The Recursion Method (Lecture Notes in Physics monographs)

Lecture Notes in Physics New Series m: Monographs Editorial Board H.Araki Research Institute for Mathematical Sciences Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan E. Brezin Ecole Normale Superieure, Departement de Physique 24, rue Lhomond, F-75231 Paris Cedex 05, France J. Ehlers Max-Planck-Institut fUr Physik und Astrophysik, Institut fUr Astrophysik Karl-Schwarzschild-Strasse 1, D-85748 Garching, FRG U. Frisch Observatoire de Nice B. P. 229, F-06304 Nice Cedex 4, France

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v. S. Viswanath

Gerhard Muller

The Recursion Method Application to Many-Body Dynamics

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors V. S Viswanath Gerhard Muller Department of Physics The University of Rhode Island Kingston, RI 02881-0817, USA Email: [email protected]

ISBN 3-540-58319-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58319-X Springer-Verlag New York Berlin Heidelberg CIP data applied for. 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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 -Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heide1berg 1994 Printed in Germany

Typesetting: Camera-ready by the authors SPIN: 10080272 55/3140-543210 - Printed on acid-free paper

PREFACE AND ACKNOWLEDGMENTS This monograph is the product of our exploration of the recursion method as a general calculational technique for the study of quantum and classical many-body dynamics - a method of considerable versatility with an enormous potential for refinement and further development. What had accumulated in our minds and on our desks as a collection of facts about the recursion method and as an assortment of new results for a variety of applications, was not originally intended to be published in any form other than that of a series of research papers. However, our work on projects involving the recursion method took on a new dimension when one of us (GM) was invited to present a lecture series on this subject in Switzerland and in Germany during the fall of 1991. The first lecture series was given at the Ecole Polytechnique Federale in Lausanne, as part of the Troisieme Cycle de Physique des Universites Romandes. The second lecture series, given at the Institut fUr Physik der Universitat Dortmund, was sponsored by the Graduiertenkolleg Festkorperspektroskopie. Support from both institutions is gratefully acknowledged. The book may be divided into two parts of roughly equal size. The first part (Chapters 1 to 9) describes the different representations and formulations of the recursion method and its accessories, including various techniques of continuedfraction analysis, and recursive algorithms for the computation of ground-state wave functions. It contains numerous illustrations and simple applications. The second part (Chapters 10 and 11) is intended to illustrate the usefulness and importance of the recursion method in relation to other analytical and computational techniques as applied to two areas of current research in many-body dynamics. It reports recent results which we have obtained in a series of collaborative projects with several colleagues and students. We owe many thanks to H. Beck of the University of Neuchatel for suggesting and organizing the first lecture series, and to his collaborators at the time, M. B. Cibils (now in Lausanne) and Y. Cuche (now in Florence), for their kind assistance in many respects. No fewer thanks go to H. Keiter, U. Brandt, and W. Weber, for their kind hospitality at the University of Dortmund and, especially, to J. Stolze (now in Bayreuth), for organizing the second lecture series. Substantial parts of the manuscript were outlined and drafted at the Institut fUr Theoretische Physik at the University of Basel, where GM spent the larger part of his sabbatical leave from URI in the stimulating and pleasant environment around H. Thomas and his collaborators. Thank you very much for the kind hospitality! The research done in Basel was supported in part by the Swiss National Science Foundation and the research done at URI by the US National Science Foundation. Most of the computations were carried out on supercomputers at the National Center for Supercomputing Applications, University of TIlinois at UrbanaChampaign. Very special thanks go to M. H. Lee at the University of Georgia for having kindled our curiosity about the recursion method in the first place, and for his critical reading of the manuscript and his invaluable suggestions for improvements. We are much indebted to M. P. Nightingale, J.-M. Liu, S. Zhang, and Y. Yu of

VI

URI, to H. Leschke of the University of Erlangen-Niimberg and to N. Srivastava of Thinking Machines Corporation, for their contribution to the recent work portrayed in this book, and/or for their feedback from reading parts of the manuscript. Our warmest thanks go to Joachim Stolze of the University of Bayreuth, who came to URI for one year in the fall of 1992, and also to Markus Bohm of the University of Erlangen-Niimberg, who stayed at URI for one month in the spring of 1993. We have enjoyed very rewarding collaborations with these two colleagues and friends on several projects involving the recursion method. These collaborations have produced results of crucial importance for the shaping of the second part of the monograph. It would not have been possible to keep this project on track without the institutional support secured by S. S. Malik, Chairman of the Physics Department at URI, during a turbulent three years. The technical assistance provided by our systems manager, S. Pellegrino, has greatly facilitated the writing of this book and the computational research reported therein. Finally, we would like to express our thanks to the editor, Prof. W. BeiglbOck, for his most valuable advice, and to his coworkers at Springer-Verlag, Ms. S. Landgraf and Ms. B. Reichel-Mayer, for their expert help in the production of the camera-ready manuscript. Kingston, June 1994

V.S. Viswanath Gerhard Muller

CONTENTS INTRODUCTION 1-1 Calculational Techniques in Condensed Matter Theory 1-2 Recursion Method Applied to Many-Body Dynamics 1-3 Fonnalism and Goals 1-4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 6 9

2 LINEAR RESPONSE AND EQUILffiRIUM DYNAMICS . . . . . . . . . .. 11 2-1 Response Function and Generalized Susceptibility 11 2-2 Fluctuation-Dissipation Theorem 13 2-3 Moment Expansion 15 3 LIOUVILLIAN REPRESENTATION 3-1 Quantum Fonnulation . . . . . . . . . .. 3-2 Classical Fonnulation 3-3 Orthogonal Expansion of Dynamical Variables. . . . . . . . . . . .. 3-4 Relaxation Function and Spectral Density 3-5 Recursion Method and Moment Expansion 3-6 Generalized Langevin Equation 3-7 Projection Operator Fonnalism . . . . . . . . . . . . . . . . . . . . . . .. 3-8 Retarded Green's Functions

17 17 18 19 21 22 24 26 29

4 HAMILTONIAN REPRESENTATION 4-1 Orthogonal Expansion of Wave Functions 4-2 Structure Function 4-3 Continued-Fraction Coefficients and Frequency Moments 4-4 Lanczos Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Modified Lanczos Method . . . . . . . . . . . . . . . . . . . . . . 4-6 Conjugate-Gradient Method . . . . . . . . . . . . . . . . . . . . . 4-7 Steepest-Desceht Method . . . . . . . . . . . . . . . . . . . . . . . 4-8 Comparative Perfonnance Test . . . . . . . . . . . . . . . . . . . 4-9 Green's Functions: Spectral and Continued-Fraction Representations

32 32 34 35 37 40 42 43 45

5 GENETIC CODE OF SPECTRAL DENSITIES 5-1 Finite ~k-Sequences 5-2 Spectral Densities with Bounded Support 5-3 Spectral Densities with Bounded Support and a Gap 5-4 Spectral Densities with Unbounded Support 5-5 Spectral Densities with Unbounded Support and a Gap 5-6 Orthogonal Polynomials 6 RECURSION METHOD ILLUSTRATED 6-1 Hannonic Oscillator 6-2 Spin Waves

. . . . .

. . . . .

. . . . .

.. .. .. .. ..

47 51 51 52 56 , 57 60 62 64 64 68

VIII

6-3 Lattice Fennions 69 6-4 Quantum Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 6-5 Classical Spins 74 7 UNIVERSALITY CLASSES OF DYNAMICAL BEHAVIOR 7-1 Dynamics of the Equivalent-Neighbor XYZ Model 7-2 Fluctuation Functions and Spectral Densities for the XXZ Case 7-3 Recursion Method Applied to Equivalent-Neighbor Spin Models 7-4 Quantum Equivalent-Neighbor XYZ Model 7-5 Prototype Universality Classes 7-6 Two-Sublattice Spin Model with Long-Range Interaction 7-7 Many-Body Systems with Short-Range Interaction

76 76 78 80 82 82 87 90

8 TERMINATION OF CONTINUED FRACTIONS: ATTEMPTS AT DAMAGE CONTROL 93 8-1 Cut-Off Tennination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94 8-2 n-Pole Approximation 97 8-3 Pole Locations and Spectral Densities 98 8-4 Memory Functions and Fluctuation Functions 102 8-5 Moving Beyond Truncation 104 9 RECONSTRUCTION OF SPECTRAL DENSITIES FROM INCOMPLETE CONTINUED FRACTIONS 9-1 Model Tenninators from Model Spectral Densities 9-2 Square-Root Tenninator . . . . . . . . . . . . . . . . . . . . . . . . . .. 9-3 Rectangular Tenninator 9-4 Endpoint Singularities 9-5 ~-Tenninator................................... 9-6 Gap Tenninators 9-7 Infrared Singularities in Spectral Densities with Bounded Support 9-8 Spectral Densities with a &-Function Central Peak. . . . . . . .. 9-9 Tenninator with Matching Infrared Singularity 9-10 Compact a-Tenninator 9-11 Gaussian Tenninator . . .. 9-12 Infrared Singularities in Spectral Densities with Unbounded Support 9-13 Unbounded a-Terminator. . . . . . . . . . . . . . . . . . . . . . . . .. 9-14 Split-Gaussian Terminator 10

109 109 112 114 116 118 118 122 124 127 129 131 133 133 134

TRANSPORT OF SPIN FLUCTUATIONS AT HIGH TEMPERATURE 136 10-1 Generic High-Temperature Spin Dynamics 136 10-2 10 s=1/2 XYZ Model on Semi-Infinite Chain . . . . . . . . . .. 138

IX

10-3 Spin-1I2 XX Model: Neither Spin Diffusion nor Exponential Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10-4 Boundary Effects: Buildup of an Infrared Divergence 10-5 Boundary Effects: Crossover Between Growth Rates A.=O and A.=l 10-6 Spin-1I2 XXZ Model . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10-7 From Gaussian Decay to Exponential Decay , 10-8 Analysis of L\k-Sequences with Growth Rates near A.=1 10-9 From Exponential Relaxation to Diffusive Long-Time Tails 10-10 Sustained Power-Law Decay. . . . . . . . . . . . . . . . . . . . . .. 10-11 From Ballistic to Diffusive Transport of Spin Fluctuations .. 10-12 Boundary-Spin Spectral Densities 10-13 Spectral Signature of Quantum Spin Diffusion in Dimensions d=I,2,3 10-14 Spin Diffusion in the Classical Heisenberg Model. . . . . . .. 10-15 Is Classical Spin Diffusion Anomalous? . . . . . . . . . . . . . .. 10-16 Exponential DecayVersus Long-Time Tails 10-17 Anomalous Exponent or Non-Asymptotic Effect? 10-18 Experimental Evidence for Anomalous Spin Diffusion . . . .. 10-19 Two Kinds of Computational Errors 10-20 q-Dependent Correlation Function 10-21 Power Law Long-Time Tail with Logarithmic Correction .. 10-22 Effective Exponent 10-23 Effect of Exchange Inhomogeneities 11 QUANTUM SPIN DYNAMICS AT ZERO TEMPERATURE 11-1 ID s=1I2 XY Model with Magnetic Field. . . . . . . . . . . . .. 11-2 Product Ground State of the ID Spin-s XYZ Model with Magnetic Field 11-3 Conditions for the Existence of Linear Spin Waves 11-4 Resonances with Intrinsic Width 11-5 Finite and Infinite Bandwidths. . . . . . . . . . . . . . . . . . . . .. 11-6 Limitations of Single-Mode Picture 11-7 Spin Dynamics at Tc=O Critical Point: Exact Results for the Transverse Ising Model and the XX Model 11-8 Long-Time Asymptotic Expansions 11-9 Structure Functions and Their Singularities . . . . . . . . . . . .. 11-10 Structure Functions Reconstructed by Continued-Fraction Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-11 Dynamic Structure Factors Szz(q,ro)n and Szz(q,ro)xx: Two-Particle Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-12 Dynamic Structure Factor S;xx(q,ro)xx: Continued-Fraction Analysis . . . . . . . . . . . . . . . . . . . . . .. 11-13 ID s=1I2 XYZ Model: Ground State and Excitation Spectrum

138 140 142 145 147 147 149 151 154 155 158 164 164 167 167 168 171 173 175 176 178 183 183 186 187 188 189 191 193 194 196 199 202 204 206

x 11-14 ID s= 112 XXZ Model: Criticality and Long-Range Order , 11-15 Excitation Spectrum of the ID s=1I2 XXZ Ferromagnet: Spin Waves and Bound States 11-16 Excitation Spectrum of the ID s=1I2 XXZ Antiferromagnet: Spinons and Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-17 Equal-Time Correlation Functions 11-18 Dynamic Correlation Functions . . . . . . . . . . . . . . . . . . . .. 11-19 Continuum Approximation (Luttinger Model) 11-20 Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . .. 11-21 Weak-Coupling and Strong-Coupling Regimes 11-22 Infrared Singularities in SIJI.I(1t,ro) and ~Il(ro) . . . . . . . . . . .. 11-23 Reconstruction of Szz(1t,ro) (Weak-Coupling Analysis) 11-24 Reconstruction of Sxx(q,ro) (Strong-Coupling Analysis) 11-25 Strong-Coupling Reconstruction of SU(ro) 11-26 2D s= 1/2 XXZ Antiferromagnet 11-27 Dynamic Structure Factors Sxx(1t,1t,ro) and Szz(1t,1t,ro) . . . . .. 11-28 ID Spin-1 Heisenberg Antiferromagnet with Uniaxial Exchange and Single-Site Anisotropy " 11-29 Dynamically Relevant Excitation Gaps 11-30 Dispersion and Line Shapes BffiLIOGRAPHY

211 212 214 217 218 219 222 225 227 230 234 237 239 241 244 245 248 251

1 INTRODUCTION I-I Calculational Techniques in Condensed Matter Theory The toolbox of an experienced condensed matter theorist is divided into two major compartments, each one crowded with calculational techniques. One of the two compartments bears the label universal tools, the other one precision instruments. The universal-tools compartment contains an assortment of general methods for the calculation of observable quantities of interest in condensed matter physics. Among them are general methods for a particular purpose. General methods for the calculation of dynamic correlation functions which are applicable to arbitrarily selected model systems, for example, belong in that compartment. Also found there are multi-purpose methods with a wide range of applicability. Universal tools mus~ have a certain robustness against conditions that may invalidate their applicability. However, they are not meant to yield exact results on such a wide territory of applications. The general calculational or computational techniques that are commonly used in condensed matter theory may be categorized as follows: o Methods with extrinsic limitations, such as computer simulations, Green's function methods, the recursion method, or finite-size studies. In these methods, the limitations are set by the amount of calculational effort or computational power invested in them. o Methods with intrinsic limitations, such as mean field theory, linear spinwave theory (harmonic approximation), the random-phase approximation (within the framework of Green's function methods), the n-pole approximation (within the framework of the recursion method), have built-in limitations that cannot be overcome within their own respective scope. The precision instruments, stored in the other compartment of the theorist's toolbox, are a collection of special methods. They have been designed for the exact solution of particular problems. The small but precious collection of exactly solved models in statistical mechanics and solid state physics is their main source of origin. Once a challenging problem has been solved by a special method, it is by no means guaranteed that the solution can be reproduced by a general method. Nevertheless, it is usually instructive and illuminating to test the performance of universal tools on problems that have previously been solved by specially designed precision instruments. Many a general method has its root in a special method designed for the exact solution of a specific problem. What makes it a general method are applications to problems of a similar nature, where it is subject to intrinsic or extrinsic limitations. For example, mean field theory may be regarded as a special method for the solution of certain model systems with long-range interaction, and spin-wave theory derives its legitimacy from those special situations in which anharmonicities can be ignored in magnetic excitations. For a thorough study of a broadly defined topic we need universal tools and precision instruments, i.e. general methods with their wide range of applicability and special methods that provide deeper insight for particular circumstances. We must combine systematics in breadth with systematics in depth in order to gain the

2

Chapter 1

best possible understanding of the topic under scrutiny. The use in isolation of (i) general methods with severe intrinsic limitations or (ii) special methods applicable only under highly non-generic circumstances is likely to invite misleading conclusions. Systematics in both directions is key to an understanding of the least accessible territory. Exact solutions are usually out of reach except for particular circumstances and by special methods. The particular circumstances are always describable in terms of a simplification of the problem. There are basically two types of simplifications that may bring an exact solution to within reach: o Simplifications due to a special type of interaction between the degrees of freedom. The free-particle limit or special types of infinite-range interaction are obvious examples. o Simplifications due to a special state of a model system which otherwise exhibits generic behavior. A typical example is the ordered ground state of the Heisenberg ferromagnet. Do these particular, simplifying circumstances translate into an improved performance of applicable general methods as well? The answer depends on the specifics of the general method under consideration. (i) Green's function methods pose the problem of approximating the infinite hierarchy of equations of motion in a controlled and systematic way. That is notoriously difficult even for weakly coupled degrees of freedom. However, in the noninteracting limit, that hierarchy reduces to a closed set of equations, from which the exact solution can readily be extracted. Simplifications due to a special state of the system do not, in general, result in a more tractable hierarchy of equations of motion. The reason is that the state of the system remains unspecified in the hierarchy of equations of motion. (ii) In the recursion method, the properties of generic systems manifest themselves in highly complex patterns exhibited by the sequences of continued-fraction coefficients, as we shall see. For noninteracting degrees of freedom, the amount of simplification in those sequences is comparable to that in the Green's function approach. However, the recursion method is decidedly better equipped to handle situations in which the simplification is due to a special state of the system. The reason is that the specification of the state has its impact on every continuedfraction coefficient as it is evaluated in the recursive calculational procedure.

1-2 Recursion Method Applied to Many-Body Dynamics The link between almost any kind of experiment that probes dynamical properties of condensed matter systems and the calculational techniques used for studying equilibrium dynamics of quantum or classical many-body systems is the linear response theory (see Fig. 1-1). Its core consists of (i) the fluctuation-dissipation theorem in the most general form and (ii) the implications of the causality principle including the Kramers-Kronig dispersion relations.

Section 1-2

3

RECURSION METHOD

Figure 1-1: Linear response theory is the indispensable link between dynamics experiments

in condensed matter physics and the recursion method for applications to many-body dynamics or any other calculational technique for the study of systems in thermal equilibrium.

4

Chapter 1

The recursion method is one of several general methods for the calculation of dynamic correlation functions for model systems in equilibrium states. 1 The purpose of this book is to give a detailed description of the recursion method in its diverse representations and formulations (see Fig. 1-1). The emphasis will be on the presentation of a user's guide for a wide range of applications in many-body dynamics. We hope to convince the reader that the recursion method is a userfriendly yet powerful calculational tool. Technically, the core of the recursion method is the orthogonal expansion of a specific quantity. If the goal is to determine a time-dependent autocorrelation function of the system. It enters the recursion method via the initial condition 1'l'(0» = A lC\)o> for the solution I'l'(t» of the SchrMinger equation. In the Liouvillian representation, the equilibrium state of the system is specified, in general, by a stationary density operator. Depending on the application, that density operator may represent any eigenstate of the system or any suitable equilibrium ensemble. Here the specification of the state enters the recursion method in the form of an inner product that is needed for the execution of the orthogonal expansion of operator A. It is straightforward to transcribe the Liouvillian representation of the recursion method from the quantum formulation to the classical formulation. The dynamical variable A(t) is then a function in phase space. Replacing the quantum Liouvillian operator L u (commutator) by its classical counterpart L cl (Poisson bracket), transforms the iieisenberg equation into Hamilton's equation. The historical roots of the recursion method can be found in research areas as diverse as statistical mechanics, nuclear physics, solid state physics and applied

o

IThe recursion method has been used for other purposes not discussed in this work (Haydock 1980, Pettifor and Weaire 1985, Czycholl and Ponischowski 1988].

Section 1-2

5

mathematics, but for the types of applications emphasized in this work, it is squarely based on the projection operator formalism [Zwanzig 1961, Mori 1965]. Liouvillian representation

i"~ I'l'(t»

cIA = iLA dt

L qu

= .!.[H,

"

Hamiltonian representation

= H!'l'(t»

dt

], L cl = i{H, }

A(t) = eiLtA(O) 00

=L

k=O

I'l'(t»

=

e-iHtlh 1'l'(0» 00

Cit}fk

=:LDk(t)lf,;> k=O

f k+1 = iLfk - ...

lfk+1> = Hlf,;> - ...

f o = A(O)

lfo>

=

1'l'(0»



= A1

Table 1-1: Some characteristic quantities that distinguish the two main representations of

the recursion method. The Liouvillian representation will be further discussed in Chapter 3 and the Hamiltonian representation in Chapter 4.

Carrying out the program of the recursion method in the Liouvillian representation for the time evolution of a dynamical variable A(t) is equivalent to setting up and solving the generalized Langevin equation for that variable. This connection is especially useful for the interpretation of various approximation schemes commonly invoked in applications of the recursion method in terms of more or less familiar phenomenological models of dynamical systems on a contracted level of description (stochastic processes). The Hamiltonian representation of the recursion method grew out of attempts to solve an important problem in solid state physics - the determination of electronic densities of states under special circumstances, (i) situations in which band theory is unreliable, such as in strongly correlated systems, and (ii) situations in which band theory is inapplicable, such as in systems without spatial periodicity because of impurities, vacancies or amorphous structure. The local-environmentapproach to the electronic structure of solids [Haydock, Heine and Kelly 1972] introduced the recursion method to solid state theory as a powerful alternative to band theory.

6

Chapter 1

Expanding the Green's function which determines the desired density of states (or spectral density, in dynamics applications) into a continued fraction is equivalent to the tridiagonalization of the associated model Hamiltonian in a particular orthogonal basis. The latter task had already been a familiar one for many years [Lanczos 1950] and continues to play a significant role in the numerical diagonalization of large matrices. The body of mathematics that bears on the various implementations of the recursion method is quite extensive. Apart from the theory of continued fractions, we should also mention the close links to the classical moment problem and to the theory of orthogonal polynomials. How to reduce practically any problem to one dimension, was the title of the first talk (given by D.C. Mattis) at the 1980 International Conference on Physics in One Dimension. That title refers, of course, to the recursion method as an instrument by means of which a large set of possible problems in condensed matter physics can be reduced to a pseudo-ID problem, known as the chain model associated with the original problem. The chain model, characterized by the tridiagonal Hamiltonian produced by the recursion method, is isomorphic to a chain of local degrees of freedom with nearest-neighbor coupling only, "the archetypal normal mode problem in ID" [Mattis 1981]. This may tell us that ID model systems can be constructed to exhibit the most complex structures found in condensed-matter problems. But that is only of limited use as long as no isomorphism between the physical interpretations of such structures can be found.

1-3 Formalism and Goals Our presentation of the recursion method is preceded, in Chapter 2, by a brief account of the main results of linear response theory. This serves a twofold purpose: o It provides definitions for all the dynamical quantities that are of relevance in the context of the recursion method, and it clarifies some of their general properties. o It establishes the vital link between dynamics experiments and theoretical approaches (including the recursion method) to equilibrium dynamics under clearly defined assumptions (see Fig. 1-1). Having thus laid the foundation for both the formal development of the recursion method and the experimental relevance of its results, we shall proceed with the former in some detail in Chapters 3 to 9. The main lines of formal development are mapped out in Fig. 1-2. The Liouvillian representation of the recursion method will be introduced in Chapter 3 both in its quantum and its classical formulation (Secs. 3-1 to 3-4). The two main goals are the following: o We elucidate the physical meaning of the particular dynamical quantity which is the direct result of the recursion method: the relaxation function in the continued-fraction representation. o We provide a user's guide for the calculation of continued-fraction coefficients by the recursion method in that representation.

Section 1-3

7

An alternative method, which yields essentially the same data, is the moment method. In Sec. 3-5 we discuss how it is related to the recursion method.

Reconstruction of Spectral Densities

Figure 1·2: Overview of the main lines of formal development of the recursion method as presented in Chapters 3 to 9 for applications to many-body dynamics.

8

Chapter 1

The last three Sections of Chapter 3 branch out from the main line of formal development. In Sec. 3-6 we present a formal derivation· of the generalized Langevin equation for the dynamical problem at hand. The main benefit of that exercise will be drawn in Chapter 8, where phenomenolo$ical approximations of memory functions will be discussed. In Sec. 3-7 we turn our attention to the projection operator formalism, the original formulation of the recursion method. Our goal there is to deepen the understanding of what the physical consequences of an orthogonal expansion are and of how such a scheme could be motivated on grounds derived from physics rather than formal procedure. Section 3-8 introduces retarded Green's functions, principally for comparative studies (in Chapters 6 and 11) with the recursion method. In Chapter 4 the Hamiltonian representation of the recursion method is introduced along lines parallel to Chapter 3. The goal remains the same, but the recursive scheme proceeds via a different path and yields information on a different but related dynamical quantity (Secs. 4-1 to 4-3). For almost all dynamics applications of the recursion method in this representation, the equilibrium state of interest is the ground state of the system. Except for rare circumstances, the wave function of the exact ground state is not known analytically. It must be determined numerically for finite systems. Hence practically all applications of the recursion method in the Hamiltonian representation are preceded by the computation of the ground-state wave function. Recursive methods are recommended for that task. A survey of such methods including a comparative study of their performance is found in Secs. 4-4 to 4-8. The data that come out of the computational implementation of either representation of the recursion method are, in their most condensed form, a sequence of non-negative numbers {.!\}, ~, ... }. They are the continued-fraction coefficients of a certain relaxation function. That relaxation function determines the spectral density in whose properties we are primarily interested. These properties are, for example, the bandwidth iik is finite, the size of the gap if one is present, the singularity structure, and the detailed shape of the spectral-weight distribution. All this information is encoded in the .!\k-sequence, but only a finite number of coefficients is, in general, explicitly known. The retrieval of quantitative information on the above mentioned properties of spectral densities from incomplete .!\ksequences is clearly the most challenging task in typical applications of the recursion method. One popular but unsophisticated approach to this task is based on the use of an artificially truncated continued fraction combined with a more or less fancy way to convert a finite set of &-functions into a pseudo-eontinuous spectrum. We shall see that this type of analysis has little merit in many-body dynamics. However, one variant of this approach, the n-pole approximation, has played a significant role in the early development of the recursion method (Chapter 8). In this work we aim considerably higher than that. Our approach is based on a quantitative analysis (carried out in Chapter 5) of the finite .!\k-sequence (the data of the recursion method) prior to any attempt Of terminating the continued fraction. The key to any successful reconstruction of a spectral· density is that certain properties of the associated .!\k-sequence can be extracted with sufficient

9

Section 1-4

reliability from the limited number of coefficients that are known. We may call those properties the implicit information extractable from the sequence Ll!, Ll2 , ... , LlK in addition to the explicit information contained in the values of the first K coefficients. The amount of implicit information which can be inferred depends, of course, on the number of coefficients which are available and on the degree of complexity of the sequence under investigation. We shall see that under not too unfavorable circumstances it is possible to make quantitative predictions about bandwidths, gaps, singularity structures etc (Chapter 9). In addition to that we can determine the decay laws of the spectral density at high frequencies from the growth rates of the Llk-sequence. This decay law may serve as the basis for a definition of universality classes of dynamical behavior, a concept that will be discussed extensively in Chapter 7. The growth rate itself determines the convergence properties of the continued fraction and the analyticity properties of the spectral density. This is indispensible information for the construction of a termination function. The termination function will be constructed such that the completed continued fraction takes into account all the available explicit and implicit information from the known continued-fraction coefficients (Chapter 9).

1-4 Applications Almost all applications of the recursion method presented in this book will be in the general area of spin dynamics with emphasis on ID quantum spin models. Quantum spin chains are, in fact, physically realized in quasi-ID magnetic insulators for a large variety of spin models [Steiner, Villain and Windsor 1976; Willett, Gatteschi and Kahn 1985]. Typically these are compounds in which the magnetic ions are arranged in chains along one crystallographic axis such that the exchange interaction between ions within a chain is very much stronger than the interaction between ions belonging to different chains. Two of the most important experimental techniques which probe dynamical properties of quasi-ID magnetic insulators very directly are neutron scattering and NMR. Their connection to dynamic two-spin correlation functions are the following: o The inelastic scattering cross section for magnetic scattering of neutrons is directly proportional to the dynamic structure factor 2

d (J dQdro

o

oc

S ( 1.11.1

~

q,roj

=

-iqnf- dteiCiJt<SI.I(t)Sl.lI+n>'

1~ N LJ e I,n

I

(1 1) .

-00

In situations where a hydrogen atom is crystallographically located suitably close to the (electron) spin chain, the NMR spin-lattice relaxation rate liT! of this proton is dominated by the local fluctuations of the unpaired electrons of the nearest magnetic ion. It is proportional to the frequencydependent autocorrelation function at the nuclear Larmor frequency ~:

10

Chapter I +00

_1 Tt

cc

Sllll(~) = 2.E jdteioot.'<sj(t)sj> . N

I

(1.2)

-eo

Dynamics of 10 quantum spin models is the field of research where the authors of this work have been most active in recent years, which explains their choice of applications. However, the reader will find it straightforward to translate the methods of analysis designed for the applications presented here into equally powerful calculational tools within hislher own field of research. Applications of the recursion method will be presented on three levels of complexity and sophistication. On each level they serve a different purpose: o The simplest applications are to such toy models as the harmonic oscillator, linear spin waves, free fermions, classical and quantum spin clusters,... These applications (starting in Chapter 6) are used solely for didactic reasons, serving the purposes of (i) illustrating the "mechanics" of the recursion method and (ii) illuminating the links between the recursion method and other calculational techniques. o The first batch of applications to models with nontrivial spin dynamics will be presented in the first 9 Chapters, where the focus is on the development of the recursion method as a calculational technique. Only passing attention will be given to the underlying physics. o The bulk of applications to models of considerable interest in former and current research in spin dynamics will be presented in Chapters 10 and 11. Here the focus will be prinrarily on the physical phenomena discussed and less so on the methodological aspects. In these last two Chapters we shall portray the strengths and limitations of various general methods in the contexts of two major areas of research in spin dynamics. The theme of Chapter 10 is spin dynamics at high temperature. For all practical purposes, high temperature means T = 00 in this context. The main question of interest is an old one: What types of processes govern the transport of spin fluctuations in the absence of instantaneous spatial correlations? This question will be illuminated from different angles with results for classical and quantum spin models obtained by a number of calculational techniques including exact analysis, rigorous bounds, simulations, and the recursion method. Finally, in Chapter 11, we shall explore the fascinating world of lD quantum spin dynamics at zero temperature. At the center of attention will be the 10 s = 1/2 XYZ model and its special cases. We shall discuss the multi-faceted interrelations between the properties of (i) the ground state of the system, (ii) the excitation spectrum, and (iii) the spectral densities. These properties have been investigated by exact analysis, Green's function techniques in the fermion representation, field-theoretic approaches, finite-chain analysis, and the recursion method.

