to the ideal gas, not all particles are in the condensate. For T = 0 this condensate depletion is completely due to quantum fluctuations. With increasing temperature both the superfluid density and the condensate decrease and vanish eventually at the critical temperature T = Tc. That the melting of the condensate is continuous shows that the phase transition is of second order. We plot our results for the superfluid fraction in Fig. 10.15 and for the condensate in Fig. 10.16. For small temperatures, we also show the corresponding result obtained in the framework of Bogoliubov theory [22] (dashed lines). This approximation assumes a gas of non-interacting quasiparticles (phonons) with dispersion relation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 þ ~ ðpÞ ¼ 2knp ð10:37Þ p4 : It is valid in the regime with small temperatures T  Tc and small interaction strength an1/3  1. For a detailed discussion of Bogoliubov theory and the calculation of thermodynamic observables in this framework we refer to ref. [23]. Our curves for the superfluid fraction match the Bogoliubov result for temperatures T=n2=3 .1 in all three cases I, II, and III. For larger temperatures there are deviations as expected. For the condensate density, there is already notable a deviation Fig. 10.15 Superfluid fraction of the density nS/n as a function of the temperature T/n2/3 for the cases I, II, and III. For small T/n2/3 we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines)

1.0

0.8

0.6

0.4

0.2

0.0

2

4

6

8

10.1

Bose–Einstein Condensation in Three Dimensions

Fig. 10.16 Condensate fraction of the density nC/n as a function of the temperature T/n2/3 for the cases I, II, and III. For small T/n2/3 we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines)

133

1.0

0.8

0.6

0.4

0.2

0.0

2

4

6

8

at small temperatures for case III with an1=3  0:1: This is also expected, since Bogoliubov theory gives only the first order contribution to the condensate depletion in a perturbative expansion for small an1/3. For temperatures slightly smaller than the critical temperature Tc one expects that the condensate density behaves like  2b 2 Tc  T ð10:38Þ nc ðTÞ ¼ B Tc with b = 0.3485 the critical exponent of the three-dimensional XY-universality  0 u  0 is the  0 where u class [12]. Indeed, the condensate density is given by nC ¼ u expectation value of the boson field which serves as an order parameter in close ~ in a ferromagnet. Equation (10.38) is comanalogy to, e.g. the magnetization M patible with our findings, although our numerical resolution does not allow for a precise determination of the exponent b. With our method we can also calculate the correlation length n. For temperatures T \ Tc one distinguishes between the Goldstone correlation length nG and the radial correlation length nR. While the former is infinite, n1 G ¼ 0, the latter is finite for T \ Tc. It is also known as the ‘‘healing length’’, given by 1 o2 U  n2 q R ¼ 2kq0 ¼ 2  q 0 A o

ð10:39Þ

and diverges only close to the phase transition. In the symmetric regime for T [ Tc there is only one correlation length n1 ¼ m ¼ A1 oU o q ; which also diverges for T?Tc. From the theory of critical phenomena one expects close to Tc the behavior  m  Tc  T for T\Tc nR ¼ fR Tc ð10:40Þ  m T  T c n ¼ fþ for T [ Tc : Tc

134 Fig. 10.17 Correlation length nRn1/3 for T \ Tc and n n1/3 for T [ Tc as a function of the temperature T/n2/3 for the cases I, II, and III

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Many-Body Physics

400

300

200

100

0

2

4

6

8

The critical exponent m = 0.6716 [12] is again the one of the three-dimensional XY- or O(2) universality class. We plot our result for the correlation length in units of the interparticle distance nRn1/3 for T \ Tc and nn1/3 for T [ Tc as a function of the temperature T/n2/3 in Fig. 10.17.

10.1.7.2 Entropy Density, Energy Density, and Specific Heat The next thermodynamic quantity we investigate is the entropy density s and the entropy per particle s/n. We can obtain the entropy as  op  s¼ : ð10:41Þ oT l We compute the temperature derivative by numerical differentiation, using flows with neighboring values of T and show the result in Fig. 10.18. For small temperatures our result coincides with the entropy of free quasiparticles in the Bogoliubov approximation (dashed lines in Fig. 10.18). As it should be, the entropy per particle increases with the temperature. For small temperatures, the slope of this increase is smaller for larger concentration c.

Fig. 10.18 Entropy per particle s/n as a function of the dimensionless temperature T/n2/3 for the cases I, II, and III. For T/n2/3 \ 5 we also plot the results obtained within the Bogoliubov approximation (dashed lines)

1.5

1.0

0.5

0.0

2

4

6

8

10.1

Bose–Einstein Condensation in Three Dimensions

Fig. 10.19 Specific heat per particle cv as a function of the dimensionless temperature T/n2/3. The dashed lines show the Bogoliubov result for cv which coincides with our findings for small temperature. However, the characteristic cusp behavior cannot be seen in a mean-field theory

135

3.0 2.5 2.0 1.5 1.0 0.5

2

4

6

From the entropy density s we infer the specific heat per particle,  T os  cv ¼ ; n oT n

8

ð10:42Þ

as the temperature derivative of the entropy density at constant particle density. Using the Jacobian, we can write  os  oðs; nÞ oðs; nÞ oðT; lÞ ¼ : ð10:43Þ ¼ oT n oðT; nÞ oðT; lÞ oðT; nÞ For the specific heat this gives T cv ¼ n

      ! os  os  on  on  1  : oT l ol T oT l ol T

ð10:44Þ

Our result for the specific heat per particle is shown for different scattering lengths in Fig. 10.19. While this quantity is positive in the whole range of investigated temperatures, it is interesting to observe the cusp at the critical temperature Tc which is characteristic for a second order phase transition. This behavior cannot be seen in a mean-field approximation, where fluctuations are taken into account only to second order in the fields. Only for small temperatures, our curve is close to the Bogoliubov approximation, shown by the dashed lines in Fig. 10.19. In fact, close to Tc the specific heat is expected to behave like  a  Tc  T for T\Tc ; c v  b1  b2 Tc ð10:45Þ  a þ T  Tc for T [ Tc ; c v  b1  b2 Tc with the universal critical exponent a of the three-dimensional XY universality class, a = -0.0146(8) [12]. The critical region, where the law cv*|T - Tc|-a

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holds, may be quite small. Our numerical differentiation procedure cannot resolve the details of the cusp. In the grand canonical formalism, the energy density  is obtained as  ¼ p þ Ts þ ln:

ð10:46Þ

We plot p/(n5/3) as a function of temperature in Fig. 10.20 and the energy density /(n5/3) is plotted in Fig. 10.21. We have normalized the pressure such that it vanishes for T = l = 0. Technically we subtract from the flow equation of the pressure the corresponding expression in the limit T = l = 0. This procedure has to be handled with care and leads to an uncertainty in the offset of the pressure, i.e. the part that is independent of T/n2/3 and l/n2/3. For zero temperature, the pressure is completely due to the repulsive interaction between the particles. For nonzero temperature, the pressure is increased by the thermal kinetic energy, of course. For the energy and the pressure we find some deviations from the Bogoliubov result already for small temperatures in cases II and III. These deviations may be partly due to the uncertainty in the normalization process described above. For

Fig. 10.20 Pressure in units of the density p/n5/3 as a function of temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines)

6 5 4 3 2 1

2

Fig. 10.21 Energy per particle /n5/3 as a function of temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines)

4

6

8

4

6

8

12 10 8 6 4 2

2

10.1

Bose–Einstein Condensation in Three Dimensions

137

weak interactions an1/3 = 0.01 as in case I, the Bogoliubov prediction coincides with our result.

10.1.7.3 Compressibility The isothermal compressibility is defined as the relative volume change at fixed temperature T and particle number N when some pressure is applied jT ¼ 

  1 oV  1 on  ¼ : V op T;N n op T

ð10:47Þ

Very similar, the adiabatic compressibility is   1 oV  1 on  jS ¼  ¼ V op S;N n op s=n

ð10:48Þ

where now the entropy S and the particle number N are fixed. Let us first concentrate on the isothermal compressibility jT. To evaluate it in the grand  canonical formalism, we have to change variables to T and l. With op=oln;T ¼ n and   op=on ¼ nol=on one obtains T

T

jT ¼

 1 on  : n2 ol T

ð10:49Þ

This expression can be directly evaluated in our formalism by numerical differentiation with respect to l. The approach to the adiabatic compressibility is similar. Using again the Jacobian we have  1 on  1 oðn; s=nÞ jS ¼ ¼ n op s=n n oðp; s=nÞ 1 oðn; s=nÞ oðl; TÞ : ¼ n oðl; TÞ oðp; s=nÞ

ð10:50Þ

We need, therefore,     ! oðn; s=nÞ 1 on  os  on  os  ¼  oðl; TÞ n ol T oT l oT l ol T and also

ð10:51Þ

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Many-Body Physics

    op  os  op  s on   ol T oT l ol T n oT l      op  os  op  s on   þ oT l ol T oT l n ol T    ! os  s on  s2 on  ¼ 2 þ : oT l n oT l n2 ol T

oðp; s=nÞ 1 ¼ oðl; TÞ n



ð10:52Þ

  on  os  In the last equations we used the Maxwell identity oT ¼ ol : Combining this we l T find    2    on  os  on  ol T oT l  oT l jS ¼  ð10:53Þ    : os  on  2 on  n2 oT  2sn þ s oT l ol T l   Since os=oT l ¼ ðo2 p=oT 2 Þl we need to evaluate a second derivative numerically. We plot the isothermal and the adiabatic compressibility in Figs. 10.22 and 10.23.

Fig. 10.22 Isothermal compressibility jT n5/3 as a function of temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines)

7 6 5 4 3 2 1 0

Fig. 10.23 Adiabatic compressibility jS n5/3 as a function of temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines)

2

4

6

8

2

4

6

8

3

2

1

0

10.1

Bose–Einstein Condensation in Three Dimensions

139

For the isothermal compressibility the temperature dependence is qualitatively different than in Bogoliubov theory already for small temperatures, while there seem to be only quantitative differences for the adiabatic compressibility. The perturbative calculation of the compressibility is difficult since it is diverging in the non-interacting limit an1/3?0.

10.1.7.4 Isothermal and Adiabatic Sound Velocity The sound velocity of a normal fluid under isothermal conditions, i.e. for constant temperature T is given by  1 op  2 vT ¼ : ð10:54Þ M on T We can obtain this directly from the isothermal compressibility Mv2T ¼ ðnjT Þ1

ð10:55Þ

as follows from (10.47). We plot our result for v2T in Fig. 10.24, recalling our units 2M = 1 such that v2T stands for 2Mv2T : This plot also covers the superfluid phase where the physical meaning of v2T is partly lost. This comes since the sound propagation there has to be described by more complicated two-fluid hydrodynamics. In addition to the normal gas there is now also a superfluid fraction allowing for an additional oscillation mode. We will describe the consequences of this in the next section. For most applications the adiabatic sound velocity is more important then the isothermal sound velocity. Keeping the entropy per particle fixed, we obtain  1 op  v2S ¼ ð10:56Þ M on s=n and, therefore, Fig. 10.24 Isothermal velocity of sound as appropriate for single fluid v2T =n2=3 ¼ 1=ðjT n5=3 Þ as a function of the dimensionless temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines)

10 8 6 4 2 0

2

4

6

8

140 Fig. 10.25 Adiabatic velocity of sound as appropriate for single fluid v2S =n2=3 ¼ 1=ðjS n5=3 Þ as a function of the dimensionless temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines)

10

Many-Body Physics

30 25 20 15 10 5

2

Mv2S ¼ ðnjS Þ1 :

4

6

8

ð10:57Þ

Our numerical result is plotted in Fig. 10.25. Again the plot covers both the superfluid and the normal part, but only in the normal phase the object v2S has its physical meaning as a sound velocity.

10.1.7.5 First and Second Velocity of Sound For temperatures 0 \ T \ Tc there are two components of the gas: the superfluid and the normal part. It was shown by Landau [24] that this leads to two-fluid hydrodynamics with two distinct velocities of sound c1/2 corresponding to different kinds of excitations. The main reason for the existence of two sound velocities is that the entropy flow is carried only be the normal component while the particle flow (or equivalently mass-flow) is carried by both the normal and the superfluid part. The continuity equation for the conserved particle number reads ~ ~ ot n þ r j ¼ 0;

ð10:58Þ

where ~ j ¼ nN~ vN þ nS~ vS is the (complete) particle number current and ~ vN ; ~ vS are the velocities of the normal (nN) and superfluid (nS) parts of the density, n = nN + nS. The conservation equation for the entropy reads ~ ðsv ~N Þ ¼ 0: ot s þ r

ð10:59Þ

vS : To close the We work in linear order in an expansion in the velocities ~ vN and ~ set of hydrodynamic equations for small ~ vN ; ~ vS we need the equations for momentum conservation ~ p ¼ 0; j þr Mot~ and for the change in the superfluid velocity

ð10:60Þ

10.1

Bose–Einstein Condensation in Three Dimensions

~ l ¼ 0: Mot~ vS þ r

141

ð10:61Þ

The last equation guarantees that the superfluid flow remains irrotational, ~ ~ r vS ¼ 0: From the combination of (10.58) and (10.68) one obtains Mo2t n ¼ Dp:

ð10:62Þ

To linear order in ~ vS and ~ vN one infers from the combination of (10.60) and (10.61) 2

n ~ ðv ~N  ~ nS r vS Þ ¼  ot ðs=nÞ: s

ð10:63Þ

We recover s/n = const. for nS = 0 as appropriate for the disordered phase. Similarly, the combination of (10.60) and (10.61) gives ~l  r ~p ~N  ~ vS Þ ¼ nr MnN ot ðv ~ T: ¼ sr

ð10:64Þ

The last equation uses the relation ~ p ¼ sr ~ T þ nr ~l r

ð10:65Þ

which follows directly from the differential of p, dp ¼ s dT þ n dl:

ð10:66Þ

Combining now (10.63) and (10.64) yields the analogue of (10.62). Mo2t ðs=nÞ ¼

s2 nS DT: n 2 nN

ð10:67Þ

One next makes an ansatz for the thermodynamic variables in the form p ¼ p0 þ dp; n ¼ n0 þ dn;

T ¼ T0 þ dT s=n ¼ s0 =n0 þ dðs=nÞ;

ð10:68Þ

where p0, T0, n0 and s0 are constant in space and time whereas dp, dT, dn, and d(s/n) are small and vary like sin[p(x - ct)]. We use dT and dn as independent variables, with   op  op  dp ¼ dT þ  dn; oT n on T   ð10:69Þ oðs=nÞ  oðs=nÞ  dðs=nÞ ¼ dT þ dn; oT n on T in order to obtain from (10.62) and (10.67) the wave equation

142

   s22 nS Mc2 oðs=nÞ oT n nN n  op   oT n

; ;

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Many-Body Physics

 !   Mc2 oðs=nÞ dT on T ¼ 0:  op dn Mc2  on T

ð10:70Þ

As a condition for possible sound velocities c one obtains "

#   op  s 2 nS T s2 nS T op  þ ¼ 0: ðMc Þ  ðMc Þ þ 2 on s=n n2 nN cv n nN cv on T 2 2

2

ð10:71Þ

This relation uses  oðs=nÞ  cv ¼ T oT n

ð10:72Þ

  "   # oðs=nÞ  cv oT  op  op  ¼  : on T T op n on T on s=n

ð10:73Þ

as well as

The latter relation follows from     op  op  op  oT  ¼ þ on s=n on T oT n on s=n

ð10:74Þ

 oT  oðT; s=nÞ oðT; s=nÞ oðT; nÞ ¼ ¼ on s=n oðn; s=nÞ oðT; nÞ oðs=n; nÞ  oðs=nÞ  T ¼ : on T cv

ð10:75Þ

together with

With these ingredients one can now solve (10.71) for the first and second velocity of sound. The numerical results as a function of temperature are shown in Figs. 10.26 and 10.27. We also show there the prediction from Bogoliubov theory for T?0 (short solid lines). For c21 the agreement with our findings is rather good, although there are some deviations for strong interactions as in case III. For c22 our numerical determination becomes unreliable for T/n2/3 \ 1 since c22 is dominated by the term (s2 nS T)/(n2 nN cv) in (10.71). In the limit T?0 the quantities s, nN, and cv also go to zero so that the numerical value for c22 is sensitive to the precise way how this limit is approached.