2 LINEAR RESPONSE AND EQUILffiRIUM DYNAMICS 2-1 Response Function and Generalized Susceptibility Dynamical properties of a many-body system specified by a Hamiltonian Ho in thermal equilibrium are most frequently expressed in terms of time-dependent correlation functions of the form
-- - 1 T r[ e -~Ho eiHotlhA le -iHotlhA 2]

o

Zo

'

(2.1)

where Zo = Tr[exp(-~Ho)] is the partition function and ~ = l/kBT the inverse temperature. Here AI' A 2 are two operators in whose time-delayed correlations we are interested. In any dynamical experiment, the system Ho is necessarily perturbed, typically by some kind of time-dependent radiation field b(t). This implies that one actually p~obes the dynamical properties of a nonequilibrium system. That system is characterized by the Hamiltonian (2.2)

where the external field b(t) couples to the variable represented by operator B of the system Ho. As a consequence, the expectation value of any operator A, which is a constant o in thermal equilibrium, turns into a time-dependent quantity o reflects the response of the system to the perturbation. In general, the response depends on the external field b(t) in a complicated nonlinear manner. In the case of weak perturbations, however, one can argue that its dominating contribution is linear in the external field b(t), +00

(2.3)

provided that an expansion in powers of b(t) is valid. This linear relation between cause and (time-delayed) effect serves as the definition of the (linear) response function XAB(t). The linearity implies that XAB(t) itself does not depend on the field b(t). The function XAB(t) is bounded and satisfies the causality condition, XAit)=O for t
12

Chapter 2

i _dA = -[Ho,A] dt '!'l dB i (it = l;" [HoB] ,

~ A(t) = eiHotl'hAe-iHotl'h , ~

B( )

iHotl'h -iHotl'h Be t =e ,

(2.4)

and in part by the density operator via dp ~

t

= -~[Hl(t),P] ~

= Po

p(t)

'!'l

+

~Jdtlb(tl)[B(t'),P(tl)], '!'l

(2.5)

-00

where Po = Zo-lexp(-~Ho) characterizes the state of the system in equilibrium. The leading term in a systematic approximation of the integral equation (2.5), t

p(t)-Po

= ~Jdtlb(tl)[B(t'),Po]

(2.6)

,

used for the evaluation of the expectation value o indeed yields a result of the form (2.3) with the response function given by the following explicit expression: XAB(t-t I) =

~EXt-t I )<[A(t),B(t I )]>0 '!'l

.

(2.7)

This is the Kubo formula. The a-function ensures causality. Note that XAit) depends only on equilibrium properties of the system. It provides the crucial link between equilibrium dynamical properties of a many-body system and the dynamical response of that system to experimental probes, which necessarily perturb the equilibrium. l In the following, we discuss some general properties of the response function and related quantities for the special case A=B. It is useful and instructive to express the response function XAA(t) X(t) as the sum of two functions:

=

X(t)

= Xl (t)

+ iX" (t) .

(2.8)

The two functions are defined by their symmetry properties, (real, symmetric) ,

Xl (-t) = Xl (t)

X" (-t)

=

-X" (t)

(imaginary, antisymmetric) ,

(2.9a) (2.9b)

and have been named the reactive part and the dissipative part of the response function, respectively. They are expressible in terms of equilibrium expectation values as follows:

·0·

lHenceforth we drop the subscript in the (time-independent) Hamiltonian of the system under investigation and in equilibrium expectation values.

Section 2-2

'X'(t)

1

.

2

2~

= -['X(t)+'X(-t)] = _ l sgn(t)<[A(t),A]>,

13

(2.10)

!VII (t) = 2.[!V(t) _!V( -t)] = _1 <[A(t),A]> . A 2i A A 2~

(2.11)

We note that the response function is fully detennined by either its reactive or dissipative part alone: (2.12) 'X(t) = 2ifX.t)'X" (t) = 2Q:t)'X' (t) . The Fourier transform of the response function defines the generalized susceptibility

-J

X(~) = dtei~t'X(t)

(2.13)

.

Under very mild assumptions for the response function, X(~) is analytic in the upper-half complex ~-plane. For real frequencies we express the generalized susceptibility in terms of a symmetric real part and an antisymmetric imaginary part as follows: X(ro)

=

lim X(oo+ it)

= X' (00)

+ iX" (00)

(2.14)

£~o+

with

= .!.[X(oo) +X( -00)] = X' (-00)

X' (00) X" (00)

2

=

;i[X(oo)-X(-oo)]

=

(2.15a)

,

(2.15b)

-X" (-00) .

A direct consequence of the causality property of the response function are the Kramers-Kronig dispersion relations,

J -

-

II_J

X' (00) = ..!.. do:l X 1t

(w) , 0:1-00

-

I _J

X" (00) = -..!.. J~ 1t

-

0:1-00

.

(2.16)

2-2 Fluctuation-Dissipation Theorem Kubo's formula (2.7) establishes the connection between the dynamical response of the system to weak external perturbations and the internal fluctuations as they occur during the time evolution of the system in thermal equilibrium. It is a relation between the fluctuation function, ci>(t) == {<[(A(t)-

= {<[A(t),AJ...>-2

,

(2.17)

14

Chapter 2

and the dissipative part (2.11) of the response function. This relation is most conveniently expressed in terms of the correlation function,2 (2.18)

S(t) == -2 ,

as follows:

1../1 (t) = _1 [S(t) -S( -t)] 2"

,

cIl(t) = ':[S(t) +S( -t)] . 2

(2.19)

We note that the three functions in (2.19) have the following transformation properties under time reversal (for real t):

1.." (-t) = -1.." (t) = [1.." (t)]* ell(-t) = cIl(t) = [~(t)]* S(-t)

,

(2.20b)

,

= S(t-i"~) = [S(t)]*

(2.20a)

.

(2.2Oc)

l.."(t) is an imaginary, antisymmetric function and ~(t) a real, symmetric function, whereas S(t) is, in general, a complex function; its real part is symmetric and its imaginary part antisymmetric. However, the imaginary part of S(t) disappears for two special situations: (i) for general quantum Hamiltonian systems at infinite temperature (~=O); (ii) for general classical Hamiltonian systems ("=0) at arbitrary temperature. The structure function

f

+00

S(co) '"'

dteiOltS(t) ,

(2.21)

is always a real function. It satisfies the condition of detailed balance, S( -00)

= e -~lIroS(oo)

,

(2.22)

a direct consequence of (2.2Oc). The spectral density cI>(oo), which is the Fourier transform of the fluctuation function, and the dissipation function X"(oo), the dissipative part of the generalized susceptibility, can be expressed in terms of the structure function as follows: cI>(oo) '"' .!..(l +e-~lIro)S(oo),

2

X" (00)

= _1(l-e-Pllro)S(co) .

2"

(2.23)

Evidently cI>(oo) is symmetric and X"(oo) is antisymmetric. Equations (2.23) are known under the name fluctuation-dissipation theorem, specifically the direct relation

2At the risk of being slightly ambiguous, we continue the common practice of using the term "correlation function" also for the expectation value alone, as in (2.1).

Section 2-3

15

(2.24)

(co) = 'l\coth(..!..P'I\CO)X" (co) . 2

Hence the dissipation effects resulting from an interaction with a weak external force are determined by the natural fluctuations existing in the system at equilibrium [Callen and Welton 1951]. In the limit '1\-+0, Eq. (2.24) turns into the so-called classical fluctuation-dissipation theorem

2kBT //

(2.25)

(CO)cl = - - X (co).

co

For T=O, Eq. (2.24) describes the zero-point fluctuations (co) = 'l\sgn(co)x"(co) of a quantum system. In the classical relation (2.25), the fluctuations die out completely as T-+0. We may remark here that (2.24) is frequently used to introduce quantum fluctuations at the end of an otherwise purely classical calculation of the response function.

2-3 Moment Expansion It is generally assumed that the correlation functions S(t) of quantum and classical Hamiltonian systems are entire functions in the complex t-plane. However, a formal proof of this property exists only for 10 quantum many-body systems with shortrange interaction [Araki 1969], and there is circumstantial evidence (to be discussed later) that the correlation functions of some otherwise well-behaved classical Hamiltonian many-body model systems with short-range interaction do not share this property. However, except for rare pathological cases (one example will be demonstrated later), any singularities in the complex t-plane tend to stay away from the real axis (in particular from the origin). Hence, under almost all circumstances, the correlation function can be expressed as a Taylor series,

s(t) =

EM n=O

n

(2.26)

(-it)n ,

n!

and the procedure for the determination of the coefficients Mn is well defined. They are the frequency moments of the structure function: (2.27) We use the Heisenberg equation of motion, A = (-i/h)[A,H], in (2.27) to express them in terms of expectation values of n-fold commutators of the dynamical variable with the Hamiltonian: M- n

1 = -<[... [[A,H],H], ... ,H]A> 'l\n

.

(2.28)

16

Chapter 2

At T = 00, where S(t) is a real, symmetric function, we have M2k+! = 0 and the M2k can be written more compactly in terms of a product of two k-fold commutators [Roldan, McCoy, and Perk 1986]: _

M2k

_ (-l)k - < [...[A,H], ...,H][...[A,H], ...,H]> .

-

(2.29)

'l\2k

For a classical Hamiltonian system, where we have M2k+! = 0 at arbitrary temperatures, we use Hamilton's equation of motion, A = {A,H}, in (2.27) to arrive at the result

M- 2k

= (-1) k <( ... {{A,H},H},...,H}A>

(2.30)

for the even moments in terms of a 2k-fold Poisson bracket. Equations (2.19) imply that the fluctuation function, cI>(I), and the dissipative part of the response function, X"(I), can be expressed in terms of subsets of expansion coefficients (2.28): 4>(1) =

.2k

L M2k (-'1) 00

t

11

1~ 00

(1)

=

,

(2.3 la)

(2k)!

k"()

-

l;"tQM2k +!

(

-,1)2k+! •

(2k+l)!

(2.31b)

In fact, the even and odd moments (2.28) can be written somewhat more concisely as

-

1

M 2k = _<[[... [[A,H],H], ...,H],A]+> ,

(2.32a)

1 . M 2k +! = - - < [ [... [[A,H],H], ...,H],A]>

(2.32b)

2'l\2k

2'l\2k+l

in terms of 2k or 2k+l commutators and an anticommutator or an additional commutator. The direct reconstruction of the functions S(oo), «1>(00), or X"(oo) from the moments Mn is well known to be an ill-conditioned problem, in general, and worse if the Mn grow faster than n! for large n; in that case the above mentioned functions are no longer uniquely determined by their moments (more about that in Sec. 5-4). We shall see that many of the practical difficulties which plague the moment method can be circumnavigated by the recursion method in combination with a continued-fraction analysis.

3 LIOUVILLIAN REPRESENTATION Liouvillian dynamics provides a direct link between classical and quantum mechanics. It provides the language for a direct transcription of the recursion method from a quantum formulation to a classical formulation. At the same time, it is the basis of a particular representation of the recursion method, which we call the Liouvillian representation. A strikingly elegant and user-friendly formulation of this representation has resulted from Lee's important developmental work [Lee 1982, 1983].

3-1 Quantum Formulation The dynamical system is specified by a quantum Hamiltonian H and a dynamical variable (operator) A in whose time evolution we are interested. The goal is to determine the dynamic correlation function
= i[H,A] = iIA

(3.1)

,

where the commutator (3.2)

L == [H, ]

is the quantum Liouvillian operator, a Hermitian super-operator in this case. The formal solution of (3.1) reads A(t)

= eiLtA(O)

(3.3)

.

The recursion method requires, in addition to H and A, the specification of an inner product for operators in the Hilbert space of H and A: (A,B) (A,TlB)

= (B,A)· , = TI(A,B) ,

(A,A) ~ 0 , (A,B+C)

= (A,B)+(A,C)

.

(3.4)

The choice of the inner product determines the type of dynamic correlation function obtained in the end. A particular choice is the Kubo inner product, ~

(A,B)

= ~ fdA,<eARA te-ARB> -

,

(3.5)

Po where = Tr[e-~HA]/Z, Z = Tr[e-~H], P= l/kBT. It was designed such that the

Fourier transform of the associated fluctuation function (A(t),A(O» satisfies the classical fluctuation-dissipation theorem (2.25) also for quantum systems at arbitrary

INote that we have set h=1 everywhere except in Chapters 1 and 2.

18

Chapter 3

temperatures. The quantity ~(A,B) is then a static susceptibility. A more general scalar product has the form (A,B)

1~

= _jdA.g(A.)<eAHA te-AHB> ~o

-

,

(3.6)

with the weight function g(A.) satisfying the conditions g(A.) ;;:: 0,

g(~ -A.) = g(A.)

1~

,

- jdA.g(A.)

~o

=1

(3.7)

The Kubo inner product (3.5) corresponds to a constant weight function, g(A.)=1. An important alternative choice is based'on the weight function

1 = _~[O(A.) +O(~ -A.)]

g(A.)

2

.

(3.8)

The resulting inner product, (A,B)

= .!.. 2

- ,

(3.9)

is more readily evaluated than (3.5) at ~>O including the important case of zero temperature. The recursion method with this inner product yields the standard spectral density <1>(0), which satisfies the general fluctuation-dissipation theorem (2.24). In the infinite-temperature limit (~~), the inner product (3.6) is the same for bQth choices of g(A.) : (A,B)

=

- .

(3.10)

3-2 Classical Formulation In the classical case, the dynamical system is specified by a Hamiltonian function H(ql ,···,qn;Pl,·..,Pn) = const and a dynamical variable A(ql ,...,qn;Pl ,... ,Pn) = A(t). The time evolution of A(t) is determined by Hamilton's equation of motion,

_dA = -{H,A} = iLA dt

,

(3.11)

where

L= i{H, }= it(aH ~ _aH~) j=l

aqj apj

(3.12)

apj aqj

is the classical Liouvillian operator (again Hermitian), now operating on classical dynamical variables. The canonical coordinates qj' Pj are special dynamical variables which satisfy the fundamental Poisson brackets

Section 3-3

{qj,qj} = {Pj'p) = 0

, {qj'Pj} = Ojj .

19

(3.13)

For the repeated evaluation of Poisson brackets {H, } as required by the recursion method, it is sometimes more convenient to express the energy function H and the dynamical variable A in terms of a set of elementary but non-canonical dynamical variables ul'''''um , if the Poisson brackets of these variables {uj'Uj } = Pij(ul"",u m )

(3.14)

have a sufficiently simple structure. The inner product which produces the classical fluctuation function as defined in Chapter 2 is the following : (A,B)

=

- <8> ,

(3.15)

where the expectation value denotes the phase-space average

= ~Jdnqdnpe-~H(q,p)A(q,P) .

(3.16)

It can be interpreted as the classical limit of the general quantum inner product (3.6), Le. the limit in which the operators A, B and H turn into classical functions of the phase-space coordinates. 3-3 Orthogonal Expansion of Dynamical Variables At the core of the Liouvillian representation of the recursion method is the orthogonal expansion of the (quantum or classical) dynamical variable under scrutiny [Lee 1982]: A(t) =

L Ck(t)fk .

(3.17)

k=O

For a classical system, the A are an orthogonal set of functions in phase space, for a quantum system an orthogonal set of operators. In both cases they span a Hilbert space of (generally) infinite dimensionality. The Liouville operator acts on the vectors A (functions or operators) in this Hilbert space. The orthogonal expansion (3.17) is then carried out in two steps: #1 Determine a particular orthogonal basis {fk} in the Hilbert space of dynamical variables by applying the Gram-Schmidt procedure with the Liouvillian L as the generator of new directions. #2 Insert the expansion (3.17) into the equation of motion - the Heisenberg equation (3.1) for quantum systems or Hamilton's equation (3.11) for classical systems - to obtain a set of differential equations for the timedependent coefficients Ck(t). For step #1 we note that the general inner product (3.6) between the vectors A and iLA vanishes for arbitrary A:

20

Chapter 3

(A,iIA)

=0 .

(3.18)

This simplifies the Gram-Schmidt orthogonalization considerably and leads to the following set of recurrence relations for the vectors A:

A+I

(3.l9a)

= iLlk + ~k!k-I' k =0,1,2, ...

~k =

(/k,fk) , k=I,2,... (/k-I,fk-I)

(3.l9b)

with f I = 0 and 10 =A. The first three iterations are illustrated in Table 3-1. We shall see that the sequence of numbers ~k contains all the information necessary for the reconstruction of the fluctuation function (A(t),A(O)) or the associated spectral density.

10 "

A

(initial condition)

ilfo .. 11 -. (/0/1) .. (/o,ilfO>. .. 0

ilfl .. 12 - Al/o -. (/lJ~ .. (/1,ilfl)+AI(/IJO> .. 0 -. (/oJ~ .. (/0.ilf1) +A1(/oJO>

if

.. -(ilfOJ 1) +AtifoJO> .. 0

A (/IJI) I .. (/oJO>

ilf2 .. 13 -A,/I - r ,to -. (/2J3) .. (/2.ilf~ +~(/2JI) +r2(/2JO> .. 0 -. (/lJ~ .. (/1.ilf~+~(/lJl)+r2(/IJO> • -(ilf1J 2) +Az(/I J 1) .. 0

if Az .. (/2J2) (/lJI)

-. (/oJ~ • (/o.ilf~ +~(/OJI) +r2(/oJo>

.. (/IJ2)+r2(/OJO> .. 0

if r 2 .. 0

.Table 3-1: The first three iterations of the orthogonalization procedure carried out in detail.

In step #2 we insert the orthogonal expansion (3.17) into the equation of motion (3.1) or (3.11). The differential operator acts on the Ck(t), and the Liouvillian acts on the A with the result given by (3.19). Term-by-term comparison of the coefficients of each vector Ik in the equation of motion then yields the following set of coupled linear differential equations for the functions Ck(t): Ck(t) = Ck_l(t) - ~k+lCk+l(t),

k=0,1,2,...

(3.20)

Section 3-4

21

with C_1(t) == 0, Ck(O) = 0k,O' Unlike the vectors!k in (3.19), the Cit) cannot be detennined recursively. If our goal is to detennine the fluctuation function of the dynamical variable A(t), then it suffices to know just one of the functions Ck(t):

C (t)

o

=

(A(t) ,A(O» (A(O) ,A(O»

=

«I>(t) .

«1>(0)

(3.21)

This relation follows directly from the orthogonal expansion (3.17). Throughout this book, the inner product (3.9) will be used for quantum systems and (3.15) for classical systems. The last expression in (3.21) is the (normalized) fluctuation function with «I>(t) as defined in Chapter 2.

3-4 Relaxation Function and Spectral Density A formal solution of the coupled differential equations (3.20) can be obtained by Laplace transform. The functions (3.22)

Ciz) == jdte-ztCk(t)

o satisfy the set of algebraic equations zciz) - 0k,O

= ck-I(Z)

- L\k+lck+l(z),

k=0,1,2,...

(3.23)

with c_ 1(z) == O. In order to solve these equations, we rewrite (3.23) for k=O in the form co(z)

1

= ---~ C1(z) z + lll-A

(3.24)

co(z)

and express the ratio ck(z)lck_I(Z) in the denominator recursively (for k=1,2,...) in terms of ck+I(Z)lck(z), again from (3.23). The result is the function co(z) in the continued-fraction representation: co(z)

1

= ---.....,....-L\l

Z + ----,..-

(3.25)

~

z+-Z + ...

The functions cI(Z), c 2(z), ... can then be determined recursively from (3.23), once co(z) has been detennined. The result (3.25) demonstrates the claim made earlier that the L\k-sequence alone fully determines the correlation function under investigation. The function co(z) is, ,named relaxation function. It represents the Laplace transform of the normalized fluctuation function (3.21). The latter is thus recovered by the inverse Laplace transform of (3.25),

22

Chapter 3

Co(t)

= ~ jdzez1co(z) 21tz c

,

(3.26)

where the integral is performed along the path C parallel to the imaginary axis in the complex z-plane as depicted in Fig. 3-1. In most applications, we wish to determine the Fourier transform of Co(t) rather than the function Co(t) itself. The normalized spectral density,

J

"-


(3.27)

dteiC01Co(t) ,

can be recovered directly from the relaxation function as follows:
= tim 29t[co(£ -iro)]

.

(3.28)

£-+0

~[z]

tc e ~[']

8t[z]

Figure 3-1: The complex frequency plane in two different notations. The variable z is primarily used in the context of the Liouvillian representation of the recursion method (Chapter 3) and the variable ~=iz primarily in the context of the Hamiltonian representation (Chapter 4). The real frequencies (in the physical sense) are located on the vertical axis: 0>=9t[~]=-5[z). The dashed line denotes the path of integration in the inverse Laplace transform (3.26).

3-5 Recursion Method and· Moment Expansion There are alternative ways to calculate the ~k-sequence for specific fluctuation functions of a given model system. One such method is based on the moment

Section 3-5

23

expansion (see Sec. 2-3). The normalized fluctuation function can be expanded in a Taylor series, C (t)

o

00

=~

k

t 2k

(-1) M

~ (2k)!

(3.29)

2k

with Mo::!. The coefficients M 2k are the frequency moments of the (normalized) spectral density, (3.30) and can be expressed as an inner product involving a 2k-fold commutator (quantum system),

1

M 2k = _ ( [... [[A ,Hl,Hl,...,Hl,A)/(A ,A) , 2'h 2k

(3.31a)

or a 2k-fold Poisson bracket (classical system), 1 k M 2k = _(-1) ({ ... { {A,H},H}, ...,H},A)/(A,A) . 2

(3.31b)

The information content of the frequency moments M 2k, k=1,...,K, is identical to that of the !:lk' k=1, ...,K, and one set of numbers is readily expressed in terms of the other. The !:lk' for example, can be expressed in terms of Hankel determinants whose elements consist of moments M 2k [Dupuis 1967; Roldan, McCoy, and Perk 1986; Grosso and Pastori Parravicini 1985]. However elegant, determinants of large numbers are not suitable for the numerical implementation of this transformation. There exist several algorithms for the numerical conversion of one set of numbers into the other. Some procedures are more susceptible to numerical instabilities than others [Gordon 1968]. The following algorithm, which is a product of the recursion method, has proven to be fairly robust against numerical instabilities. It is derived from Eqs. (3.23) in conjunction with the asymptotic expansion, Co(Z)

= L (-l)kM2k z -(2k+l)

,

(3.32)

k=O

the Laplace transform of (3.29). The transformation is described by a simple set of recurrence relations (see Table 3-2 for some explicit results). o Forward direction: For a given set of moments M2k, k=O,I,... ,K with Mo=l, the first K coefficients L\z are determined by (n) _

M2k

-

M (n-l) 2k

!:In-l

M (n-2) 2k-2

!:In-2

'

_ M(n) 2n

!:l n

(3.33)

24

o

Chapter 3

for k=n,n+l,...,K and n=1,2,...,K, and with set values M~2)=M2k' A_I=~=l, (-ILO MZk - • Reverse direction: For a given set of An=Mff)/, n=l,...,K, and A_I=~=l, the moments MfJ),>=M2n result from the relations,

(n-l) _ M2k -

A

"""n-I

M(n) 2k

An-I M (n-Z) + - - 2k-Z '

(3.34)

An - Z

for n=k,k-l,...,l and k=1,2,...,K and with set values Mtl)=O.

MZ = Al

M4

M6

Al = Mz

M

= Al (AI +Llz)

4 Az = _-M z Mz

= Al [(AI +Az)z+LlzA3l

A3 =

M6IMz -M4 M4 MiMz-Mz Mz

Table 3-2: The first 3 expansion coefficients of (3.32) expressed in terms of the first 3 continued-fraction coefficients of (3.25) and vice versa.

These simple transformation relations establish the link between the recursion method and the various techniques in use for the analysis of moment expansions. The computational effort of calculating the moments (3.31) is similar to that of calculating the same number of Ak's by the recursion method. However, we shall see that working with continued-fraction coefficients has at least two advantages: (i) the Ak's are more suitable for the direct reconstruction of the spectral density (1)0«(0); (ii) information on specific properties of the spectral density, such as bandwidths, gap sizes, edge singularities, infrared singularities, and decay laws at high frequencies can be extracted directly from the Ak-sequence.

3-6 Generalized Langevin Equation In the context of transport theory, the formalism of the recursion m~thod may serve as a link between microscopic model systems and phenomenological theories bas~ on a contracted level of description. The classical relaxator, specified by the

Langevin equation, A(t) +

~(t) = F(t) 't

,

(3.35)

is a prototypical example. In this simple system, the quantity described by the dynamical variable A is subject to a linear damping, specified by the relaxation time

Section 3-6

25

't, and by a B-correlated random force F(t) (white noise). The resulting fluctuation function is then a pure eXRonential, CO(t)=e- t/t , and the spectral density a Lorentzian, o(00)=2't/(1 +002't ). Now consider the 'generalized Langevin equation, t

A(t) + JdtlI;(t-tl)A(t l ) = F(t) .

(3.36)

o In the context of a phenomenological theory, the memory function I;(t) contains essentially the entire model specification. The associated random force F(t) cannot be chosen freely but must satisfy the consistency c6nditions imposed by the fluctuation-dissipation theorem. Remarkably, the phenomenologically motivated generalized Langevin equation provides, under very general circumstances, a rigorous description of quantum or classical many-body systems. For any given microscopic (quantum or classical) Hamiltonian model system, the generalized Langevin equation for the dynamical variable of our choice can be derived within the formalism of the recursion method [Dupuis 1967, Lee 1983]. Let us define the two functions, c1(z) L(Z) ;: A1- co(z)

,

ck(z)

biz);: - _ , k=I,2, ... co(z)

(3.37)

where co(z) can be determined from first principles in the continued-fraction representation (3.25), and the ck(Z) for k~1 can be inferred recursively from Eqs. (3.23). We rewrite these latter equations in the form ZCO(Z) - 1 + L (z)co(z)

=0 ,

(3.38a) (3.38b)

The inverse Laplace transforms of these functions then satisfy the equations t

to(t) + JdtlI;(t-tl)Co(tl) = 0 ,

(3.39a)

o t

tk(t) + JdtlI;(t-tl)Ck(t l )

= Bk(t),

k= 1,2,...

(3.39b)

o From Eqs. (3.39) and the expansion (3.17), we obtain the generalized Langevin equation (3.36) for the dynamical variable A(t). In this context, the memory function I;(t) and the random force F(t) are expressed in terms of microscopic properties of the model. However, we must keep in mind that both Ut) and F(t) are not explicitly known prior to the solution of the dynamical problem. The functions Bit), k=1,2, ... can be interpreted as the coefficients of an orthogonal expansion of the random force,

26

Chapter 3

F(t)

= E Bit)fk

, Bk(O)

k:}

= ak,}

,

(3.40)

in the same basis lftl as previously used for the dynamical variable. Note that F(t) has zero projection onto f o' which guarantees that there is no correlation between the dynamical variable at time zero and the random force at any later time: (F(t),A(O)) = O. In phenomenological theories, the random force is usually specified by its fluctuation function. The latter is equal to the first coefficient of the orthogonal expansion (3.40): (F(t) ,F(O)) (F(O) , F(O))

(3.41)

3-7 Projection Operator Formalism Having introduced up front a modern and user-friendly formulation of the recursion method, we now wish to have a closer look at the way it was originally designed by Zwanzig [1961] and Mori [1965]. This earliest formulation is known under the name projection operator formalism or memory function formalism. Today it can be regarded as a particular formulation of the recursion method in the Liouvillian representation. From a computational point of view, it is an unwieldy tool that has discouraged many a potential user through the years. Its main attractive feature is that it provides a physical motivation for the orthogonal expansion which is at the basis of any variant of the recursion method. It was the hope of the inventors of the memory function formalism that by means of a few orthogonal projections the most essential features of the overwhelmingly complex many-body dynamics could be separated from the rest. That hope remained unfulfilled in all but the simplest applications. However, the pioneering effort led the way to more successful methods of. analysis and modes of representation. Following Forster's insightful account [Forster 1975] of the memory function formalism, we represent the (classical or quantum) dynamical variables in terms of bras, , and identify the brackets with the inner product (A,B) introduced in Secs. 3-1 and 3-2. Likewise, matrix elements of any Hermitian operator such as the Liouvillian L have an obvious interpretation: == (A,LB) = (LA, B). In this notation, the dynamical quantity of interest the (normalized) fluctuation function for the variable A(t) - reads C (t)

o

= =



=

(3.42)



and its Laplace transform, the relaxation function (3.22), can be expressed as follows:

1 co(z) = _1_ . z+iL

(3.43)

Section 3-7

27

This last expression is the one to be determined recursively by a succession of subdivisions of the full many-body dynamics into an increasing number of components that can be dealt with explicitly and a remainder which will be replaced by a phenomenological model at the end of the calculation. The subdivision will be accomplished by a sequence of projection operators onto onedimensional subspaces of the Hilbert space traversed by the vector 1A(t». Denoting the initial condition by lA> = IA(O» == Ifo> as in Sec. 3-3, the first projection operator to be used and its complement are

Po ==

I

lfo>--
(3.44)

Qo == I-Po'



When we write L = LPo + LQo and use Dyson's identity,

_1_ = ~ _ ~y_l_ X+Y X X X+Y

(3.45)

with X == z+iLQo ' Y == iLPo, expression (3.43) can be split into two terms as follows:

co(z)

= <101

.1 Ifo> + <101 .1 iLP0_1._Ifa>. z+ILQo z+ILQo z+IL

(3.46)

The first term is readily evaluated by expansion into a geometric series,

1 ( -i) ( -i)2 <101-[1 + -LQo + -LQoLQo + ... ]lfa> z z z2

I z

= -

'

(3.47)

while inserting Po from (3.44) into the second term yields

<10'

?

iLlfo>co(z). (3.48)

= ----.".------z - _1_<101 z iD/a>

(3.49)

1 iLlfo>_I_ z + iLQo z + iL

= <101

z + ILQo

Equation (3.46) with these results yields the expression

co(z)

1



z + iLQo

The matrix element in the denominator can be brought into a more suitable form by adding and subtracting the unity operator and by exploiting some properties of the projection operator Qo :

?

<101 ~ iLl/a> = z +ILQO z +ILQO

= -
.1 iQoLlfa> . z+IQoLQo

(3.50)

28

Chapter 3

The relaxation function can thus be expressed as (3.51) in terms of another unknown function I

1



z+ IL,

I: ,(z) = -- ,

=

(3.52)

=

where !f,> iQoL!fri>, L, QoLQo' The function I:,(z) is called the memory function associated with the relaxation function co(z). It is the Laplace transform of the function f(t) which specifies the generalized Langevin equation (3.36) for the dynamical variable A(t). In fact, (3.51) is equivalent to (3.38a). At this point it is important to note that the memory function I:, (z) of the original problem can be reinterpreted as the (as yet non-normalized) relaxation function of a new dynamical problem. This new problem is specified by a new initial condition !f,> and a new Liouvillian L,. The operator L, does no longer contain the full many-body dynamics. It is obtained from the original Liouvillian by having it sandwiched between a pair of projection operators Qo' These projection operators act like filters; they absorb that part of the full many-body dynamics which has been dealt with explicitly in the course of this first projection. The explicit information is contained in the normalization constant of the memory function I:, (z) as we shall see. Carrying out another projection means repeating the calculational steps that led from (3.46) to (3.52) for the new; slightly reduced dynamical problem. The result is the old memory function in the role of anew (still not normalized) relaxation function

I: ,(z)

(3.53)

z +I:lz)

expressed in terms of its own memory function

I: 2(z)

1

1



z+ IL2

= --

(3.54)

with ~, = /, !f2> = iQ,L,!f1>, Lz = Q,L,Q,. Repeating this cycle over and over again produces a continued fraction which, after n iterations, is terminated by the nth-level memory function I:n(z): 1

co(z) = - - - - - - ; - - - - ~,

Z

+-------z

+ ...

(3.55)

Section 3-8

29

The coefficients {Ll l , ..., ~-d reflect the explicit dynamical information extracted by means of the first n projections. The last coefficient, Lln , is part of the function ~n(z).

Each iteration adds a layer of projection operators around the original Liouvillian. If n is sufficiently large, so it has been argued, all distinctive spectral properties of L will be filtered out and incorporated explicitly into the relaxation function via the continued-fraction coeficients Llk • What remains of L is then believed to be indistinguishable from a source of white noise. The associated memory function ~n(z) would then be a constant, equal to the inverse relaxation time l/'tn of a classical relaxator process, a process described by. the simple Langevin equation (3.35). It turns out that this type of reasoning is not really valid except for special circumstances. The best known and most frequently quoted example is the relaxation function associated with the density fluctuations in a normal fluid as can be probed by light scattering [Forster 1975]. These fluctuations are governed by two processes - heat diffusion and sound waves - giving rise to the well-known three-peak structure in the corresponding spectral density (the Rayleigh peak at ro=O and one Brillouin peak on either side). These structures are recovered by the projection operator formalism after one or two projections, respectively. We shall return to that theme in Chapter 8, in the context of our discussion of the n-pole approximation.

3-8 Retarded Green's Functions In the context of this book, the Green's function method can be regarded as another general method for the calculation of dynamic correlation functions of many-body systems [Elk and Gasser 1979, Lovesey 1980, Doniach and Sondheimer 1974]. As in the case of the recursion method, the application of the Green's function method to nontrivial models poses challenging tasks of approximation and interpretation. It is not a priori clear which of the two general methods is likely to produce a more reliable result in a given application where neither method can be carried through without resort to some sort of approximation. A number of comparative applications will be presented later for the twofold purpose of (i) testing their performance under specific circumstances or (ii) gaining complementary pieces of information on a complicated dynamical problem. Our goal in this Section is to establish the formal connections between the fundamentals of the Green's function formalism and the Liouvillian representliltion of the recursion method. We begin with the definition of the two varieties of the retarded Green's function, G±(t-t') :; <±

= -i<:Xt-t')<[A(t),B(t')]±> ,

(3.56)

where the choice between commutator (-) or anticommutator (+) is dictated soleley by calculational convenience, not by the statistics of the underlying degrees of freedom (bosons or fermions). Note that the response function as given by the Kubo

30

Chapter 3

formula (2.7) is equal to minus the commutator variety of the retarded Green's function (3.56): XAB(t-t')

= -<-

(3.57)

.

The retarded Green's function satisfies the equation of motion, i ;t<±

= B(t-tl)<[A,B]±>+«[A(t),HL;B(t l »>±,

(3.58)

which is derived from (3.56) and the Heisenberg equation (3.1) for the operator A(t). The equation of motion for the frequency-dependent Green's function +00

(3.59) is then algebraic in nature, ± m<
= <[A,B]±>

±

+ «[A,HL;B»ro·

(3.60)

The difficulty in solving that equation lies in the fact that it is just the first link of an infinite chain of coupled equations in all but the very simplest applications. Except for noninteracting degrees of freedom, the hierarchy of equations generated by (3.60) is infinite and can, therefore, not be closed in general. For frequencies I; in the upper half complex plane, the retarded Green's functions G+(I;) and G_(I;) can be expressed in terms of the familiar functions S(ro) (structure function), «1>(00) (spectral density) or X"(oo) (dissipation function) as (3.61) or, with (2.23), as 11_

2f_21tdoo ~ . 1;-00 +00_

Gjl;) =

(3.62)

These relations are known under the name spectral representation of the Green's functions. The inverse relations of (3.62) read «1>(00)

= -lim S[G+(oo+ie)] £ ..... 0

,

X" (00) = -limS[G_(oo+ie)]. (3.63) £ ..... 0

The connection to the Liouvillian representation of the recursion method is then established by the following relation between the relaxation function (3.25) or (3.43) and the Green's function G+(I;) for A=B:

Section 3-8

G+(~) = -2i$(t=O)cO(Z) = -2i z+IL

,

31

(3.64)

where ~=iz, and $(t=0) = is the initial value of the fluctuation function, expressed in tenns of the inner product as used in Sec. 3-7.

4 HAMILTONIAN REPRESENTATION This alternative representation of the recursion method is designed for the study of dynamic correlation functions of a quantum Hamiltonian system in its ground state.} An important initial task in most practical applications is therefore the determination of the ground-state wave function ICPO> of the system. There exist quite a few computational methods for that purpose. Some of them are based on the Lanczos algorithm for the tridiagonalization of large matrices [Lanczos 1950]. That algorithm can, in fact, be regarded as one of the roots from which the recursion method has originated. The connection will be established and discussed in Secs. 4-4 and 4-9. Two widely used recursive algorithms for the determination of groundstate wave functions - the modified Lanczos method and the conjugate-gradient method - will be presented in Secs. 4-5 to 4-8 along with a comparative performance test. In the meantime, for our description of the Hamiltonian representation of the recursion method, the assumption is that we know the groundstate wave function for a system of given size.

4-1 Orthogonal Expansion of Wave Functions If our goal is to determine, for a given quantum Hamiltonian H and its ground-state wave function Icpo>, the normalized correlation function,



So(t) = ---:-~,..,..-

(4.1)

of the dynamical variable represented by the Hermitian operator A, then we can accomplish that task along two different avenues. We either determine the fluctuation function (3.21) via the Liouvillian representation of the recursion method and then convert the result into the correlati?n function using the fluctuation-dissipation theorem, or we determine the correlation function (4.1) directly via the Hamiltonian representation of the recursion method as described in the following [Gagliano and Balseiro 1988]. In order to simplify the expressions for the dynamical quantities as produced in the Hamiltonian representation, we consider henceforth the Hamiltonian ii = H-Eo' whose ground-state energy has been shifted to zero. The time evolution of the wave function,

",,(t»

= A( -t)lcpo> ,

(4.2)

is governed by the Schrodinger equation,

}Any other eigenstate of the Hamiltonian would be acceptable too, but for stationary states that do not correspond to thermal equilibrium some properties of dynamic correlation functions are different from those discussed in Chapter 2.

33

Section 4-1

d

i_I'I'(t» dt

= H1'I'(t» .

(4.3)

The recursion method is based on an orthogonal expansion of that quantity:

1'I'(t»

= L Dk(t)!fy ,

(4.4)

k=O

where {!fy} is an orthogonal basis in the Hilbert space of the Hamiltonian operator H. Carrying out the expansion involves again two steps: #1 Determine the orthogonal basis {!fy} in the Hilbert space of wave functions via Gram-Schmidt with initial condition !fo> = A1o> and with the Hamiltonian as the generator of new directions. #2 Insert the expansion (4.4) into the equation of motion - here the SchrOdinger equation (4.3) - to obtain a set of differential equations for the timedependent coefficients Dk(t).

lto>

= AI~o>

It, >

=

Hlto>-aolto>

lt2>

=

HI!,> -a,lt,> -b,Yo>

(initial condition)

-+



-+

-ao

=



-

lt3>

=

2

-

2

= -a, -b,

Hlt2>- a2lt2>-b2Y,> - c2lto>



= -a, -b,

= -b,2 -+

ao =

if

=0

=0

if

2 b , -

=0

if

a, =

-+



-+

= 0

if

b2

-+

= 0

if

~=

=0



if c2 = 0

2



=--



Table 4-1: The first three iterations of the orthogonalization process generated by the Hamiltonian H.

The first three iterations of step #1 are illustrated in Table 4-1. Successive orthogonal vectors !ft> are produced by the following set of recurrence relations:

34

Chapter 4

(4.5 a)

(4.5b)

2

bk

=

= ,

(4.5c)

k=1,2,...,

<;{k-l!fk-l>

=

with !f. 1> 0 and !fo> A1cJlo>. It will turn out that the double sequence of numbers {ak' bi} is all that is needed for the reconstruction of the dynamical quantity of interest, the correlation function S(t). In step #2 the Schrodinger equation (4.3) is applied to the orthogonal expansion (4.4). The differential operator acts on the Dk(t) and the Hamiltonian on the !f~. The result is the following set of coupled linear differential equations for the functions Dit): iD k(t) = D k_1(t) +

a~k(t)

k =0,1,2,...

+ bk2+1Dk+l(t),

(4.6)

with D_ 1(t) == 0, Dk(O) = ~k.O' The first one of these functions is equal to the (normalized) correlation function we set out to determine: 2



_

S(t) _ = So(t) .

- -

S(O)

(4.7)

4-2 Structure Function The differential equations (4.6) are converted by Lap1ace. transform, dk(S)

f

= dtei~tDit)

(4.8)

,

o

into algebraic equations

(S -ak)dk(s)

-

i~k.O = d k- 1(S)

+ b;+ldk+l(S),

k=0,1,2,...

(4.9)

with d_ 1(S) == O. This set of equations can be solved for do(S) in the continued-fraction re~resentation by manipulations similar to those used in Sec. 3-4 for expression (3.25):

2Except for a slight mismatch in definition if ># O. 3We shall refer to do(~) by the name relaxation function, a term also used for the function Co(z) in Sec. 3-4. There is little danger of confusion in any given context.

Section 4-3

35

i

do(~)

2

b1

~ - ao -

(4.10)

2

b2

~ - a1 -

~ - a2

-

The normalized structure function, +00

(4.11) is then recovered directly from (4.10) via the relation So(oo)

= lim 2~[do(oo+iE)]

(4.12)

£-+0

4-3 Continued-Fraction Coefficients and Frequency Moments We recall from Chapter 2 that the structure function So( (0) is real (and equal to zero for ro
=L 00

Do(t)

n=O

•

M (-It) n

n

(4.13)

,

n!

the coefficients M2k of the real, symmetric part are identical to the coefficients M2k in the expansion (3.29) of the fluctuation function Co(t), determined via the Liouvillian representation of the recursion method. The odd moments M2k+1 can be derived from the even moments M2k via the Kramers-Kronig relations, provided the full set is known. In Sec. 3-5 we have already provided an algorithm for the conversion of the M 2k, the moments of the spectral density «1>0(00), into the continued-fraction coefficients !J.k of the function co(z), and vice versa. If we wish for the to convert the double sequence of continued-fraction coefficients ak' function do(~) as produced by the Hamiltonian representation of the recursion method into a single sequence !J.k for use in the continued-fraction analysis proposed later, then we need another algorithm - one for the conversion of the ak' into the M n and vice versa. We have derived such an algorithm from equations (4.9) applied to the asymptotic expansion

bi

bi

do(~)

= iL Mn~-(n+l)

,

(4.14)

n=O

which is obtained from the power series (4.13) by Laplace transform. The result is a set of recurrence relations, similar to (3.33) and (3.34), between the continued-

36

Chapter 4

bi

fraction coefficients ak' and the moments M n . These relations are most conveniently expressed in terms of two arrays of auxiliary quantities L k(n) and M k(n): o Forward direction: Given a set of moments Mo=l, M 1,...,M2K+I' the continued-fraction coefficients aO, ...,aK and bI,...,bi are obtained by initializing Mk(O) = (-l)kM

k'

for k

L

(0)

k

= (-l)k+1 M

k+1 '

(4.l5a)

= O, ... ,2K and then applying the recurrence relations (n-1) M(n) _ L(n-1) _ L(n-l)Mk k

-

k

--C;;:O' M _

n-I

(4.l5b)

n 1

L (n) le

M(n-I)

=

k

M(n-I) n-I

=

for k n,...,2K-n+l (in two successive inner loops) and n loop). The resulting continued-fraction coefficients are b:

o

= M~n),

an

= _L~n) "

n=O,...,K.

(4.l5c)

= 1,... ,2K (outer (4.l5d)

Reverse direction: Now the proper initialization is 2 M n(n) = bn'

L(n) = -a n n'

=0 K n- ,..., ,

(4.l6a)

(where b~ = b:1 == 1), and Mt1)

= 0,

k=O, ...,2K+l.

(4.l6b)

The recurrence relations to be carried out for n = O,... ,min(K,2K-J) (inner loop) andj = O,... ,2K+l (outer loop) are 2

bn (n-I) (n) 2 (n) M n+j+1 =bL. n n+J + -2-M n+j ,

(4.l6c)

bn - I L (n) _ M(n+l) an M(n) n+j+1 n+j+l - 2 n+j+l

(4.16<1)

bn

The moments are then given by Mn

= (_l)nM~O),

n=O,...,2K+l.

(4.l6e)

Explicit results for the transformation of the first four coefficients are given in Table 4-2. The relationship between the data of the two representations of the recursion method is thus completely established. No matter which one of the two approaches

Section 4-4

37

turns out to be the most practical one or perhaps the only one feasible, we can always express the data (or part of them) as a single sequence of coefficients Ilk' This Ilk -sequence represents a very compact code for the dynamical quantity under investigation. In later Chapters we show how to extract useful information on the spectral densities directly from its Ilk-sequence.

MI M2 M3

M4

= ao

ao

= a o2 +b l2

2

bl

22 = (ao3 + 2aob l + b l a l )

= bl2 [ao2

2 2 ~ + al + aoal + b l + b2 322 + ao[ao + 2aob l + b l all

=MI

= M 2 -M I2 3

M3 -M I

al =

2

b2 =

2

M 2 -M I

M 4 -M I M 3

-[

2

M 2 -M I

- 2M I

- M2

3 M 3 _M M I -M3 + 2M][ I 3 + Mll I 2 2 M 2 -M I M 2 -M I

Table 4-2: The first four expansion coefficients of (4.14) expressed in terms of the first two pairs of continued-fraction coefficients of (4.10) and vice versa.

4-4 Lanczos Algorithm We now return to the task that must be carried out at the beginning of most applications of the recursion method in the Hamiltonian representation: the determination of the ground-state energy Eo and the ground-state wave function IcI>o> of the system in whose T=O dynamical properties we are interested. For that purpose we wish to use a calculational technique that uses the available storage capacity and CPU time allocation most effectively. The Lanczos algorithm for the tridiagonalization of matrices has been a milestone in the design of such techniques [Lanczos 1950]. The formulation of the Lanczos algorithm presented in the following produces an orthonormal basis along with the elements of a tridiagonal matrix [Haydock 1980]. For a given Hamiltonian operator H and normalized inital state luo>, the first three iterations of the Lanczos scheme proceed as spelled out in Table 4-3. The algorithm is described by the following set of recurrence relations:

(4. 17a)

38

Chapter 4

luk+1>

for k

::

1

-[(H-ak)lu~

bk+1

-bkluk-l» ,

(4. 17b)

=0,1,2,... and with bo =0 and initial state luo>.

bllu l>

= HI"o> -CJoIUQ> -+ bl<"olu l > = -tJo~UQ> .. 0 2

-+ bI

if tJo =

= b; = (H-a~IUQ>,

bl

=+r;;[

1 -+ lu l > = -(H-ao)lUQ> bl

b21"2> = Hlul> -allu l > -bllUQ> -+ b2 = -ol-bl = -a1 -b1 2 -+ b2 ~1"2> 2

-+ b2

= +b l2 -bl -bl

= -bl2

-+ 1"2>

1 = -[(H-a1)lu l > -b1luy]

b2

b31"3> :: HI"2> -~I"2> -b2Iu l > -c2luy -+ b3 = -~ -b2 -c2 .. -b2 :: b2 -0 -b2 .. 0 -+ b3~1"3> :: ~IHluy -~ -b2 :: 0 . if a2 = :: +b2 -b2 -b2<"2IHlul>

-+ bi ::

-bi 1 b3

-+ 1"3> :: -[(H~)I"2>-b21"2>]

Table 4-3: The first three iterations of the Lanczos algorithm for the tridiagonalization of the Hamiltonian H in an orthonormal basis that starts with the given vector luo>'

The Hamiltonian H in this orthonormal basis is tridiagonal and represents the chain model mentioned previously (Sec. 1-2) for the underlying physical system - an effectively ID model Hamiltonian characterized by a site-specific local energy and a bond-specific hopping term:

Section 4-4

39

ao b I 0 0 b I a I b2 0 H=

Vi> = Hl/o>-aolfQ>

1f2>

0

b2 a2 b3 ...

0

0

b3 a3

(4.18)

...

= -2ao+a; d . -+ - = 0 If ao = --;:-;-::-_ ao k~> -+ = 0 -+

= HI/I> -allfl > -bIYO> 2 -+ = +a I <111f1> +b: -2a I -2b[ d da l

-+ - -+

=0

~ =0 2 db 1

. If a l

=....:.,..,..,....:;...

if b[ =

Table 4-4: The first three iterations of Lanczos' original scheme of tridiagonalizing a Hamiltonian matrix H. For given initial vector Ifo>' the algorithm produces the matrix elements {sk' b/} of (4.18) along with the orthogonal but not yet normalized basis {If?}.

The main advantage of the Lanczos tridiagonalization over other methods which accomplish the same task, such as the Householder method, is its very low

40

Chapter 4

memory requirements. The input consists of the first basis vector luo> and the operator rules for the recurrence relations (4.17). The orthonormal basis {Iu~} is generated along with the matrix elements {a", bi} and needs to be stored only if necessary for further calculations. Once the Hamiltonian has been brought into tridiagonal form, its eigenvalues can be determined by standard methods. For the computation of one or several low-lying eigenstates, the bisection method (for eigenvalues) combined with the method of inverse iterations (for eigenvectors) has proven to be very useful [Cullum and Willoughby 1985, Nishimori and Taguchi 1986]. In the context of the recursion method, however, the main significance of the Lanczos algorithm is a different one. Step #1 in the Hamiltonian representation is nothing but the Lanczos algorithm slightly disguised. The difference between the two sets of recurrence relations (4.5) and (4.17) (see also Tables 4-1 and 4-3) is that the orthogonal basis of the former is not normalized. We have lu~ = IA>l = Alcj>o> rather than for the determination of eigenstates. The relationship between tridiagonal Hamiltonian matrices and Green's functions in the continuedfraction representation will be further discussed in Sec. 4-8. Finally, we should like to point out that Lanczos' original formulation of his in the algorithm invokes a minimization condition for the coefficients a", recurrence relations (4.5) rather than the orthogonality condition used in Table 4-1. The two conditions are obviously equivalent. In Table 4-4, we display the first three steps of the Lanczos scheme as based on minimization criteria. In Secs. 4-5 to 4-7 we report a comparative study of algorithms (also based on minimization principles) for the computation of ground-state energy eigenvalues and eigenvectors of large Hamiltonian matrices [Nightingale, Viswanath, and Muller 1993].

bi

4-5 Modified Lanczos Method In an effort to adapt the standard Lanczos algorithm for a more direct computation of the ground-state energy and wave function of a Hamiltonian system without the intermediate step of tridiagonalization, a modified version which accomplishes precisely that was developed by Dagotto and Moreo [1985]. The modified Lanczos method, which happens to be a special case of the iterative Lanczos method discussed much earlier by Karush [1951], was first applied in a lattice-gauge-theory context and later introduced by Gagliano et al. [1986] to condensed matter applications. It has since been widely employed for the study of ID and 2D quantum spin models and models of strongly correlated electronic systems. The idea behind the method is to embed the standard Lanczos recursive cycle within another recursive cycle. The outer cycle terminates the inner one after one iteration and resets the initial condition. In practice, the two cycles make up a single loop consisting of two iterative steps. The loop is started by an initial step and terminated by a user supplied convergence criterion. Initial step: Select a (normalized) trial vector 1'1'0> for the ground state of the system, which is specified by a Hamiltonian H. 1'1'0> must have a nonzero

Section 4-5

41

projection onto the true (but unknown) ground-state wave function 14>0>. Different choices of starting vectors 1'1'0> will result in different rates of convergence. Iterative step #1: Given the kfh approximate vector l'Py . apply one cycle of the standard Lanczos algorithm to generate a vector Iyy which is orthonormal to 1'1'y:

'yy =

(H -k)I'I' y

,

(4.19)

2

Vk-i where k == <'I'~H"I'I'y. Iterative step #2: Construct a new vector l'I'k+l> in the subspace spanned by l'Py and Iyy such that it minimizes the energy. Here it is obtained by diagonalizing the 2x2 matrix with elements

(4.20)

The lower eigenvalue reads (4.21) where £k

= k and 3

2-

3

k- 3kk +2k

2(i)312

(4.22)

The associated eigenvector

l'Pk +l > =

1'1'y-vkly y

--===---

Jt+v;

(4.23)

will be the starting vector of the next iteration. Termination: Since vk in (4.22) is non-negative, the sequence Eo'£I"" is monotonically decreasing. Under normal circumstances, it converges toward the exact value of the ground state energy Eo, and the sequence 1'1'0>. 1'1'1>' ... converges toward the exact ground-state wave function 14>0>. We recommend the use of the following convergence criterion: 2

2

k-k - - - - :2 - - < k where £ is comparable to machine precision.

£,

(4.24)

42

Chapter 4

In the modified Lanczos algorithm, memory must be allocated for the simultaneous storage of three vectors, l'Py , H1'P y , H21'Py , in the kth iteration. Each iteration involves two matrix multiplications.

4-6 Conjugate-Gradient Method The conjugate-gradient method has long been known in the context of minimizing functions of several variables [Hestenes and StiefeI1952]. It was designed such that for quadratic functions in n variables the algorithm is guaranteed to converge after n steps. That property has the effect that even for more general functions the rate of convergence is typically enhanced considerably as compared, for example, to the steepest-descent (SD) method. 4 In the context of an eigenvalue problem HIx> = E1x>, the same approach can be taken for the minimization of the Rayleigh quotient R <xIHIx>/<xIx>, as was first suggested and demonstrated by Bradbury and Fletcher [1966]. The fact is that the minimum (maximum) value of R is equal to the lowest (highest) eigenvalue of H. The conjugate-gradient method represents an efficient recursive algorithm for the minimization of the Rayleigh quotient. It has proven to be a reliable computational tool in statistical mechanics, notably in the context of the transfer operator approach [Nightingale 1990]. Its algorithm is formulated here, like the modified Lanczos method, in terms of an initial step followed by two iterative steps forming a loop that is terminated by a user supplied convergence criterion [Nightingale, Viswanath, and Muller 1993]. Initial step: Select a trial vector Ixo> (not necessarily normalized) with nonzero projection onto the ground state wave function 1<1>0>. Iterative step #1: Given the ~ approximate vector Ixy , apply the gradient to the Rayleigh quotient

=

R _ <xklHIxy k - <xklx y to generate a vector Ig y which is orthogonal to Ix y

Ig y == VR k

=

2

<xklxy

(4.25)

:

[HIxy-Rklx y ]

(4.26)

Iterative step #2: Construct a new vector of the form Ixk+1>

= Ix y

+ (J,k1py

,

(4.27)

where (4.28)

40etailed discussions of the conjugate-gradient and steepest-descent methods in that context are found, for example, in Golub and Van Loan [1983] and in Press et al. [1986].

Section 4-7

43

and uk_1 = l for k~l (u_ I = 0), by minimizing the Rayleigh quotient Rk+ I of (4.27) with respect to the real parameter a.k. The condition dRk+I/dClk = 0 leads to the quadratic equation (4.29) with coefficients (4.30a) (4.30b) (4.3Oc) The larger one of the two solutions of (4.29) minimizes R k+ l • Termination: The sequence RI' Rz' ... of minimized Rayleigh quotients converges toward the exact lowest eigenvalue Eo of H, and the sequence Ix l >, Ixz>, ... of vectors converges (after normalization) toward the corresponding eigenvector 1<1>0>. The convergence criterion corresponding to (4.24) reads <Xklx J?>

----=--Z Rk

< 4£ .

(4.31)

Note that the vector IpJ?> (unlike IgJ?» is in general not orthogonal to 1xJ?>. The second term in (4.28), uk_IIPk_I>' has the effect of stabilizing the direction of the path in the Hilbert space toward the minimum of the Rayleigh quotient. This enhances the rate of convergence as will be demonstrated later. What is the most economical implementation of the conjugate-gradient method? The answer depends on whether the most valuable (or most limited) resource is (a) available CPU time or (b) accessible memory. Implementation (a) requires only a single matrix multiplication per iteration (one less than modified Lanczos) but needs memory for the simultaneous storage of four vectors (one more than modified Lanczos). Implementation (b), by contrast, requires two matrix multiplications per iteration and storage space for three vectors, just like the modified Lanczos method does. The sequence of computations in one iteration of the two implementations of the conjugate-gradient method as dictated by the above mentioned minimum requirements are summarized in Table 4-5.

4-7 Steepest-Descent Method The steepest-descent method is based on the same principle as the conjugategradient method - the minimization of the Rayleigh quotient for a one-parameter vector. But unlike the latter the former constructs the new vector Ixk+ l > from the previous one and the gradient of its Rayleigh quotient R k alone, (4.32)

44

Chapter 4

Results of (k_l)'h iteration:

0 Ixt>, Hlxt>, lPt.l>' <x!Jxt>, <x,)HIx,,>, , R" l(a)

vector addition: Igt>=(2/<x!Jxt>)[Hlxy-R"Ixt>]

l(b)

vector addition: Igt>=(2!<x!Jxt»[Hlxt>-R!Jxt>] , overwrites Hlxt>

2

inner product: , u"_l

3

vector addition: IPt>=-lgt> + u"-llp"-l>

4(a)

inner products: <x"IPt>, ,

4(b)

inner products: <x"IPt>,

5

matrix multiplication: Hlpt> ' overwrites Igt>

6(a)

inner product:

6(b)

inner products: <x~HlPt>,



7

quadratic equation: <X"

8

vector addition: Ix"+l>=lxt> + <X~Pt>, overwrites Ixt>

-

9(a)

vector addition: HIx"+l>=Hlxt> + <xtlflpt> • overwrites Hlxt>

9(b)

matrix multiplication: HIx"+l> • overwrites Hlpt>

10

inner products: <x"+11x"+1>' <x"+lIH1x"+l>' R"+l

Table 4·5: Sequence of computations to be performed during the J(h iteration of the conjugate-gradient method. Implementation (a) involves one matrix multiplication and requires memory for four vectors; implementation (b) involves one additional matrix multiplication but requires memory for only three vectors. The same sequence of computations, but with several simplifications, applies to the steepest-descent method discussed in Sec. 4-7 [from Nightingale, Viswanath, and MOller 1993].

and thus ignores the directional information from the previous iterations that is contained in the vector IPt> used in the conjugate-gradient method. Nevertheless,

Section 4-8

45

the general structure of the algorithm remains the same as in the conjugate-gradient case. Implementations (a) and (b) detailed in Table 4-5 are still applicable, but with some obvious simplifications. None of them reduces the number of matrix multiplications per iteration or the number of vectors to be stored simultaneously. It is fairly obvious that the modified Lanczos and steepest-descent methods are equivalent. For Ipy =-Igy , the coefficients (4.30) of the quadratic equation that determines the parameter (J,k in (4.32) can be expressed in terms of <X~y and <.H">k == <xklH"'lxyl<xklxy . The results for a vector Ixy , which, for mere convenience, is normalized read: (4.33a) (4.33b) (4.33c) yielding the parameter value (J,k

Vk

=

(4.34)

V
Hence the vectors (4.23) and (4.32) with (4.33) differ only in normalization. Likewise, the Rayleigh quotient Rk+1 is equal to the eigenvalue Ek+ 1 given in (4.21).