10.1

Bose–Einstein Condensation in Three Dimensions

Fig. 10.26 First velocity of sound c21 =n2=3 as a function of the dimensionless temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the prediction from Bogoliubov theory for T?0 (short solid lines)

143

30 25 20 15 10 5 0

Fig. 10.27 Second velocity of sound c22 =n2=3 as a function of the dimensionless temperature T/n2/3 for the cases I (crosses), II (dots), and III (stars). We also show the prediction from Bogoliubov theory for T?0 (short solid lines). For T/n2/3 \ 1 our numerical determination becomes unreliable, since c22 is dominated by a ratio of terms that vanish for T?0

2

4

6

8

8

6

4

2

0

2

4

6

We observe that (10.71) can be written as

 nS Ts2 ðMc2 Þ2  Mv2S þ ðMc2 Þ ðn  nS Þcv n2 nS Ts2 þ Mv2 ¼ 0; ðn  nS Þcv n2 T

8

ð10:76Þ

with the single fluid isothermal and adiabatic sound velocities vT and vS given by (10.54) and (10.56). This shows that c coincides with vS in the disordered phase where nS = 0. An intuitive form of the wave equation can be written as ~ o2t dn ¼ v2T Ddn þ ðv2S  v2T ÞDdT; o2t dT~ ¼ ðv2S  v2T þ v2 ÞDdT~ þ v2T Ddn;

ð10:77Þ

with Mv2 ¼

s 2 nS T ; n2 ðn  nS Þcv

dT~ ¼

dT g

ð10:78Þ

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Many-Body Physics

and   T oðs=nÞ  oT  g¼ ¼ : cv on T on s=n

ð10:79Þ

For fluctuations of dn and dT~ only vT, vS and v matter. In the limit T?0 one observes v2S ! v2T such that the fluctuations dn are governed by the isothermal sound velocity vT. On the other hand, the the velocity v characterizes the dynamics of a linear combination of dT~ and dn.

10.2 Superfluid Bose Gas in Two Dimensions We now come to the many-body properties of the Bose gas model (6.1) in the case of two spatial dimensions. Although the formalism is the same as in threedimensions (only one number is changed!) the physical properties are quite different. Many quantities have a logarithmic scale-dependence in two-dimensions and the effect of fluctuations is more important. As discussed in Sect. 6.2, it is important to realize that experimental observables are associated with a characteristic momentum scale kph and probe the flowing action Ckph : We start with the investigation of the flow equations at zero temperature and come then to the phase diagram at nonzero temperature which is governed by the Kosterlitz–Thouless phase transition.

10.2.1 Flow Equations at Zero Temperature In this section we investigate the many body problem for a nonvanishing density n at zero temperature T = 0. A crucial new ingredient as compared to the vacuum discussed in Sect. 9.1 is the nonzero superfluid density nS ¼ q0 ðkph Þ:

ð10:80Þ

For interacting bosons at zero temperature the density n and the superfluid density nS are equal. In contrast, the condensate density is given by the unrenormalized order parameter  1 ðkph Þq0 ðkph Þ: 0 ðkph Þ ¼ A nC ¼ q

ð10:81Þ

Due to the repulsive interaction, nC may be smaller than n, the difference n - nC being the condensate depletion. (In the limit k?0 we have n = nS = nC.) To obtain a nonzero density at temperature T = 0 we have to go to positive chemical potential l [ 0. At the microscopic scale k = K the minimum of the effective potential U is then at q0,K = l/k [ 0.

10.2

Superfluid Bose Gas in Two Dimensions

145

The superfluid density q0 is connected to a nonvanishing ‘‘renormalized order parameter’’ u0 , with q0 ¼ u0 u0 : It is responsible for an effective spontaneous breaking of the U(1)-symmetry. Indeed, the expectation value u0 points out a direction in the complex plane so that the global U(1)-symmetry of phase rotations is broken by the ground state of the system. Goldstone’s theorem implies the presence of a gapless Goldstone mode, and the associated linear dispersion relation ~j accounts for superfluidity. The Goldstone physics is best described by x  jq using a real basis in field space by decomposing the complex field u ¼ u0 þ p1ffiffi ðu1 þ iu2 Þ: Without loss of generality we can choose the expectation 2

value u0 to be real. The real fields u1 and u2 describe then the radial and Goldstone mode. respectively. For l = l0 the inverse propagator reads in our truncation  2  ~ Sp0 p þ Vp20 þ U 0 þ 2qU 00 ; 1  G ¼A : ð10:82Þ ~ Sp0 ; p2 þ Vq20 þ U 0 Here ~ p is the momentum of the collective excitation, and for T = 0 the frequency obeys x = -ip0. In the regime with spontaneous symmetry breaking, q0(k) = 0, the propagator for q = q0 has U0 = 0, 2qU00 = 2k q0 = 0, giving rise to the linear dispersion relation characteristic for superfluidity. This strongly modifies the flow equations as compared to the vacuum flow equations once k2  2kq0. For n = 0 the flow is typically in the regime with q0(k) = 0. In practice, we have to adapt the initial value q0,K such that the flow ends at a given density q0(kph) = n. For kph  n1/2 one finds that q0(kph) depends only very little on kph. As mentioned above, we will often choose the density to be unity such that effectively all length scales are measured in units of the interparticle distance n-1/2. In contrast to the vacuum with T = q0 = 0, the flow of the propagator is nontrivial in the phase with q0 [ 0 and spontaneous U(1) symmetry breaking. In  V; S for a renormalized Fig. 10.28 we show the flow of the kinetic coefficients A; or macroscopic interaction strength k = 1. The wavefunction renormalization

Fig. 10.28 Flow of the  (solid), kinetic coefficients A S (dashed), and V (dasheddotted) at zero temperature T = 0, density n = 1, and vacuum interaction strength k=1

1.0

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Many-Body Physics

 increases only a little at scales where k  n1=2 and saturates then to a value A  [ 1: As will be explained below, we can directly infer the condensate depletion A  at macroscopic scales. The coefficient of the linear s-derivative from the value of A S goes to zero for k?0. The frequency dependence is then governed by the quadratic s-derivative with coefficient V, which is generated by the flow and saturates to a finite value for k?0. The flow of the interaction strength k(k) for different values of k = k(kph) is shown in Fig. 10.29. While the decrease with the scale k is only logarithmic in vacuum, it becomes now linear k(k)* k for k  n1/2. It is interesting that the ratio k(k)/k reaches larger values for smaller values of kK.

10.2.2 Quantum Depletion of Condensate As k is lowered from K to kph, the renormalized order parameter or the superfluid density q0 increases first and then saturates to q0 = n = 1. This is expected since the superfluid density equals the total density at zero temperature. In contrast, the

Fig. 10.29 Flow of the interaction strength k(k) at zero temperature T = 0, density n = 1, for different initial values kK. The dotted lines are the corresponding graphs in the vacuum n = 0. The vertical line labels our choice of kph. The lower plot shows k(k)/k for the same parameters, demonstrating that k(k)* k for small k

2.0

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0.5

10

8

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0

2

0

2

70 60 50 40 30 20 10

10

8

6

4

2

10.2

Superfluid Bose Gas in Two Dimensions

Fig. 10.30 Condensate depletion (n - nc)/n as a function of the vacuum interaction strength k. The dashed line is the Bogoliubov k for result ðn  nc Þ=n ¼ 8p reference

147

1

0.1

0.01

0.001

0.0001

0.001

0.01

0.1

1

 flows to a smaller value q 0 ¼ q0 =A 0 \q0 : As argued in bare order parameter q Sect. 10.1, the bare order parameter is just the condensate density, such that   1  0 ¼ q0 1   ð10:83Þ n  n C ¼ q0  q A is the condensate depletion. Its dependence on the interaction strength k is shown in Fig. 10.30. For small interaction strength k the condensate depletion follows roughly the Bogoliubov form n  nc k : ¼ 8p n

ð10:84Þ

However, we find small deviations due to the running of k which is absent in Bogoliubov theory. For large interaction strength k 1 the deviation from the Bogoliubov result is quite substantial, since the running of k with the scale k is more important.

10.2.3 Dispersion Relation and Sound Velocity We also investigate the dispersion relation at zero temperature. The dispersion relation x(p) follows from the condition det G1 ðxðpÞ; pÞ ¼ 0

ð10:85Þ

where G-1 is the inverse propagator after analytic continuation to real time p0?ix. As was shown at the end of Sect. 7.1 the generation of the kinetic coefficient V by the flow leads to the emergence of a second branch of solutions of ~Þ and (10.85). In our truncation the dispersion relation for the two branches xþ ðp ~Þ are x ðp

148

10

0 ~Þ ¼ @ x ðp  

Many-Body Physics

1 2 S2 ~ þ kq0 Þ þ 2 ðp V 2V 1 2 S2 ~ þ kq0 Þ þ 2 ðp V 2V

2

1 2 2 ~ þ 2kq0 Þ  2~ p ðp V

!1=2 11=2 A :

ð10:86Þ

In the limit V?0, S?1 we find that the lower branch approaches the Bogoliubov result qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 þ 2kq0 Þ while the upper branch diverges x+?? and thus disappears p2 ðp x ! ~ from the spectrum. The lower branch is dominated by phase changes (Goldstone mode), while the upper branch reflects waves in the size of q0 (radial mode). In principle, the coupling constants on the right-hand side of (10.86) also ~j: Since an external momentum provides an depend on the momentum p ¼ jp ~j we can approximate the jp ~j-dependence by using infrared cutoff of order k  jp on the right-hand side of (10.86) the k-dependent couplings with the identification ~j: Our result for the lower branch of the dispersion relation is shown in k ¼ jp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 þ 2kq0 Þ for comFig. 10.31. We also plot the Bogoliubov result x ¼ ~ p2 ðp parison. For small k our result is in agreement with the Bogoliubov result, while we find substantial deviations for large k. Both branches x+ and x- are shown in Fig. 10.32 on a logarithmic scale. Since we start with V = 0 at the microscopic ~Þ ! 1 for jp ~j ! K: scale K we find xþ ðp The sound velocity cS can be extracted from the dispersion relation. More ~Þ precisely, we compute the microscopic sound velocity for the lower branch x ðp ox as cS ¼ op at p = 0. In our truncation we find 6 5 4 3 2 1 0 0.0

0.5

1.0

1.5

2.0

Fig. 10.31 Lower branch of the dispersion relation x-(p) at temperature T = 0 and for the vacuum interaction strength k = 1 (solid curve), k = 0.5 (upper dashed curve), and k = 0.1 (lower dashed curve). The units are set by the density n = 1. We also show the Bogoliubov result for k = 1 (upper dotted curve) and k = 0.5 (lower dotted curve). For k = 0.1 the Bogoliubov result is identical to our result within the plot resolution

10.2

Superfluid Bose Gas in Two Dimensions

149

Fig. 10.32 Dispersion relation x-(p), x+(p) at temperature T = 0 and for vacuum interaction strength k = 1 (solid), k = 0.5 (long dashed), and k = 0.1 (short dashed). The units are set by the density n = 1

20 15 10 5 0 5 10

10

c2S ¼

8

S2

6

4

2

2kq0 : þ 2kq0 V

0

2

ð10:87Þ

Our result for cS at T = 0 is shown in Fig. 10.33 as a function of the interaction strength k. For a large range of small k we find good agreement with the Bogoliubov result c2S ¼ 2kq0 : However, for large k or result for cS exceeds the Bogoliubov result by a factor up to 2.

10.2.4 Kosterlitz–Thouless Physics 10.2.4.1 Superfluidity and Order Parameter At nonzero temperature and for infinite volume, long range order is forbidden in two spatial dimensions by the Mermin–Wagner theorem. Because of that, no proper Bose–Einstein condensation is possible in a two-dimensional homogeneous Bose gas at nonvanishing temperature. However, even if the order parameter Fig. 10.33 Dimensionless sound velocity cS/n1/2 as a function of the vacuum interaction strength (solid). We also show the Bogoliubov pffiffiffiffiffiffiffiffiffiffi result cS ¼ 2kq0 for reference (dashed)

4

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vanishes in the thermodynamic limit of infinite volume, one still finds a nonzero superfluid density for low enough temperature. The superfluid density can be considered as the square of a renormalized order parameter q0 ¼ ju0 j2 and the particular features of the low-temperature phase can be well understood by the physics of the Goldstone boson for a phase with effective spontaneous symmetry breaking [25]. The renormalized order parameter u0 is related to the expectation 0 ¼ u  20 by a  0 and, therefore, to the condensate density q value of the bosonic field u  at wave function renormalization, defined by the behavior of the bare propagator G zero frequency for vanishing momentum  1=2 u  0; u0 ¼ A

 q0 ; q0 ¼ A

 1 ðp ~2 : ~ ! 0Þ ¼ Ap G

ð10:88Þ

While the renormalized order parameter q0(k) remains nonzero for k?0 if T \ Tc  vanishes since A  diverges with the anomalous 0 ¼ q0 =A the condensate density q g  dimension, A  k : After restoring dimensions the relation 0 q 2  ~Þ ~ p!0 ~ p Gðp

q0 ¼ lim

ð10:89Þ

is the Josephson relation [26]. 0 ¼ 0 and a The strict distinction between a zero Bose–Einstein condensate q nonzero superfluid density q0 [ 0 for nonzero temperature 0 \ T \ Tc is valid only in the infinite volume limit of a homogeneous system. For a finite size of the  system, as atoms in a trap, the running of AðkÞ is effectively stopped at some scale kph. There are simply no collective modes with wavelength larger than the size of  the system, whose fluctuations would be responsible for a further increase of A:  ph both q 0 and q0 are nonzero for T \ Tc, and the distinction With a finite A between a Bose-Einstein condensate and superfluidity is no longer relevant in  ph Þ can be large, however, such that the condensate practice. For large systems Aðk density can be suppressed substantially as compared to the superfluid density. In any case, there is only one critical temperature Tc, defined by q0(T \ Tc) [ 0.

10.2.4.2 Critical Temperature The flow equations permit a straightforward computation of q0(T) for arbitrary T, once the interaction strength of the system has been fixed at zero temperature and density. We have extracted the critical temperature as a function of k = k(kph) for different values of kph. The behavior for small k, Tc 4p ¼ lnðf=kÞ n

ð10:90Þ

is compatible with the free theory where Tc vanishes for kph?0 and with the perturbative analysis in Ref. [27–29]. We find that the value of f depends on the choice of kph. For kph = 10-2 we find f = 100, while kph = 10-4 corresponds to

10.2

Superfluid Bose Gas in Two Dimensions

Fig. 10.34 Critical temperature Tc/n as a function of the interaction strength k. We choose here kph = 10-2 (circles), kph = 10-4 (boxes) and kph = 10-6 (diamonds). For the last case the bound on the scattering length is 4p  0:78: We also k\ lnðK=k Þ

151

3.0

2.5

2.0

ph

show the curve

Tc n

4p ¼ lnðf=kÞ

(dashed) with the MonteCarlo result f = 380 [27–30] for reference

1.5

1.0 0.0

0.2

0.4

0.6

0.8

1.0

f = 225 and kph = 10-6 to f = 424. In Fig. 10.34 we show or result for Tc/n as a function of k for these choices. We also plot the curve in (10.90) with the MonteCarlo result f = 380 from Ref. [30–33]. We find that Tc vanishes for kph?0 in the interacting theory as well. This is due to the increase of f and, for a fixed microscopical interaction, to the decrease of k(kph). Since the vanishing of Tc/n is only logarithmic in kph, a phase transition can be observed in practice. We find agreement with Monte-Carlo results [30] for small k if kph =K  107 : The dependence of Tc/n on the size of the system k-1 ph remains to be established for the Monte-Carlo computations. The critical behavior of the system is governed by a Kosterlitz–Thouless phase transition. Usually this is described by considering the thermodynamics of vortices. In Refs. [34, 35] it was shown that functional renormalization group can account for this ‘‘nonperturbative’’ physics without explicitly taking vortices into account. The correlation length in the low-temperature phase is infinite. In our picture, this arises due to the presence of a Goldstone mode if q0 [ 0. The system is superfluid for T \ Tc. The powerlike decay of the correlation function at zero frequency  ~Þ  ðp ~2 Þ1þg=2 Gðp

ð10:91Þ

 As long as k2  ~ is directly related to the running of A: p2 the bare propagator obeys approximately ¼ G

1 ;  ~2 AðkÞp

  kg : AðkÞ

ð10:92Þ

p2 ; the effective infrared is given by ~ p2 instead of k2, and Once k2  ~ qffiffiffifficutoff ffi   ~ therefore, AðkÞ gets replaced by A p2 ; turning (10.92) into (10.91). For large q0 the anomalous dimension depends on q0 and T, g = T/(4p q0).