4-8 Comparative Pedormance Test Consider the 10 s=112 Heisenberg antiferromagnet N

H

= lE 8 (81+1

(4.35)

1=1

with periodic boundary conditions. The ground state for even N is known to be a singlet (Sor=0), with wave number k=O for even NI2 and k=1t for odd N12. We employ a type (b) implementation of the conjugate-gradient and steepest-descent algorithms and use translationally invariant basis vectors in the invariant subspace Sf = 0. The initial trial wave function used in both cases is a translationally invariant linear combination of Neel states:

Ixo>

=

_1[li-l-i...-l->+(-I)N12 I-l-i-l-...i>l .

{2

(4.36)

For this comparative study we have run the programs for systems of four different sizes, N=12, 14, 16, 18. Figure 4-1 shows the logarithm of the relative deviation of the eigenvalue estimate Rk from the asymptotic value Roo' plotted versus the number of iterations. The data points connected by solid lines have been obtained with the conjugate-gradient method and those connected by dashed lines with the steepest-

46

Chapter 4

descent method. The magnitude of the slope is a measure for the rate of convergence. The faster convergence of the conjugate-gradient algorithm is quite evident. Not only does it require less than half the number of iterations in comparison to the steepest-descent method, that number increases considerably more slowly with system size too. The fact is that the conjugate-gradient algorithm makes the sequence of wave vectors IxI >, Ix z>,... progress on a more direct path toward the exact ground-state wave function than the modified-Lanczos/steepest-descent algorithms do. A strong indicator of the directness of that path is the sequence of angles (4.37)

k=1,2, ...

0,-------------------------, o n=14 o n=16 t:.

n=lB

-12+--------r--------r1---lI1-~-~----_l

o

10

20

30

40

k Figure 4-1: Logarithm of the relative deviation of the eigenvalue estimates Rk from the asymptotic value R~, plotted versus k (the number of iterations) for the 1D s= 1/2 Heisenberg antiferromagnet. The value of R~ has been approximated by our best estimate, which satisfies the convergence criterion (4.31) to within machine precision. The data points connected by solid lines have been obtained from the conjugate-gradient method and those connected by dashed lines from the steepest-descent method. Data for three different system sizes are shown.

Section 4-9

47

between the vectors IPk_l>' Ipy of successive iterations. In the steepest-descent/modified Lanczos method successive directions of the path toward the exact ground state are orthogonal to one another: <'Yk-I1yy =0 and
e

e

90

75

60

45+------,-----,------,-----,------1 25 o 10 15 20 5

k Figure 4-2: Sequence of angles (4.37) between the directions of successive steps in the path toward the exact ground state as produced by the conjugate-gradient method for the case N=18 of the 1D 5=1/2 Heisenberg antiferromagnet (full circles). The steepestdescent/modified Lanczos method yields an angle of 90° between any two successive directions (open circles).

4-9 Green's Functions: Spectral and Continued-Fraction Representations The goal here is the same as in Sec. 3-8, namely to establish the link between Green's functions and the recursion method, now in the Hamiltonian representation. We shall discuss two different representations of Green's functions, both of which play a significant role in computational approaches to many-body dynamics. Consider, as in Secs. 4-1 to 4-3, a model Hamiltonian ii = H-Eo' whose ground-state energy has been shifted to zero. The zero-temperature correlation

48

Chapter 4

function for the dynamical variable A(t) (Hermitian operator) can then be expressed as follows:

s(t) :: <epolA(t)Alepo>

:: <wole-iHthIlO> '

(4.38)

where hllO> = Alepo>, and lepo> is the ground-state wave function. Upon Laplace transform, 00

fdtei~ts(t) :: <Wol~I'I'O> o

::

~-H

<Wol'l'o>do(~)

(4.39)

we arrive at the relaxation function do(~)' the quantity investigated most directly by the recursion method. At the same time we recognize (4.39) to have the structure of a Green's function,

G(~)

::



=

~-H

-ido(~)

(4.40)

,

for the same state now expressed in terms of the normalized wave function luo> = 1'1'0>1<'1'01'1'0>112. Inserting 1 = L).,lep).,><ep).,1 yields the spectral representation of the Green's function:

G(~)

::

L ).,

<ep).,luo> :: ~ -E).,

1

L

<epolAAlepo> ).,

l<ep olAlep).,>1

2

.

(4.41)

~ -E).,

The spectral representation of the structure function is then a set of &-functions, obtained from the Green's function as follows: S(ro) :: -2<ep olAAlepo> limS[G(ro+ie)] :: 21tL l<ep olAlep).,>1 20(ro-E).,). £~o

).,

(4.42)

This representation has played an important role e.g. in the finite-size analysis of the T=O dynamics of quantum spin chains [Muller et al. 1981]. In those applications, expression (4.42) was evaluated from the complete set of eigenfunctions lep).,> and eigenvalues E)., determined by numerical diagonalization of H. For a system with a dynamically relevant subspace of finite dimensionality, the same set of 0functions (4.42) results via (4.12) from a finite continued fraction. In Sec. 4-4 we have presented the Lanczos algorithm as one of several iterative methods for the computation of the ground-state wave function of a model Hamiltonian. For any given normalized initial state luo>, the recurrence relations (4.17) produce an orthonormal basis of the invariant subspace in which luo> is located and a tridiagonal Hamiltonian matrix in that basis. For the eventual determination of the ground-state wave function the vector luo> can be chosen arbitrarily as long as it has a nonzero projection onto the ground state. For the particular initial state luo> = <wol'l'0>-1I21'1'O>, where 1'1'0> =Alepo> and lepo> is the ground state of H previously determined by the same method, the Lanczos algorithm produces a tridiagonal matrix (4.18) whose elements are at the

Section 4-9

49

same time the continued-fraction coefficients of the Green's function (4.40) or of the relaxation function (4.10). The connection between the tridiagonal Hamiltonian (4.18) and the continued-fraction representation of G(~) is readily established if we recognize that the resolvent (4.40) is in essence the OO-element of the inverse of the matrix previously determined by the Lanczos algorithm in tridiagonal form [Haydock 1980, Grosso and Pastori Parravicini 1985]: 1 - 1 = = [~l-H]~

G(~)

~-H

.

(4.43)

For a finite tridiagonal matrix fi, that matrix element can be expressed as the ratio of two determinants as follows:

G(~)

=

Tl(~)

.

(4.44)

To(~)

Here To(~) is the determinant of the full matrix ~l-fi, whereas Tl(~) has the first row and column omitted. Because of the tridiagonal structure of the matrix ~l-fi, these determinants can be evaluated by means of simple recurrence relations. Denoting by Vk(~) the determinant comprising the first k rows and columns of ~l-fi, we have (4.45a) (4.45b) (4.45c) and, for the general case, the recurrence relations

Vk+l(~) = (~-ak)Vk(~) -b;Vk_l(~)'

k =0,1,2, ... ,N

(4.46)

with Vl(~) =0 and Vo(~) = 1. Once the determinant To(~) = VM:~) of~l-fihas been composed in this way, these recurrence relations can be inverted and brought into the form

Tk(~)

_ Y_

-=-......,..",.,.. Tk+l(~)

~

ak

-b 2 Tk+2(~) k+l '

Tk+l(~)

k=0,1,2, ...

(4.47)

Inserting (4.47) into (4.44) recursively, yields the Green's function in the continuedfraction representation

50

Chapter 4

G(~) ~

b

-a _

o

~

Z 1

Z bz

(4.48)

-a 1 - - ; ; - - ~

-a z - ...

which is equivalent to the relaxation function (4.10). The original assumption that the matrix ~l-fj is finite can be abandoned without consequences other than that the infinite tridiagonal matrix yields an infinite continued fraction.

5 GENETIC CODE OF SPECTRAL DENSITIES It is tempting to invoke this term from biology to characterize the relationship between the recognizable information stored in the ~k-sequence and the properties of the associated spectral density. The analogy has some validity in three aspects: o The ~k-sequence is a code of retrievable information on specific properties of the spectral density (band structure, singularity structure, decay laws for (O~oo etc). o The ~k-sequence is generative in nature. It produces the relaxation function in the continued fraction representation, which, in turn, determines the spectral density. o The ~k-sequence leaves room for contingencies if convergence criteria are violated. The exact shape of the spectral density is then no longer determined uniquely by the continued-fraction coefficients. The main goal of this Chapter is to familiarize the reader with some of the patterns found in ~k-sequences that are readily translated into specific properties of the associated spectral densities. In subsequent applications we shall see that these properties are relevant (sometimes crucial) for the interpretation of the underlying physics. In addition to this primary goal, we wish to elucidate some interesting mathematical relationships between ~k-sequences and spectral densities, which are quite illuminating even if they have only little bearing on typical condensed matter applications.

5-1 Finite

~k-Sequences

The most elementary situation occurs if an application of the recursion method for the evaluation of a spectral density terminates spontaneously and thus produces a finite sequence, ~!' ... , ~K' Technically, the algorithm of the recursion comes to a natural stop when the basis generation procedure (3.19) yields a vector A+! of zero norm. This implies that ~K+! = 0, which terminates the continued-fraction representation (3.25) of the relaxation function at the K h level: co(z)

1

= --------~!

Z + --------

z

~

+ ------

z

+

(5.1)

~K

+-

z

Likewise, if the recurrence relations (4.5) produce the vector !fK+!> = 0, the implication is that bk+! = 0, which terminates the continued fraction (4.10). That function in turn has been shown (in Sec. 4-3) to be related to a function of the form (5.1).

52

Chapter 5

Expression (5.1) is a rational function Px:<.Z)/qK+I(Z). The K+ 1 roots of the polynomial qK+I(Z) are all located on the imaginary z-axis. In general, K is odd, and the K+1 poles of co(z) correspond to L (K+1)/2 distinct frequencies 001, [=l, ...,L, which govern the time evolution of the function Co(t). The spectral density inferred via (3.28) from (5.1) then consists of L pairs of O-functions,

=

L

<1>0(00)

= 1&.E ut[B(ro-OOt) +B(ro+cot)]

(5.2)

,

t=1

and the fluctuation function is multiply periodic: L

Co(t) =

.E utcos(cott) .

(5.3)

t=1

In other words, the first frequency, 001, in the time evolution of Co(t) contributes one coefficient, AI' to the continued-fraction representations of co(z). Every additional frequency, OOt' [= 2,...,L, contributes two additional coefficients, Ak , k = 2,...,K, K 2L-1. For £=1, we have the function Co(t) cos(OOlt). Its Laplace transform

=

=

(5.4)

2

001 Z +Z

=

is characterized by a single continued-fraction coefficient, Al 0012. For £=2 we consider the fluctuation function Co(t) = [cos(00It)+cos(002t)]l2. Its Laplace transform, expanded into a continued fraction, indeed terminates at the third level and yields the coefficients Al

1

2

2

1

2 . 2

= -2(~ +~), Az = -2(0J] +m.:.) -z

'll.:ll·

(,t

~ --Z

- -2 -2 . , A3 001 +~

2('--I--Z :?:' (,t =-. 22

(5.5)

001 +~

If one of the L frequencies OOt happens to be zero, the total number of continuedfraction coefficients is reduced by one to K = 2L-2. Hence, an even number of nonzero Ilk's indicates that one of the frequencies is zero. For an illustration of this fact, consider the example with two frequencies. The coefficient A3 in (5.5) goes to zero if either 00 1 or ~ vanishes.

5-2 Spectral Densities with Bounded Support Under most circumstances, dynamic correlation functions of many-body systems have a continuous component in the spectral-weight distribution. Any such spectral density is described by an infinite Ak-sequence. No spontaneous termination of the recurrence relations (3.19) or (4.5) takes place in this situation. The simplest category of Ak-sequences characterizing functions <1>0(00) with continuous spectralweight distributions are those which converge, as k~oo, to some finite value A...

Section 5-2

53

This property implies that the spectral density Cl»o(c.o) has bounded support with all the spectral weight of the continuous part confined to the interval on the frequency axis. Moreover, the endpoint frequency c.oo is determined by the asymptotic value of the L\k-sequence:

[-roo, roo]

L\oo =

I 2

(5.6)

4C% .

More detailed properties of the spectral density, e.g. information on the singularity structure, can be extracted from the Ilk and liP correction terms in an asymptotic expansion of the L\k-sequence, as will be discussed in the following. Spectral densities with no singularities except at the endpoints of the band are characterized by L\k-sequences which converge uniformly toward the asymptotic value (5.6). For a demonstration, consider a model spectral density of the form

-(21l+1)

2

Cl» (c.o\

o?

= 1tc%

B(l/2,I+P)

( 2 _cJ)1l Ic.oI< A 1 C% ; -C%, ... >- .

(5.7)

Its L\k-sequence is known in closed form [Magnus 1985]:

~k(k+2P)

L\ - ~~n--=:-:--=-~ k -

(5.8)

(2k+2p -1)(2k+2p +1)

The first two terms of its asymptotic expansion read

_ 1 2 1-4p2 L\k - -C%[l + - - + ... ] . 4

4k 2

(5.9)

Note that the square of the exponent p, which specifies the singularity at the band edge in the spectral density, is determined solely by the amplitude of the leading liP correction in (5.9). Hence, sequences that converge from above toward L\. have p2<1I4, while convergence from below implies P~1I4. This is illustrated by the two cases shown in Fig. 5-1 of the model spectral density (5.7) and the corresponding L\k-sequences. The special case p=1I2 of the spectral density (5.7) is represented by the completely uniform sequence

roo

L\l = ~ = ... =

1 2 4

(5.10)

-C% '

and the case P=-1I2 by the almost uniform sequence L\l =

1 2

-C%, 2

~ = L\3 = ... =

1 2

-C% . 4

(5.11)

The fluctuation functions for these two cases are Bessel functions, Co(t) = Jo(root) + J2(c.oot) and Co(t) = Jo(c.oot), respectively. The value p2 = 114 pertains to any sequence with L\k=L\. for all

k~.

54

Chapter 5

A k

A k

(J=1

, \

foI~4

foI~4

(J=o

.......... ---.- ....

\

k

;-...

3 ........

k

0

t9t

------1

1-----1 1 1 1 1 1 1 1 I I

1 1 1 1 1 1 1 1 1 I

0

-(,10

(,10

CJ

Figure 5-1: Two examples of spectral densities with compact support and endpoint singularities only: expression (5.7) with ~=1 (solid line) and ~=O (dashed line). The A/C sequences (5.8) for the two cases are displayed in the insets.

The first correction tenn, (l-4~2)/4~, in (5.9) is by no means an exclusive property of the model spectral density (5.7). It is conjectured to be a universal property of spectral densities which have bounded support and no singularities except at the band edges. In later Chapters we shall demonstrate how endpoint singularity exponents can be determined from the analysis of Llk-sequences pertaining to problems of interest in many-body dynamics. Now consider spectral densities which have compact support on the interval [-000, roo] and an additional singularity at ro=O (infrared singularity). They are characterized by Llk-sequences which converge in an alternating approach toward the asymptotic value (5.6). For a demonstration of this characteristic property, we consider again a simple model spectral density for which the continued-fraction coefficients have been determined in closed fonn [Magnus 1985]: 2 «1>o(ro)

=

-(a+213 +1)

1t0\)

l+a

loola(ot-cJl,

B(_,I+~)

2

IroI:S;O\),

a,~>-I.

(5.12)

Section 5-2

L\2k =

~

4~k(k+~)

, ~+l =

(4k+2 +a-l)(4k+2~ +a+I)

55

~(2k+a+1)(2k+2~ +a+I) (4k+2~ +a+ 1)(4k+2~ +a+3)

(5.13)

The first three terms of the asymptotic expansion are the following:

{i;

= .!.C%[l - (_l)k~ 2

1-4~2+(-I)k2a(2~+a)

+

+ ...].

M2

U

(5.14)

It is conjectured that at least the leading correction term, (-l)ka/2k, is universal in the sense that it is valid for a large class of spectral densities with bounded support and an infrared singularity. Note that if the L\2k_l converge from below toward l:1.o and the L\2k from above, this is the signature of a divergent infrared singularity (aO). This relationship is illustrated in Fig. 5-2 for two cases of the model spectral density (5.12). The determination of singularity exponents from the analysis of L\ksequences produced by the recursion method will be demonstrated in later Chapters for several applications.

Ak ~-------.., a--1/2. fJ=O

Ak ~-------., a=I, fJ=1

k

/ I I I I

I I I I

/

/

/

/

/

... / I .

k

-,,

,

\

\

\

\

/

\

\

\

\

\

\

\ \

\

\ / \/

/

o

cv

I

/ /

I

I

/

... -,,

\

\

\

\

\

\ \ \ \ \ \ \ \ \ \

"'0

Figure 5-2: Two examples of spectral densities with compact support and an infrared singUlarity in addition to the endpoint singularities: expression (5.12) with a=-1/2, ~O (solid lines) and a=~=1 (dashed lines). The ak-sequences for the two cases are displayed in the insets.

56

Chapter 5

5-3 Spectral Densities with Bounded Support and a Gap An additional degree of complexity is introduced, if we allow for a gap centered at eo=<> in a spectral density of bounded support. For any such function, the ~2k-l and the ~ converge toward different asymptotic values as k ~ 00. The prototype for that category of spectral densities is the one whose ~k-sequence is periodic with period two: (5.15) The continued-fraction representation of the relaxation function co(z) specified by the coefficients (5.15) can be terminated by the same function at level two: co(z) ::: - - - - - ~o

(5.16)

Z + -~-­ z + ~ecO(z)

This expression amounts to a quadratic equation for co(z). The physically relevant solution reads 1 [

co(z) ::: _ . 2~e

2(~o+~e) +z 2 +

(~ _~)2e

(5.17)

0

Z2

and the spectral density inferred via (3.28),

with (5.19) This prototype spectral density is depicted in Fig. 5-3 for a case with ~o>~e (solid lines) and a case with ~o<.Ae (dashed lines). In both cases the function $0(00) has a continuous part confined to the frequency intervals oornin < 1001 < oornax and with square-root cusp singularities at the band edges. However, if ~e>~o' then the spectral density has also a ~-function central peak. When we set ~e=~o' the gap goes to zero, the central peak disappears, and (5.20) reduces to (5.7) with ~=1/2. Now consider the function (5.20) representing a spectral density of that category. For any such function, the ~-1 and the ~ converge toward different asymptotic values, ~~o) and ~e), respectively. If the central peak is present (A > 0), then the ~k-l converge toward the lower

Section 5-4

asymptotic value (A~o) where

= AL )

and the ~k toward the higher value (A~e)

AL = {(Ofuax -Ofuin)2, An = {(Ofuax + Ofuin)2 ;

57

= An), (5.21)

otherwise (A=O) we have A~o) = AH and A~e) = AL" In either case, the rate of convergence depends on the function F(ro).

~k , . . . . - - - - - - - - - ,

~k , . . . . - - - - - - - - - - ,

~

1\

o

e

1\ o

1\

e

-Col

max

-Col

•

mm

o

k

CoI

min

Col

max

Figure 5·3: Prototype spectral density (5.18) with bounded support and a gap for the two cases !J.q>!J. e (solid lines) and !J.o
5-4 Spectral Densities with Unbounded Support In many-body dynamics, spectral densities with bounded support are realized only under exceptional circumstances, for example, when the relevant degrees of freedom are noninteracting or when selection rules screen out all but a very limited class of excitations. Under normal circumstances, spectral densities of many-body systems have unbounded support, Le. their spectral weight extends to ro~±oo. Such spectral densities are characterized by Ak-sequences which grow to infinity as k~oo. The growth rate of any such Ak-sequence is defined as the exponent A. which characterizes its average power-law growth,

58

Chapter 5

A.

k

(5.22)

kt.. ,

-

asymptotically for large k. In any given realization, the A.k may scatter considerably about the mean growth curve. These deviations determine the detailed shape of the spectral density, specifically its singularity structure. However, the growth rate alone determines the decay law of the spectral density,

$o(ro) = exp( -ofII..) ,

(5.23)

asymptotically for large ro [Magnus 1985, Lubinsky 1987]. In Chapter 7 we shall discuss the role of the growth rate of A.k-sequences for the identification of universality classes of dynamical behavior. The limiting decay law with growth rate A.=O corresponds to spectral densities with bounded support, which we have discussed in Secs. 5-2 and 5-3. A.k-sequences with linear growth rate (A.=I) are a common occurrence in quantum many-body dynamics, as we shall see in the applications discussed later on. This type of behavior is exhibited by the following model spectral density:

$ o(ro)

=

IX

21t ro e -oh~ . Cl\!r( a + I) Cl\!

(5.24)

2

The corresponding model fluctuation function is a degenerate hypergeometric function, eo(t)

1 1 2 2 = $ (a+l - ; - ; --%1 ) .

2

2

(5.25)

4

The A.k-sequence for this model spectral density is known in closed form: ~-l =

I 2

'2Cl\!(2k-1 +a), A.2k =

I 2

'2 CJt(2k)

.

(5.26)

In this simple pattern, the (average) linear growth of A.k determines the Gaussian decay of $o(ro) at large ro, the slope of the line ~ determines the frequency unit COo in (5.24), and the vertical displacement of the ~-l from that line determines the exponent a of the infrared singularity (see Fig. 5-4 for two specific cases of that model spectral density and its A.k-sequence). Growth rates 1..>1 are not uncommon in many-body dynamics, especially in classical systems with nonlinear dynamics. Consider the model spectral density,

21t IA.c%

ro

IX

' $o(ro) = --,.----1 exp( -lro!Cl\!l v ",) , n A(l +a)] Cl\!

(5.27)

2

of which (5.24) is a special case. We know the frequency moments of (5.27) in closed form,

Section 5-4

= ~n A(l +a +2k)]tr[ A(1 +a)]

M 2k

2

2

59

(5.28)

,

but not the ak-sequence. However, the latter can be generated numerically from moments by means of the transformation formula (3.33).

fo k . . - - - - - - - - - _ _ , ,

a=-1/2 k

k

;'

--

,'"

...

;';'

/

/

/

/

;'

... - ....,

\

I

\

\

\

\

I

\ \

;'

\

,

//

I

I

/

;'

...-....,,

\

I

\

I

\

\

\ "

o

"

...........

_-

"'0

CA)

Figure 5-4: Model spectral density (5.24) with unbounded support and Gaussian decay at high frequencies for the two cases a=-1/2 (solid lines) and a=2 (dashed lines). The Aksequences (5.26) for the two cases are displayed in the insets [adapted from Viswanath et al. 1994].

For the special case A.=2 of (5.27), which corresponds to a quadratic growth rate of the ak-sequence and an exponentially decaying spectral density, we obtain the following expression for the fluctuation function: cos[(1 +a)arctan(Cl\>t)]

Co(t)

= ----=----(1 +

ott

(5.29)

2)(0. ...1)/2

For <X=O, this reduces to a Lorentzian, whose spectral density is a simple exponential: ,cI»o(ro) = _l_e -lro/C%' •

1

1

+ott

2

Cl\>

(5.30)

60

Chapter 5

However, the simplest pattern with quadratic growth rate, the uniformly quadratic dFsequence, (5.31)

belongs to the functions Cl> o( (0)

7t 7t00 = _sech(_),

Cl\!

2Cl\!

= sech(Cl\!t)

Co(t)

.

(5.32)

For fluctuation functions CoU) which are entire functions of (complex) time the growth rate A. of the d k-sequence is expressible as

A. = 2(p -1)/p

(5.33)

in terms of the growth order p. The latter quantity specifies the growth of Co(t) for large imaginary times [Roldan, McCoy, and Perk 1986]: Co(it) - exp(t p )

•

(5.34)

A proof that Co(t) is entire exists only for fairly special circumstances [Araki 1969]. Note that the growth rate A.=2 represents the limiting case of infinite growth order (p=oo). dk-sequences with growth rates Q2 do not represent fluctuation functions which are entire functions of t. One case in point is the function Co(t) in (5.32), for example. It is a realization of A.=2 and evidently not entire. For Q2, the continued fraction representation (3.25) does no longer converge; the moment problem does not have a unique solution; the spectral density cannot be reconstructed uniquely from the d k-sequence. The difference between spectral densities with the same d ksequence is, in general, an oscillating function.

5·5 Spectral Densities with Unbounded Support and a Gap The presence or absence of a gap in a given spectral 'density of a many-body system may serve as an important indicator of a specific type of ordering in the system and provide crucial clues on the nature of the excitations (quasi-particles) which govern the dynamics of that system. In Sec. 5-3 we have already learned how to recognize the presence of a gap in spectral densities with bounded support and how to determine its size by analyzing the dk-sequence produced by the recursion method. How do we recognize by the same methods the presence of a gap in spectral densities with unbounded support? What are the characteristic patterns in the associated dk-sequences that indicate the presence of a gap, and what are those that indicate the presence of a 0(00) central peak in the gap? Consider the model spectral density cI>0(00)

= 27tAB(00) + 2{i (l -A)e(lool-O)e -(ICOI-n>2tot. '%

(5.35)

It has unbounded support and a gap of width 20 centered at 00=0. For A=O and 0:0, expression (5.35) reduces to a pure Gaussian, whose dk-sequence grows linearly

Section 5-5

61

with k, !:ik = roo2k12, as discussed in Sec. 5-4. For the investigation of the effect of a gap and the effect of a central peak in the gap on that !:ik-sequence, we have detennined the moments of (5.35) in closed fonn [Viswanath et al. 1994], k (

M 2k = 21t(1-A)L

m=O

)

2md-(k-m)

2k Ob 2m 2m

(2m-l)!! (5.36)

+ 2{i(1-A)E ( 2k l02(k-m)-lroo2m +1m !, k=I,2, ... m.() 2m+l

J-

from which the !:ik-sequence can be derived numerically by means of the transfonnation fonnula (3.33). The results for two cases are plotted in Fig. 5-5. The effect of the gap is to split the !:ik-sequence into two subsequences !:i2k and !:i2k_l that still grow (roughly) linearly, but with different slopes. In the absence of a central peak, the !:i2k_l grow more steeply than the !:i2k (solid lines, left inset). If the ~ grow more steeply, this is an indicator that a central peak is present (dashed lines, right inset).

1\ ,----------::>1

k

k

-0

o

o

c.J

Figure 5-5: Model spectral density (5.35) with unbounded support and a gap for the two cases A=O, roo=2!l (solid lines) and A=1/2, roo=20 (dashed lines). The 4 k-sequences for the two cases are displayed in the insets [adapted from Viswanath et al. 1994].

62

Chapter 5

The same effect of a gap and a central peak within the gap can be observed even more clearly in a class of model spectral densities with growth rate 1..=2. Consider the two Jacobi elliptic functions .... (c)

q.oo

_

(t) - cn(C%t,K),

.... (d)

q.oo

_

(t) - dn(C%t,K) ,

(5.37)

which we assume to play the role of (normalized) fluctuation functions of some hypothetical dynamical system. For Kbc )(oo) and 4>bd)(oo) consist of infinite sets of ~functions at equidistant frequencies. For large 00, the spectral weight of these individual lines approaches zero exponentially, in agreement with (5.23) for 1..=2. The discreteness of the spectrum implies an energy ga~. Consider first the function 4>bc (00). The size of the gap is 41t divided by the period of cn(root,K). There is no central peak. The ~k-sequence for that spectral density is known in closed form:

141-1

= olo(2k-1)2,

141

= olor(2k)2 .

(5.38)

In a plot ~1c) versus 0 we have again two lines with different slopes similar to the example with growth rate 1..=1 depicted in Fig. 5-5 (inset left). The main difference of the spectral density 4>bd)(oo) with respect to 4>bc )(oo) is that the former has one additional line at ro=O - the central peak in the gap. The consequences for the ~k-sequence are very similar to what we have already observed in the context of Fig. 5-5:

~-1

= olor(2k-l)2,

~

= olo(2k)2 .

(5.39)

tJ4P

It is now the subsequence which takes off more steeply in a plot ~1d) vs 0. All combined, the analysis of the ~k-sequence as obtained by the recursion method may enable us to make a reliable prediction about the existence of a gap and the presence of a central peak within the gap.

5-6 Orthogonal Polynomials It is appropriate to conclude this Chapter, which is all about the relationship between patterns in the sequence of continued-fraction coefficients and properties of the spectral densities, with yet another such relationship. For that discussion we use the concepts developed in Chapter 4, specifically the double sequence of numbers {ak> b~}. They play the role of continued-fraction coefficients in the relaxation function (4.10) or the role of matrix elements in the tridiagonal Hamiltonian (4.18) as produced by the Lanczos algorithm. We recall that for a given Hamiltonian H and a given (normalized) initial state 1"0>, the {ak> b~}-sequence of the (normalized) structure function 5 0(00) in whose properties we are interested, is determined iteratively by means of the recurrence relations (4.17). The same set of recurrence relations can now be turned around and be used for given {ak' b~} to generate a complete set of orthogonal polynomials Pk(oo),

Section 5-6

1

Pk+1(ro) = -[(ro-ak)Pk(ro) -b~k-l(ro)], k=0,1,2, ...

bk+1

63

(5.40)

with initial conditions P_1(ro)=0, Po(ro)=1 [Grosso and Pastori Parravicini 1985]. The structure function So(ro) associated with the double sequence {ak' b~} plays the role of a weight function in both the orthogonality and completeness conditions: +00

(5.41)

So(ro)L Pn(ro)Pn(cJ)

= 21tO(ro-cJ)

.

(5.42)

n

Table 5-1 lists some of the better known sets of orthogonal polynomials along with the associated structure function and continued-fraction coefficients. Orthogonal Polynomials

Frequency Interval

Tchebycheff 1sI kind: Tn(ro)

[-1, 1]

Tchebycheff 2nd kind: Un(ro)

[-1, 1]

Legendre: Pn(ro)

[-1, 1]

Laguerre: Ln(ro)

[0, 00]

Hermite: Hn(ro)

[-00, 00]

Structure Function

2

~ 4~ 1t fXl -loo)

Continued-Fraction Coefficients ak = 0

b~ = (l +Ok,I)/4 ak = 0

b~ = 1/4 ak = 0

b~ = ~/(4~-1) 21t e -Q)

ak = 2k+l

b~ =~ {41te;d

ak = 0

b~ = k/2

Table 5-1: Selected sets of orthonormal polynomials and associated weight functions (structure functions 80(00)) generated by the recurrence relations (5.40) from specific double sequences {ai<' ~} [adapted from Grosso and Pastori Parravicini 1985].

6 RECURSION METHOD ILLUSTRATED The goal of this Chapter is to introduce the non-expert reader to the practicalities of the recursion method and to demonstrate its elegance and aesthetic appeal. All model systems considered in this first round of applications are exactly solvable by one or several methods including the recursion method. We begin with the ultimate overkill of the perennial toy model of physics teachers - the harmonic oscillator. Then we move on to more complex models, but we analyze them only for special situations associated with simple dynamical behavior. The focus here is entirely on the use of calculational techniques. In later Chapters we shall return to some of these same model systems and analyze their complex dynamical properties under more general circumstances. 6-1 Harmonic Oscillator The Hamiltonian of the quantutn harmonic oscillator, expressed in terms of momentum and displacement variables or in terms of boson creation and annihilation operators reads (6.1)

where

q =

1

J2mroo

(a t +a). p •

[a,a t ]

= 1,

iJ 7

[q,p]

=i

(6.2)

(a La) ,

(6.3)

.

We wish to evaluate the structure function

f dte

+00

S(o» ==

iOlt

(6.4)


of the displacement coordinate q(t). We perform this simple task along six distinct paths. Four of them employ the recursion method, the remaining two use retarded Green's functions. Path #1: The Hamiltonian representation of the recursion method can be used for the direct calculation of the structure function (6.4) at T=O. The ground state of (6.1) has zero boson occupancy and is denoted by 10>. The recurrence relations (4.5) terminate spontaneoulsy during the first step, .

!fo> = qlO> =

J

1

2mroo

11>,

ii = H -Eo = rooa t a

,

(6.5a)

Section 6-}

65

(6.5b)

aO

2

(6.5c)

= '%' If}> = 0, hI = 0 ,

and yields the relaxation function (6.6) from which we infer the structure function S(ro)

= lim 29t[dO(ro+ie)] = ~5(ro-,%) m,%

E~O

.

(6.7)

Path #2: For the calculation of the same structure function of the quantum harmonic oscillator at arbitrary temperature, we employ the Liouvillian representation of the recursion method. The most direct path toward that goal uses the inner product (3.9). The first iteration of the recurrence relations (3.19) is carried out as follows:

10

'=

(6.8a)

q,

(fo,/o) = +} = _1_ 0 2m,% 2m,%

+2n B)

,

(6.8b)

where nB=[exp(~roo)_l]-1 is the Bose distribution. Here the spontaneous termination occurs during the second iteration:

I 1 '=

iLlo

= i[H,q] = !!... ,

(6.9a)

m

12 = iLII +/).tfo = ~[H,p] +o:tq = 0, ~ = 0 . m .

(6.9c)

The resulting relaxation function reads co(z)

1 ill = --..".. = -{-+ --} ,%2 2 iz-,% iz+,% z +Z

The spectral density follows from (3.28),

.

(6.10)

66

Chapter 6

= 2
<<1>(0))

£~o

= _1t_ O +2nB){0(00-000) 2mOOo

(6.11) + 0(00+000)} •

and the desired structure function in turn from (2.23): S(oo)

=

2

1 +e -pm

«1>(00)

1t = -(O +nB)o(oo-OOo)

+ nBo(oo+OOo)] . (6.12)

mOOo

For T~O. this expression reduces to the previous result (6.7). Path #3: The same task can be accomplished somewhat more indirectly if we use the Kubo inner product (3.5) instead of (3.9). For the first two vectors/o and/I. we then obtain

1

1

~mOOo

pm

(fOJO)K = - - 2 • (fI JI)K = ~ ,

(6.13)

replacing (6.8b) and (6.9b). The relaxation function cO(Z)K is identical to (6.10), but the spectral density «I>(oo)K

= (foJo) lim 29t[co(£ -ioo)K] 1tk T

£~o

= _B_{ 0(00-000) 2

(6.14)

+ 0(00+000)}

mOOo has not the same T-dependence as (6.11). In order to arrive at the structure function (6.12), we must first determine the dissipation function from (6.14) via the classical fluctuation-dissipation theorem (2.25),

x"

(00) = _1t_{0(00-000) - 0(00+000)} ,

(6.15)

2mOOo

and then apply the quantum mechanical fluctuation-dissipation theorem (2.23). Path #4: Now we reinterpret expression (6.1) as the energy function of the classical harmonic oscillator and determine the structure function (6.4) via the Liouvillian representation of the recursion method in its classical formulation (see Sec. 3-2) [Florencio and Lee 1985]:

10 = q,

(fo,/o)

=
(6. 16a)

'

~mOOo

11 = iLlo = {q,H} = (JH = .!!... , (fI J I) = _1_ , a I = ~ , (Jp

m

~m

12 = iLII +aI/o = -!.{p,H} +~q = O. ~ = 0 m

.

(6.16b)

(6.16c)

Section 6-1

67

The resulting relaxation function is the same as that obtained for the quantum harmonic oscillator - expression (6.10). For a classical system, the spectral density and the structure function are equal. Hence we obtain:
7tk T

= S(oo) = _8_[0(00-<%) m<%2

+ 0(00+<%)] .

(6.17)

This result is also recovered from the quantum results (6.1)) and (6.12) for the quantum harmonic oscillator by taking the classical limit.! Path #5: Returning to the quantum harmonic oscillator, let us determine the structure function (6.4) by solving the equation of motion of the retarded Green's function «q;q»~ as defined in Sec. 3-8. Equation (3.60) employed hierarchically produces a closed set of two equations,

~<
=

..!-<
(6.18a)

m

r':I<
2

-

(6.18b)

.

im<%<
From the solution «q;q»~

1

I - -r-- l 2m<% ':1-'% ':I +<%

1/m

1

=- - = --{-r~2 _~

(6.19)

the dissipation function (6.15) is derived via (3.63) and the structure function (6.12) in turn via (2.23). Path #6: If we use instead the retarded Green's function «q;q»t, Eq. (3.60), used hierarchically, produces another closed set of two equations,

r':I< r

i

+

+ -<
m

,

2 + = -lm<%<
(6.20b)

+.

':I<
(6.20a)

The solution

<~ = {_I_ ~2_~

~-<%

+ _1_ l

~+<%

(6.21)

in combination with the result (6.8b) yields the fluctuation function (6.11) via (3.63), and from that the structure function (6.12) via (2.23).

1For that purpose factors of " must be reinserted where they belong in the quantum expressions (see Chapter 2).

68

Chapter 6

6-2 Spin Waves Consider the ID s= 1/2 Heisenberg model, specified by the Hamiltonian N

H

= -lE S "SI... 1

(6.22)

1=1

with periodic boundary conditons. Even though this model is completely integrable i.e. exactly solvable ~ the Bethe ansatz [Bethe 1931], its dynamical properties are far from being fully understood. Exact results for dynamic correlation functions of this model exist only for T=O, where the degree of complexity for the ferromagnet (l>{») reduces to that of the harmonic oscillator. The two-spin correlation function <Sl(t)Si+1> is then governed by linear spin-wave excitations alone. The determination of the dynamic structure factor

Sxx(q,ro)

=E

e iqn

n

......

f

(6.23)

dteirot<st(t)SI:n>

in the ferromagnetic ground state, l«!lo> = IF> = liit....i>, with all spins aligned in the positive z-direction is then just another simple exercise. The ground-state energy is Eo = -N/4. In the Hamiltonian representation of the recursion method, the following steps lead to the desired result:

!fo> = SqXI F>, Sx= q

l~iqlsx

(6.24a)

-LJ e l '

1N l

1

(6.24b)

= '4' ao = roq = l(1-cosq) ,

(6.24c)

Sxx(q,~

= 21t 0 (CO-roq)

(6.24d)

.

In the Liouvillian representation, different steps lead to the same result:

(6.25a)

= r;;;LJ 1 ~eiql{SZSY (e iq -l) f 1 = I'[HSx] 'q I 1"'1 VN I

+

SYSZ (l-e iq )} I 1...1

(625b)

,.

(6.25c)

Section 6-3

«I».u(q,ro)

69

= 2(fo,fo) lim 9t[co(£ -iro)] £~o

=

(6.25d)

~ {O(ro-roq)+O(ro+roq)} , 7t

(6.25e)

S.u(q,ro) = ZO(ro-roq ) .

This exact solution is readily generalized to spin quantum numbers s> 112 and to higher-dimensional lattices. Now suppose we wish to determine the structure function S.u(q,ro) or the spectral density «I».u(q,ro) for the Heisenberg antiferromagnet (J is known to decay algebraically as -(-1)nln for large distances [Luther and Peschel 1975]. o No spontaneous termination takes place in the application of the recursion method, implying that the structure function S.u(q,ro) for fixed wave number now has a continuous spectrum. Any serious attempt to deal with these complications would employ one of the algorithms presented in Secs. 4-4 to 4-7 for the determination of the true (finitesize) ground-state wave function and then proceed in the Hamiltonian or the Liouvillian representation of the recursion method. In Sec. 4-8 we have already reported how to execute the first part of that calculation for the model system at hand. How best to carry out the remaining tasks will be discussed in Chapter 11.

6-3 Lattice Fermions Consider the ID s=1I2 XXZ model, specified by the Hamiltonian N

H

= -E [J(StSI:1

+

1=1

S/SI~I)

+

(6.26)

J.J/SI~d .

It is well known that this model is equivalent to a system of spinless lattice fermions [Lieb, Schultz, and Mattis 1961; Katsura 1962] H

= E £o(k)a1a k k

+

E

_1 V(q)p(q)p(-q) , 2N q

£o(k) = -Jcosk, V(q) = -2Jzcosq , p(q) =

E ala

(6.27a)

k+q'

(6.27b)

k

The mapping between these two representations of the same system is provided by the Jordan-Wigner transformation,

70

Chapter 6

t [.11tL-Ja ~ ta ] S1+ == S1x +1·S1y .. a,exp j j

,

j<1

(6.28)

S,- == S/-iS/ .. exp[-i1tEa]aj ]a/, jd

combined with a spatial Fourier transform. For even N, the discrete fermion wave numbers are [McCoy, Barouch, and Abraham 1971] 21t N

N 2

I 2

k .. ±_(n+_) , n "0,1,2, ...,_-1 21tn N k .. 0,±_,1t , n "1,2, ...,_-1

N

2

N (_ even) , 2

(6.29a)

N odd) . (_ 2

(6.29b)

The dynamic structure factor Su.(q,ro) is then essentially the space-time Fourier transform of a fermion density correlation function. That function is readily evaluated in the noninteracting limit Jz=O [Niemeijer 1967], e.g by standard Green's function methods [Katsura, Horiguchi, and Suzuki 1970]. This is a useful exercise for the purpose of comparing the "mechanics" of the Green's function method and the recursion method. At the same time this formulation is also a convenient starting point for a perturbative approach to the weakly interacting case, as will be discussed in Chapter 11. The retarded Green's functions most directly related to the function Su.(q,ro) are the following:

G±(k,q,~)

..

E «a1ak+q;a:,ak'_?>~ .

(6.30)

k'

The equations of motion for the two functions G+ and G-;- read:

~E <~ k'

..

E <[a1ak+q,akt,ak' -q]±> k'

~

t

t

±

.. L-J «[akak+q,H];ak,ak' -?>~ k'

.

(6.31)

The expectation value of the (anti-)commutator on the right-hand side can be expressed in terms of the fermion occupation number nk =
E <[a1ak+Q,a:,ak'_Q]+> .. nk +nk+Q-2n~k+Q .

(6.32b)

k'

For the noninteracting case (Jz=O), the Green's function on the right-hand side of (6.31) is a reproduction of (6.30),

Section 6-3

-

t

t

±

71

(6.33)

= [£o(k+q) -£o(k)]E «akak+q;aklakl-?>~ ,

k'

and thus closes the equation of motion. The solutions for the two cases are (6.34a)

G -
=

n -n k

k+q

(6.34b)

,

1; - [£o(k+q) -£o(k)]

from which the spectral density and the dissipation function are derived according to (3.63): (6.35a)

Xz/

(q,ro)

= -lim.!.E ~[G-
(6.35b)

E-+oN k

The dynamic structure factor Szz(q,ro) (structure function) is then derived in turn via the fluctuation-dissipation theorem (2.23). Explicit closed-form expressions for the function Szz(q,ro) are readily obtained for T=O and T=oo. At infinite temperature, the fermion occupancy is nk = 1/2 independent of k. The Green's function (6.34b) and the dissipation function (6.35b) are identically zero. For this case, the dynamical structure factor must be derived from the Green's function (6.34a) via the spectral density (6.35a): Szz(q,ro)

= ~ E>(2JIsin(q/2)1-lcoI) V4J 2sin2(q/2) -cl-

(T=oo) .

(6.36)

At zero temperature we have nk=E>(cosk). For this particular ferrnion occupancy, we can write n k +nk+q -2n~k+q

= (n k -nk+q)2

(6.37)

.

Both Green's functions (6.34) then yield the result Szz(q,ro) = 2 E>(lroI-Jlsinql)E>(2JIsin(q/2)1-lroI)

V4J

2sin 2(q/2)

-cl-

(T=O)

(6.38)

along the two paths specified. The exact result (6.36) for T=oo can be derived with equal ease by means of the recursion method in the Liouvillian representation. The initial vector to be used for the determination of Szz(q,ro) is

72

Chapter 6

10

z 1 ~ t = Sq = - L aka k+q .

(6.39)

INk

The first few iterations can be carried out by hand:

(6.40a)

(6.40b)

(6.4Oc)

12 = i[HItl- ~/o The pattern of the emerging

~l

=

= ...,

~k-sequence

if2/ 2)

=

2J 4sin

i '.. .

4

(6.40d)

is a familiar one:

2J 2sin2! , ~=~3="'= J2 sin2! . 2 2

(6.41)

It bears the signature, according to Chapter 5, of the spectral density (5.7) with ~1~ .

o( (0)

=

2

VC%2_ ot

, I roI

~ C% = 2Jlsin!1 2

,

(6.42)

and yields via (2.23) the dynamic structure factor (6.36). For T=O, the analysis proceeds along the same lines except that a different inner product is to be used: ifn.!n) = , where the ground state Icjlo> of the noninteracting ferrnion system is the half-filled band. This makes the evaluation somewhat tedious for general q. However, for q=1t it is easy to verify that the emerging ~k-sequence is exactly the same as in the T=oo case discussed previously. Hence we obtain the same normalized spectral density (6.42) for q=1t, but a different dynamic structure factor, namely (6.38), for q=1t. We shall return to the case of arbitrary q in Chapter 9.

Section 6-4

73

6-4 Quantum Spins Exact spectral densities for the quantum spin model (6.26) at infinite temperature can be obtained by the recursion method, if we use it in the original spin representation. Let us consider the more general ID s=I12 XYZ model N

H = -

L [J~tSI:l

+

1=1

JyS/SI~1

+ J .J/SI~d .

(6.43)

It takes less than 150 lines of FORTRAN code to write a computer program which calculates high-precision numerical values of the /i/ s for spin autocorrelation functions of this model at T=oo. The computation is carried out in the Liouvillian representation of the recursion method. The input information consists of (i) the parameter values Jx,Jy,Jz of the XYZ model, (ii) the operator 10 which specifies the structure function to be investigated, (iii) the number K of iterations to be performed. The output is the set of values for ~I ""'~K' Owing to the property (Sf)2 = 114 of spin-I12 operators, the vectors Ik arising from the recurrence relations (3.19) have the following general structure (for lo=sg):

Ik

M(k)

=

I=+k

L am(k) IT IT

(st)lCm1a(k)

.

(6.44)

I=-k a=xyz

m=1

with ~Ia = 0, 1. Here M(k) denotes the number of distinct terms in the kth iteration of the orthogonal expansion. Knowledge of the numbers M(l), ...,M(k) is important for an estimate of storage requirements in the (k+ 1)th iteration. Since the Sf have zero trace, the evaluation of the norms of expressions like (6.44) is very simple: M(k) ifk,fk) =

L

[a m(k)]2

m=1

.

(6.45)

For an illustration we have determined (on the basis of 16-digit floatingpoint arithmetic) the first 15 ~k's for the autocorrelation function <So(t)SQ> of the isotropic XY model (Jx=Jy=J, Jz=O). We find the sequence

~k

=

!.-J 2k, 2

k=I,2,...,15,

(6.46)

to within 1 part in 1011 (see also [Florencio and Lee 1987]). The number of distinct terms in the vector I k increases from M(O)=1 to M(l5)=3956. We readily recognize in (6.46) a special case of the model ~k-sequence (5.26). The associated spectral density and its Fourier transform are pure Gaussians:

<1>0(00) = 2{it e-fd1J 2 J

,

Co(t) = e-J2t2/4 .

(6.47)

This is the exact result [Brandt and Jacoby 1976; Capel and Perk 1977]. We shall return to the dynamics of the quantum XYZ model with more results in subsequent Chapters.

74

Chapter 6

6-5 Classical Spins The recursion method in the Liouvillian representation is equally useful for the study of classical spin models at infinite temperature as it has proven to be for quantum spin models. Consider the classical XYZ model, specified by the energy function (6.43), with the spin variables now interpreted as classical vectors of unit length, SI

= (st,S/.S/) = (sinefoscl>l,sine~incl>l,cosel)

,

(6.48)

and the dynamics determined by the Poisson brackets of classical angular momentum variables a

~

_s:

~

Y

{SI ,SI'} - u/l'L.£a~ySI .

(6.49)

y

The vectors ik generated by the recurrence relations (3.19) in the classical case are also of the form (6.44) for io=Sf, but with arbitrary non-negative integer Kmla ' All inner products are phase-space integrals as defined in (3.16). For the classical spins (6.48), the phase manifold is a product of unit spheres, and a pair of canonical coordinates for SI are

(6.50) All nonvanishing inner products are multi-spin equal-time correlation functions. At T=oo these expectation values conveniently factorize, and each factor can be evaluated analytically in closed form [Liu and MUller 1990]: x 2K y 2K, z 2~ (2Kx -1)!!(2~ -I)!!(2Kz -I)!! «SI) '(SI) (SI) > = (2 +2 +2 + 1)" . Kx~Kz ..

(6.51)

All expectation values in which an odd power of Sf occurs are zero. In this introductory context, let us briefly look at the spin autocotrelation function <st(t)st> for special cases of the classical XYZ -dimer, H

.c;' x x = -J~l S2

y y z z - Jl1 S2 - J~l S2 .

(6.52)

The simplest case pertains to parameter values Jx=Jy=O, Jz=1 (X dimer). For the apprentice's convenience, we have reproduced in Table 6-1 the steps of the first three iterations. The pattern of this sequence is very simple and readily recognized: 2

!:!. _ k -

2

k (2k -1)(2k + 1) ,

(6.53)

It is the special case ~=O, 0l0=1, of the model sequence (5.8). For these parameter values, the model spectral density (5.7) reduces to the function

2A different application of the dk-sequence (6.53) was reported by Lee and Hong [1984] in the 3D Sawada model.

Section 6-5

et> o((0) = 1t eo -lroI)

75

(6.54)

.

The associated spin autocorrelation function <Sj(t)Sj> is then a spherical Bessel function:

= sint .

eo(t)

(6.55)

t

Our second example is the XX dimer, characterized by the parameter values (1x=Jy= I, Jz=O). The dynamics of that model belongs to an entirely different

category. This is reflected, for example, in the L\k-sequence of the same spin autocorrelation fucntion <Sj(t)Sj>. It does not converge to a finite value, but instead grows to infinity quadratically (shown in Fig. 7-4 below). That dramatic change in pattern is attributable, as we shall see, to the nonlinear nature of the underlying dynamics. An extensive discussion of the different categories of dynamical behavior - we call them universality classes - will be given in Chapter 7. In Chapter 9 we shall discuss a third case of (6.52), the XXX dimer (1x=Jy=Jz)' for the demonstration of yet another feature of dynamical behavior.

10

=

st

I1

=

iLlo = -S/S2

~ (/1 ,/1) = «SIY)2(S2z)~ =

12

=

zLII +L\I!0

.

x

1

x2

~ (/0'/0) = «SI) > = -

3

z

z2

1

"9

~ L\I =

1

"3

1 x 3

= -SI (S2) +_SI

z ~ - -«SI) 2 X2(S2) z ~ + -«SI 1 )~ = ~ (/2 '/2) = «SI )2(S2)

3

4 135

9

4 15

~~=-

13

=

. zLI 2 +L\~I

y

S z3

y

2

9 15

= SI ( 2) --SI

z6

z 2

ys

18 15

y

2S z4

81 225

~ (/3'/3) = «SI) (S2) >-_«SI) ( 2) >+-«

S IY)2(S 2z)~ = - 4

525

9 ~ L\3 = 35 Table 6-1: The first three iterations of the recurrence relations (3.19) for the classical X dimer, specified by the energy function (6.52) with J;FJrO, Jz=1.

7 UNIVERSALITY CLASSES OF DYNAMICAL BEHAVIOR In most studies of dynamic correlation functions the focus is on their long-time asymptotic behavior and on the singularity structure of the associated spectral densities. These properties reveal important information on the nature of the physical processes which govern the dynamics of the system under given circumstances. However, the analysis of the same dynamic correlation functions from a quite different perspective can be equally useful and revealing. The focus there is on the properties of spectral densities at high frequencies, specifically their decay law, expressible as in (5.23), in terms of a characteristic exponent A.. The information contained in A. about the underlying dynamical processes is in some sense complementary to that inferred from the singularity structure of spectral densities. Since the characteristic exponent A. is equal to the growth rate of the ~k­ sequence for the spectral density as defined in (5.22), the recursion method is the ideal calculational technique for such dynamical studies. However, we have yet to unlock the dynamical information contained in the characteristic exponent A. for general situations. In an effort to gain an intuitive understanding for the connection between decay laws of spectral densities and other dynamical properties, we report here a study which employs this type of analysis for a particular, exactly solvable model: the equivalent-neighbor XYZ model [Liu and Muller 1990]. Its dynamical properties depend strongly on the symmetry of the exchange interaction, and the analysis of dynamic correlation functions can be carried out to a considerable extent. The results of that study can be used as the basis for a classification of dynamical behavior in terms of the characteristic exponent A.. 7-1 Dynamics of the Equivalent-Neighbor XYZ Model Consider an array of N spins interacting via some model-specific spin-pair coupling of uniform strength r. In order to ensure that the free energy is extensive, the coupling strength must be scaled like r = llN. In this scaling regime, the equivalent-neighbor spin model is a microscopic realization of mean-field theory, but it has no longer any intrinsic dynamics. The right-hand side of Hamilton's equation for individual classical spins, dS/dt = -SI x aHlaSI, vanishes in the limit

N~oo.

A nontrivial intrinsic dynamics (for N~oo) can be restored, at least in the paramagnetic phase, if the spin coupling is scaled differently: r = l1N1I2. Timedependent correlation functions at T=oo are then meaningful and interesting quantities. They are the object of investigation in what follows. The two scaling regimes are best understood by noting that the thermodynamic properties of the equivalent-neighbor spin model are governed by the mean value of the magnetization vector (which is the basis of Landau theory), whereas the dynamical properties are determined by the fluctuations about the mean value.

77

Section 7-1

The Hamiltonian of the classical equivalent-neighbor XYZ model reads N

E

H = __1_ 2.jN jj=l.j#j

[J~tS/

+ J,f/S/ + JzS/S/J .

The equations of motion for the classical spin variables a j

dS _ J <:it - yCJySj(3

(7.1)

Sf are

sy I y (3 (3 y - J(3CJ(3 j - .jN(JySj Sj -J(3Sj Sj )

(72) .

with aPr= cycl(xyz). The collective spin variable CJ

I

N

= -ESj

(7.3)

.jN j=1

represents the vector of instantaneous magnetization fluctuation and has unit meansquare length, = 1. The N 1I2-terms in (7.2) disappear in the limit N~oo. The dynamical problem then reduces to a Hamiltonian system with two degrees of freedom. Summing Eqs. (7.2) over all sites i, dividing by N 1I2 and taking the limit N~oo yields the equations of motion for the collective spin: dCJ

<:ita

= (Jy-J(3)CJ yCJ(3' apy=cycl(xyz).

(7.4)

They are equivalent to Euler's equations for the free asymmetric top. The solutions of (7.4) depend on the symmetry of the exchange coupling in (7.1): XXX model (Jx=Jy=Jz): This model has full rotation~l symmetry in spin space. Consequently the collective spin is stationary, CJ = const. XXZ model (Jx=Jy#z): In the presence of uniaxial exchange anisotropy, the vector CJ(t) describes a harmonic motion, a uniform rotation about the symmetry axis. o XYZ model (Jx<Jy<Jz): In the generic case (with biaxial anisotropy), the vector CJ(t) still undergoes periodic motion, but the time evolution is anharmonic, expressible in terms of Jacobi elliptic functions. For a given solution of CJ(t) and in the limit N~oo, the equations of motion (7.2) for individual spins turn into linear first-order ODEs with time-periodic coefficients:

o o

a dS j

<:it

(3

= JyCJy(t)Sj

y

- J(3CJ(3(t)Sj , apy=cycl(xyz);

(75)

.

The solutions of (7.5) again depend on the symmetry of the model: o XXX model: Every individual spin S;
78

Chapter 7

o

XYZ model: The time evolution of individual spins Sj(t) is quasiperiodic and anharmonic, characterized by two fundamental frequencies, their multiples, sums and differences. In summary, the infinite-N equivalent-neighbor XYZ model is an integrable classical Hamiltonian system with linear or nonlinear dynamics, depending on the symmetry of the exchange interaction. For finite N, integrability is, in general, destroyed by the N 1I2 corrections in (7.2). The time evolution is then intractably complicated for the general XYZ case. It is nonlinear even for the case with XXZ symmetry. Only for the special XXX case, it stays linear if N is finite. However, as N~oo, the many-body dynamics of the classical equivalent-neighbor XYZ model turns effectively into a physical ensemble of nonlinear Hamiltonian systems with two degrees of freedom, the collective spin 0' and one of the individual spins Sj'

'-2 Fluctuation Functions and Spectral Densities for the XXZ Case In a recent study [Liu and Muller 1990], the spin autocorrelation functions and their spectral densities for the equivalent-neighbor XXZ model (Jx=ly#z) at infinite temperature were calculated exactly and evaluated in closed form. The results for the collective spin 0' read:

3<0'x
= eXP(-~(1-llt2), 3 = I

4>~(ro)a = I~~~Iz exJcd 2)' 4>~(ro)a Cl ~ (l-lz)

,

= 21tO(ro)

(7.6)

,

(7.7)

where C=:(3/21t)ln. The single-spin autocorrelation functions and their spectral densities are given by the following expressions:

(7.8b)

Section 7-2

79

(7.9a)


+ 4nCcJ exp( -3cJl2J 2) .

J3

(7.9b)

The function
2nC

__2

2

<1»0 (co)s = __ exp( -3OTlJz ) Jz

,

(7.10)

and (7.8a) reduces to 3<Sj(t)Si> = exp(-J/?16). In fact, the long-time asymptotic decay of <Sj(t)Si> is Gaussian throughout the regime J.;>2J of easy-axis anisotropy. In the regime O<Jz<2J, Jz#, the spectral density cI»Q(co)s has a singularity at ro=O of the form -co21n(co), implying that the correlation function decays algebraically for long times, <Sj(t)Si> - (3. In the limit Jz=O, a stronger singularity in the spectral density makes its appearance, -lcol, resulting in a slower long-time asymptotic decay of the associated correlation function, <Sj(t)Si> - (2. Figure 7-1 shows the spectral density
80

Chapter 7

10.0

8.0 6.0

8.0

J=O

0.00

--.. 3

'"

Jz=J

Jz=O

4.0

0.25

0.50

2.0

,

I

,

0.0

g

6.0

0.0

0.2

0.4

'--" 0

>&

4.0

2.0

o.0 --t-<:""'-"""-,.-,.--r--r---r----r--r--.----,--f-"'-r---.-.,-----.----,--,.-.,.--,.--,.-,r--.--l 0.0

0.5

1.0

0.0

0.5

1.0

Figure 7-1: Spectral density «I>~X(co)sof tile classical equivalent-neighbor ><XZ model at T=oo. The curves represent the exact result (7.9a) for six different values of uniaxial anisotropy, here parametrized as J = sin(ng), Jz = cos(ng). The inset shows the same function for parameter values g = 0.29, 0.28, 0.27, 0.26, approaching the XXX model (g = 0.25), for which case (7.9a) reduces to (7.9b) [from Liu and Muller 1990].