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10.2.4.3 Superfluid Fraction Another characteristic feature of the Kosterlitz–Thouless phase transition is a jump in the superfluid density at the critical temperature. However, a true discontinuity arises only in the thermodynamic limit of infinite volume (kph?0), while for finite systems (kph [ 0) the transition is smoothened. In order to see the jump, as well as essential scaling for T approaching Tc from above, our truncation is insufficient. These features become visible only in extended truncations that we will briefly describe next. 2 For very small scales kT  1; the contribution of Matsubara modes with frequency q0 = 2pTn, n = 0, is suppressed since nonzero Matsubara frequencies act as an infrared cutoff. In this limit a dimensionally reduced theory becomes valid. The long distance physics is dominated by classical two-dimensional statistics, and the time dimension parametrized by s no longer plays a role. The flow equations simplify considerably if only the zero Matsubara frequency is included, and one can use more involved truncations. Such an improved truncation is indeed needed to account for the jump in the superfluid density. In Ref. [35] the next to leading order in a systematic derivative expansion was investigated. It was found that for k  T the flow equation for q0 can be well approximated by ot q0 ¼ 2:54 T 1=2 ð0:248 T  q0 Þ3=2 hð0:248 T  q0 Þ:

ð10:93Þ

We switch from the flow equation in our more simple truncation to the improved flow equation (10.93) for scales k with k2/T \ 10-3. We keep all other flow equations unchanged. A similar procedure was also used in Ref. [36]. In Fig. 10.35 we show the flow of the density n, the superfluid density q0 and 0 for different temperatures. In Fig. 10.36 we plot our the condensate density q result for the superfluid fraction of the density as a function of the temperature for different scales kph. One can see that with the improved truncation the jump in the superfluid density is indeed found in the limit kph?0. Figure 10.37 shows the

Fig. 10.35 Flow of the density n (solid), the superfluid density q0 (dashed), and the 0 (dotcondensate density q ted) for chemical potential l = 1, vacuum interaction strength k = 0.5 and temperatures T = 0 (top), T = 2.4 (middle) and T = 2.8 (bottom). The vertical line marks our choice of kph. We recall n = q0 for T = 0 such that the upper dashed and solid lines coincide

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Superfluid Bose Gas in Two Dimensions

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Fig. 10.36 Superfluid fraction of the density q0/n as a function of the dimensionless temperature T/n for interaction strength k = 0.5 at different macroscopic scales kph = 1 (upper curve), kph = 10-0.5, kph = 10-1, kph = 10-1.5, kph = 10-2, kph = 10-2.5 (bottom curve). We plot the result obtained with the improved truncation for small scales (solid) as well as the result obtained with our more simple truncation (dotted). (The truncations differ only for the three lowest lines.) Fig. 10.37 Condensate frac0 =n as a tion of the density q function of the dimensionless temperature T/n for interaction strength k = 0.5 at macroscopic scale kph = 10-2 (solid curve) and kph = 10-4 (dashed curve). For comparison, we also plot the superfluid density q0/n at kph = 10-2 (dotted). These results are obtained with the improved truncation

1.0 0.8 0.6 0.4 0.2 0.0 0.0

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2.5

3.0

0 =n and the superfluid density fraction q0/n as a function of condensate fraction q T/n. We observe the substantial kph dependence of the condensate fraction, as well as an effective jump at Tc for small kph. We recall that the infinite volume limit 0 ¼ 0 for T [ 0. kph = 0 amounts to q The Kosterlitz–Thouless description is only valid if the zero Matsubara frequency mode (n = 0) dominates. For a given nonzero T this is always the case if the the characteristic length scale goes to infinity. In the infinite volume limit the characteristic length scale is given by the correlation length n. The description in terms of a classical two-dimensional system with U(1) symmetry is the key ingredient of the Kosterlitz–Thouless description and holds for n2T1. In the infinite volume limit this always holds for T \ Tc or near the phase transition, where n diverges or is very large. For a finite size system the relevant length scale becomes k-1 ph if this is smaller than n. Thus the Kosterlitz–Thouless picture holds only for T [ k2ph.

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For very small temperatures T \ k2ph one expects a crossover to the characteristic behavior near a quantum critical phase transition, governed by the quantum critical fixed point. The crossover between the different characteristic behaviors 2 for T [ k2ph and T\kph can be observed in several quantities. As an example we 0 for low T (close to the may take Fig. 10.35 and compare the flow of q0 and q T = 0 curve) or large T (other curves).

10.3 Particle–Hole Fluctuations and the BCS–BEC Crossover In this chapter we discuss the many-body properties of the BCS-BEC crossover model in (6.14). We treat with our method the whole crossover phase diagram but concentrate the discussion on the effect of particle–hole fluctuations. These fluctuations give rise to a the first nontrivial correction to BCS theory on the fermionic side of the crossover. For small negative scattering length their effect can be included in a perturbative setting as was shown by Gorkov and Melik-Barkhudarov [37].

10.3.1 Particle–Hole Fluctuations The BCS theory of superfluidity in a Fermi gas of atoms is valid for a small attractive interaction between the fermions [38–40]. In a renormalization group setting, the features of BCS theory can be described in a purely fermionic language. The only scale dependent object is the fermion interaction vertex kw. The flow depends on the temperature and the chemical potential. For positive chemical potential (l [ 0) and small temperatures T, the appearance of pairing is indicated by the divergence of kw. In general, the interaction vertex is momentum dependent and represented by a term Z kw ðp01 ; p1 ; p02 ; p2 Þ Ckw ¼ ð10:94Þ p1 ;p2 ;p01 ;p02

w1 ðp01 Þw1 ðp1 Þw2 ðp02 Þw2 ðp2 Þ in the effective action. In a homogeneous situation, momentum conservation restricts the expression in (10.94) to three independent momenta, kw  dðp01 þ p02  p1  p2 Þ: The flow of kw has two contributions which are depicted in Fig. 10.38. The first diagram describes particle–particle fluctuations. For l [ 0 its effect increases as the temperature T is lowered. For small temperatures T B Tc,BCS the logarithmic divergence leads to the appearance of pairing, as kw??.

10.3

Particle–Hole Fluctuations and the BCS–BEC Crossover

155

Fig. 10.38 Running of the momentum dependent vertex kw. Here ~ ok indicates derivatives with respect to the cutoff terms in the propagators and does not act on the vertices in the depicted diagrams. We will refer to the first loop as the particle–particle loop (pp-loop) and to the second one as the particle–hole loop (ph-loop)

In the purely fermionic formulation the flow equation for kw has the general form [41–44] ok kaw ¼ Aabc kbw kcw ;

ð10:95Þ

with a, b, c denoting momentum as well as spin labels. A numerical solution of this equation is rather involved due to the rich momentum structure. The case of the attractive Hubbard model in two-dimensions, which is close to our problem, has recently been discussed in [45]. The BCS approach concentrates on the pointlike coupling, evaluated by setting all momenta to zero. For k ?0, l0 ?0, T ?0 and n ?0 this coupling is related the scattering length, 1 kw ðpi ¼ 0Þ: In the BCS approximation only the first diagram in Fig. 10.38 is a ¼ 8p kept, and the momentum dependence of the couplings on the right-hand side of (10.95) is neglected, by replacing kaw by the pointlike coupling evaluated at zero momentum. In terms of the scattering length a, Fermi momentum kF and Fermi temperature TF, the critical temperature is found to be Tc  0:61ep=ð2akF Þ : TF

ð10:96Þ

This is the result of the original BCS theory. However, it is obtained by entirely neglecting the second loop in Fig. 10.38, which describes particle–hole fluctuations. At zero temperature the expression for this second diagram vanishes if it is evaluated for vanishing external momenta. Indeed, the two poles of the frequency integration are always either in the upper or lower half of the complex plane and the contour of the frequency integration can be closed in the half plane without poles. The dominant part of the scattering in a fermion gas occurs, however, for momenta on the Fermi surface rather than for zero momentum. For non-zero momenta of the ‘‘external particles’’ the second diagram in Fig. 10.38—the particle–hole channel—makes an important contribution.

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Setting the external frequencies to zero, we find that the inverse propagators in the particle–hole loop are ~~ p1 Þ2  l ; Pw ðqÞ ¼ iq0 þ ðq

ð10:97Þ

~~ Pw ðqÞ ¼ iq0 þ ðq p20 Þ2  l:

ð10:98Þ

and

p20 ; there are now values of the loop Depending on the value of the momenta ~ p1 and ~ momentum ~ q for which the poles of the frequency integration are in different half planes so that there is a nonzero contribution even for T = 0. To include the effect of particle–hole fluctuations one could try to take the full momentum dependence of the vertex kw into account. However, this leads to complicated expressions which are hard to solve even numerically. One, therefore, often restricts the flow to the running of a single coupling kw by choosing an appropriate projection prescription to determine the flow equation. In the purely fermionic description with a single running coupling kw, this flow equation has a simple structure. The solution for k1 w can be written as a contribution from the particle-particle (first diagram in Fig. 10.38, pp-loop) and the particle–hole (second diagram, ph-loop) channels 1 1 ¼ þ pp-loop þ ph-loop: kw ðk ¼ 0Þ kw ðk ¼ KÞ

ð10:99Þ

Since the ph-loop depends only weakly on the temperature, one can evaluate it at T = 0 and add it to the initial value kw(k = K)-1. Since Tc depends exponentially on the ‘‘effective microscopic coupling’’  1 keff ¼ kw ðk ¼ KÞ1 þ ph-loop ; ð10:100Þ w;K  1 any shift in keff results in a multiplicative factor for Tc. The numerical value w;K of the ph-loop and, therefore, of the correction factor for Tc/TF depends on the precise projection description. Let us now choose the appropriate momentum configuration. For the formation of Cooper pairs, the relevant momenta lie on the Fermi surface, ~ p21 ¼ ~ p22 ¼ ~ p102 ¼ ~ p202 ¼ l ;

ð10:101Þ

and point in opposite directions ~ ~2 ; ~ ~20 : p1 ¼ p p10 ¼ p

ð10:102Þ

p10 unspecified. Gorkov’s approximation This still leaves the angle between ~ p1 and ~ uses (10.101) and (10.102) and projects on the s-wave by averaging over the angle p10 : One can shift the loop momentum such that the internal between ~ p1 and ~

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Particle–Hole Fluctuations and the BCS–BEC Crossover

157

~þ~ propagators depend on ~ q2 and ðq p1  ~ p10 Þ2 : In terms of spherical coordinates the first propagator depends only on the magnitude of the loop momentum q2 ¼ ~ q2 ; ~1  ~ p10 Þ2 while the second depends additionally on the transfer momentum ~p2 ¼ 14 ðp ~1  ~ and the angle a between ~ q and ðp p10 Þ; ~þ~ ðq p1  ~ p10 Þ2 ¼ q2 þ 4~ p2 þ 4 q ~p cosðaÞ :

ð10:103Þ

Performing the loop integration involves the integration over q2 and the angle a. p01 translates to an averaging over ~p2 : The averaging over the angle between ~ p1 and ~ Both can be done analytically [46] for the fermionic particle–hole diagram and the result gives the well-known Gorkov correction to BCS theory, resulting in Tc ¼

1 ð4eÞ1=3

Tc;BCS 

1 Tc;BCS : 2:2

ð10:104Þ

p1 ; In our treatment we will use a numerically simpler projection by choosing ~ p10 ¼ ~ 0 0 and ~ p2 ¼ ~ p2 ; without an averaging over the angle between ~ p1 and ~ p1 : The size of ~ p2 ¼ ~ p21 is chosen such that the one-loop result reproduces exactly the result of the pffiffiffi Gorkov correction, namely ~ p ¼ 0:7326 l: Choosing different values of ~p demonstrates the dependence of Tc on the projection procedure and may be taken as an estimate for the error that arises from the limitation to one single coupling kw instead of a momentum dependent function.

10.3.2 Bosonization In Sect. 6.3, we describe an effective four-fermion interaction by the exchange of a boson. In this picture the phase transition to the superfluid phase is indicated by the vanishing of the bosonic ‘‘mass term’’ m2 = 0. Negative m2 leads to the spontaneous breaking of U(1)-symmetry, since the minimum of the effective potential occurs for a nonvanishing superfluid density q0 [ 0. For m2 C 0 we can solve the field equation for the boson u as a functional of w and insert the solution into the effective action. This leads to an effective fourfermion vertex describing the scattering w1 ðp1 Þw2 ðp2 Þ ! w1 ðp10 Þw2 ðp20 Þ kw;eff ¼

h2 ~1 þ ~ iðp1 þ p2 Þ0 þ 12 ðp p2 Þ 2 þ m 2

:

ð10:105Þ

To investigate the breaking of U(1) symmetry and the onset of superfluidity, we first consider the flow of the bosonic propagator, which is mainly driven by the fermionic loop diagram. For the effective four-fermion interaction this accounts for the particle-particle loop (see r.h.s. of Fig. 10.39). In the BCS limit of a large microscopic m2K the running of m2 for k?0 reproduces the BCS result [38, 39, 40].

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Fig. 10.39 Flow of the boson propagator

The particle–hole fluctuations are not accounted for by the renormalization of the boson propagator. Indeed, we have neglected so far that a term Z kw w1 w1 w2 w2 ; ð10:106Þ ~ s;x

in the effective action is generated by the flow. This holds even if the microscopic pointlike interaction is absorbed by a Hubbard–Stratonovich transformation into an effective boson exchange such that kw(K) = 0. The strength of the total interaction between fermions kw;eff ¼

h2 ~1 þ ~ iðp1 þ p2 Þ0 þ 12 ðp p2 Þ 2 þ m 2

þ kw

ð10:107Þ

adds kw to the piece generated by boson exchange. In the partially bosonized formulation, the flow of kw is generated by the box-diagrams depicted in Fig. 10.40. We may interpret these diagrams and establish a direct connection to the particle–hole diagrams depicted in Fig. 10.38 on the BCS side of the crossover and in the microscopic regime. There the boson gap m2 is large. In this case, the effective fermion interaction in (10.107) becomes momentum independent. Diagrammatically, this is represented by contracting the bosonic propagator. One can see, that the box-diagram in Fig. 10.40 is then equivalent to the particle–hole loop 2 investigated in Sect. 10.3.1 with the pointlike approximation kw;eff !  mh 2 for the fermion interaction vertex. As mentioned above, these contributions vanish for T = 0, l \ 0 for arbitrary ~ pi : Indeed, at zero temperature, the summation over the

Fig. 10.40 Box diagram for the flow of the four-fermion interaction

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Particle–Hole Fluctuations and the BCS–BEC Crossover

159

Matsubara frequencies becomes an integral. All the poles of this integration are in the upper half of the complex plane and the integration contour can be closed in ~2 ¼ p ~20 ; jp ~1 j ¼ p1 ¼ ~ p10 ¼ p the lower half plane. We will evaluate ok kw for ~ pffiffiffi ~ p ¼ 0:7326 l; as discussed in the Sect. 10.3.1. For l [ 0 this yields a nonvanishing flow even for T = 0. Another simplification concerns the temperature dependence. While the contribution of particle–particle diagrams becomes very large for small temperatures, this is not the case for particle–hole diagrams. For nonvanishing density and small temperatures, the large effect of particle–particle fluctuations leads to the spontaneous breaking of the U(1) symmetry and the associated superfluidity. In contrast, the particle–hole fluctuations lead only to quantitative corrections and depend only weakly on temperature. This can be checked explicitly in the pointlike approximation, and holds not only in the BCS regime where T/l  1, but also for moderate T/l as realized at the critical temperature in the unitary regime. We can, therefore, evaluate the box-diagrams in Fig. 10.38 for zero temperature. We note that an implicit temperature dependence, resulting from the couplings parameterizing the boson propagator, is taken into account. After these preliminaries, we can now incorporate the effect of particle–hole fluctuations in the renormalization group flow. A first idea might be to include the additional term (10.106) in the truncation and to study the effects of kw on the remaining flow equations. On the initial or microscopic scale one would have kw = 0, but it would then be generated by the flow. This procedure, however, has several shortcomings. First, the appearance of a local condensate would now be indicated by the divergence of the effective four-fermion interaction kw;eff ¼ 

h2 þ kw : m2

ð10:108Þ

This might lead to numerical instabilities for large or diverging kw. The simple picture that the divergence of kw,eff is connected to the onset of a nonvanishing expectation value for the bosonic field u0 , at least on intermediate scales, would not hold anymore. Furthermore, the dependence of the box-diagrams on the center of mass momentum would be neglected completely by this procedure. Close to the resonance the momentum dependence of the effective four-fermion interaction in the bosonized language as in (10.107) is crucial, and this might also be the case for the particle–hole contribution. Another, much more elegant way to incorporate the effect of particle–hole fluctuations is provided by the method of bosonization [47–49], see also Chap. 3. For this purpose, we use scale dependent fields in the average action. The scale dependence of Ck[vk] is modified by a term reflecting the k-dependence of the argument vk [48, 49] ok Ck ½vk  ¼