Note that the long-time aymptotic behavior of these correlation functions depends on the amount of uniaxial exchange anisotropy, whereas the decay law of the spectral densities at high frequencies is always Gaussian, independent of the model parameters. The Gaussian decay of spectral densities represents one of four different universality classes of dynamical behavior that are realized in the context of the general equivalent-neighbor XYZ model. We shall see that they can be interpreted in terms of basic notions of classical dynamics. Before we proceed with a discussion of these universality classes (in Sec. 7-5), we must adapt the recursion method to the special requirements of equivalent-neighbor models. 7-3 Recursion Method Applied to Equivalent-Neighbor Spin Models At first glance, it seems that the infinite-range interaction in equivalent-neighbor spin models makes them inaccessible to any useful analysis by the recursion method except for small N: The opposite is true. The metamorphosis as N~oo of the equivalent-neighbor XYZ model into a physical ensemble of two-body systems can

Section 7-3

81

be exploited to boost the computational efficiency of the recursion method substantially. We start out with the formulation of the method for classical spins as outlined in Sec. 6-5 and rewrite the Hamiltonian (7.1) in the form I ~ I ~ ~ a2 H = - - - L.J JaMaMa + - - L.J L.J Ja(Sj) ,

2{N a=XYZ

2{N a=xyz j=l

(7.11)

where N

Ma

= Lst,

(7.12)

a.=xyz.

j=l

The Poisson brackets for the two sets of variables {stJ.,S!} = fJjjLEa[3ySl,

Sf and Ma are

{Sja,M[3} = LEa[3ySl '

y

y

(7.13)

{Ma ,M[3} = LE a [3yMy . y

In the recursion method as applied to the spin autocorrelation functions <Sf(t)Sf>, all inner products to be evaluated have the following general structure: (7.14)

Hence all terms except the leading (N-independent) one represent finite-size corrections. It turns out that only the Ma-terms in (7.11) contribute to the dynamics of the infinite system. Any surviving contribution to the vector A in the orthogonal expansion (3.17) has the general form N-(mx+my+mz)l2st(Mx)mX(MytY(Mz)mZ .

(7.15)

All nonvanishing inner products, expanded in inverse powers of N, factorize to leading order: «Sja)2

IT

y=xyz

N-m., My2.m.y>

=

«Sja)~ IT N-mY <My2.m.y>[1 +O(N-1)]

(7.16)

y=xyz

with (7.17) Exactly the same inner products are obtained from a physical ensemble of the 2-degrees-of-freedom system consisting of a spin Sj of unit length coupled parasitically (Le. with no dynamical feedback) to an autonomous I-spin system, a spin cr of unit rms length driven by the Hamiltonian (7.18)

82

Chapter 7

This representation is particularly suitable for evaluating !:ik-sequences of the infinite-N equivalent-neighbor XYZ model by means of the recursion method.

7-4 Quantum Equivalent-Neighbor XYZ Model Through minor modifications, the recursion method can be adapted to the dynamics of the quantum equivalent-neighbor XYZ model, specified by Hamiltonian (7.1) or (7.11) with the Si now representing spin-s operators. We employ the quantum Liouville equation (3.1) instead of its classical counterpart (3.11), replace the symplectic structure (6.49) by the commutator algebra for quantum spins, [Sia,S!] = iOi/E£apySl '

(7.19)

y

and use (for T=oo) the inner product (A,B) = Tr(AB). In spite of the structural similarity of the elements which go into the classical and quantum versions of the recursion method, the resulting dynamics is, in general, quite different for the two cases. However, the equivalent-neighbor XYZ model is atypical in this respect. All inner products have the general structure (7.14) in both the quantum and the classical cases, and the dynamics of the infinite system is determined by the leading term alone. All terms in I k which contribute to leading order in (7.14) contain commuting operators only - operators pertaining to different sites of the array. The net result is that the coefficients !:ik of <S:'(t)S? for the infinite quantum equivalent-neighbor XYZ model differ from those of its classical counterpart only by a multiplicative conStant (which depends on s), amounting to a different time scale in the dynamical correlation functions. This explains the observation that the results of Liu and Muller [1990] for dynamic correlation functions of the classical equivalent-neighbor XXZ model reported in Sec. 7-2 are fully consistent with the results previously obtained by Lee, Kim and Dekeyser [1984] for the quantum spin1/2 counterpart of that model (see also Dekeyser and Lee [1991]).

7-5 Prototype Universality Classes In integrable classical dynamical systems, the growth rate A. of the !:ik-sequences as defined in (5.22) for specific autocorrelation functions is basically dominated by two factors: Factor A depends on whether the equations of motion are linear or nonlinear. Each harmonic mode contributes exactly one o-function to the spectral density (at 00>0), whereas each anharmonic mode contributes an infinite set of &-functions, at frequencies with no upper bound. In nonlinear systems, factor A is governed by the large-m decay law of the line intensities for single modes. Factor B is governed by the distribution of fundamental frequencies pertaining to individual linear or nonlinear modes. That distribution depends sensitively on whether the size of the system is finite or infinite and (for infinite systems) on whether the interaction range is finite or infinite.

o

o

Section 7-5

83

The effect of each factor on the large-ro decay law of the spectral density is expressible in tenns of a distribution function: factor A by the spectral-weight distribution Pn(n) of individual modes and factor B by the distribution Po.(O) of fundamental frequencies of these modes. The large-ro decay law of the spectral density is then obtained from these distributions by the following construction: 00

(7.20)

(ro-nO) .

o

0

For distributions with asymptotic decay laws of the fonn P n(n) - exp( -n Cl) , Po. - exp( -03)

,

(7.21)

the large-ro decay law of the resulting spectral density as obtained from (7.20) is given by


=

2(a. +~) .

(7.22)

a.~

In the context of the classical equivalent-neighbor XYZ model, factors A and B produce a total of four different decay laws of spectral densities, characterized by four different integer-valued growth rates A of the associated Llk-sequences. These four universality classes of dynamical behavior are summarized in Table 7-1. They can be identified as the special or limiting cases A(OO,oo) = 0, A(oo,2) = 1, A.(1,oo) = 2, A(l,2) = 3. In the following, we discuss realizations of each case. type of dynamics linear linear nonlinear nonlinear

Pn(n)

l>(n-1) l>(n-l)

-e -n -e -n

size of system finite infinite finite infinite

A

Po.(O)


bounded support

bounded support

0

-e -ri

_e-fJi

1

bounded support

_ e--ro

2

-e -ri

-e -om

3

Table 7·1: Large-co asymptotic decay law of the spectral density ct>o(co) and growth rate A. of the associated Ak-sequence for the four different universality classes of dynamical behavior realized in the classical equivalent-neighbor XYZ model. The four classes arise as the product of the two factors A (type of dynamics) and B (size of system). each represented by a distribution for which there are two distinct realizations [adapted from Liu and Muller 1990].

84

o

Chapter 7

Bounded support (A=O): Quite generally, .:lk-sequences with zero growth rate are realized in linear dynamical systems with a finite number of degrees of freedom. The finite-N XXX case is such a system. The .:lk-sequence for the case N=2 is plotted in Fig. 7-2. It converges toward a finite asymptotic value .:loo in an alternating approach. This situation is represented by the first row of Table 7-1 and by the limiting case (a.,~) (00,00) of (7.22). The exact spectral density will be reconstructed in Chapter 9 from the .:lk-sequence.

=

2.0

2-spin XXX model a a

<So (t)S. > 1

1

1.5

1.0

0.5

o. 0 +-__r_--.---.----"T---,.-r--.,.---r-~_r___r___r__r____r---,....__.,.__-r--._-l 18 o 2 6 8 10 12 14 16 4 k Figure 7-2: Sequence of continued-fraction coefficients !:J.k versus kfor the spin autocorrelation function <8MB? at T=oo of the XXX dimer, (6.52) with J;=J.,;:Jj=1, which is equivalent to (7.1) for N=2 and J;=J Jj=2 112• The sequence converges toward the asymptotic value

!:J._=1.

o

r

Gaussian decay (A.=1): .:lk-sequences with linear growth rate (A.=1) are common in many-body systems with linear dynamics. In the context of the infinite-N XXZ case, the Gaussian decay of the spectral densities (7.9) arises, via factor B, from linear moc:les with a Gaussian frequency distribution, a property dictated by the central limit theorem. In Fig. 7-3 we show several .:lk-sequences (plotted versus k) for <Sj(t)S1> at various values of J, Jz' all of which show indeed linear growth rate. This situation is represented by the second row of Table 7-1 and described by (7.22) with (a.,~) (00,2). Only the sequence for J=O is purely linear, .:lk=kJ/13, representing expression (7.10). Throughout the regime O<J<Jz the.:lk oscillate about the

=

Section 7-5

85

line kJ//3 (shown dashed in Fig. 7-3). These oscillations persist as k~oo if 1j2<J<Jz and thus determine the singularity structure of the spectral density (7.9a) and the power-law long-time tail of the correlation function (7.8a). For O<J<Jj2, on the other hand, the oscillations damp out as k~oo, which is illustrated for two cases in the inset to Fig. 7-3. These ak-sequences describe spectral densities with no power-law singularities and correlation functions with no power-law long-time tail. 0.05

40.0 t')

"..lI:

I ..lI:
30.0 0.00

~

0.0

<J

20.0

0.5

1.0

k -1/2

10.0

o.0 ---t-''r--T'-,-..,..-,r-r--.-r-r--,-..,..-,r-r--.-r-r-,-...,.......,,...,...-.-r-r--,-..,..-,r-r--.-r-r---r-l 5 30 o 10 15 20 25 k Figure 7-3: Sequence of continued-fraction coefficients Ak versus k for the single-spin autocorrelation function <SM~> at T=oo of the infinite-N equivalent-neighbor XXZ model for three different sets of parameter values. For 0<J!.J:r=1, the Ak-sequences oscillate about the (dashed) line kl3. The inset shows the deviations Ak-kl3 versus /(1/2 for two cases in which the oscillations die out as k--+oo [from Liu and Muller 1990].

o

Exponential decay (A.=2): Consider the spin autocorrelation function <Sj(t)S> for the XXZ case with N=2, specifically the 2-spin model with lz=O. For this case, the anharmonic motion of the spin components Sf was analyzed for arbitrary initial conditions in a quite different context [Srivastava et al. 1988]. The solutions are expressible in terms of Jacobi elliptic functions. These functions have the property that their line intensities decay exponentially fast at high frequencies (factor A). For finite systems, the distribution of fundamental frequencies has compact support (factor B). This situation is represented by the third row of Table 7-1 and by (7.22)

86

Chapter 7

with (a.,~) = (l,oo), yielding an exponentially decaying spectral density and a ~k-sequence with quadratic growth rate. Figure 7-4 shows a plot of ~k versus k?- for the first 18 continued-fraction coefficients of this case determined by our computational procedure. The observed growth rate is perfectly consistent with A.=2. 80.0-,-----------------------,

2-spin XX model x x 60.0

<So (t)S. > 1

1


20.0

o.0 ""-"r-T'"...,.....,r-r-....--r-r-.,...-,-r--r-r-.--.,.......,......,.-r-r-'--r-T'"...,.....,--r....--r--r-r-r-.--,J 50 o 300 100 150 200 250 Figure 7-4: Sequence of continued-fraction coefficients Ak versus ~. 1c=1 ,...,18, for the spin autocorrelation function <SMSf> at T=oo of the classical two-spin XX model, (6.52) with Jr Jy=:1. J;t=0. which is equivalent to (7.1) for N=2 and Jr J-2 112 , J;t=0. The quadratic growth rate (A.=2) for this function is demonstrated by the excelfe"nt fit of the regression line [from Liu and MOller 1990j.

o

Stretched exponential decay (A.=3): For realizations of this last universality class, consider the collective-spin autocorrelation function of the XYZ case with N=oo, specifically the function for lx=(l +y)/2, ly=(l-y)/2, lz=O and four different values of the parameter y. For the two cases with uniaxial symmetry, y=O and "(=1, the dynamics is linear, whereas for the two intermediate cases with biaxial symmetry, "(=1/4 and "(=3/4, it is nonlinear and describable in terms of Jacobi elliptic functions. In all of these cases, the distribution of fundamental frequencies of individual (linear and nonlinear) modes is characterized by a Gaussian decay (factor B), but for the nonlinear cases, the large-m behavior of the spectral density is further governed by the exponential decay of line intensities at multiples of the

Section 7-6

87

fundamental frequencies (factor A). These situations are represented, for the linear and nonlinear cases, respectively, by the second and fourth rows in Table 7-1 and by (7.22) with exponent values (a,~) equal to (00,2) and (1,2). Figure 7-5 shows for the four specified cases !:1k versus k in a log-log plot as determined computationally by the recursion method. The switch from linear to nonlinear dynamics causes the growth rate to jump from 1..= I to

1..=3.

1000.0

• J x =0.B75 J y =0.125 • Jx =0.625 J y =0.375

100.0

10.0

o J x =1.0 Jy=o. o J x =0.5 J y =0.5

1.0

0.1 +-----.------,--..----,---,r--r----.--.-.-----..-----r-----; 40 6 8 10 20 2 4 1

k Figure 7-5: Log-log plot of sequences l:1 k versus k for the collective-spin autocorrelation function
7-6 Two-Sublattice Spin Model with Long-Range Interaction Here we wish to present another exactly solvable spin model for analysis in the context of dynamical universality classes: the two-sublattice XYZ model with uniform inter-sublattice interaction and zero intra-sublattice interaction,

88

Chapter 7 NA NB

H =

1 ~ ~

x x

.r;:

Y Y

Zs z]

--L..t L..t [J~I SI' + Jfl SI' + J~I I'

(7.23)

,

{NI=II'=I

where N = NA+NB is the total number of spins [Liu and Muller 1991]. The equations of motion for individual spins read

dS

a

_ _I

dt

= JYO'Yr;: 13 JJU I

a

dSl ' dt

with (a~'Y)

= Jy

0'1sl13,

-

(7.24a)

J 13 O'I3B SI Y 1=, 1 ..., NA '

(7.24b)

13

- JI3O'ASI~' I' =1, ...,NB

=cycl(xyz). The collective-spin variables NA

NB

1 1 O'A=-ESI'O'B=-ESI'

{N 1=1

(7.25)

{N I' =1

represent the vectors of instantaneous subIattice magnetization fluctuations. Summing Eqs. (7.24) over all sublattice sites and dividing by N I12 yields the equations of motion for the two sublattice spins, a

dO'B

Y

13

dt = JYO'AO'B

-

J

13

Y

13O'AO'B

(7.26)

with (a~'Y) =cycl(xyz). These equations describe the nonlinear rotational dynamics of an effective two-spin model,

ii = - E

a=xyz

JaO'~O'~ .

(7.27)

The integrability of this two-body problem was proven through explicit construction of a 2nd independent integral of the motion [Magyari et al. 1987], -

I

=-

Y Y ~ 1 2 a2 a2 JaJ 13 O'AO'B + L..t -Ja[(O'A) +(O'B) ] . al3y =cycl(xyz) a=xyz 2

~

L..t

(7.28)

For given solutions O'it) and O'it), Eqs. (7.24) for individual spins turn into a set of linear and decoupIed vector equations with time-dependent coefficients. A complete set of N independent integrals of the motion in involution for the twosublattice XYZ model (7.23) consists of two invariants which govern the time evolution of the two vectors O'A and O'B and of N-2 = (Nr l)+(NB-1) invariants which govern the time evolution of the individual spins in arrays A and B. The first two invariants are iI and I, and the remaining N-2 can be selected as follows [Srivastava et al. 1988]:

Section 7-6

It =

:E S (S k' k<1

1/ =

:E

1=2, ...,NA ,

I' =2, ...,NB

SI"Sk"

k'
89

(7.29a)

(7.29b)

.

It is interesting to compare these properties of the two-sublattice XYZ model (7.23) with those of the equivalent-neighbor XYZ model (7.1). The latter is not completely integrable except for N=2 or N=oo. Only the fully isotropic case (Jx=Jy=Jz' XXX model) is completely integrable for arbitrary N. For that case, a set of N independent integrals of the motion can be chosen as follows [Srivastava et al. 1988]: N

11 =

:E Skz , k=1

(7.30)

11 = S (SI +S2 +'" +SI_I)' 1=2, ... ,N .

What makes the model nonintegrable for finite N>2 in the presence of anisotropy are the N 1I2-terms in the equations of motion (7.2). In spite of its complete integrability, the determination of dynamic correlation functions for the two-sublattice XYZ model can be quite involved. Explicit expressions have been worked out only for the fully isotropic XXX case (Jx=Jy=Jz=J) [Liu and Muller 1991]. The T=oo autocorrelation function for the sublattice spins reads <<J it)'<J A>

= <<J it)·<J B> = .: +..!.(I_':J2t2)e-J2t2/3 3

3

3

.

(7.31)

The associated spectral density is then the sum of a B-function at ro=O and a Maxwellian spectral-weight distribution: o(oo)cr

= 41t B(oo) 3

+

'!:...J41t/3 cd- e-3ol/4J2 2 J3

.

(7.32)

According to our classification of dynamical behavior, the spectral density (7.32) of the two-sublattice XXX model belongs to the universality class N=1 (Gaussian decay). The spectral densities of the equivalent-neighbor XXX model belong to the same universality class. But the two models (7.1) and (7.23) part company when we reduce the rotational symmetry by introducing a uniaxial anisotropy (Jx=Jy#z). For the spectral densities of the equivalent-neighbor XXZ model, the universality class stays the same (N=I, Gaussian decay), whereas it changes to N=3 (stretched exponential decay) for those of the two-sublattice XXZ model. When we further reduce the symmetry by introducing a biaxial anisotropy (e.g. O<Jx<Jy<Jz)' we find that both models belong to the same universality class again (N=3). The remaining two prototype universality classes, N=O (bounded support) and N=2 (exponential decay), are realized in either model for finite N. In both models the reason for any change in universality class of dynamical behavior

90

Chapter 7

is associated with a switch between finite N and infinite N or a switch between linear dynamics and nonlinear dynamics. ,-, Many-Body Systems with Short-Range Interaction The results of this study suggest that the four different universality classes of dynamical behavior observed in the context of the classical equivalent-neighbor XYZ model may serve as prototypes for a classification of the dynamics of general classical and quantum many-body systems. The exponent values A.=0,1,2,3, which are realized in this somewhat artificial dynamical model, then play a role similar to the classical exponent values in the theory of phase transitions, which are also realized by the equivalent-neighbor model now interpreted as mean-field approximation to a thermodynamic system with short-range interactions. Studies of critical phenomena have convincingly demonstrated that the values of critical-point exponents of model systems with short-range interaction are determined by more subtle properties than is suggested by mean-field theory. Likewise, the growth rates which characterize dynamic correlation functions of many-body systems with short-range interactions call for an interpretation which transcends the classification used in Sec. 7-5. The purpose of the following remarks is to highlight some observations made on classical and quantum spin models with short-range interactions [Liu and Muller 1990]. (i) Dynamic correlation functions of generic quantum many-body systems appear to be characterized by ilk-sequence with growth rates in the vicinity of A.=1. (ii) One of very few quantum many-body systems for which nontrivial dynamic correlation functions can be evaluated is the 10 s=1I2 XX model, N

H = -

E J(S/SI:I 1=1

+

SlSI~I) .

(7.33)

For that model, growth rates A.=1 and A.=O are both realized (Secs. 6-3 and 6-4). The former characterizes the correlation functions with· maximum degree of complexity for that model, while the latter occurs as a result of special circumstances, which impose exceptionally stringent selection rules on transition rates. (iii) Some evidence for growth rates 121 in quantum many-body systems does exist, e.g. for the spin autocorrelation functions at T=oo of the 10 s= 1/2 XXX model (Heisenberg model), specified by Hamiltonian (6.22). An extrapolation based on the analysis of the first 15 nonzero frequency moments (they determine ilk up to k(15) suggests a growth rate 1..>1.18 for that case [Roldan, McCoy and Perk 1986]. Further results for quantum spin models at T=oo will be presented in Chapter 10. (iv) Dynamic correlation functions of generic classical many-body systems with short-range interactions and nonlinear dynamics appear to have growth rates in the vicinity of A.=2. This is in marked contrast to their quantum counterparts. (v) Figure 7-6 shows in juxtaposition the first 7 continued-fraction coefficients ilk versus k (in a log-log plot) for the spin autocorrelation function <Sj(t)Sj> of the 10 spin-1I2 XX model (7.33) and its classical counterpart. While the sequence of the

Section 7-7

91

quantum model is an exact realization of A= I, the sequence of the classical model suggests a growth rate A.=2. (vi) The number of continued-fraction coefficients which we have been able to compute so far for various cases of the ID classical XYZ model is somewhat too small to detect or rule out deviations from the borderline growth rate A=2 with some .confidence. The dk-sequence of <Sj(t)Si> for the XX model suggests 1..>2 (see Fig. 7-6), while the corresponding sequence for the Heisenberg model (XXX case) indicates "-<2. Further work with more computational power is needed.

3.0

x

x

1

1

<So (t)S. > ID classical XX model

2.0

/

/

/

/

/

/

/

/

/

. /

/

/ /

/

/

/

/

/

/

/

/

/

/

/

/

/

/ / ;\=2

;\=1

0.0

- 1. 0 +--r--r----.----r---r---r-r----r---,----,--,-r-.--..--,---..--..----.--....--! 2.0 0.0 0.5 1.5 1.0

In(k) Figure 7-6: Log-log plot of sequences li.k versus k, k=1, ...•7, for the spin autocorrelation function <S'Ms'f> at T=oo of the 10 classical and quantum spin-1/2 XX model (7.33) with .1=1. The sequence for the quantum model is exactly known, li.k =k/2. The (solid) regression line determined for li. 1 ,••. ,li.7 of the classical model has slope 1..=2.19. Lines with slope 1..=1 and slope 1..=2 are shown dashed [from Liu and Muller 1990].

(vii) Time-dependent correlation functions with singularities are not unheard of in

otherwise well-behaved classical many-body systems with Hamiltonian dynamics. For the case of the completely integrable logarithmic Heisenberg model [Ishimori 1982; Haldane 1982; Papanicolaou 1987], N

H =

-L In(l +S i'Si+l) , i=l

(7.34)

92

Chapter 7

such singularities even make it to the real t-axis, at least for finite N. The exact T=oo autocorrelation function for N=2 reads: Co(t)

= -!.. 2

+

-!..(1 +-!"t 2)cost - t(~ +_I_t 2)sint 2

6

6

12

+ t 2(1 +_1_t 2 )ci(t)

12

= t 2(1 + _1_ t 2)lnt + regular terms .

12

This function is no longer describable in terms of .1k-sequences.

(7.35)

8 TERMINATION OF CONTINUED FRACTIONS: ATTEMPTS AT DAMAGE CONTROL In almost all practical applications of the recursion method, we are confronted with a problem that cannot be ignored. The implementation of one or the other version of this calculational technique enables us to determine only a limited number of continued-fraction coefficients ~1""~K before we run out of computational power (CPU time or memory space). This produces an incomplete continued fraction (3.25) for the relaxation function co(z). How much worth is an incomplete continued fraction? We might say, it is as much worth as an (incomplete) K-term asymptotic expansion (3.32) of the same function co(z) or a K-term short-time expansion (3.29) of the (normalized) fluctuation function Co(t). However, we must not ignore the following difference: An incomplete short-time expansion is a well-defined function. It represents an approximation to the function Co(t) within a certain radius of convergence. An incomplete continued fraction, by contrast, is in itself a meaningless object, somewhat like a fraction with the numerator or the denominator missing. The use of incomplete continued fractions as they invariably emerge from nontrivial applications of the recursion method depends on an acceptable scheme for the artificial completion of these nonentities. The objectives underlying this process must be the following: (i) Terminate the incomplete continued fraction artificially to make it a meaningful mathematical object - a function co(z). (ii) Use a termination scheme that does not violate the general constraints imposed on the relaxation function co(z) (causality, analyticity etc). (iii) Select a termination scheme which transforms the incomplete continued fraction into the best possible approximation to the relaxation function co(z) based on the maximum amount of information that can be extracted from the sequence of numbers ~1""'~K' Achieving the third goal looks like a formidable challenge. If an infinite ~k­ sequence is necessary to fully determine the function co(z), then knowledge of the first K coefficients ~K does not add up to any measurable amount of information in the worst possible scenario, and the recursion method would then be hopelessly inadequate. The power of the recursion method derives from the very peculiar way in which the information contained in the ~k-sequence is translated into properties of the associated spectral density. In the following, we discuss a number of different strategies which have been used to achieve the goals (i)-(iii) stated above. They range from acts of sheer desperation to more or less sophisticated attempts at controlling and minimizing the damage done by artificially terminating a continued fraction. In Chapter 9 we shall propose a particular scheme of implementing the recursion method specifically designed for the reconstruction of spectral densities from incomplete continued fractions.

94

Chapter 8

8-1 Cut-OtT Termination The crudest way of tenninating an incomplete continued fraction is by means of setting the first unknown coefficient equal to zero, aK+1 O. The relaxation function is then a finite continued fraction

=

co(z)

1

= ---~-­

a1

z+----:--~ z+----z+...

(8.1)

a

... +...!.. z

with K +1 poles on the imaginary axis. It yields a multiply periodic fluctuation function Co(t) of the form (5.3) and a spectral density 4>0(00) of the form (5.2), which consists of (K+l)12 pairs of ~functions. For a dynamic correlation function which in reality has a continuous spectrum this is worse than a caricature. The cut-off termination has many advocates among the users of the recursion method. Their argument in justification of its use may be paraphrased as follows (e.g. for the case of an interacting quantum spin system): Finite systems generate only a finite number of frequencies in the time evolution. The relaxation function of any such system can be rigorously expressed in terms of a finite continued fraction. Hence the error caused by the cut-off termination cannot be more serious than that of a typical finite-size effect. Its impact weakens as the number of coefficients used in (8.1) increases. Even relatively small systems yield hundreds if not thousands of poles along the imaginary z-axis. If the spectral density is evaluated, via (3.28), but with E kept nonzero, the hundreds or thousands of &-functions are spread into Lorentzians and Itlimic a continuous spectrum. For the right choice of E, these Lorentzians overlap sufficiently to bring forth a spectral-weight distribution that reflects some of the major features of the function which it is intended to approximate.

Unfortunately, it must be said that the argument is flawed, and the analysis of results based on this approach runs a considerable risk of mlsinterpretation. The problems with the cut-off tennination are best understood after we have identified and distinguished the following specifications: N number of particles or degrees of freedom (system size); L number of frequencies in the time evolution (dimensionality of the dynamically relevant Hilbert subspace of the Liouvillian); K (=2L-l) number of ak in the finite continued fraction (8.1); J number of k which are unaffected or only little affected by the finite size of the system. Our observation for generic quantum spin systems is that L and K increase exponentially in N, while J increases proportional to some power of N. In a system with nearest-neighbor interaction on a d-dimensionallattice, we have 21+1 .. N Ud. For an illustration consider the ID s=1I2 XX model (7.33) with periodic boundary conditions. We have computed ak-sequences of the T=co spin autocorrelation function <S[(t)S[>, for different system sizes N. We find that the number K(N) of nonzero continued-fraction coefficients does indeed increase very rapidly with

a

N:

95

Section 8-1

K(2)

= 1,

= 8,

K(4)

K(6)

= 36,

K(8) > 80 .

(8.2)

For given N, the !:J..k's are unaffected by the finite size of the system for k up to l (8.3)

J(N) = N-1 .

Beyond that mark, the !:J..k's start to deviate at first gradually and then erratically from the sequence (6.46), !:J..k=kJ212, which pertains to the infinite system. This is illustrated in Fig. 8-1. 8 10

1.1 N=oo

8

6

~


8 4 2 0

~4
I

,/

A Pt ,I 'V--.,

\il

'N''''~~""

0

20

I

I

I

I

I

/

N=8

\ \ \

40

Ii{

2

"

I \ I \ I \ I \ I \ I I \ I \

N=8

k

\

\ \

\ \

~,

\

\ \

\

\

,

N=6

'G----e--

_e--

\

\C!t'"........a,, '''0...

........ N=4

'&...... O-+--r----r--...,..--r----r--...,..--r----r--.::.::"'i'--r----r--r--r--l

o

2

4

8

6

10

12

14

k Figure 8-1: Ak-sequence plotted versus k for the spin autocorrelation function <S':(t)S':> at

T=oo of the 10 5=1/2 XX model (7.33) with .1=1. Data for finite chains of three different sizes (open circles, dashed lines) are shown along with the sequence Ak = k/2 pertaining to the infinite chain (full circles, solid lines). The inset shows the Ak-sequence for the case N=8 up to k=60.

Let us have a closer look at the case N=8. The inset to Fig. 8-1 shows highprecision numerical values of !:J..k up to k=60. The !:J..k-sequence for the infinite system (N=oo) follows the dashed line. Suppose we wish to use the data extracted from the finite system (N=8) to reconstruct the spectral density for the infinite system (N=oo) as best as can be done. How many coefficients from Fig. 8-1 should lThis number is twice the estimate given above. The windfall is attributable to special circumstances, in this case the free-fermion nature of the dynamical system (see Sec. 6-3).

96

Chapter 8

we use before we cut off the continued fraction? Should we stop at k =l(N) =N-l =7? Or should we use as many as we have been able to determine reliably (k up

to 80 at least), even if we know that most of them strongly deviate from the coefficients which determine the exact spectral density of the infinite system? Figure 8-2 shows the spectral density reconstructed from (8.1) via (3.28) with £=0.05 with the lik's for N=8. The four dashed curves represent results produced at different cut-off levels. For low cut-off levels (main plot), the individual poles of the finite continued fraction are conspicuously present, and convergence toward any smooth function seems remote, let alone convergence toward the exact result (6.47) (solid line). The results shown in the inset use a much larger number of coefficients. For frequencies roll? 0.8, the two curves are sufficiently similar to suggest that some kind of convergence is taking place. However, that should not be confused with convergence toward the exact result (solid line). The latter cannot be achieved in this way on the basis of the N=8 data alone. Only the first ten or so lik's are close to the sequence (6.46).

Figure 8-2: The dashed lines show attempts to reconstruct the spectral density of <SMSf'> at T=oo for the 10 8=1/2 XX model (7.33) with .1=1 from a finite continued fraction (8.1) by cut-off (at level K) and smear-out (with £=0.05). Note the different scales used in the main plot and in the inset. All f1 ks used are for a system of N=8 spins. The solid line represents the exact result (6.47) for the infinite system (N=oo).

Section 8-2

97

One might attempt to improve the appearance of the numerical results by increasing the broadening of the spectral lines, Le. by choosing a larger value for E. Moving the frequency axis further away from the line of poles indeed makes the artificial structures in the dashed curves of Fig. 8-2 less pronounced. The probiem is that they tend to melt away along with any recognizable feature characteristic of the exact result. Ironically, for the case at hand, the exact result (a pure Gaussian) is itself the most featureless bump imaginable! The bottom line of all this is that the need for circumspection along the lines pointed out in this Section must not be ignored in applications of the recursion method to quantum many-body dynamics.