Z

  1 dCk 1 ð2Þ ok vk þ STr Ck þ Rk o k Rk : 2 dvk

ð10:109Þ

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 and For our purpose it is sufficient to work with scale dependent bosonic fields u keep the fermionic field w scale independent. In practice, we employ bosonic fields  k ; and u  k with an explicit scale dependence which reads in momentum space u  k ðqÞ ¼ ðw1 w2 ÞðqÞok t ; ok u

ð10:110Þ

 k ðqÞ ¼ ðwy2 wy1 ÞðqÞok t: ok u

In consequence, the flow equations in the symmetric regime get modified   u ðqÞok t ; ok  ð10:111Þ h ¼ ok  hu  P k

 ok kw ¼ ok kw u  2 hok t:

ð10:112Þ

k

Here q is the center of mass momentum of the scattering fermions. In the notation of (10.105) we have q = p1 + p2 and we will take ~ q ¼ 0; and q0 = 0. The first term on the right hand side in (10.112) gives the contribution of the flow equation  k : The second term comes from the explicit scale which is valid for fixed field u  k : The inverse propagator of the complex boson field u  is denoted dependence of u 2 2    by Pu ðqÞ ¼ Au Pu ðqÞ ¼ Au ðm þ iZu q0 þ ~ q =2Þ; cf. (6.16). We can choose ok t such that the flow of the coupling kw vanishes, i.e. that we have kw = 0 on all scales. This modifies the flow equation for the renormalized Yukawa coupling according to     m2  ok h ¼ ok h  ok kw  ; ð10:113Þ 2h k k u u   with ok hu the contribution without bosonization and ok kw u given by the box k

k

diagram in Fig. 10.40. Since kw remains zero during the flow, the effective fourfermion interaction kw,eff is now purely given by the boson exchange. However, the contribution of the particle–hole exchange diagrams is incorporated via the second term in (10.114). In the regime with spontaneously broken symmetry we use a real basis for the bosonic field 1  ¼u  0 þ pffiffiffi ð u u2 Þ; u1 þ i 2

ð10:114Þ

 0 is chosen to be real without loss of generality. The where the expectation value u  1 and u  2 then describe the radial and the Goldstone mode, respecreal fields u tively. To determine the flow equation of  h; we use the projection description pffiffiffi ok  h ¼ i 2X1

d d d ok Ck ; du2 ð0Þ dw1 ð0Þ dw2 ð0Þ

ð10:115Þ

R with the four volume X ¼ T1 ~x : Since the Goldstone mode has vanishing ‘‘mass’’, the flow of the Yukawa coupling is not modified by the box diagram (Fig. 10.40)

10.3

Particle–Hole Fluctuations and the BCS–BEC Crossover

161

in the regime with spontaneous symmetry breaking. We emphasize that the nonperturbative nature of the flow equations for the various couplings provides for a resummation similar to the one in (10.99), and thus goes beyond the treatment by Gorkov and Melik-Barkhudarov [37] which includes the particle–hole diagrams only in a perturbative way. Furthermore, the inner bosonic lines h2 =Pu ðqÞ in the box-diagrams represent the center of mass momentum dependence of the fourfermion vertex. This center of mass momentum dependence is neglected in Gorkov’s pointlike treatment, and thus represents a further improvement of the classic calculation. Actually, this momentum dependence becomes substantial— and should not be neglected in a consistent treatment—away from the BCS regime where the physics of the bosonic bound state sets in. Finally, we note that the truncation (8.26) supplemented with (10.106) closes the truncation to fourth order in the fields except for a fermion-boson vertex kwu wy wu u which plays a role for the scattering physics deep in the BEC regime [50] but is not expected to have a very important impact on the critical temperature in the unitarity and BCS regimes.

10.3.3 Critical Temperature To obtain the flow equations for the running couplings of our truncation (8.26) we use projection prescriptions similar to (10.115). The resulting system of ordinary coupled differential equations is then solved numerically for different chemical potentials l and temperatures T. For temperatures sufficiently small compared to the Fermi temperature TF = (3p2n)2/3, T/TF  1 we find that the effective potential U at the macroscopic scale k = 0 develops a minimum at a nonzero field value q0 [ 0;oq Uðq0 Þ ¼ 0: The system is then in the superfluid phase. For larger temperatures we find that the minimum is at q0 = 0 and that the ‘‘mass parameter’’ m2 is positive, m2 ¼ oq Uð0Þ [ 0: The critical temperature Tc of this phase transition between the superfluid and the normal phase is then defined as the temperature where one has q0 ¼ 0;

oq Uð0Þ ¼ 0

at

k ¼ 0:

ð10:116Þ

Throughout the whole crossover the transition q0?0 is continuous as a function of T demonstrating that the phase transition is of second order. In Fig. 10.41 we plot our result obtained for the critical temperature Tc and the Fermi temperature TF as a function of the chemical potential l at the unitarity point with a-1 = 0. From dimensional analysis it is clear that both dependencies are linear, Tc, TF*l, provided that non-universal effects involving the ultraviolet cutoff scale K can be neglected. That this is indeed found numerically can be seen as a nontrivial test of our approximation scheme and the numerical procedures as well as the universality of the system. Dividing the slope of both lines gives Tc/TF = 0.264, a result that will be discussed in more detail below. We emphasize

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10

Fig. 10.41 Critical temperature Tc (boxes) and Fermi temperature TF = (3p2n)2/3 (triangles) as a function of the chemical potential l. For convenience the Fermi temperature is scaled by a factor 1/5. We also plot the linear fits Tc = 0.39l and TF = 1.48l. The units are arbitrary and we use K = e7

Many-Body Physics

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.0

0.2

0.4

0.6

0.8

that part of the potential error in this estimates is due to uncertainties in the precise quantitative determination of the density or TF.

10.3.4 Phase Diagram The effect of the particle–hole fluctuations shows most prominently in the result for the critical temperature. With our approach we can compute the critical temperature for the phase transition to superfluidity throughout the crossover. The results are shown in Fig. 10.42. We plot the critical temperature in units of the

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

2

0

2

4

6

Fig. 10.42 Dimensionless critical temperature Tc/TF as a function of the inverse concentration c-1 = (akF)-1. The black solid line includes the effect of particle–hole fluctuations. We also show the result obtained when particle–hole fluctuations are neglected (dotted-dashed line). For comparison, we plot the BCS result without (left dotted line) and with Gorkov’s correction (left dashed). On the BEC side with c-1 [ 1 we show the critical temperature for a gas of free bosonic molecules (horizontal dashed line) and a fit to the shift in Tc for interacting bosons, DTc*c (dotted line on the right). The black solid dot gives the QMC results [51–53]

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Particle–Hole Fluctuations and the BCS–BEC Crossover

163

Fermi temperature Tc/TF as a function of the scattering length measured in units of the inverse Fermi momentum, i.e. the concentration c = akF. We can roughly distinguish three different regimes. On the left side, where c1 .  1, the interaction is weakly attractive. Mean field or BCS theory is qualitatively valid here. In Fig. 10.42 we denote the BCS result by the dotted line on the left (c-1 \ 0). However, the BCS approximation has to be corrected by the effect of particle–hole fluctuations, which lower the value for the critical temperature by a factor of 2.2. This is the Gorkov correction (dashed line on the left side in Fig. 10.42). The second regime is found on the far right side, where the interaction again is weak, but now we find a bound state of two atoms. In this regime the system exhibits Bose–Einstein condensation of molecules as the temperature is decreased. The dashed horizontal line on the right side shows the critical temperature of a free Bose–Einstein condensate of molecules. In-between there is the unitarity regime, where the two-atom scattering length diverges (c-1 ? 0) and we deal with a system of strongly interacting fermions. Our result including the particle–hole fluctuations is given by the solid line. This may be compared with a functional renormalization flow investigation without including particle–hole fluctuations (dot-dashed line) [21]. For c?0- the solid line of our result matches the BCS theory including the correction by Gorkov and Melik-Barkhudarov [37],   Tc eC 2 7=3 p=ð2cÞ ¼ e  0:28ep=ð2cÞ : TF p e

ð10:117Þ

In the regime c-1 [ - 2 we see that the non-perturbative result given by our RG analysis deviates from Gorkov’s result, which is derived in a perturbative setting. On the BEC-side for very large and positive c-1 our result approaches the critical temperature of a free Bose gas where the bosons have twice the mass of the fermions MB = 2M. In our units the critical temperature is then Tc;BEC 2p ¼  0:218: 2 TF ½6p fð3=2Þ2=3

ð10:118Þ

For c?0+ this value is approached in the form Tc  Tc;BEC aM c 1=3 ¼ jaM nM ¼ j : Tc;BEC a ð6p2 Þ1=3

ð10:119Þ

Here, nM = n/2 is the density of molecules and aM is the scattering length between them. For the ratio aM/a we use our result aM/a = 0.718 obtained from solving the flow equations in vacuum, i.e. at T = n = 0, see Sect. 9.2. For the coefficients determining the shift in Tc compared to the free Bose gas we find j = 1.55. For c1 J 0:5 the effect of the particle–hole fluctuations vanishes. This is expected since the chemical potential is now negative l \ 0 and there is no Fermi surface any more. Because of that there is no difference between the new curve with particle–hole fluctuations (solid in Fig. 10.42) and the one obtained when

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particle–hole contributions are neglected (dot-dashed in Fig. 10.42). Due to the use of an optimized cutoff scheme and a different computation of the density our results differ slightly from the ones obtained in [21]. In the unitary regime ðc1  0Þ the particle–hole fluctuations still have a quantitative effect. We can give an improved estimate for the critical temperature at the resonance (c-1 = 0) where we find Tc/TF = 0.264. Results from quantum Monte Carlo simulations are Tc/TF = 0.15 [51–53] and Tc/TF = 0.245 [54]. The measurement by Luo et al. [55] in an optical trap gives Tc/TF = 0.29 (+ 0.03/ - 0.02), which is a result based on the study of the specific heat of the system.

10.3.5 Crossover to Narrow Resonances Since we use a two channel model (6.14) we can not only describe broad resonances with h2K ! 1 but also narrow ones with h2K ! 0: This corresponds to a nontrivial limit of the theory which can be treated exactly [56, 57]. In the limit hK?0 the microscopic action (6.14) describes free fermions and bosons. The essential feature is, that they are in thermodynamic equilibrium so that they have equal chemical potential. (There is a factor 2 for the bosons since they consist of two fermions.) For vanishing Yukawa coupling hK the theory is Gaussian and the macroscopic propagator equals the microscopic propagator. There is no normalization of the ‘‘mass’’-term m2 so that the detuning parameter in (6.14) is m = lM(B - B0) and m2 ¼ lM ðB  B0 Þ  2l:

ð10:120Þ

To determine the critical density for fixed temperature, we have to adjust the chemical potential l such that the bosons are just at the border to the superfluid phase. For free bosons this implies m2 = 0 and thus 1 1 2 1 ha : l ¼ lM ðB  B0 Þ ¼  2 16p

ð10:121Þ

In the last equation we use the relation between the detuning and the scattering length a¼

h2 : 8plM ðB  B0 Þ

ð10:122Þ

The critical temperature Tc is now determined from the implicit equation

Z d3 p 2 2 þ 1 2 ¼ n: ð10:123Þ 2 1 ð2pÞ3 eTc ðp~ lÞ þ 1 e2Tc~p  1

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Particle–Hole Fluctuations and the BCS–BEC Crossover

165

While the BCS-BEC crossover can be studied as a function of B - B0 or l, (10.122) implies that for h2K ! 0 a finite scattering length a requires B?B0. For all c = 0 the narrow resonance limit implies for the phase transition B = B0 and therefore l = 0. (A different concentration variable cmed was used in [56, 58], such that the crossover could be studied as a function of cmed in the narrow resonance limit, see the discussion at the end of this section.) For l = 0 (10.123) can be solved analytically and gives Tc ¼ TF

!2=3 pffiffiffi 4 2 pffiffiffi  0:204: 3ð3 þ 2Þp1=2 fð3=2Þ

ð10:124Þ

This result is confirmed numerically by solving the flow equations for different microscopic Yukawa couplings hK and taking the limit hK ? 0. In Fig. 10.43, we show the critical temperature Tc/TF as a function of the dimensionless Yukawa pffiffiffiffiffi coupling hK = kF in the ‘‘unitarity limit’’ c-1 = 0 (solid line). For small values of pffiffiffiffiffi the Yukawa coupling, hK = kF .2 we enter the regime of the narrow resonance limit and the critical temperature is independent of the precise value of hK. The numerical value matches the analytical result Tc =TF  0:204 (dotted line in pffiffiffiffiffi Fig. 10.43). For large Yukawa couplings, hK = kF J40; we recover the result of the broad resonance limit as expected. In between there is a smooth crossover of the critical temperature from narrow to broad resonances. We use here a definition of the concentration c = akF in terms of the vacuum scattering length a. This has the advantage of a straightforward comparison with experiment since a-1 is directly related to the detuning of the magnetic field B - B0, and the ‘‘unitarity limit’’ c-1 = 0 precisely corresponds to the peak of the resonance B = B0. However, for a nonvanishing density other definitions of the concentration parameter are possible, since the effective fermion interaction kw,eff depends on the density. For example, one could define for n = 0 a ‘‘in medium scattering length’’  a ¼ kw;eff =ð8pÞ; with kw,eff = -h2/m2 evaluated for T = 0 but n = 0 [56]. The corresponding ‘‘in medium concentration’’ cmed ¼ akF would differ from our definition by a term involving the chemical potential, resulting in a shift of the location of the unitarity limit if the latter is defined as c-1 med = 0. While Fig. 10.43 The critical temperature divided by the Fermi temperature Tc/TF as a function of the dimensionless pffiffiffiffiffi Yukawa coupling hK = kF -1 for c = 0 (solid line). One can clearly see the plateaus in the narrow resonance limit (Tc =TF  0:204, dotted line) and in the broad resonance limit (Tc =TF  0:264, dashed line)

0.26

0.24

0.22

0.20 0.1

1

10

100

1000

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for broad resonances both definitions effectively coincide, for narrow resonances a precise statement how the concentration is defined is mandatory when aiming for a precision comparison with experiment and numerical simulations for quantities as Tc/TF at the unitarity limit. For example, defining the unitarity limit by c-1 med = 0 would shift the critical temperature in the narrow resonance limit to Tc/TF = 0.185 [56].

10.4 BCS–Trion–BEC Transition In this section we discuss briefly the many-body physics for the SU(3) symmetric model in (6.18) that describes three fermion species close to a common Feshbach resonance. No proper many-body calculation for this model has been performed so far. However, based on the knowledge of the BCS-BEC crossover physics and the vacuum calculation presented in Sect. 9.3.1 we can already infer some qualitative features of the quantum (zero temperature) phase diagram. For increasing density the chemical potential increases compared to the vacuum chemical potential shown in Fig. 10.44. The BEC phase occurs for small density for a1 [ ðac2 Þ1 : Due to the symmetries of the microscopic action, the effective potential for the bosonic field depends only on the SU(3) 9 U(1) invariant combination q ¼ uy u ¼ u1 u1 þ u2 u2 þ u3 u3 : It reads for small l - l0, with l0 the vacuum chemical potential,

0.1 0 0.1 0.2 0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 10.44 Chemical potential or binding energy per atom l = E in vacuum (n = T = 0) as a function of the inverse scattering length (a/a0)-1 for three fermion species at a common Feshbach resonance. The solid line gives the energy per atom of the trion with respect to the fundamental fermion (zero energy line). The dashed line is the energy of the dimer per atom. Here a0 is the Bohr radius which we also use as the ultraviolet scale K = 1/a0. Parameters are chosen to correspond to 6Li in the (mF = 1/2, mF = -1/2)-channel [59]

10.4

BCS–Trion–BEC Transition

UðqÞ ¼

167

ku 2 q  2ðl  l0 Þq; 2

ð10:125Þ

where we use the fact that the term linear in q vanishes for l = l0, i.e. m2u ðl0 Þ ¼ 0: The minimum of the potential shows a nonzero superfluid density q0 ¼

2ðl  l0 Þ 2 ; Uðq0 Þ ¼  ðl  l0 Þ2 ; ku ku

ð10:126Þ

oUðq0 Þ 4 ¼ ðl  l0 Þ ¼ 2q0 : ol ku

ð10:127Þ

with density n¼

The factor two is expected since every boson contributes two fermions to the density. The superfluid density is twice the renormalized order parameter, nS = 2q0. At zero temperature the density equals the superfluid density n = nS in the BEC regime. For the BCS phase for a \ ac1 (ac1 \ 0), one has l0 = 0. Nonzero density corresponds to positive l - l0. In this region the contribution of fermionic atom fluctuations to the renormalization flow drives m2u always to zero at some finite kc, with BCS spontaneous symmetry breaking (q0 [ 0) induced by the flow for k \ kc. Both the BEC and BCS phases are therefore characterized by superfluidity with a nonzero expectation value of the boson field u0 6¼ 0 with q0 ¼ u0 u0 : As an additional feature to the BCS-BEC crossover for a Fermi gas with two components, the expectation value for the bosonic field u0 in the three component case also breaks the spin symmetry of the fermions (SU(3)). Due to the analogous in QCD this was called ‘‘color superfluidity’’ [60]. For any particular direction of u0 a continuous symmetry SU(2) 9 U(1) remains. According to the symmetry breaking SU(3) 9 U(1)? SU(2) 9 U(1), the effective potential has five flat directions. For two identical fermions the BEC and BCS phases are not separated, since in the vacuum either l0 = 0 or m2u ¼ 0: There is no phase transition, but rather a continuous crossover. For three identical fermions, however, we find a new trion phase for ð1Þ ð1Þ ðac1 Þ1 \a1 \ðac2 Þ1 : In this region the vacuum has l0 \ 0 and m2u [ 0: The atom fluctuations are cut off by the negative chemical potential and do not drive m2u to zero, such that for small density m2u remains positive. Adding a term mu2 q to the effective potential (10.125) we see that the minimum remains at q0 = 0 as long as m2u [ 2ðl  l0 Þ: No condensate of bosons occurs. The BEC and BCS phases that show both extended superfluidity through a spontaneous breaking of the SU(3) 9 U(1) symmetry, are now separated by a phase where u0 ¼ 0, such that the SU(3) 9 U(1) symmetry remains unbroken (or will be only partially broken). Deep in the trion phase, e.g. for very small |a-1|, the atoms and dimers can be neglected at low density since they both have a gap. The thermodynamics at low

168

10

Many-Body Physics

density and temperature is determined by a single species of fermions, the trions. In our approximation it is simply given by a noninteracting Fermi gas, with fermion mass 3M and chemical potential 3(l - l0). Beyond our approximation, we expect that trion interactions are induced by the fluctuations. While local trion interactions *(v*v)2 are forbidden by Fermi statistics, momentum dependent interactions are allowed. These may, however, be ‘‘irrelevant interactions’’ at low density, since also the relevant momenta are small such that momentum dependent interactions will be suppressed. Even if attractive interactions would induce a ditrion condensate, this has atom number six and would, therefore, leave a Z6 subgroup of the U(1) transformations unbroken, in contrast to the BEC and BCS phase where only Z2 remains. Furthermore, the trions are SU(3)-singlets such that the SU(3) symmetry remains unbroken in the trion phase. The different symmetry properties between the possible condensates guarantee true quantum phase transitions in the vicinity of ac1 and ac2 for small density and T = 0. We expect that this phase transition also extends to small nonzero temperature. While deep in the trion phase the only relevant scales are given by the density and temperature, and possibly the trion interaction, the situation becomes more complex close to quantum phase transition points. For a  ac1 we have to deal with a system of trions and atoms, while for a  ac2 a system of trions and dimers becomes relevant. The physics of these phase transitions may be complex and rather interesting. We finally comment on the precision of our computation. At nonzero density, our truncation may be improved by including in the two-body sector a four fermion coupling  ðwy wÞ2 , which may be partially bosonized in favor of a running  h, see Sect. 10.3. Also the momentum dependence of the interactions  kuw or kw may be resolved beyond the pointlike approximation. Furthermore we expect an improvement from including a second atom-dimer scattering channel  kAD uy uwy w: These improvements are expected to change the quantitative values of ac1, ac2, l0, and s0, but not the qualitative situation. We believe that already the present truncation will yield a reliable picture of the qualitative properties of the phase diagram, once it is extended to nonzero density and temperature.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Wetterich C (2008) Nucl Phys B 802:368 Berges J, Tetradis N, Wetterich C (2002) Phys Rept 363:223 Papenbrock T, Wetterich C (1995) Z Phys C65:519 Andersen JO, Strickland M (1999) Phys Rev A 60:1442 Canet L, Delamotte B, Mouhanna D, Vidal J (2003) Phys Rev B 68:064421 Canet L, Delamotte B, Mouhanna D, Vidal J (2003) Phys Rev D 67:065004 Bervillier C, Juttner A, Litim DF (2007) Nucl Phys B 783:213 Sachdev S (1999) Quantum phase transitions. Cambridge University Press, Cambridge Uzunov DI (1981) Phys Lett A 87:11

References 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. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

169

Wetterich C (2008) Phys Rev B 77:064504 Wilson KG, Fisher ME (1972) Phys Rev Lett 28:240 Pelissetto A, Vicari E (2002) Phys Rept 368:549 Andersen JO (2004) Rev Mod Phys 76:599 Arnold P, Moore G (2001) Phys Rev Lett 87:120401 Kashurnikov VA, Prokof’ev NV, Svistunov BV (2001) Phys Rev Lett 87:120402 Baym G, Blaizot J-P, Holzmann M, Laloë F, Vautherin D (1999) Phys Rev Lett 83:1703 Blaizot J-P, Méndez-Galain R, Wschebor N (2006) Phys Rev E 74:051116 Blaizot J-P, Méndez-Galain R, Wschebor N (2006) Phys Rev E 74:051117 Ledowski S, Hasselmann N, Kopietz P (2004) Phys Rev A 69:061601 Hasselmann N, Ledowski S, Kopietz P (2004) Phys Rev A 70:063621 Diehl S, Gies H, Pawlowski JM, Wetterich C (2007) Phys Rev A 76:021602 Bogoliubov NN (1947) J Phys (Moscow) 11:23 Pitaevskii L, Stringari S (2003) Bose–Einstein condensation. Oxford University Press, Oxford Landau LD (1941) J Phys USSR 5:71 Wetterich C (1993) Z Phys C 57:451 Josephson BD (1966) Phys Lett 21:608 Popov VN (1983) Functional integrals in quantum field theory and statistical physics. Reidel, Dordrecht Fisher DS, Hohenberg PC (1988) Phys Rev B 37:4936 Holzmann M, Baym G, Blaizot J-P, Lalo F (2007) Proc Natl Acad Sci USA 104:1476 Prokof’ev N, Ruebenacker O, Svistunov B (2001) Phys Rev Lett 87:270402 Prokof’ev N, Svistunov B (2002) Phys Rev A 66:043608 Sachdev S, Demler E (2004) Phys Rev B 69:144504 Sachdev S (1999) Phys Rev B 59:14054 Gräter M, Wetterich C (1995) Phys Rev Lett 75:378 Gersdorff G, Wetterich C (2001) Phys Rev B 64:054513 Krahl H, Wetterich C (2007) Phys Lett A 367:263 Gorkov LP, Melik-Barkhudarov TK (1961) Sov Phys JETP 13:1018 Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 106:162 Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 108:1175 Cooper LN (1956) Phys Rev 104:1189 Aoki KI (2000) Int J Mod Phys B 14:1249 Salmhofer M, Honerkamp C (2001) Progr Theor Phys 105:1 Metzner W (2005) Progr Theor Phys Suppl 160:58 Ellwanger U, Wetterich C (1994) Nucl Phys B 423:137 Strack P, Gersch R, Metzner W (2008) Phys Rev B 78:014522 Heiselberg H, Pethick CJ, Smith H, Viverit L (2000) Phys Rev Lett 85:2418 Gies H, Wetterich C (2002) Phys Rev D 65:065001 Gies H, Wetterich C (2002) Acta Phys Slov 52:215 Pawlowski JM (2007) Ann Phys (N. Y.) 322:2831 Diehl S, Krahl HC, Scherer M (2008) Phys Rev C 78:034001 Bulgac A, Drut JE, Magierski P (2006) Phys Rev Lett 96:090404 Bulgac A, Drut JE, Magierski P (2008) Phys Rev A 78:023625 Burovski E, Prokof’ev N, Svistunov B, Troyer M (2006) Phys Rev Lett 96:160402 Akkineni VK, Ceperley DM, Trivedi N (2007) Phys Rev B 76:165116 Luo L, Clancy B, Joseph J, Kinast J, Thomas JE (2007) Phys Rev Lett 98:080402 Diehl S, Wetterich C (2006) Phys Rev A 73:033615 Gurarie V, Radzihovsky L (2007) Ann Phys (NY) 322:2 (2007), january Special Issue 2007 Diehl S, Wetterich C (2007) Nucl Phys B 770:206 Bartenstein M, Altmeyer A, Riedl S, Geursen R, Jochim S, Chin C, Denschlag JH, Grimm R, Simoni A, Tiesinga E, Williams CJ, Julienne PS (2005) Phys Rev Lett 94:103201 Honerkamp C, Hofstetter W (2004) Phys Rev B 70:094521

Chapter 11

Conclusions

In this thesis we applied the method of functional renormalization to the theoretical description of ultracold quantum gases. We derived flow equations for various physical systems, namely for a Bose gas with approximately pointlike, repulsive interaction, for a Fermi gas with two hyperfine components in the BCS–BEC crossover and for a Fermi gas with three components. The implications of these flow equations are investigated both in the few-body and the many-body regime. For repulsive bosons with approximately pointlike interaction we find an upper bound for the scattering length a. Quantum fluctuations lead to a screening of the interaction and imply that the scattering length is of the order of the inverse microscopical momentum scale a . K1 in three dimensions. In two dimensions, the scale dependence of the interaction strength is logarithmic and the bound is correspondingly weaker. At low temperatures we find that Bogoliubov theory gives a surprisingly accurate account of many properties of an interacting Bose gas although it treats fluctuations only to quadratic order and neglects for example the scale dependence of the interaction strength. On the other hand, perturbative extensions beyond Bogoliubov theory are plagued with infrared divergences. Our renormalization group treatment solves these problems and is free of infrared problems. It is also possible to apply our method to the case of stronger interactions. However, one should keep in mind that very large ratios of the scattering length to the interparticle distance n-1/3 are only possible if the microscopic scale K is of the same order as n1/3. In this case, the details of the interaction become important and universality might be lost. We also calculate different thermodynamic observables at nonvanishing temperatures. This includes for example the critical temperature which is larger for an interacting gas compared to a free Bose gas. The shift in the critical temperature is linear in the concentration parameter, DTc =Tc  an1=3 : The region around the phase transition is mainly governed by the critical exponents of the three-dimensional XY universality class. In principle, these critical exponents can also be calculated from S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_11, Ó Springer-Verlag Berlin Heidelberg 2010

171

172

11

Conclusions

our flow equations. However, they are obtained also from the simpler flow equations of the classical O(N) model which are studied in much detail in the literature. Instead, we calculated the thermodynamic quantities in the full range from zero temperature to the phase transition. Besides the superfluid and the condensate fraction this includes the correlation length (or healing length), the entropy and energy density, the pressure, the specific heat, the isothermal and the adiabatic compressibility. All these quantities are well described by Bogoliubov theory for small temperatures and interaction strength while deviations are found away from this limit. This is also the case for the first and second velocity of sound. The calculations concerning the Bose gas in three dimensions could be improved in different directions. One might use a regulator function that acts as an ultraviolet cutoff also in the space of Matsubara frequencies. This should improve the flow equation for the pressure and its derivatives such as density and entropy density. One could also improve the approximation by going to the next order in a systematic derivative expansion. It would also be interesting to study the flow equation for the effective potential using a method different from an expansion around the (scale-dependent) minimum. With a numerical solution of the partial differential equation for U it might be possible to derive also some predictions for attractive interactions between two atoms corresponding to a negative scattering length. Finally, an interesting extension would be to consider gases trapped by some confining potential. Much of the formalism developed in this thesis can be transferred from the homogeneous space to this more complicated case using a local density approximation. One assumes that the relevant length scale for the fluctuations is small compared to the extension of the trap and works with a ~Þ ¼ l0  Vðx ~Þ where Vðx ~Þ position dependent chemical potential according to lðx is the potential. Besides the thermodynamic observables, it would be interesting to study hydrodynamic excitations and their dependence on temperature and density. We studied the flow equations for the homogeneous Bose gas also in two spatial dimensions. The formalism is pretty similar—one has to adapt a single number, only. The physical properties, on the other side, are very different. The effect of fluctuations is more important than in three dimensions and many quantities have a logarithmic scale dependence. In the low temperature regime, the emergence of a quadratic (Matsubara-) time derivative from quantum fluctuations plays a more important role. Nevertheless, we find at small temperatures that some physical observables such as the quantum depletion of the condensate and the velocity of sound are described by Bogoliubov theory to good approximation. There are sizable deviations for large temperatures, however. Our calculations show, that a second branch in the dispersion relation becomes relevant at small frequencies. In contrast to the sound mode which describes fluctuations in the phase of the oder parameter, the second branch describes fluctuations in its absolute value. The thermal phase diagram of the two-dimensional Bose gas is governed by the Kosterlitz–Thouless phase transition. The flow equation method is especially valuable here, since the dependence of various quantities on the infrared scale is crucial. For a homogeneous system of infinite extension, long-range order and Bose–Einstein condensation in the strict sense are forbidden by the

11

Conclusions

173

Mermin–Wagner theorem. Nevertheless, one observes superfluidity at small temperatures. For a finite system the size of the probe sets an infrared cutoff scale. No fluctuations with momenta smaller then this scale contribute and the Mermin–Wagner theorem does not apply any more. Our investigations show, that the difference between superfluidity and long-range order looses its importance for systems with finite extension or for observables that are connected with a momentum scale that acts as an effective infrared cutoff. The superfluid density  q0 and the condensate density nC ¼ q 0 are connected by the wavenS ¼ q0 ¼ A  When the infrared cutoff scale kph is lowered, the function renormalization A: condensate density as the square of the order parameter, decreases. At the same  grows (and goes to infinity for time, the wavefunction renormalization factor A kph ? 0). The superfluid density nS remains positive for small enough temperatures. For the dependence of the critical temperature on the interaction strength k we find for small k the functional form Tc/n = 4p/ln(n/k). This is in agreement with an perturbative analysis. In contrast to quantum Monte-Carlo simulations, we find that the coefficient n depends on the infrared cutoff scale kph. In the infinite size limit kph?0 we find that the critical temperature vanishes, Tc/n ? 0. At least in parts this feature is attributed to the scale dependence of the interaction strength, a feature that was not taken into account for the numerical simulations. In conclusion, a rather detailed picture of the superfluid Bose gas in two dimensions has been obtained from the flow equation method. Nevertheless, there is still room for some extensions of the theory. Most of the points discussed for the Bose gas in three dimensions apply here as well. In addition, an investigation of occupation numbers as a function of temperature and interaction strength would be of interest. Besides the bosonic systems we also investigated models for fermions. For two fermion species close to a Feshbach resonance, the flow equations in the vacuum limit show the expected dimer formation. The binding energy in the broad resonance limit matches the expectation from quantum mechanical calculations. In the many-body regime the discussion presented here concentrates on the effect of particle-hole fluctuations. On the BCS side of the crossover we recover Gorkovs correction to the original BCS theory, i.e. a suppression of the critical temperature by a factor *2.2. We extend the calculation by Gorkov and Melik-Barkhudarov to the whole crossover regime including the unitarity point where the scattering length diverges. At this point we can also compare our estimate of the critical temperature in units of the Fermi temperature, Tc/TF & 0.264, to the result from Monte-Carlo simulations Tc/TF & 0.15. The deviations are rather large and may have different reasons. To calculate the ratio Tc/TF one has to determine for given chemical potential both the critical temperature and the density. (In our units the latter is connected to the Fermi temperature by TF = (3p2n)2/3.) Both steps are equally important—and nontrivial. For example, would we estimate the density by the corresponding formula in the non-interacting case, the resulting ratio would be Tc/TF & 0.165. This shows that a reliable method to calculate the density is crucial—both for the flow equation method and the Monte-Carlo simulations.