8-2 n-Pole Approximation An alternative way to replace the relaxation function of a dynamical problem by a finite continued fraction is presented by a method known under the name n-pole approximation. This scheme was originally designed to establish a link between many-body dynamics and certain simple phenomenologically motivated macroscopic dynamical models (to be discussed in Sec. 8-4). In that particular context, the npole approximation turned out to be quite useful. However, its subsequent adaptation to applications in low-temperature spin dynamics and to the dynamics of strongly correlated electronic systems proved to be much less satisfactory. In the following discussion of the n-pole approximation, the goal is to describe and understand its nature and limitations in the general framework of the recursion method. The relaxation function in the n-pole approximation of a given dynamical problem has the form ~I

Z +------;---

~

z+-----z+ ...

(8.4)

~n-I 1

+--

z+t n

where t n plays the role of a relaxation time. Here the sequence of continuedfraction coefficients ~I''''''~K' with K = n-l have been generated by some implementation of the recursion method. However, the recursion method itself does not yield a relaxation time t n to be used for a best approximation of the relaxation function by a continued fraction terminated as in expression (8.4). Several formulas have been proposed (each on somewhat shaky grounds) for expressing the relaxation time t n in terms of known continued-fraction coefficient ~K' K < n-l [Lovesey and Meserve 1972; De Raedt 1979; Pires 1988]. In the n-pole approximation the generally complicated singularity structure of the function co(z) is replaced by n poles in the complex z-plane. All poles have 9t[z]
98

Chapter 8

approximation (8.4) is identical to the cut-off termination (8.1) at level K=n-l. For very small relaxation times, on the other hand, we can write .1n - 1

1

z+_1

1: n- 1

(8.5)

1: n

Thus for 1:n~O we have 1:n_1~oo, and the n-pole approximation (8.4) reduces again to the cut-off approximation of (8.1) but now at level K=n-2. In the following, we shall briefly discuss the dynamical properties of n-pole spectral densities for n = 1,2,3. These are the most widely used cases. They are important in a variety of contexts. We start from the following explicit expressions for the relaxation functions: co(z)

co(z)

=

=

1: 1

(8.6a)

(n=1)

1: lz+ 1 1: 2Z + 1

1:~2+Z+.111:2

(8.6b)

(n=2)

(n =3) .

(8.6c)

8-3 Pole Locations and Spectral Densities The locations of the n poles are determined by the roots of the degree-n polynomials in the denominator of expressions (8.6). For the first two cases we have the following explicit results (in terms of the variable l;=iZ=ro+ie): . I;

p

I;p = __ [1 l_'

21: 2

i = -1:

(8.7a)

(n=1)

1

±

J1-4.111:;]

(n =2)

.

(8.7b)

The bottom part of each plot in Figs. 8-3 through 8-5 represents the complex 1;plane with 3[1;] in vertical direction and 9t[I;] in horizontal direction. The lines with arrows in that part of each plot indicate the path of the poles in the I;-plane for fixed .11""'~-1 and increasing values of 1:n . The general picture is the following: for small 1:n we have n-l poles near the physical frequency axis 3 [1;]=0. As 1:n increases, these poles tend to move away from that axis, while one additional pole is moving in from below. After some more or less complicated motion at intermediate values of 1:n , all n poles move toward the physical frequency axis as

Section 8-3

99

'tn~oo. The pole locations in the S-plane determine the shape of the n-pole spectral density. Explicit expressions for these functions are readily derived via (3.28) from the results (8.6): cI>o( 00)

(n = 1)

(8.8a)

(n=2)

cI>o(oo) =

2LlI~'t3

['t3ro(oT-LlI-~)]2 + (oT-Ll I)2

(n =3) .

(8.8b)

(8.8c)

They are plotted in the top part of Figs. 8-3 through 8-5 for particular values of'tn • The pole locations are indicated by the circles on the S-plane underneath. Let us discuss the three cases individually. n=l: The spectral density (8.8a) (depicted in Fig. 8-3) is a pure Lorentzian. Its width is proportional to lI't I , the distance of the single pole from the real axis. 5.0.,......._-----,--------, T 1=2.0

4.0

3'

3.0

~o

0&

2.0 1.0


o ], -0.5

" -1.0+-~.,......._~....._~+-~.__~,__.----j

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

Figure 8-3: The top part shows the 1-pole spectral density (8.8a) as a function of ro for fixed relaxation time t 1=2.0. The bottom part shows the pole location (circle) in the lower half complex ~-plane of the associated relaxation function (8.6a). The arrow indicates the path of the pole for increasing values of t 1•

n=2: For small relaxation times 't2, both poles lie on the negative imaginary S-axis. The spectral density, expression (8.8b), again consists of a single central peak, such

100

Chapter 8

as the one shown in Fig. 8-4(a). Its width is proportional to the distance between the real ~-axis and the pole nearer to it, -~1't2 . The main difference between the function (8.8b) for small 't2 and the Lorentzian (8.7a) is that the former decays more rapidly to zero in the wings. For the Lorentzian, only the zeroth frequency moment (3.30) is nondivergent, =Mo=l, whereas the function (8.8b) has one more nondivergent moment. Note that its value, (8.9)

is independent of't2. Hence the 2-pole result (8.8b) satisfies one sum rule exactly, but violates all those associated with higher-order frequency moments. For increasing values of 't2, the two poles move closer to each other on the negative imaginary ~-axis, at 't2 = ~~/2/2 they coalesce and then separate by moving away from that axis in opposite directions. The spectral density (8.8b) changes qualitatively in the process from a single central peak into a two-peak structure, like the one shown in Fig. 8-4(b). The peak locations on the frequency axis are determined by the real parts of the pole locations, .. /

(8.10)

2

±~ = %If~l -4't~

In the limit 't2~oo, the two poles hit the real ~-axis, and the spectral density reduces to a pair of o-functions. This is one of the situations described in Sec. 5-1. 3.0-,------,--------, 6.0

3'

(b)

(a) 2.0

4.0

~o

0&

1.0

2.0

0.0

l:l

l:l

o

o

~ -1.6

:;; -1.6

"'"

"'"

-3.2+-~.---~..--~I__""""'c---"'---r~---l

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-3.2+-~._~,__~I__.____,r___'___r~__j

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

Figure 8-4: The top part shows the 2-pole spectral density (8.8b) as a function of ID for fixed parameter &1=1.0 and two different values of the relaxation time: (a) 't2=O.3, (b) 't2=1.0. The bottom part shows the pole locations (circles) in the lower half complex ~-plane of the associated relaxation function (8.6b). The arrows indicate the paths of the poles for increasing values of 't2•

101

Section 8-3

n=3: For small relaxation times t 3, expression (8.6c) has one pole on the negative imaginary l;-axis far below the real axis and a pair of poles off the imaginary l;-axis near the real axis. The associated spectral density is dominated by the latter two poles. It has a double-peak structure such as shown in Fig. 8-5(a). For t 3=O expression (8.8c) reduces to: (8.11)

As t 3 is made larger, the pole on the negative imaginary l;-axis moves closer to the origin, while the other two poles loop away from that axis in opposite directions. This has the consequence that the spectral density turns into a 3-peak structure as shown in Fig. 8-5(b). In the limit t3~oo, expression (8.8c) reduces to a set of three o-functions: (8.12a)

et>o(oo) = 21tAo0(O) + 1tA I [O(oo-01) + 0(00+01)] , A2

2

001 = ~I +A2 ' A o

4.0

3.0

3'

~o

Al +A2 '

A,-1.0

3.0

3'

2.0

~o2.0

0&

1.0

1.0

0.0

0.0

\.1

~J

~!

0:

0:

c:>.

(8.12b)

Al +A2

(b)

(8)

\.J 0

Al

4.0

A,-1.0

0&

~

Al

'0

-0.5

~

c:>.

-0.5

"

"

-1.0 -3.0

-2.0

-1.0

0.0 (J

1.0

2.0

3.0

-1.0 -3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

(J

Figure 8-5: The top part shows the 3-pole spectral density (B. Bc) as a function of co for fixed parameters A1=1.0, A:!=1.0, and two different values of the relaxation time: (a) 't3=O.707, (b) 't3 =1.25. The bottom part shows the pole locations (circles) in the lower half complex ~-plane of the associated relaxation function (B.6c). The arrows indicate the paths of the poles for increasing values of 't3 .

102

Chapter 8

The 3-pole spectral density (8.8c) satisfies two exact sum rules,

= M 2 = aI'

= M 4 = a1(a1+~) ,

(8.13)

independent of the relaxation time 't3 .

8-4 Memory Functions and Fluctuation Functions The relaxation function co(z) of any dynamical problem that can be formulated in the framework of the recursion method is expressible, via (3.38a), in terms of a memory function as 1

co(z)

z +:t(z)

:t (z) == (dte -ztt (t) .

~'

(8.14)

The (normalized) fluctuation function Co(t) then satisfies the deterministic equation 00

..!Co(t) + fdt I t (t-t I )Co(t I) = 0 , dt 0

(8.15)

while the associated dynamical variable A(t) satisfies a stochastic equation of the generalized Langevin type: 00

~(t) dt

(8.16)

+ fdtlt(t-tl)A(tl) = F(t) . 0

All this was demonstrated in Sec. 3-6. From the perspective of a contracted-level description, F(t) is a random force in the sense that no correlations exist between F(t) and A(t') for «t. From the perspective of the underlying microscopic dynamics, on the other hand, there is nothing random about F(t). In the framework of the recursion method, both quantities A(t) and F(t) are determined in the form of orthogonal expansions (3.17) and (3.40), respectively. The main point of interest here is the following: the n-pole approximation establishes a connection between a dynamical problem analyzed by the recursion method on a microscopic level of description and certain simple phenomenological models used for the description of the same systems from a macroscopic perspective in terms of phenomeno10gical1y motivated (generalized) Langevin equations. In the simplest case n=l, the relation (8.14) in conjunction with the result (8.6a) yields the memory function :t(z)

= ~ = const, 't 1

t(t)

= ~O(t) 't 1

.

(8.17)

This memory function with no memory reduces (8.16) to the ordinary Langevin equation,

Section 8-4

103

(8.18)

A(t) + ..!.-A(t) = F(t) .

't l

Any effects of retardation are excluded. Consistency requires that F(t) is &correlated (white noise). Equation (8.15) turns into the differential equation for the classical relaxator, .

Co(t) + -

1

't l

(8.19)

Co(t) = 0 .

The fluctuation function in the I-pole approximation is thus always a pure exponential,

= e -tIt 1

(8.20)

and the associated spectral density a pure Lorentzian - expression (8.8a) depicted in Fig. 8-3. In this approximation, all dynamical systems are equivalent to Brownian motion. The case n=2 is known under the name relaxation time approximation because the retardation effect in the generalized Langevin equation is characterized by a pure exponential, (8.21) Equation (8.14) with this memory function is equivalent to the differential equation for the damped harmonic oscillator,

.. Co

+

1· -Co 't 2

+

(8.22)

Al Co = 0 .

The solution with the (mandatory) initial conditions Co(O)=I, Co(O)=O depends (for given AI) on the value of't2. Long relaxation times produce underdamped motion, and short relaxation times overdamped motion. The resulting fluctuation functions for the two cases read: Co(t)

= e -tl2t 2[COS(rot)

Co(t)

= e -tl2t 2[cosh(nt)

1

-

+ _ _ sin(rot)], 2rot 2

+

1 2Qr 2

sinh(nt)],

-

CO

./ -2 == VAI -4't 2

n- ==V./4't 2-2 -AI

> 0,

> O.

(8.23)

(8.24)

The associated spectral density (8.8b) is plotted in Fig. 8-4(a) and Fig. 8-4(b) for the two cases, respectively. The pattern within the framework of the n-pole approximation is quite evident: The (n-1)-pole fluctuation function Co(n-I)(t;A I""'£\_2''tn_l ) determines the structure of the n-pole memory function t(n)(t;AI,...,An_I,'tn) by the relation

104

Chapter 8

(8.25) For n=3 the memory function is one of the two damped harmonic oscillator functions (8.23-24). Such memory functions give rise to spectral densities of the type given by the expression (8.8c) and plotted in Fig. 8-5(a) and Fig. 8-5(b). The prototype physical application in which the n-pole approximation performs reasonably well pertains to light scattering from an isotropic fluid. The Rayleigh peak (heat diffusion) and the two Brillouin peaks (sound waves) that govern the scattering cross-section can be interpreted as a linear combination of relaxation functions with one and two poles, respectively. These functions are a direct result of the linearized Navier-Stokes equations [Forster 1975]. The 3-pole approximation was invoked by Lovesy and Meserve [1972] for the interpretation of the magnon peaks and the diffusive central peak that had been observed by inelastic neutron scattering from quasi-ID Heisenberg antiferromagnetic materials.

8-5 Moving Beyond Truncation The n-pole approximation and the cut-off termination only use the information which is explicitly contained in the first n coefficients !:ik' Consider, for example, the finite sequence !:ik= kP/2, k=1,...,15, which we have computed for the T=oo spin autocorrelation function <Si(t)S[> of the 10 s=1I2 XX model (7.33). We may close our eyes to any attempt of pattern recognition, use only these explicitly determined coefficients in the continued fraction, and then terminate it by cut-off or impose the n-pole approximation, which is almost the same. The resulting spectral density will inevitably be a poor approximation to the Gaussian (6.47). We have illustrated this in Sec. 8-1. Adding one or two explicitly known coefficients, typically obtained at great computational expense, does not yield a significant improvement of the result. It is important that we also use the information implicitly contained and recognizable in the finite!:ik-sequence. In the above example, one crucial piece of implicit information is that the !:ik-sequence has growth rate A.=1 as defined in Sec. 5-3. More detailed (and perhaps less certain) implicit information would be the expectation that all !:ik's continue to fall exactly onto the line kJ212. If we want to move beyond truncation in our attempt to reconstruct relaxation functions from finite!:ik-sequences we must develop ways to incorporate such implicit information into the approximation scheme. The terminating relaxation time 'tn of the n-pole approximation is, in general, a very inadequate instrument for that purpose. Moving beyond truncation means replacing the relaxation rate lI'tn in (8.4) by a termination function rn(z) which incorporates to the fullest extent possible everything we know with some confidence about the !:ik-sequence :

105

Section 8-5

1 co(z) '" - - - - - - - -

61 z+-------z+ ... 6 K -1 +-....,....=---..,-

(8.26)

r

Z +6K ~z)

The termination function r ~z) is also known under the name K!h-order memory function. In the remainder of this Chapter we briefly discuss some widely used termination schemes which employ termination functions of some sort. All this is just a prelude to Chapter 9, where we present what we think is a particularly powerful method of terminating a continued fraction. All three approximation schemes discussed in the following have basically two ingredients [Roldan, McCoy, and Perk 1986]: (i) the explicit information contained in a finite number of moments M 2k or continued-fraction coefficients 6 k ; (ii) the use of a termination function r ~z), which amounts to an extrapolation of the finite 6 k-sequence.

3.0

2.0

--- Kal0 - - K=15

~~ 1.0

2.0 ......... 3 .........

I1 I1 )1

i I

0.0 0

N NO

5

'9'

10

k

1.0

",'"

//

--

4/

.&."'/./.

-:/"

~

0.0 0.0

~--

","

1.0

2.0

CJ

Figure 8-6: Spectral density ctl~Z(
106

Chapter 8

Stationary-memory-function approximation [Nagao and Miyagi 1976]: Consider a finite L\k-sequence whose trend it is to converge towards a finite value for increasing k. The example shown in the inset to Fig. 8-6 pertains to the spin autocorrelation function <Sj(t)S]> at T=O of the 10 s=1I2 XX model (7.33). For such a sequence, the following approximation scheme is guaranteed to converge (albeit perhaps slowly): set ~

= L\K

(8.27)

for all m > K .

In practice, this scheme is implemented by the termination function,

r~z) = ~[J%+z2 -z]. %= 4L\K' C%

(8.28)

to be used in the continued fraction (8.26). This termination function is equivalent to an infinite continued fraction with (8.27) satisfied, as is readily shown by a construction like the one used in Sec. 5-3 for the determination of the exact relaxation function of a periodic L\k-sequence. The main plot of Fig. 8-6 shows the spectral density derived from (8.26) with (8.28) inserted at two different stages for the example at hand. Convergence toward the exact result (solid line) is slow but real. The stationary-memory-function approximation scheme is designed for any relaxation function whose L\k-sequence converges to a finite value, L\oo' However, the method breaks down if, for example, the ~k and the ~k-l converge toward different values or if the L\k tend to increase indefinitely. Gaussian-memory-function approximation [Bennett and Martin 1965; Morita 1975]: Now consider the finite L\k-sequence shown in Fig. 8-7, which has been obtained from the recursion method applied to the spin autocorrelation function <Sj(t)S/> of the 10 s=1I2 XX model once again but now at T=O. The average growth is linear in k (growth rate A.=1). We recall that a purely linear L\k-sequence is associated with a purely Gaussian fluctuation function, (8.29) whereas a constant L\k-sequence is associated with the sum of two Bessel functions: (8.30) The Laplace transform of (8.30) is precisely the termination function (8.28) used for the stationary-tQOmory-function approximation. By analogy, the Gaussianmemory-function approximation uses the Laplace transform of (8.29),

.[it Z2/~ r~z) = _e erfc(zlOb), C%

2

C% =2L\K+l

'

(8.31)

as the termination function in (8.26). Here again, the last explicitly used coefficient, L\K+l' determines the parametrization. The implementation of the Gaussian-memoryfunction approximation is equivalent to the following extrapolation of the explicitly known finite L\k-sequence:

107

Section 8-5

8.0

I£l

7.0 6.0 I

<1

4.0 3.0 2.0

I I

~

5.0 ~

I I I

I I

I I

I

I I I

I

I I

• I

I

I I I

,

/

i-

,

/

/

j

/

/

! •

o Gaussian-memory-function approximation at K"'3 '" Linear-extrapolation approximation at K"'5

1.0

o. 0 -+--,-------r---.--....---,.----r---r---r---,r----r---,--.,....-~ o 1 2 3 4 5 6 7 8 9 10 11 12 13 k Figure 8-7: dk-sequence pertaining to the spectral density ~X«(J») at T=O of the 10 5=1/2 XX model (7.33) with ,),,1 (circles). It was computed by means of the Hamiltonian representation of the recursion method. The squares and triangles represent the effective extrapolations of that sequence imposed by two different approximation schemes. Also shown is the linear regression line d k '" O.574k (forced through the origin) for the 13 data points.

set !!J.K +m

'"

m!!J.K + l

for all m> 1 .

(8.32)

Figure 8-7 shows the m-dependence of the extrapolated coefficients !!J.K +1 for the case where the Gaussian-memory-function approximation is implemented at K+l=4 (short-dashed line). What strikes the eye is a serious mismatch in slope between the linear growth of the new coefficients extrapolated according to the recipe (8.32) and the slope of the average linear growth of the exact coefficient. This mismatch becomes worse for larger K, where the approximation is supposed to be better. The trick which is justifiable in the case of the stationary-memory-function approximation cannot be transcribed without major modifications to Gaussian-type memory functions. The Gaussian-memory-function approximation is manifestly not convergent, not even for a pure Gaussian. Linear-extrapolation approximation [Morita 1975]: An alternative way to extrapolate a !!J.k-sequence with linear growth has been employed in a number of applications. It also uses a Gaussian-type memory function, but one with two parameters, given by the last two explicitly used exact coefficients, !!J.K, !!J.K+1• The

108

Chapter 8

prescription for extrapolating the ~k-sequence which is equivalent to such a twoparameter memory function reads: (8.33) This extrapolation procedure is implemented at level K=5 in the ~k-sequence shown in Fig. 8-7 (long-dashed lines). It is clear that the linear-extrapolation approximation is beset by essentially the same problems as the previous scheme. For a pure Gaussian, the extrapolation (8.33) is exact, but that is the exception. Whenever we have (~K+r ~K) < 0, which occurs not infrequently, (8.33) loses all legitimacy. It would be preferable to use an approximation whose effective extrapolation of the ~k-sequence follows the linear regression line of the explicitly known coefficients (shown solid in Fig. 8-7). We shall return to this suggestion in Chapter 9. The termination procedures discussed in this Chapter seem to operate under the erroneous assumption that in approximating x
r

r

r

9 RECONSTRUCTION OF SPECTRAL DENSITIES FROM INCOMPLETE CONTINUED FRACTIONS The central theme of this Chapter is the art of terminating a continued fraction [Viswanath and MUller 1990]. The diverse facets of the recursion method which we have developed in previous Chapters are now combined into a new method for the construction of termination functions - a method which transcends the limitations of the calculational schemes portrayed in Chapter 8. Our method surely has its own limitations, and we shall not shy away from pointing them out. But its simplicity and versatility makes it quite powerful, and its enormous flexibility allows for customized fine tuning in individual applications. All this will be demonstrated in extensive tests here in Chapter 9 and then in applications to quantum spin dynamics at high temperature (Chapter 10) and low temperature (Chapter 11).

9-1 Model Terminators from Model Spectral Densities The key property of our calculational scheme is that it incorporates the explicit and the implicit information contained in an incomplete Llk-sequence to the fullest extent possible into the reconstructed spectral density. The method is best described in terms of three different relaxation functions, each expanded into a continued fraction down to level K. The exact relaxation function co(z) = - - - - - - - - - -

Ll 1

Z + ---------

z

+ ...

(9.1)

Ll

_ K 1

+---Z + LlKr~z)

is the one we wish to determine, here expressed in terms of explicitly known continued-fraction coefficients Ll1,...,LlK, determined recursively, and an unknown termination function r ~z). The model relaxation function

co(z)

=

1

_ Ll 1

Z + ---------

z

+ '"

(9.2)

LlK _1

+----Z +

LlKr ~z)

is determined via Hilbert transform from a model spectral density $0(0)) with a known sequence '&1''&2'''' of model coefficients. The function $0(0)) is designed to reflect all the implicit information that can reliably be extracted from the incomplete

110

Chapter 9

ak-sequence - information on bandwidths, gaps, singularity exponents, decay laws etc. Ways and means to recognize that information were discussed in Chapter 5. The reconstructed relaxation function co(z)

=- - - - - - - - - z

al

+ ---------

z

+ ..,

aK _1 +-----

(9.3)

Z + aKr ~z)

combines the explicit and the implicit information consistently and to the largest extent possible, namely the known coefficients al,...,aK from (9.1) and the (approximately) matching termination function t ~z) from (9.2). In practical applications, the construction of coW is implemented along the steps outlined in the flow chart of Fig. 9-1 and discussed in the following: The input data consist of a finite set of continued-fraction coefficients {a.,...,aK ,} which have been produced by the recursion method, either directly by the recurrence relations (3.19) or indirectly by the recurrence relations (4.5). This is the explicit information to be incorporated into (9.3). Next we pi<:k ~ model spectral density eI>o(OO) such that the associated model sequence {a.,~, ... } reproduces and extrapolates the patterns_that have been identified in the input data {a.,~, ...,aK}' The function «1>0(00) must be specified in the form of an analytic expression, simple enough that its frequency moments {MO.M2,... } as defined in (~.3Q) can be determined exactly. In some cases the model coefficients {al'~''''} may be known exactly too, in other cases they must be calculated numerically from the exact moments as shown in Sec. 3-5. From the selected model spectral density $0(00) we determine the model relaxation function co(z) by Hilbert transform

o o

o

co(z)

o

1 = _. 21t1

Jdoo _- _. .

..".

-00

«1>0(00) ro-IZ

(9.4)

For most of the model spectral densities used in this work, the associated model relaxation function can be determined analytically. However, for the present purpose it is sufficient to know the values of that function on a line of points Z = £-ioo with 0<£«1 in the complex frequency plane. An array of such values may be determined with sufficient accuracy from a numerical implementation of the Hilbert transform (9.4). The tennination function t ~z) is then determined (only for a set of points on the line z=£-iro with 0<£«1) by expanding the model relaxation function co(z) into a continued fraction. This is carried out iteratively by means of the relation

Section 9-1

-riz) = -=-1[1 -_-- - z], r Ll k

111

(9.5)

k=1,2, ...,K

k - 1(z)

with f'o(z) == co(z). The function extracted from the input data.

f'x
contains the implicit information

Pick analytic expression based on implicit information extracted from the d r

Exact values

Exact or numerical values

Analytic expression or numerical results for z =

f k(£ -jro)

Numerical values at z =

£-jCJ)

£-jCJ)

Data produced by recursion method

Reconstructed spectral density

Figure 9·1: Implementation of the proposed method for the reconstruction of a spectral density from an incomplete sequence of continued-fraction coefficients and a judiciously selected model spectral density.

o

The explicit information contained in the input data is now needed for the determination of the approximate relaxation function co(z) by executing the iterative relations

112

Chapter 9

r- k- 1(Z) =

o

1

z+b.l k(Z)

,

k=K,K-l, ...,l

(9.6)

with f x0(00)"29t[cO(£ -ic.o)] , 0<£« 1 .

(9.7)

In expressions (9.1-3), the termination can be carried out at any level for which the b.k's are explicitly known. The expectation is that the quality of the approximation improves systematically as more and more model coefficients, ~k' are replaced by exact ones, b.k , provided their dominant patterns match sufficiently. The strengths and limitations of our method are best demonstrated and evaluated on a case by case basis. In the remainder of this Chapter we shall introduce a selection of model terminators for use in test runs on exactly solved models and in applications to realistic models of current interest in condensed matter physics. In the process of these tests and applications, we shall see ample opportunity for further development and fine-tuning of the method. The model terminators are named after some characteristic property of the associated model spectral density.

9-2 Square-Root Terminator Consider an application of the recursion method in which the b.k-sequence produced tends to converge, for increasing values of k, toward a finite value b.oo ' The simplest termination procedure which is consistent with that observation (implicit information) employs the square-root terminator, constructed from the model spectral density 4>0(00) =

~J~-cJEXCi\)-lool) Ci\)

,

(9.8)

whose model ~k-sequence is constant,

b. 1

-

1

2

= ~ = ... = -Ci\) 4

(9.9)

(see Sec. 5-2). The associated model relaxation function is (9.10)

Section 9-2

113

When expanded into the continued fraction (9.2), this particular relaxation function yields a termination function f ~z) that is the same at all levels K: r~z)

= co(z),

K=O,I,2,...

(9.11)

This function with the parameter 010 set equal to the value 2ii!.!2 is to be inserted into expression (9.3) along with the explicitly known coefficients .11,.12,... ,.1K , for the reconstruction of the spectral density 0(0l). In a first application, we consider the almost uniform .1k-sequence:

.11 =

~~,

.12 = .13 = ... =

4

~~ 4

.

(9.12)

Changing the value of the first coefficient .1k alone may have a strong impact on the shape of spectral density. The relaxation function generated by (9.12) is obtained from (9.3) with the termination function (9.11) inserted at level K=I:

1

co(z) =

2

= ------

U 2-

Z+4~rl(Z)

~

uV~+Z2 + (2-u)z

(9.13)

The associated spectral density has a continuous part of the form

4uJ~-cJ

cI>~(0l) = -~---e(~-10l1) .

(9.14)

u2~ +4(1-u)cJ

The bandwidth is independent of u. The endpoint singularity is a square-root cusp (~=1I2) for all u>O with one exception. For u=2, expression (9.14) turns into c

cI>0(0l) =

2

J~-cJ

e(~ -lOll)

(9.15)

with a square-root divergence (~=-1/2) at the band edge. The spectral-weight distribution within the band [-Olo, Olo], depends very strongly on u, Le. on .11, For certain values of u, the spectral density contains, in addition to the continuous part (9.14), a ~-function contribution outside the band. It arises from a pole in (9.13) on the imaginary z-axis. We shall return to this topic with a more detailed analysis in Chapter 11, in the context of an application where the continuum contribution and the pole contribution of (9.13) representjree and bound states, respectively, of a ID spin system (Sec. 11-23).1 For an entirely different illustration of how the square-root terminator operates, consider the classical 2-spin model (6.52). For one particular case the recursion method yielded the .1r sequence (6.53):

I The ak-sequence (9.12) was also reported to be relevant for the dynamics of a 20 electron gas [Lee and Hong 1985].

114

Chapter 9

/1k

k2

(9.16)

= "":":":~-:-=-:--:-:(2k -1)(2k + I)

We can reconstruct the spectral density (6.54), which is encoded in that sequence, by means of the square-root terminator. The parameter value COo=1 is set by the asymptotic value /1.. = 1/4. The reconstruction is carried out by taking the first K exact coefficients from (9.16), inserting them along with the termination function (9.11) into expression (9.3) and evaluating the spectral density cl>o(ro) via (9.7). The results for K=O,1,2,1O are displayed in Fig. 9-2(a) together with the exact result (6.54). The curve for K=O is the model spectral density (9.8). Notice that only few coefficients are needed to produce a fairly accurate approximation of the step function.

9-3 Rectangular Terminator This last spectral density which we have reconstructed with the square-root terminator can be used as the basis for an alternative terminator applicable under the same general circumstances. The building blocks of this new terminator, which we call the rectangular terminator, are the model spectral density -

1t

-EXCOo -lOll) COo

,

(9.17)

I ~In__ Z +i.COo = _.- ) ,

(9.18)

<1>0(00) = the model relaxation function co(z)

2lCOo

Z-/COo

and the model sequence of continued-fraction coefficients

_

=

/1

k

r.h,2

"1Y"

(2k-I)(2k+I)'

(9.19)

They all depend on a single parameter, COo, which determines the bandwidth of the spectral density to be reconstructed. The /(tb-level termination function as derived iteratively from (9.18) via (9.5) now strongly depends on K. The square-root terminator Was an exception in that respect. For the rectangular terminator, the JChlevel termination function can be expressed in closed form as the ratio of two associated Legendre functions:

rx
+ 1/2

-1/2

zI r-22"

VCOo+Z-

(9.20)

i!.;/~2 zlJ~ +Z 2 However, our method does not require that we have explicit analytic expressions for f'k(Z),

Section 9-3

115

4.0;---~

1.0

(a)

O.O-t-----r------,-----,--------f!'--' 0.00 0.25 0.50 1.00 0.75

K=l

4'01===~~

3.0

1.0

(b)

O.O+-----...-----...-----...------T""--' 1.00 0.25 0.50 0.75 0.00

Figure 9-2: (a) Reconstruction of the spectral density (6.54) (dashed line) from the model spectral density (9.8) with coo=1 and the first K exact continued-fraction coefficients (9.16). The curve for K=O is the model spectral density itself. In (b), the roles of the model spectral density and the spectral density to be reconstructed are interchanged. For the evaluation of the spectral densities we have used (9.7) with £=0.001.