174

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Conclusions

Apart from the density, other improvements of our calculation are conceivable. The inclusion of a vertex kuw, which describes the scattering of a fermionic atom with a dimer, is work in progress. On the BEC side of the crossover and in the vacuum limit it leads to a reduction in the ratio aM/a [1]. One expects that the inclusion of this term in the many-body calculation slightly reduces the critical temperature on the BEC side and in the crossover regime. Another improvement concerns the rebosonization method. The calculations performed so far employ the approximate flow equation proposed in Ref. [2]. Using the exact flow equation derived in Chap. 3 might improve the result. Once the truncation has reached a satisfactory quantitative precision, it would be interesting to study different thermodynamic observables. For a comparison to experimental investigations it is also necessary to transfer the formalism to the trapped Fermi gas. With increasing experimental precision, the goal of a accurate test for non-perturbative methods seems to be in reach. The knowledge of the phase diagram is much less developed for the Fermi gas with three components than for the two-component case. Before quantitative questions can be answered for this system, one has to settle the qualitative features. In this thesis we investigated a simple SU(3) symmetric model for fermions with equal properties (except for hyperfine-spin, of course) close to a common Feshbach resonance. As expected, we find that the two-body problem is equivalent to the one for two species. In contrast, the three-body problem is now governed by Efimov physics. In our model and within our approximation scheme we recovered the infinite tower of three-body bound states (trimers) at the resonance and for energies close to the free atom threshold. In the renormalization group picture the Efimov effect shows as a limit cycle scaling. With our simple truncation we find a scaling parameter s0 & 0.82 while a calculation that takes the full momentum dependence of vertices into account confirms Efimovs value s0 & 1.00624. More important, for the phase diagram then the tower of bound states is the state with lowest energy and the corresponding chemical potential. In a region -1 a-1 \ a-1 c1 \ a c2 with ac1 \ 0 and ac2 [ 0 we find that the many-body ground state is dominated by trions (the trimers with lowest energy). The chemical potential in this region is lower than in the two-component case. The mechanisms that lead to BCS superfluidity for a ? 0- or BEC-like superfluidity for a ? 0? do not work in this region. While SU(3) symmetry is broken spontaneously in the BCS and BEC phases, this is not the case in the trion phase. We therefore expect -1 true and rather interesting quantum phase transitions around the points a-1 c1 and ac2 (‘‘BCS–Trion–BEC transition’’). In addition to the SU(3) symmetric model we also investigate a model for three fermion species where this symmetry is broken explicitly by different positions and widths of the resonances. Although the Efimov tower of bound states is not expected for three distinct resonances, one still finds three-body bound states provided the scattering lengths are large enough. We apply our method to the case of 6Li which is currently of experimental interest. The trion is not stable and can decay into lower lying bound states (possibly the two-body bound states of the nearby Feshbach resonance). Introducing a decay width Cv as an additional

11

Conclusions

175

parameter to be fixed from experiment, we can explain the form of the observed three-body loss coefficient as a function of magnetic field. The calculations performed for the three-component Fermi gas so far are a first step towards a more detailed investigation of the interesting phase diagram. A proper many-body calculation at zero temperature could shed more light on the quantum phase transitions while the inclusion of temperature effects will allow for an investigation of the complete phase diagram. Functional renormalization provides a very good method for this purposes. Note that since three-body effects are important, it is not possible to work with mean-field theory. Although one might first investigate the SU(3) symmetric model, it would also be interesting to consider 6Li where comparison to future experiments seems possible. Besides the application to concrete models, we also considered some more formal aspects of the renormalization group method. We derived a generalization of the flow equation to scale-dependent composite operators. In the new equation, the usual renormalization flow with one-loop form is supplemented by an intuitive tree-level term. One application of this generalized flow equation is a scaledependent translation from ultraviolet to infrared degrees of freedom as discussed in [2, 3]. It may also provide a method to treat momentum dependence of vertices. Indeed, one could introduce scale-dependent composite operators for different ‘‘scattering channels’’. The difficult problem of treating momentum-dependent vertices would be reduced to the momentum dependence of propagators for composite operator fields. This might be simpler in some respects and also allows to trace the renormalization group flow in the regime with spontaneous symmetry breaking.

References 1. Scherer M (2007) Few-body scattering and four-fermion interaction in BCS-BEC crossover physics. Diploma thesis, Universität Heidelberg, Germany 2. Gies H, Wetterich C (2002) Phys Rev D 65:065001 3. Gies H, Wetterich C (2002) Acta Phys Slov 52:215

Chapter 12

Appendices

12.1 Appendix A: Some Ideas on Functional Integration and Probability The wave-particle duality is discussed since the early days of quantum mechanics. ~; tÞ: Schrödingers equation [1] is a non-relativistic equation for a wave function wðx On the other side this wave function, or more precisely its modulus square ~; tÞj2 ; can be interpreted as the probability density to detect a particle at the jwðx point ~ x at time t. This was first pointed out by Born [2]. In this sense an electron for example has properties of both a particle and a field. In the modern understanding of quantum field theory the situation is basically still the same. The formulation of the theory is mainly in terms of wave functions or fields, but one still needs the interpretation in terms of probabilities to ‘‘find a particle in a certain state’’ to bring the formalism into contact with experiments. The formulation of quantum field theory using the functional integral [3, 4] is quite close to formulations of statistical field theory [5, 6, 7]. The far reaching parallels in both formalisms were important for many achievements, for example in the development of the renormalization group theory [8, 9]. One might ask whether this duality between quantum and statistical field theory has a deeper physical origin and whether it can help us for a better understanding of the wave– particle duality. One might hope that some difficult points in the foundations of quantum theory related to the collapse of the wave function could be clarified. What prevents us from using our good knowledge and intuition about statistical field theory to investigate these questions is at the same time the most important difference between the two formalisms. In statistical field theory one calculates expectation values, correlation functions, etc. by taking the sum over all possible field configurations u weighted by the real and positive semi-definite probability measure

S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses, DOI: 10.1007/978-3-642-14113-3_12, Ó Springer-Verlag Berlin Heidelberg 2010

177

178

12 Appendices

p½u ¼ eS½u :

ðA:1Þ

For a local theory in d dimensions the action S½u is given as an integral over the Lagrange density Z ðA:2Þ S½u ¼ dd xL where L is a function of u and its derivatives. In quantum field theory, the corresponding measure under the functional integral eiS½u

ðA:3Þ

is complex and can therefore not be interpreted as a probability. In the following we show, that the formalism of quantum field theory can be reformulated such that the weighting factor under the functional integral is real and discuss possible consequences for our understanding of what ‘‘particles’’ are.

12.1.1 Functional Integral with Probability Measure In this section we reconsider the functional integral formulation of quantum field theory and formulate an representation with a (quasi-) probability distribution. Let us start with a simple Gaussian theory X S¼ ua ðPab þ idab Þub : ðA:4Þ a;b

We use here an abstract index notation where e.g. a stands for both continuous variables such as position or momentum and internal degrees of freedom. We assume that u is a bosonic field. We included in (A.4) a small imaginary part  i to make the functional integral convergent and to enforce the correct frequency integration contour (Feynman prescription). Although e is usually taken to be infinitesimal, we work with an arbitrary positive value here and take the limit  ? 0+ only at a later point in our investigation. For simplicity, we will often drop the abstract index and use a short notation with e.g. S ¼ u ðP þ iÞu:

ðA:5Þ

The operator P is the real part of the inverse microscopic propagator. As an example we consider a relativistic theory for a scalar field where P reads in position space   ð4Þ lm o o 2 Pðx; yÞ ¼ d ðx  yÞ g m : ðA:6Þ oyl oym

12.1

Appendix A: Some Ideas on Functional Integration and Probability

Another example is the nonrelativistic case with   o 1 ~2 ð4Þ r : þ Pðx; yÞ ¼ d ðx  yÞ i oy0 2M y

179

ðA:7Þ

For a Gaussian theory the microscopic propagator coincides with the full propagator. The latter is obtained for general actions S from iGab ¼ hua ub ic ¼ ðiÞ3

ðA:8Þ

d d W½J dJa dJb

with eiW½J ¼ Z½J ¼

Z

DueiS½uþi

R

fJ  uþu Jg

:

ðA:9Þ

x and b ¼ ðy0 ; ~Þ y the object Gab can be interpreted as the probability For a ¼ ðx0 ; ~Þ amplitude for a particle to propagate from the point ~ y at time y0 to the point ~ x at time x0. More general, one might label by a some single-particle state jua i at time ta and with b the state jub i at time tb. The propagator Gab describes then the probability amplitude for the transition between the two states. However, the description of an actual physical experiment involves the transition probability given by the modulus square of the probability amplitude (no summation over a and b) qab ¼ jGab j2 ¼ Gab Gab :

ðA:10Þ

This transition probability can also directly be obtained from functional derivatives of Z Z ~ 1 ; J2  ¼ Du1 Du2 eiS½u1  eiS ½u2  Z½J ðA:11Þ R R i fJ1 u1 þu1 J1 g i fJ2 u2 þu2 J2 g e e : We note that (A.11) contains twice the functional integral over the field configuration u. One may also write this as a single functional integral over fields that depend in addition to the position variable ~ x on the contour time tc which is integrated along the Keldysh contour [10, 11]. For our purpose it will be more convenient to work directly with the expression in (A.11). For hui ¼ hu i ¼ 0 we can write qab ¼

1 d d d d ~ Z½J1 ; J2 : Z~ dðJ1 Þa dðJ1 Þb dðJ2 Þa dðJ2 Þb

~ 1 ; J2  ¼ Z½J1 Z  ½J2  and This is immediately clear since Z½J

ðA:12Þ

180

12 Appendices

iGab ¼ ðiÞ2 iGab ¼ i2

1 d d Z½J1   Z½J1  dðJ1 Þa dðJ1 Þb

1 d d Z  ½J2 : Z  ½J2  dðJ2 Þa dðJ2 Þb

ðA:13Þ

for hui ¼

i d i d Z½J ¼ 0: Z½J ¼ hu i ¼  Z dJ Z dJ

ðA:14Þ

Usually one obtains qab by first calculating Gab and then taking the modulus square thereof. The way we go here seems to be more complicated from a technical point of view, but has the advantage that it will allow for an intuitive physical interpretation. ~ 1 ; J2 : This object plays a similar role as the partition We first concentrate on Z½J function in statistical field theory. In some sense it is a sum over microscopic states weighted with some ‘‘probability’’. However, in contrast to statistical physics, the summation does not go over states of a system at some fixed time t but over field configurations that depend both on the position variable ~ x and the time variable t. The summation seems to go over a even larger space since the functional integral appears twice Z Z Du1 Du2 ðA:15Þ so that the configuration space seems to be the tensor product of twice the space ~; tÞ: In addition the that contains the field configurations in space-time uðx ‘‘probability weight’’ 

eiS½u1  eiS

½u2 

ðA:16Þ

is not positive semi-definite and has even complex values in general. This last two features (‘‘doubled’’ configuration space and missing positivity) prevent us from interpreting quantum field theory in a similar way as statistical field theory. An idea to overcome these difficulties is to partially perform the functional integral in (A.11). For this purpose we make a change of variables of the form 1 1 u1 ¼ pffiffiffi / þ pffiffiffi v; 2 2 1 1 u2 ¼ pffiffiffi /  pffiffiffi v; 2 2

1 1 J1 ¼ pffiffiffi J/ þ pffiffiffi Jv ; 2 2 1 1 J2 ¼ pffiffiffi J/ þ pffiffiffi Jv : 2 2

ðA:17Þ

~ this gives then For Z Z~ ¼

Z

D/ v½/; Jv ei

R

fJ/ /þ/ J/ g

ðA:18Þ

12.1

Appendix A: Some Ideas on Functional Integration and Probability

181

with v½/; Jv  ¼

Z

pffiffi pffiffi  DveiS½ð/þvÞ= 2 eiS ½ð/vÞ= 2 R    ei fJv vþv Jv g :

ðA:19Þ

We note that v[/, Jv] as a functional of / and Jv is real. This follows from comparison with the complex conjugate together with the change of variables v ? -v. If it is also positive, we can interpret this object as a probability for the field configurations /(x). We call v the functional probability for the field configuration /. Before we discuss the general properties of v[/, Jv] in more detail, we consider it explicitly for a Gaussian action S½u as in (A.4). In that case we can perform the functional integral Z     v½/; Jv  ¼ Dvef/ /þv vg eif/ Pvþv P/g 

 eifJv vþv 



ðA:20Þ

Jv g

1



¼ e/ / e ðJv þ/



PÞðJv þP/Þ

:

The last line holds up to a multiplicative factor that is irrelevant for us here. For Z~ we are left with Z R   ~ / ; Jv  ¼ D/v½/; Jv ei fJ/ /þ/ J/ g Z½J Z   1  2 2 ðA:21Þ ¼ D/e / ðP þ Þ/ eifJ/ /þ/ J/ g 1



 e fJv P/þ/



PJv g 1 Jv Jv

e

:

For J/ = Jv = 0 the integrand in (A.21) is strictly positive. Equation (A.21) can therefore be interpreted in a similar way as the partition function in statistical field theory. The probability measure is 1 

v ¼ e  /

ðP2 þ2 Þ/

:

ðA:22Þ

We can distinguish three different classes of field configurations /. In the simplest case the norm vanishes, X j/j2 ¼ / / ¼ /a /a ! 0: ðA:23Þ a

The functional probability for this case is of order 1. The second class contains field configurations where the norm is nonzero, / / 6¼ 0; but where / satisfies the on-shell condition, i.e. /*P2/ = 0. The functional probability for this case is of order e-e (for / /  1). Finally, in the third class the norm is nonzero and the field configuration is off-shell, i.e.

182

12 Appendices

/ /  1;

/ P2 /  1:

ðA:24Þ

The functional probability in (A.22) for this case is only of the order e1= : This shows that off-shell configurations are strongly suppressed in the limit  ! 0 compared to the trivial case |/| = 0 and the on-shell fields with / P2 / ¼ 0:

ðA:25Þ

However, for  [ 0 the probability for off-shell configurations is not strictly zero ~ This is in contrast to classical statistics and they give also contributions to Z: where only states that fulfill the equation of motion are included. (At nonzero temperature states with different energies are weighted according to a thermal distribution.) There are more differences between the partition function in classical statistics and the quantum partition function in (A.21). In classical statistics the averaging over some phase space is directly linked to time averaging by the ergodic hypothesis. Indeed, this hypothesis says that a mean value calculated by taking the average of some quantity over the accessible phase space is equal to the average of that quantity over a—sufficiently long—time interval. In classical statistics time plays an outstanding role. The formalism breaks space–time symmetries such as Lorentz- or Galilean symmetry explicitly. For the case of quantum field theory this point is different. The theory in the vacuum (zero temperature and density, T = n = 0) is symmetric under Lorentz symmetry (or Galilean symmetry in the nonrelativistic case). The summation over possible field configurations in (A.21) is not related to time averaging. We directly interpret it in the following way. Every physical experiment or ‘‘measurement-like situation’’ corresponds to a different microscopic field configuration /. This field configuration does not necessarily have to fulfill the on-shell condition (A.25) but has a probability v that strongly favors on-shell fields. There is, however, a subtlety in this interpretation. When the action S is given as an integral over the 3 + 1 dimensional spacetime X; then v describes the probability for a field configuration /ðx0 ; ~Þ x on this manifold X: Since we experience only one universe with one configuration one might ask why we should take the sum over different configurations weighted with some probability. To answer that question it is important to realize that our information about the field configuration /ðx0 ; ~Þ x is limited. First we can investigate only limited regions in space-time (around our own ‘‘world-line’’). Regions that are too far away either in the spatial or the temporal sense are not accessible. However, in the framework of a local field theory, the experiments in some region of space-time depend on the other (not accessible) regions only via the boundary conditions. Second, and more important, we have only access to the field configuration in some ‘‘momentum range’’. No experiment has an arbitrary large resolution and can resolve infinitely small wavelength. Therefore the true microscopic field configuration is inevitably hidden from our observation. In a Gaussian or non-interacting theory this issue seems to be not so

12.1

Appendix A: Some Ideas on Functional Integration and Probability

183

important since different momentum modes decouple from each other. In a theory with interactions this is different, however. Modes with different momenta pl (and different values of plpl) are coupled via the interaction. The microscopic regime does influence the macroscopic states. Our interpretation of (A.21) is therefore that the functional integral sums over possible microscopic configurations /(x) with probability (up to a factor) given by 1 

e  /

ðP2 þ2 Þ/

ðA:26Þ

for the Gaussian theory considered above and more general (for Jv = 0) by Z pffiffi pffiffi  ðA:27Þ v½/ ¼ v½/; Jv ¼ 0 ¼ DveiS½ð/þvÞ= 2 eiS ½ð/vÞ= 2 : In this general case, the functional probability for nonzero source terms is given by (A.19). As argued there, it is always real. This is made more explicit in the expression Z v½/; Jv  ¼ Dv cosðS1 ½/ þ v  S1 ½/  vÞ ðA:28Þ  expðS2 ½/ þ v  S2 ½/  vÞ with o pffiffiffi 1 n S1 ½u ¼ Re S½u= 2 þ pffiffiffi Jv u þ u Jv 2

ðA:29Þ

pffiffiffi S2 ½u ¼ Im S½u= 2:

ðA:30Þ

and

We note that the functional integral in (A.28) converges when S2 ½u increases with u u fast enough. For S2 ½u  u u as in our Gaussian example, the convergence is quite good. Although for arbitrary actions S½u the ‘‘probability’’ v[/, Jv] does not have to be positive, this is expected to be the case for many choices of S½u: When v[/, Jv] is negative for some choices of /, this indicates that different values for / do not directly correspond to independent physical configurations. One might come to positive definite probabilities when the space of possible fields is restricted to a physically subspace. However, v[/, Jv] as defined above can in any case be seen an quasi-probability for /. This is in some respect similar to Wigner’s representation of density matrices [12]. Let us make another comment on the case of non-Gaussian actions S½u: When S½u contains terms of higher then quadratic order in the fields u the form of the action is subject to renormalization group modifications. Usually the true microscopic action S½u is not known. Measurements have only access to the effective action C½u which already includes the effect of quantum fluctuations. (Measurements at some momentum scale k2 ¼ jpl pl j might probe the average

184

12 Appendices

action or flowing action Ck ½u [13].) The microscopic action S is connected to C by a renormalization group flow equation [13], however it is in most cases not possible to construct S from the knowledge of C: Typically many different microscopic actions S lead to the same effective action C: It may therefore often be possible that a microscopic action S exists that is consistent with experiments and allows for a positive probability v[/, Jv]. Finally we comment on the general properties of v[/, Jv]. Since it is defined as a functional integral over a (local) complex action one expects that v[/, Jv] is local to a similar degree as the the effective action C½u or the Schwinger functional W[J] defined in (A.9). For general non-Gaussian microscopic actions S½u the functional v[/, Jv] may be quite complicated and not necessarily local in the sense that it can be written in the form R  L v½/; Jv  ¼ e x v ðA:31Þ where Lv is a local ‘‘Lagrange density’’ that depends only on /(x), Jv(x) and derivatives thereof at the space–time point x. Since v[/, Jv] is similarly defined as the effective action C½u or the Schwinger functional W[J] we expect that it respects the same symmetries as the microscopic action S½u when no anomalies are present. For example, when S½u is invariant under some U(1) symmetry transformation u ! eia u; we expect that v[/, Jv] has a corresponding symmetry under the transformation / ! eia /;

Jv ! eia Jv :

ðA:32Þ

12.1.2 Correlation Functions from Functional Probabilities In this section we use the expression for Z~ in (A.21) to derive functional integral representations of some correlation functions. In the following we denote by hi the ‘‘expectation value’’ in the quantum field theoretic sense, e.g. for an operator A½u Z 1 hA½ui ¼ ðA:33Þ DueiS½u A½u: Z (A.33) In contrast, we use hhii to denote the expectation value with respect to the functional integral over /, i.e. Z 1 1  2 2 D/e / ðP þ Þ/ A½/; ðA:34Þ hhA½/ii ¼ ~ Z or more general hhA½/ii ¼

1 Z~

Z D/v½/ A½/:

ðA:35Þ

12.1

Appendix A: Some Ideas on Functional Integration and Probability

185

For the discussion of the correlation functions it is useful to express Z~ in (A.21) again in terms of J1 and J2. Using (A.17) we find Z  ~ 1 ; J2  ¼ D/e1 / ðP2 þ2 Þ/ Z½J e

p1ffi fJ1 ðPiÞ/þ/ ðPiÞJ1 g

e

p1ffi fJ2 ðPþiÞ/þ/ ðPþiÞJ2 g

2

ðA:36Þ

2

 e2fJ1 J1 þJ1 J2 þJ2 J1 þJ2 J2 g : 1









We start with the modulus square of the quantum field theoretic one-point function (no summation convention used in the following) 1 d d ~ Z  Z~ dðJ1 Þa dðJ2 Þa Z 1 1  2 2 D/e / ðP þ Þ/ ¼ Z~ " # 1 X1  ðP  iÞab /b /c ðP þ iÞca  daa  2 b;c  " # 1 X1  ðP  iÞab hh/b /c iiðP þ iÞca  daa ¼ 2 b;c 

jh/a ij2 ¼

ðA:37Þ

¼ 0: In the last line of (A.37) we used the standard property of the Gaussian distribution (A.34) hh/b /c ii ¼ ðP2 þ 2 Þ1 bc :

ðA:38Þ

Next we turn to the two-point function or ‘‘transition probability’’ 1 d d d d ~ Z½J1 ; J2   Z~ dðJ1 Þa dðJ2 Þa dðJ1 Þb dðJ2 Þb Z 1 1  2 2 D/e / ðP þ Þ/ ¼ Z~ " # 1 X1  ðP  iÞag /g /c ðP þ iÞca  daa  2 g;c  " # 1 X1  ðP þ iÞbj /j /k ðP  iÞkb  dbb  2 j;k    ¼ hqa qb i  hhqa iihhqb ii:

qab ¼

ðA:39Þ

186

12 Appendices

This expression is the (connected) two-point correlation function of the operator qa ¼

1 X ðP  iÞag /g /c ðP þ iÞca 22 c;g

ðA:40Þ

with respect to averaging over the possible field configurations /(x). Note that qa is real and positive for all field configurations /. Indeed, we can write with Py ¼ P 2   1 X  qa ¼ 2  ðP  iÞag /g   2  g

ðA:41Þ

showing this more explicit. The multiplicative normalization of q is somewhat arbitrary and could be changed by rescaling the fields according to / ! /0 ¼ c/: Note that for on-shell modes with / P2 / ¼ 0 the operator q reads 1 qa ¼ /a /a : 2

ðA:42Þ

Although the description of qab as a connected correlation function of the operators qa and qb is appealing, its meaning as a transition probability is not yet completely clear. In a typical experiment one asks for the probability to find a particle both at the space-time point y ¼ ðy0 ; ~Þ y and at the space–time point x ¼ ðx0 ; ~Þ: x We denote the probability for this by pðx \ yÞ: Quite generally, one would calculate this quantity as a sum over all field configurations / weighted by the product p½/pðxj/pðyj/:

ðA:43Þ

Here p(x|/] gives the probability for the event ‘‘particle measured at x’’ under the condition that the field configuration / is realized. The expression p[/] is the probability for the field configuration /. In combination, we find Z pðx \ yÞ ¼ D/p½/pðxj/pðyj/: ðA:44Þ Let us now compare this to our expression for qab in (A.39). If we identify a ¼ x ¼ ðx0 ; ~Þ x and b ¼ y ¼ ðy0 ; ~Þ y and neglect for the moment the second term in the last line of (A.39), we can write Z qðx; yÞ ¼ D/v½/qðxÞqðyÞ: ðA:45Þ The expressions for pðx \ yÞ and q(x, y) are proportional when the probability for the field configuration / is p½/  v½/

ðA:46Þ

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Appendix A: Some Ideas on Functional Integration and Probability

187

and the probability to find a particle at x ¼ ðx0 ; ~Þ x for the field configuration / is given by pðxj/  qðxÞ:

ðA:47Þ

The subtraction of the term hhqa iihhqb ii in (A.39) provides for the two events ‘‘particle measured at y’’ and ‘‘particle measured at x’’ not to be in a coincidence. Instead, there has to be a ‘‘causal connection’’ between them. Only in that case would we speak of ‘‘two measurements on the same particle’’. Moreover, fluctuations at different space–time points that are uncorrelated would not show the characteristics of particles at all. Let us assume for definiteness that we use a cloud chamber as a particle detector. The vapor would only condense if neighboring points in space are stimulated during a small but nonzero period of time. Stimulations at random points in space-time would not lead to the detection of a particle. The disconnected part of the two point function hhqa iihhqb ii should therefore be seen as part of the nontrivial vacuum structure in quantum field theory. To end this subsection let us comment of the general, not necessary Gaussian case. We can obtain the quantum field theoretic one-point function from jh/a ij2 ¼

1 d d ~  Þ dðJ Þ Z ~ dðJ Z 2 a 1 a

! 1 d d ¼ þ 2Z~ dðJ/ Þa dðJv Þa   d d ~ / ; Jv : Z½J   þ dðJ/ Þa dðJv Þa

ðA:48Þ

With (A.18) this gives jh/a ij2 ¼

Z 1 D/ 2Z~ " 

/a /a

# d d þ v½/; Jv : dðJv Þa dðJv Þa

ðA:49Þ

Here we used that v[/, Jv] and jh/a ij2 have to be real. The general expression for the two-point function qab is somewhat more complicated, but straightforward to obtain in an analogous way as the calculations above.

12.1.3 Conservation Laws for On-Shell Excitations Although particles are created and annihilated in quantum field theory, these processes are constraint by several conservation laws. For example, in quantum electrodynamics, the electric charge is a conserved quantum number. Electrons

188

12 Appendices

and positrons can only be created in pairs such that the total charge remains constant. In a formalism where particles are described as excitations of fields, one must show that these excitations fulfill the usual conservation constraints. In quantum field theory, conserved quantities such as charge or also energy are associated to a continuous symmetry via Noether’s theorem. However, only the combination of a symmetry together with some field equation leads to a conservation law. For example, for a field that satisfies the on-shell condition P/ ¼ ðol ol  m2 Þ/ ¼ 0

ðA:50Þ

one can easily show that the current jl ¼ iðol / Þ/  i/ ðol /Þ

ðA:51Þ

is conserved, i.e. ol jl ¼ 0: This current is directly linked to the symmetry of the action Z S½u ¼ u ðol ol  m2 Þu ðA:52Þ x

under global U(1) transformations u ! eia u; u ! eia u : As discussed in the last section, the functional probability v[/] is invariant under the same symmetries as the microscopic action S½u if no anomalies are present. This implies that there should be conservation laws associated with these symmetries for on-shell excitations, that fulfill a field equation as (A.25). We emphasize again that e.g. the current in (A.51) is not conserved for general field configurations with P/ 6¼ 0: However, if particles correspond to on-shell field excitations, the usual conservation laws are indeed fulfilled.

12.1.4 Conclusions In this appendix we discussed a (quasi-) probability representation of quantum field theory based on the functional integral. We showed for a Gaussian theory of bosonic fields that the functional integral can be reordered such that an interpretation in terms of real and positive probabilities for field configurations (‘‘functional probabilities’’) is possible. Our formalism is also applicable to the more general case of non-Gaussian microscopic actions where it may be necessary to work also with negative (quasi-) probabilities. We believe that a description using only positive probabilities is possible in many cases. However, it is not excluded that for some physical theories negative probabilities are needed. This would be highly interesting and demonstrate—once again—the extraordinariness of quantum theory. In any case the (quasi-) probability representation developed here might be useful as a theoretical tool, for example in studies of non-equilibrium quantum field dynamics.

12.1

Appendix A: Some Ideas on Functional Integration and Probability

189

The concept of functional probabilities addresses both classical field configurations and particles. The former are described by a nonzero expectation value hh/ii while particles correspond to on-shell excitations, described by the connected two-point function hh//ii  hh/iihh/ii: For quadratic microscopic actions as in (A.4) the functional probability is local ((A.22)). This does no longer have to be the case once interactions are included. For example in a perturbation theory for weak interactions it should be possible to derive explicit expressions beyond the Gaussian case. Higher order correlation functions can then be studied which might shed more light on the question of locality. Interesting features of quantum mechanics as entanglement and the implications of Bells inequalities [14] can then be studied in this framework.

12.2 Appendix B: Technical Additions 12.2.1 Flow of the Effective Potential for Bose Gas In this appendix we derive a flow equation for the effective potential for a Bose gas. We use the truncation presented at the beginning of Sect. 7.1 and specialize at a later stage to the more simple truncation in (8.3). We derive the flow equation for the effective potential by evaluating the flow equation for the average action (2.20) pffiffiffi for constant fields. Inserting a real constant field /ðxÞ ¼ q one finds for U ¼ Ck =X the flow at fixed q ot Uðq; lÞ ¼ gqU 0 þ fðq; lÞ; fðq; lÞ ¼ T

X

Z1 2vd

n 2

0

ðB:1Þ

ð2k  gðk  p  m2 Þ þ ot m2 Þ h2 x2n

2

dp pd1 hðk2  p2  m2 Þ

2

g1 þ g2 þ 2ðV1 þ qV10 Þx2n : þ ðg1 þ ðV1 þ 2qV10 Þx2n Þðg2 þ V1 x2n Þ

Here, d is the number of spatial dimensions and we use the abbreviations g1 ¼ k2  m2 þ ðZ2  1 þ 2qZ20 Þp2  ðV3 þ 2qV30 Þp4 þ U 0 þ 2qU 00 ; g2 ¼ k2  m2 þ ðZ2  1Þp2  V3 p4 þ U 0 ; h ¼ Z1 þ qZ10  ðV2 þ qV20 Þp2 ; xn ¼ 2pTn; vd ¼ ð2dþ1 pd=2 Cðd=2ÞÞ1 :

ðB:2Þ

190

12 Appendices

We dropped the arguments (q, l) at several places on the right hand side. Primes denote derivatives with respect to q. In the phase with spontaneous symmetry breaking, we have m2 ¼ ot m2 ¼ 0: The Matsubara sums over n can be carried out by virtue of the formulas 1 X

1

n¼1

an4 þ bn2 þ c

p ¼ pffiffiffiffiffi d 2c pffiffiffiffiffiffiffiffiffiffi  b þ dcoth

rffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffi !! pffiffiffiffiffiffiffiffiffiffi bd bþd p  b  dcoth p ; 2a 2a

1 X

n2 p ¼ pffiffiffiffiffi 4 2 an þ bn þ c d 2a n¼1 rffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffi !! pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi bþd bd  b þ dcoth p  b  d coth p ; 2a 2a 

ðB:3Þ with d ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2  4ac: This brings us to Z pffiffiffiffiffiffiffiffiffiffi 2 2 k m

dp pd1

fðq; lÞ ¼ 2vd

0

1  ð2k2  gðk2  p2  m2 Þ þ ot m2 Þpffiffiffi 8D  pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E BD  B þ D pffiffiffiffi  2 B  D coth pffiffiffiffiffiffi 8AT C   pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E BþD þ 2 B þ D  B  D pffiffiffiffi coth pffiffiffiffiffiffi ; C 8AT

ðB:4Þ

where we introduced A ¼ V1 ðV1 þ 2qV10 Þ; B ¼ h2 þ g1 V1 þ g2 ðV1 þ 2qV10 Þ; C ¼ g1 g2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ B2  4AC;

ðB:5Þ

E ¼ g1 þ g2 : In our simple truncation with S ¼ Z1 þ Z10 q0 ; V ¼ V1 ; Z20 ¼ V2 ¼ V3 ¼ V10 ¼ V20 ¼ V30 ¼ 0; and at l = l0, the integrand in (B.4) becomes mostly independent of the spatial momentum. The integral can than be carried out and we find

12.2

Appendix B: Technical Additions

 fðq; l0 Þ ¼ 1 

g dþ2

191

pffiffiffi 2v d dD

pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E BD  B þ D pffiffiffiffi  2 B  D coth pffiffiffiffiffiffi 8AT C   pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E BþD þ 2 B þ D  B  D pffiffiffiffi coth pffiffiffiffiffiffi ; C 8AT 

ðB:6Þ

with A ¼ V 2; B ¼ S2 þ 2Vðk2 þ U 0 þ qU 00 Þ; C ¼ ðk2 þ U 0 þ 2qU 00 Þðk2 þ U 0 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ B2  4AC;

ðB:7Þ

E ¼ 2ðk2 þ U 0 þ qU 00 Þ: That the momentum integral can be performed analytically is a nice feature of the cutoff (5.4). The limit T ! 0 is obtained by substituting the coth functions with unity. The flow of the effective potential contains a subtlety that can be seen in the limit Vi ! 0ði ¼ 1; 2; 3Þ; where we find pffiffiffiffiffiffiffiffiffiffi Zk2 m2 0 ot Uðq; lÞ ¼ gqU þ 2vd dp pd1 0

 ð2k2  gðk2  p2  m2 Þ þ ot m2 Þ   pffiffiffiffiffiffiffiffiffi  g1 g2 g1 þ g2 1  pffiffiffiffiffiffiffiffiffi coth pffiffiffi  þ : h 2h g1 g2 2 hT

ðB:8Þ

The term 1/h in the last line is not present if V1 is set to zero from the outset. If Z1 is independent of q, this term is independent of q and gives only an overall shift of the effective potential.

12.2.2 Flow of the Effective Potential for BCS–BEC Crossover Now we come to the flow of the effective potential for the BCS–BEC crossover model. Again, we evaluate the flow equation (2.20) with a field u that is constant ~; sÞ ¼ u: The expectation value of the in space and in (Matsubara-) time uðx fermionic field w vanishes due to its Grassmann property. Using the truncation in (8.26) we find the flow equation for the potential

192

12 Appendices

ok Uk jq ¼

1 2

Z 



 2 ðGku Þ11 þ ðGku Þ22 A1 u ok Au k rku

~ q0 ;q



 ðGkw Þ13 þ ðGkw Þ24  ðGkw Þ31  ðGkw Þ42

 2  ok k rkw

ðB:9Þ

The dimensionless function rku depends on y ¼ ~ q2 =ð2k2 Þ while rkw depends on 2 2 ~  l0 Þ=k : The propagators Gku and Gkw that appear in (B.9) are modified z ¼ ðq by the presence of the ultraviolett regulator ! 0 k2 rku 1 1 Gku ¼ Gu þ 0 k2 rku ðB:10Þ ! 2 r 0 k kw 1 G1 : kw ¼ Gw þ k2 rkw 0 We can now perform the summation over the Matsubara frequencies q0 = 2pTn for the bosons and q0 = 2pT (n + 1/2) for the fermions. The integration over ~ q is performed quite generally in d spatial dimensions. The result can be expressed in terms of the dimensionless variables w1 ¼

Uk0 ; k2

w2 ¼

Uk0 þ 2qUk00 ; k2

h2u q l ~ ¼ 20 ; ; l k4 k T Zu T~ ¼ 2 ; Su ¼ ; k Au

w3 ¼

D~ l¼

l  l0 ; k2

and the anomalous dimension gA u ¼ 

kok Au : Au

ðB:11Þ

The flow of the effective potential reads then after the variable change  ! q ¼ Au q  q   pffiffiffikdþ2 vd 2 ð0Þ gA u s B 1 kok Uk ¼ gAu qUk0 þ 8 2 dþ2 dSu ðB:12Þ kdþ2 vd ð0Þ lð~ lÞsF : 8 d Here, the vd is proportional to the surface of the d-dimensional unit sphere, which dþ1 d=2 p Cðd=2Þ: In particular one has v3 = 1/(8p2). is (2p)d 4vd with v1 d ¼2

12.2

Appendix B: Technical Additions

193

ð0Þ ð0Þ ð0Þ ~ Su ; gA Þ and sð0Þ ~; The threshold functions sB ¼ sB ðw1 ; w2 ; T; F ¼ sF ðw3 ; l u ~ D~ l; TÞ as well as the function lð~ lÞ used in (B.12) depend on the choice of the infrared regulator functions rku and rkw. They describe the decoupling of modes when the effective ‘‘masses’’ wj or ~ l get large. The threshold functions for the bosonic fluctuations reads

 1

2gAu dþ2

 Z1   d ð0Þ 0  gAu rku sB ¼ d dy y21 rku  yrku 0 1 2ðw1 þ w2 Þ þ y þ rku  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w1 þ y þ rku w2 þ y þ rku   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  þ NB ð w1 þ y þ rru w2 þ y þ rku =Su Þ 2

ðB:13Þ

with the Bose function NB ðÞ ¼

1 : 1

ðB:14Þ

e=T~

ð0Þ

We take it as a condition for the cutoff function rku that sB ¼ 1 for w1 ¼ w2 ¼ T~ ¼ 0: For the calculations below we choose the cutoff function (see Chap. 5) rku ðyÞ ¼ ð1  yÞhð1  yÞ which gives the particular simple expression rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ w1 1 þ w2 ð0Þ sB ¼ þ 1 þ w2 1 þ w1   pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  þ NB ð 1 þ w1 1 þ w2 =Su Þ : 2

ðB:15Þ

ðB:16Þ

The threshold function for the fermionic fluctuations is obtained similar. For a generic cutoff that addresses the spatial momentum, it reads Z 1   d ð0Þ 0 ~Þ21 rkw  zrkw dzðz þ l lð~ lÞsF ¼ d ~ l

ðz þ rkw  D~ lÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lÞ2 w3 þ ðz þ rkw  D~  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 lÞ2 :   NF w3 þ ðz þ rkw  D~ 2 Here we employ the Fermi function

ðB:17Þ

194

12 Appendices

NF ðÞ ¼

1 : þ1

ðB:18Þ

e=T~

Note that for a generic cutoff the right hand side of (B.17) does not necessarily ð0Þ factorize. In that case one migth work with a threshold function sF that also ~: depends on l ð0Þ Again it is a condition for possible cutoff functions rkw to fulfill SF;Q ¼ 1 for ~ ¼ D~ w3 ¼ l l ¼ 0: We choose the form rkw ¼ ðsignðzÞ  zÞhð1  jzjÞ:

ðB:19Þ

This implies for l = l0 and therefore D~ l ¼ 0 the simple form  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ð0Þ sF;Q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  NF 1 þ w3 1 þ w3 2 and lð~ lÞ ¼ hð~ l þ 1Þð~ l þ 1Þd=2  hð~ l  1Þð~ l  1Þd=2 : In the limit NB = NF = Taking a

ðB:20Þ

T~ ¼ T=k2 ! 0 the thermal contributions to the flow of Uk vanish, 0. derivative with respect to q on both sides of (B.12) we obtain   pffiffiffikd vd 2 0 0 00 g kok Uk ¼ gAu ðUk þ qUk Þ  8 2 1 d þ 2 Au dSu h i ð1;0Þ ð3Þ ð0;1Þ ðB:21Þ  Uk00 sB þ ð3Uk00 þ 2qUk ÞsB þ8

kd2 vd 2 ð1Þ hu lð~ lÞsF : d

Here we introduced the derivatives of the threshold functions ð1;0Þ

sB

ð0;1Þ

sB

ð1Þ

sF

o ð0Þ ¼ s ow1 B o ð0Þ ¼ s ow2 B o ð0Þ ¼ s : ow3 F

ðB:22Þ

We may devide these into contributions from quantum and thermal fluctuations ð1;0Þ

¼ ðw2  w1 ÞsB;Q þ sB;T ;

ð0;1Þ

¼ ðw2  w1 ÞsB;Q þ sB;T ;

sB sB

ð1Þ

ð1Þ

ð1Þ

sF ¼ sF;Q þ sF;T :

ð1;0Þ

ð1;0Þ

ð0;1Þ

ð0;1Þ

ðB:23Þ

12.2

Appendix B: Technical Additions

195 ð0;1Þ

ð1;0Þ

ð1Þ

For T~ ! 0 the thermal contribution vanishes sB;T ¼ sB;T ¼ sF;T ¼ 0: We (0,1) extracted a factor (w2 - w1) from the threshold functions s(1,0) B,Q and sB,Q to make explicit that these contributions vanish for w1 = w2 which holds for q = 0. For our choice of the regulator functions rk,w and rku we find the explicit expressions 1

ð1;0Þ

sB;Q ¼

3=2

4ð1 þ w1 Þ

ð1 þ w2 Þ1=2 1

ð0;1Þ

sB;Q ¼ 

4ð1 þ w1 Þ

1=2

1

ð1Þ

sF;Q ¼

;

2ð1 þ w3 Þ3=2

ð1 þ w2 Þ3=2

;

:

For T~ [ 0 the thermal fluctuations lead to the additional contributions from the bosons ð1;0Þ

ð1;0Þ

sB;T ¼ 2ðw2  w1 ÞsB;Q NB pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ ð0;1Þ sB;T

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð1 þ w1 Þð1 þ w2 Þ=Su

1 þ w2 ð0Þ sB;Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NB0 1 þ w 1 Su

¼ 2ðw2 

ð1;0Þ w1 ÞsB;Q NB

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð1 þ w1 Þð1 þ w2 Þ=Su ;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð1 þ w1 Þð1 þ w2 Þ=Su

ðB:24Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ w1 ð0Þ ð1 þ w1 Þð1 þ w2 Þ=Su : þ sB;Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NB0 1 þ w 2 Su Here, we use the derivative of the Bose function o NðÞ: o

NB0 ðÞ ¼

ðB:25Þ

Similarly, the fermionic part of the thermal contribution reads ð1Þ

ð1Þ

sF;T ¼ 2sF;Q NF

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ w3

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð0Þ  sF;Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NF0 1 þ w3 ; 1 þ w3

ðB:26Þ

with the derivative of the Fermi function NF0 ðÞ ¼

o NF ðÞ: o

ðB:27Þ

196

12 Appendices

12.2.3 Hierarchy of Flow Equations in Vacuum In this appendix we sketch the proof of the theorem mentioned at the beginning of Chap. 9. The theorem is about a hierarchy of flow equations in the vacuum limit n ! 0; T ! 0: In this limit the effective action describes few-body physics which might also be described using the formalism of quantum mechanics. We use a notation where the field / is a spinor that may contain in general both fermionic and bosonic degrees of freedom. We demand that the microscopic model is invariant under the global U(1) transformation / ! eia /;

ðB:28Þ

where the charge a is the same for all components of /. To fulfill the condition above it will sometimes be necessary to ‘‘integrate out’’ composite fields. The second premise is that the microscopic propagator for the field u is of the nonrelativistic form iq0 þ ~ q2 =ð2MÞ þ m  l: Here we work with a imaginary-time (or Matsubara) frequency q0. The mass M and the gap parameter m might be different for the different components of /. One can then proof the following Theorem The flow equation for the n-point function Gn is independent of corðnÞ relation functions Gm with order m [ n. Here we use a notation where Gn ¼ Ck ð2Þ

for n [ 2 and G2 ¼ ðCk þ Rk Þ1 is the regularized propagator. For the proof we use a basis with independent variables / and /*. We expand the flowing action in orders of field Ck ½/ ¼ Ck ð0; 0Þ þ Ck ð0; 1Þ þ Ck ð0; 2Þ þ    þ Ck ð1; 0Þ þ Ck ð1; 1Þ þ Ck ð1; 2Þ þ    þ Ck ð2; 0Þ þ Ck ð2; 1Þ þ Ck ð2; 2Þ þ    þ 

ðB:29Þ

Here we denote by Ck ði; jÞ a term that is of order i in the conjugate field /* and of order j in the field /. We choose the cutoff function to be invariant under the global U(1) transformation in (B.28). Since we expect no anomalies, the flowing action Ck ½/ is also invariant. This implies that all contributions Ck ði; jÞ vanish except for those with i = j, Ck ½/ ¼ Ck ð0; 0Þ þ Ck ð1; 1Þ þ Ck ð2; 2Þ þ   

ðB:30Þ

For simplicity we write this also as Ck ½/ ¼ Ck ð0Þ þ Ck ð2Þ þ Ck ð4Þ þ   

ðB:31Þ

The term Ck ðnÞ contains the information of the n-point function Gn. Graphically, Gn is represented by a vertex with n external lines. Half of these lines represent incoming particles and the other n/2 outgoing particles. We show this

12.2

Appendix B: Technical Additions

197

Fig. B.1 Graphical representation of the n-point function Gn

Fig. B.2 Flow equation of the n-point function Gn. On the right hand side we show only the contribution involving the correlation function Gn+2. Additional contributions to the flow from lower order function Gj with j  n are not shown

schematically in Fig. B.1. Since the flow equation has a one-loop structure, the flow of a term Ck ðnÞ (or Gn) can only have contributions from Gn+2 if we close one line as shown in Fig. B.2. There can be no contribution from Gj with j [ n + 2 since we would have to contract a second line which would lead to a two-loop diagram. We have to show that the loop shown on the right hand side in Fig. B.2, i.e. the contribution of Gn+2, vanishes. The argument uses the dependence of the n-point function on the frequency q0. With q0 we denote the imaginary (or Matsubara) frequency of an incoming particle (line with arrow pointing inwards in Figs. B.1 and B.2). One of the outgoing particles has frequency q0 þ Dq0 : For the case n = 2 q0 is just the frequency argument of the regularized propagator. We need for our proof that all poles (or cuts) of the n-point function Gn as a function of q0 are in the upper half of the complex plane. For the microscopic propagator (which equals the full propagator in vacuum according to our theorem) this is indeed the case. The two-point function G2 has the form G2 ¼

1 : iq0 þ ~ q2 þ m  l þ Rk ðq0 ; ~Þ q

ðB:32Þ

With a regulator of the simple form Rk = k2 the frequency pole is at ~2 þ m  l þ k2 Þ: q0 ¼ iðq

ðB:33Þ

For m - l [ 0 this is always in the upper half of the complex plane. For more q we take it as a condition that this feature is not general cutoff functions Rk ðq0 ; ~Þ affected.

198

12 Appendices

For k ? 0 and after analytic continuation q0 ! ix; the pole of G2 determines the dispersion relation, e.g. x ¼~ q2 þ m  l:

ðB:34Þ

This describes the energy as a function of momentum. As usually in nonrelativistic quantum theory there is a certain ambiguity in the absolute scale for energy. Using the semilocal U(1) symmetry described in Sect. 7.1, we can (formally) shift the chemical potential corresponding to a shift in the energy scale. The origin of this energy shift ambiguity is in the transition from a relativistic (inverse) propagator x2 þ ~ q2 c2 þ M 2 c4  i

ðB:35Þ

to a nonrelativistic one. In (B.35) the small imaginary part is to enforce the correct frequency integration contour (Feynman prescription). We write (B.35) as    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ M 2 c4 þ ~ ðB:36Þ q2 c2  i x þ M 2 c4 þ ~ q2 c2  i : The first bracket leads to a pole in the propagator that corresponds to antiparticles (negative frequency) while the second bracket describes particles (positive frequency). Close to the point where the second bracket vanishes, one can expand the square root and approximate the first bracket in (B.36) by a constant. One obtains 2Mc2 ðx þ Mc2 þ ~ q2 =ð2MÞ  iÞ:

ðB:37Þ

Due to the i term the frequency pole of the propagator is always in the lower half of the complex plane. The terms Mc2 in (B.37) can now be absorbed into a redefinition of the fields. This includes a simple rescaling and a shift in frequency, similar to the semilocal U(1) transformation described in Sect. 7.1. However, this transformation is not unique and the origin of the energy scale remains undetermined. One can always perform the frequency shift such that all states of a number of particles have positive energies. This holds within the nonrelativistic theory where all energies are small compared to Mc2. The transition to imaginary (or Matsubara-) frequencies x þ i ! iq0 involves an additional relabeling q0 ! q0 such that we arrive at the form of the propagator in (B.32). Now we consider the general n-point function Gn. A pole (or cut) at frequency q0 corresponds now to a state of n/2 particles with energy x = -iq0 of the considered particle. The energy per particle of such a state might be above or below the energy of n/2 free particles. In the latter case one speaks of a bound state, in the former of a resonance. In any case we can use the freedom to choose the energy scale to obtain q0 [ 0. In other words we set the energy scale such that all relevant states have positive energy. We can now come back to the proof of our theorem. From the above discussion it follows for the loop integral corresponding to Fig. B.2 that all poles and cuts are in the upper half of the complex plane. We can close the contour in the lower half, implying that the loop integral vanishes. This closes the proof.

References

199

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