116

Chapter 9

For a simple application of the rectangular terminator, consider the spin autocorrelation function <S/(t)5'? of the boundary spin So for the semi-infinite ID s=1I2 XX model at T=00.2 The ilk-sequence produced by the recursion method [Sen, Mahanti, and Cai 1991], _

ill - il2

_

_ 1

- '" - -

.

4

,

(9.21)

is a familiar one. It specifies the spectral density (9.8) with 000=1, which happens to be the model spectral density of the square-root terminator. While the use of that terminator for (9.21) would, of course, yield the exact spectral density at all levels of termination, it is the rectangular terminator which we want to test here. Thus we take the first K coefficients from (9.21) and the termination function t x
9-4 Endpoint Singularities Consider an application of the recursion method which produces a Llk-sequence that tends to converge uniformly to a finite nonzero value, iloo=ro5t4. It represents, as we know from Chapter 5, a spectral density with bounded support and no singularities in its continuous part except at the endpoints of the interval [-COo,oo0]. The squareroot terminator and the rectangular terminator are then, in general, equally suitable for its reconstruction. In many instances we are interested not only in the qualitative shape of the spectral weight distribution in «1>0(00), but also in the specific endpoint singularity, -(COo - oo)~, of that function. The long-time asymptotic behavior of the corresponding fluctuation function, in particular, is determined by these singularities. It is indeed possible to determine the exponent ~ by direct analysis of the incomplete ilk-sequence prior to the reconstruction of the spectral density. In Sec. 5-2 we have learned that the square of the exponent ~ of the endpoint singularity is determined by the amplitude of the leading correction term in the asymptotic expansion of the ilk-sequence, provided the latter converges uniformly toward a finite value iloo [Magnus 1985]:

~

1_~2

_ i l l + - - + .... ] ilk 2 4k

(9.22)

We can turn this relation around and analyze the sequence

2-rhe infinite temperature dynamics of that model, defined by Hamiltonian (7.33), will be discussed more extensively in Chapter 10.

117

Section 9-4

~~ = .!.4 _k 2[f!.kf!... _ 1].

(9.23)

It tends to converge toward the value ~2, the square of the singularity exponent. For ~2<1, either sign of ~ is possible. For an illustration of this exponent analysis, we preview once again some results for the ID s=1I2 XX model (7.33) at T=oo. The main plot of Fig. 9-3 shows the f!.k-sequences plotted vs k of the boundary-spin autocorrelation functions <SQ(t)SQ> and <S6(t)S~ for a semi-infinite chain. One sequence is constant, given by (9.21), the other converges uniformly toward the value f!...=1. The corresponding I~kl-sequences, plotted vs lIk, are shown in the inset to Fig. 9-3. One of them is constant, l~kl=1I2, the other converges toward the value 1~1=2. The former f!.ksequence describes the familiar function (9.8) and the latter the function (10.15) [Stolze, Viswanath, and MUller 1992]. That function does indeed have a quadratic cusp at the band edge.

1.0

2.0-,------------, ~


0.5

-..\l: Cl:l. 1.0

0.0

0.0

1.0

0.5

1/k 0.0

0

5

10

15

20

25

30

k Figure 9-3: Continued-fraction coefficients ti k versus k, k=1 '00.,30, for the boundary-spin autocorrelation functions <-%(1)-% > (open circles) and <~(I)~> (full circles) of the semiinfinite 10 5=1/2 XX model (7.33) (with .1=1) at T=oo. The inset shows the corresponding sequence I~~ versus 1/k for the same coefficients [adapted from Stolze, Viswanath, and MOller 1992].

118

Chapter 9

9-5 ~- Terminator Once we have determined the value of the singularity exponent ~ at least approximately by the method described in Sec. 9-4, we wish to incorporate that piece of implicit information into the terminator to be used for the reconstruction of the spectral density. In generalization ofthe square-root terminator @1=1/2) and the rectangular terminator (~=O) we therefore introduce the ~-tenninator. Its building blocks are the model spectral density (5.7),

-

-(2~+1)

21tc%

2

<1'0(00) = B(lI2,I+~)(C%-

cd ~ )

,100ISC%,

~>-l,

(9.24)

the model ~k-sequence (5.8), f! k

olcJc(k +2~)

= ....",....,,....-,,..,.....--:":"~----,...,,-'"""""'

(9.25)

(2k +2~ -1)(2k +2~ + 1)

and the model relaxation function in terms of a hypergeometric function,

_ I .R. 2 2 co(z) - - F(l/2, 1, ... +3/2, -O'\Iz ) .

z

(9.26)

The ~-terminator is an instrument of fine-tuning for the reconstruction of spectral densities with compact support and endpoint singularities that are known at least approximately. A minor problem is that the model relaxation function (9.26) is usually not found in computer libraries for ready use in applications of the ~­ terminator. The simplest means to generate it numerically on a line of points z = 3 £ - ioo turns out to be the continued fraction with coefficients (9.25). Convergence for 0<£«1 can be checked pointwise along the ro-axis by comparing the result against the model spectral density (9.24). Consider the nonuniform f!k-sequence shown in Fig. 9-3 for a first application of the ~-terminator. It pertains to the boundary-spin autocorrelation function <S5(t)Sfy at T=oo of the semi-infinite s=1/2 XX chain. The parameter values to be used, 000=2 and ~=2, are readily inferred from the asymptotic properties of that f!k-sequence and the associated ~k-sequence. The associated spectral density
9-6 Gap Terminators For applications of the recursion method in which the resulting continued-fraction coefficients have the property that the f!2k_l and the f!2k converge toward different

3Technically, this procedure resembles the cut-off termination, a method we have found inadequate for the reconstruction of spectral densities (Sec. 8-1). For the present purpose, however, its use is legitimate, because we have an unlimited supply of exact continuedfraction coefficients ,5./('

119

Section 9-6

(finite) asymptotic values, .1~ and .1:" respectively, we need to design a new type of terminator, one whose model spectral density is of the form (5.20),

-

-

-

(9.27)

4>0(00) = AB(oo) + F(oo)EXlool-Ofuin)EXOfuax -1(01)

with (9.28) The key properties of $0(00) are, according to Sec. 5-3, a finite bandwidth, a gap centered at <.0=0 and (for .1~ < .1:') a &-function central peak. The most convenient gap terminator is the one (named Gl) with the periodic model sequence A.. ~-1

=.1°

00

'

~2k

e

(9.29)

= .1

00'

Its model spectral density is the function (5.18),

4.0-r------------------------,

3.0

10

30

20

40

50

k 1.0

O.O+-------.------,.-------r----:::::.-,.---' 2'.0 0.0 1.0

Figure 9-4: Spectral density CI»~Z(ro)o at T= 0 for the boundary spin of the semi-infinite 10 s=1/2 XX model (7.33) with J =1. The solid line represents the result derived from the continued fraction (9.3) (at z =£-iro, £=0.001) with the ~-terminator employed at level K=50 as described in the text. The dashed line represents the exact expression (10.15).

120

Chapter 9

~

(li.0 -li.e ,2

2(li.:;.~'J -01' -

0001' Oo! EXlrol-~n)EXOfuax -lOll)

~~

;. ~ [Ili.:-li.:J - (li.: -li.'J]B(ro) , li.:

and its model relaxation function is given by (5.17),

CO(Z)Gl

=-

1 [

e

0

2(li.00 ;.li.;:) ;. Z

2li.:

(9.31)

2 (li.: -li.'J2

;. -~Z2

The terminator G 1 has the advantage that the termination function at any evennumbered level is equal to the model relaxation function itself,

r 2X
(9.32)

'

a property that was used to determine the function co(z) in the first place (Sec. 5-3). It is ironic that in our very first application of the recursion method to such a situation [Viswanath and Muller 1990], we couldn't think of this simplest kind of gap terminator. Instead, we proceeded from the model spectral density

ct>o(ro)G2 =

1t

C!kax -Ofuin

EXlrol-~n)EXC!kax -1001)

(9.33)

and the model relaxation function

Co(Z)G2

=

rln(Z;'~C!kaxJ _ In(Z;'~Ofuin],

. 1 2z(C!kax - ~n) L

Z -zC!kax

Z-zC!kin

(9.34)

from which the ~-level termination function must be generated via (9.5). Another minor disadvantage of the gap terminator G2 is that its model sequence is not known in closed form. It must be determined via (3.33) from the frequency moments 2k+l

2k+l

U\nax - Ofuin M2k = 2k;. 1 Ofuax - C!kin -

1

(9.35)

But these slight complications in design are no serious impediment to the use of G2 in practical applications. The above mentioned application of G2 pertains to the dynamic structure factor Szz(q,ro) at fixed wave numbers q=1CI2, 31t14 and 1t for the ID s=1I2 XX model (7.33) at T=O. The first 13 coefficients li.k for the two cases q=31t14 and q=1t are displayed in the inset to Fig. 9-5. They were determined indirectly via (4.16) and (3.33) from the coefficients ak' derived in turn by an application of the

hi

Section 9-6

121

Hamiltonian representation of the recursion method. The recurrence relations (4.5) were carried out in the spin representation of the XX model. The exact finite-size ground-state was determined by the modified Lanczos method (Sec. 4_5).4

O-+------r--'---.......,.----~---~+_---'

0.0

0.5

1.0

2.0

1.5

=

Figure 9-5: Dynamic structure factor Szz c;2n) for fixed q = rm/4, n = 2,3,4 of the 10 5=1/2 XX model (7.33) (with .1=1) at T=o. The full lines represent the result derived from the continued-fraction representation for Co(e-ko) (with £=0.001) terminated at level K=5 as explained in the text. The dashed lines represent the exact result (6.38). The inset shows the Ak-sequences for q=37t14 and Q=7t [from Viswanath and Muller

1990].

The L1k-sequence for q=1t is readily recognized to be the special case u=2 of (9.12) - the familiar signature of the spectral density (9.15) with IDo=2. That result was already derived at the end of Sec. 6-3, in the Liouvillian representation of the recursion method and for the fermion representation of the XX model. The other L1k-sequence shown in Fig. 9-5 clearly calls for a gap terminator. The Llzk-l and the L12k converge rapidly toward the limiting values a~ and a:", respectively. The cut-off frequencies IDmin and IDmax inferred from (9.28) are in precise agreement with the exactly known boundaries, Jlsinql and 2JIsin(ql2)l, respectively, of the spectralweight distribution (see Sec. 6-3). Having thus determined the values of its two parameters, (Omin and IDmax , we can take the model relaxation function (9.34), determine the xth-level termination

4For a system with N=18 spins, the coefficients !J.k up to k=13 were found to be truly size-independent for q = n and nearly size-independent for O
122

Chapter 9

function f'iz) via (9.5) and reconstruct the spectral density via (9.3) and (9.7). In the present application, a fairly low value of K is sufficient to obtain accurate results. The reconstructed spectral density $0(00), which in this case is the (normalized) dynamic structure factor Szz(q,oo)/<S'qS=q> at fixed q=rrJ2 and q=3rrJ4, based on continued fractions terminated at level K=5, is shown in the main plot of Fig. 9-5 (solid lines). The results are almost indistinguishable from the exact function (6.38) shown dashed in Fig.9-5. 5 Further applications of the gap terminators will be discussed in Chapter 11.

9-7 Infrared Singularities in Spectral Densities with Bounded Support We know from Sec. 5-2 that if the l\k-sequence under investigation approaches the limiting value L\co = 0002/4 on an alternating path from both sides, this is an indication that the spectral density has a singularity at 00=0, <1»0(00) -Iool a , in addition to the ones at the endpoints of the interval [-<00 , 000], The exponent a of the infrared singularity then determines according to (5.14) the leading-order term of the large-k asymptotic expansion of the associated l\k-sequence [Magnus 1985]:

a

l\k = l\.Jl - (-1) kk + ...] .

(9.36)

From a l\k-sequence of that type as produced by some application of the recursion method we can determine the value of a, at least approximately, by analyzing the sequence

a,

=

(-I)'{ -~J

(9.37)

which converges toward the value of the exact singularity exponent a. In some applications, the convergence may be slow and irregular due to the presence of further singularities at the endpoints and, perhaps, elsewhere in the interval [-000,000], Hence, the extrapolation must be carried out with due circumspection. Let us illustrate this exponent analysis with two simple applications in spin dynamics. Consider first the ID s=1I2 Heisenberg ferromagnet (6.22). In Sec. 6-2 we have already used the recursion method to determine the dynamic structure factor (6.23) at T=O. The result, expression (6.24d), is extremely simple. Here we wish to use the recursion method for the direct determination of the spectral density pertaining to the spin autocorrelation function <S[(t)S[>:

5The results of Fig. 9-5 should be compared with Fig. 1 of [Gagliano and Balseiro 1987]. That figure shows the results of attempts to obtain the dynamic structure factor (6.38) by two general methods - a quantum Monte Carlo method and the recursion method with cut-off termination. The data of the former method were quoted from a paper by SchCrttler and Scalapino [1986].

123

Section 9-7

«I»~(CO)

{J dq[B(co-J(l-cosq»+B(co+J(l-cosq))] +'It

=

~~

1

EX2J-lcol) VICOI(2J -Icol)

This function has a square-root divergence at co=O in addition to the endpoint singularities. The latter happen to be square-root divergences as well. The Iiksequence produced by the recursion method for «I»o(co) is displayed in the main plot of Fig. 9-6. It is alternating in character and tends to converge toward the value 6.oo=J as expected. The corresponding (lk-sequence plotted versus lIk is shown in the inset of Fig. 9-6. The points converge rather uniformly toward the value a=-1I2, in agreement with the exact result (9.38). 3.0

-0.5

-0.6 ~

cs 2.0

-0.7

-0.6

~

<J

0.0

0.1

0.2

0.3

0.4

0.5

1/k 1.0

0.0 +--,--,---,--,----,--,--,--,--,--,----,r---,-----,r--....,......--j 15 o 5 10

k Figure 9-6: Continued-fraction coefficients d k versus k, k=1 ,... ,15, for the spin autocorrelation function <SM8;> at T=O of the 1D 5=1/2 Heisenberg ferromagnet (6.22) with .1=1. The inset shows the corresponding sequence a k versus 1/k [from Viswanath and Muller 1991].

For our second example we consider the T=O spin autocorrelation function <S1(t)Sj> of the ID s=1I2 XX model (7.33). The associated spectral density «I»5Z(co) is derived from the exactly known dynamic structure factor, (6.38), by a sum over all wave numbers and can be evaluated in terms of elliptic integrals [Muller and Shrock 1984]. The spectral weight is confined to frequencies lcol < 2J and

124

Chapter 9

singularities are present at roll = 0,±1,±2. The Ak-sequence produced by the recursion method up to 1<=15 is shown in the main plot of Fig. 9-7. It has alternating character again, but now the Au:-l converge from above and the Au: from below. The implication is that the exponent a is positive. That is confirmed by the plot of the ak-sequence shown in the inset to Fig. 9-7. It tends to converge to the value a=1 (the exact value) but more slowly and irregularly than the sequence of the previous example. The irregularity is caused by the additional singularities at Icol;tO, which are, in fact, stronger than the one at 00=0. 3.0-,------------------------, 2.0,------------,

.lo:

ts

1.0

2.0 0.0

~


0.0

0.1

0.2

0.3

0.4

0.5

11k 1.0

0.0 +----,.--r----r-..----,--...---.----,,.----r----,--r---,.---r---"T'"--j 15 10 5 o

k Figure 9-7: Continued-fraction coefficients Ak versus k, k=1 ,... ,15, for the spin autocorrelation function <~(l'~> at T=O of the 10 8=1/2 XX model (7.33) with J=1. The inset shows the corresponding sequence u k versus 1/k [from Viswanath and Muller 1991].

'-8 Spectral Densities with a o-Function Central Peak For a further application of the exponent analysis from a Ak-sequence that converges alternatingly toward a finite value, consider the classical XXX dimer, i.e. two spins coupled as in (6.52) with lx=ly=lz=l. In Table 9-1 we have listed the first 12 Ak's. This sequence has already been plotted in Fig. 7-2 in a different context. From its simple pattern it is safe to conclude that the spectral density has compact support on the frequency interval IcoI ~ 2. The alternating approach of the Ak's toward the asymptotic value Aoo=1 indicates the presence of a divergent singularity at 00=0. If that singularity is a power-law divergence, -Icola, then we can

125

Section 9-8

detennine the exponent a from an extrapolation of the corresponding arsequence (9.37). Note that only integrable singularities (a> -1) are allowed by exact sum rules. The main plot of Fig. 9-8 shows ak versus Ilk. That sequence is clearly not consistent with a pennissible power-law singularity. What we see in the plot is the signature of a &-function present at ro=O in an otherwise well-behaved spectral density with compact support. k

!1k

M(k)

k

!1k

M(k)

1

2

2

2

4

9

3 5

20

7

61

5

32 49

98

6

66 49

243

7

70 99

340

8

128 99

707

9

3 4

944

10

5 4

1775

11

112 143

2244

12

174 143

3891

3" 3

3" 4

'5

Table 9-1: Exact values of the first 12 coefficients Ak pertaining to the spin autocorrelation function <81(1)'81> at T=oo of the classical XXX dimer (6.52) with J J Jt=1. M(k) denotes the number of terms of the form (6.44) that appear in the vector fk [from Liu and MOller 1990].

r r

The problem then is to filter out the information on the continuous part of the spectral density contained in the known !1k-sequence. This can be achieved, at least in some situations, as follows. Consider a (normalized) fluctuation function that decays to a nonzero value asymptotically for long times: (9.39) The corresponding spectral density consists of a B-function central peak and a continuous part: cI>o(O» = 21t4>o(oo)O(O» + [l-dVoo)]cI>6C)(o» .

(9.40)

The continuous function cI>~c)(o» is then the spectral density associated with the (normalized) fluctuation function ci>~c)(t), which decays to zero as t~oo. The frequency moments of the two spectral densities cI>o(O» and cI>&C)(O» are related to each other as follows:

126

Chapter 9

-0.5

-------

continuum + a-function

-1.0 continuum alone

-1.5

1.0

~

~

........

-2.0

~.!<: tS

0.5

-2.5 0.0 +--T---r---r---,-..--.....-....-....,..........,..-i 1.0 0.0 0.5

-3.0

1/k -3.5-+----r--,--..---.....----r--,--..---.....---,----I 1.0 0.0 0.5

llk Figure 9-8: Sequence (9.37) of approximants exk to the exponent ex plotted versus 1/k (k=1 ,...• 19 ) for the continued-fraction coefficients ~k produced by the recursion method (Liouvillian representation) for the spin autocorrelation function <Sl(~'S1> at T=oo of the classical XXX dimer (6.52) with Jj=!t=Jt=1. The inset shows the same type of plot for the corresponding exLc)·sequence with «1>0(00) = 1/2.

M

o --

M(C) -

0

-

1i..

1

, (c)

M 2k = [l-qlo(oo)]M2k '

(9.41) k=I,2, ...

The relation between the two sets of continued-fraction coefficients is not that simple, but it can be shown that the sequence 1 A k = 2(~-1 +~), k=I,2, ...

(9.42)

generated from the moments M 2k of (9.40) does not depend on the value of the constant $0(00). Hence we have A k = A~C) for all k. For our current application, the !:ik from Table 9-1 yield A k=1 independent of k. However, it is the !:i~ctsequence that we wish to know, because it is from that s~uence that we can most likely extract quantitative information on the function $oc)(ro). For a given !:ik-sequence, we can determine the sequence !:i~c) pertaining to the spectral density $6C)(ro) from (9.41) and the recurrence relations (3.33-34), but only if we know the spectral

Section 9-9

127

weight of the o-function central peak, Le. the constant 4>0(00). That infonnation, however, is usually not available. In our current application, we can proceed as follows. We determine a whole set of ~£C)-sequences from the given ~k-sequence, where each member corresponds to a different value of the unknown constant 4>0(00). From that set of ~£c)-sequenceswe generate in turn a set of cx£C)-sequences according to (9.37). None of these sequences will converge to a finite value in a plot of a£c) versus 1Ik except the one for which the o-function central peak was given the correct amount of spectral weight. For the case at hand, we have 4>0(00) = 112. The corresponding a£C)_ sequence is plotted in the inset to Fig. 9-8. It tends to converge to the value a=1, implying that 4>~C)(ro) has a linear cusp singularity at 00=0. In fact the ~£c)-sequence resulting from 4>0(00) = 112 can be written in closed fonn as follows:

~(C)

= 1+_1_,

2k-1

2k+ 1

~ = 1 __1_. 2k+ 1

(9.43)

We recognize this to represent the special case 000=2, a=~=1 of the model ~k­ sequence (5.13). It is then a simple matter to reconstruct the exact autocorrelation function Co(t) = 3<sj(t)Sf> and its spectral density: 3 -4t 2

+ _ _cos(2t) , 4

(9.44)

4t

4>0(00) = 1to(ro) + 2:lrol(4-ot)EX2-lrol) . 8

(9.45)

9-9 Terminator with Matching Infrared Singularity After having established reliable means to detect the presence of infrared singularities in spectral densities with compact support and to detennine their nature, we should like to use tenninators for their reconstruction which are based on model spectral densities with matching singularities at 00=0. The result is a considerable improvement in the quality of the reconstructed spectral density. For a demonstation of this point, we return to the ~k-sequence shown in Fig. 9-7, pertaining to the T=O spin autocorrelation function <S](t)Sj> of the 10 s=1I2 XX model (7.33). Our analysis of that ~k-sequence in Sec. 9-7 has yielded two pieces of implicit infonnation that can be legitimately incorporated into the tennination function: (i) the bandwidth 1001 ~ 000 = 2J and (ii) the exponent value a=1 of the infrared singularity. A convenient ad hoc model spectral density with these properties is the continuum part of the result (9.45): 4>0(00)

1t 2 = _lrol(C%-ot)EXC%-lrol)

%

.

(9.46)

The associated model ~k-sequence and model relaxation function are both known in closed fonn:

128

Chapter 9

1.5

3'

1.0

'-"

N NO

>&

0.5

0.0 -iC---,....----..----r---,....----,-----r----r----j-"--' 2.0 1.0 0.0

Figure 9-9: Spectral density ~~Z(o» at T=O for the 1D s=1/2 XX model (7.33) with .1=1. The full line represents the result derived from the continued fraction (9.3) (with Z = E-iro, £=0.001) tenninated at level K=15 as described in the text. The dashed line represents the exact result (11.43) [from Viswanath and Muller 1991).

(9.47)

2) ( %( %J % We thus take the model relaxation function (9.48) with parameter value -

2z

co(z) ::: -

z2 % 1+n 1+-

2z

- -

•

(9.48)

Z2

000 = 21, detenmne the termination function f 15(Z) iteratively via (9.5), insert it into expression (9.3) along with the known coefficients A1,oo.,A 15 from Fig. 9-7, and evaluate the resulting spectral density via (9.7). The result is shown in Fig. 9-9 along with the known exact result. The agreement between the two curves is not perfect but very satisfactory if one takes into account that the reconstruction is based on a mere 15 numbers. The match is best at small 00, where both the exact spectral density and the model spectral density have the same singularity exponent, a=1, previously inferred from the Ak-sequence directly. The discrepancy is somewhat larger near ro=21, where the exact spectral density has a discontinuity, whereas the model spectral density goes to zero linearly. Despite this mismatch in

Section 9-10

129

singularity exponent, the reconstructed spectral density reproduces the discontinuity fairly well. The agreement between the two curves is worst near ro=J, where the exact result has one more singularity, but the model spectral density does not. The importance of the matching infrared singularity in the terminator used for this application is best illustrated by comparing the result shown in Fig. 9-9 with that previously obtained from a square-root terminator (see Fig. 8-6).

9-10 Compact a-Terminator When confronted with the task of reconstructing a spectral density whose IJ.k sequence tends to converge on an alternating path toward a unique and finite asymptotic value, we best proceed as follows: At first we estimate the asymptotic value IJ... directly from the IJ.k-sequence and the value of the singularity exponent a from the corresponding ak-sequence (9.37). Then we construct a termination function from a model spectral density with matching bandwidth and matching infrared singularity. For the application discussed in Sec. 9-9 we had a matching terminator conveniently at hand. For arbitrary bandwidths and singularity exponents we introduce here a new terminator, the compact a-terminator. Its model sr:;tral density is a special case of (5.12) and depends on the two parameters OOO=2IJ..!2 and a: (1)0(00)

= ~(1+a)loYO>olafXO>o-IOOI) 0>0

(9.49)

.

The model ~k-sequence is a special case of (5.13),

-

IJ. k

=

(2k-1 +a)(2k+1 +a) ot(k+ai (2k -1 +a)(2k + 1 +a)

(even k)

(9.50)

(odd k)

and the model relaxation function can be expressed in terms of hypergeometric functions: CO(z)

=

;z[F(l+a,1;2+a;iO\lZ) + F(1+a,1;2+a;-iO\lz)] .

(9.51)

If we set a=O, this terminator reduces to the familiar rectangular terminator (9.1719). There are other cases for which (9.51) can be expressed in terms of more elementary functions, for example,

co(z)

=

~[~ Tc%

+~J

I!ln(l 4z 1 -JiO\lz

+

+~J]

I;ln(l C% 1 -JO\Iiz

~

(a =-2.) 2

,

(9.52)

130

Chapter 9 5.0......------~-------------------,

1.4

0.0

(a)

-0.1

4.0

~

tS

1.2

-0.2

~


-0.3

8 3.0

..--.. :3

-0.5

0.5

0.0

0&

1.0

0.8

0

5

Vk

'-"

N NO

1.0

(b)

-0.4

10

15

20

k

2.0

1.0

(c) O.O+---------r------..--------.------r'--' 0.0

0.5

1.5

1.0

2.0

Figure 9-10: (a) Continued-fraction coefficients Ak vs k, k=1, ...,20, for the spectral density
co(z) =

-':''In(~+Z2J ~

(a=l).

(9.53)

Z2

As in the case of the ~-terminator (Sec. 9-5), a simple way to evaluate the model relaxation function (9.51) for the compact a-terminator on a line of points z =e-iro is the continued-fraction expansion with coefficients (9.50). In a first application, we consider (as in Sec. 9-9) the spin autocorrelation function <Sj(t)Sj> of the ID s=1/2 XX model (7.33), but now at infinite temperature. The first 20 tJ..k's as produced the recursion method in the Liouvillian representation (Sec. 6-4) are displayed in Fig. 9-IO(a). Their alternating approach with increasing k toward the limiting value t1.,,=J suggests the presence of an infrared singularity. Since the ~ converge from above and the tJ..2k_1 from below (opposite to the behavior observed in Fig. 9-6 for the corresponding T=O result), that singularity is expected to be divergent. A quantitative analysis of the associated

Section 9-11

131

ak-sequence (9.37) confinns this as shown in Fig. 9-1O(b). The ak's tend to converge toward a value somewhere between a.--o and a.=0.12. The exact result has, in fact, a logarithmic divergence [Katsura, Horiguchi and Suzuki 1970]. However, for our test application we are not supposed to use "insider knowledge". Therefore, we employ the compact a-terminator with our best estimates, 000 2, a -0.1, for the parameter values as inferred from Figs. 9-1O(a) and 9-1O(b). This implicit infonnation together with the explicit infonnation contained in the 35 known !1k's yields the spectral density shown as solid line in Fig. 9-1O(c). It agrees very accurately with the exact result (dashed line).

=

=

9·11 Gaussian Terminator In applications of the recursion method to interacting quantum many-body systems, it is frequently observed that the !1k-sequence of a specific spectral density under investigation has a linear growth rate, A.=1, as defined in Sec. 5-4. For the reconstruction of this type of spectral density from a finite number of known !1k's, we must choose a terminator which is consistent with that very property. The simplest model spectral density which does the trick is a Gaussian, d.. ((0)~ ""'0

=_ I4it _ e -cJ/cJo .

(9.54)

Cl\)

The associated model relaxation function is

-

co(z)

(ieZ2/cJo erfc(z/Cl\», =_

(9.55)

Cl\)

and the model Ak-sequence is strictly linear in k, 1 2 !1 k = -c4} . 2

(9.56)

with the adjustable parameter 000 determining the slope. Before we proceed with an application of the Gaussian tenninator, we should like to emphasize that in our method the ~h-Ievel tennination function f ~z) (at z = -ioo) is not itself a Gaussian except for K=O. In fact, it has a very strong Kdependence. This is dictated by the requirement that the tenninator simulates the correct average growth rate of the exact !1k-sequence. This important criterion is not taken into account if the termination function itself is modeled after a pure Gaussian as is the case, for example, in the Gaussian-memory-function approximation discussed in Sec. 8-5. For a first application of the Gaussian tenninator, consider the spin autocorrelation function <8/(t)81> of the 10 s=1I2 XX model (7.33) at T=O. This function has been analyzed on a rigorous basis (see Chapter 11 for a more detailed discussion). The spectral density, «1»0(00), as detennined on the basis of that rigorous analysis [Muller and Shrock 1984] is plotted as a dashed line in Fig. 9-11. It has

132

Chapter 9

three singularities on the frequency range shown: a square-root divergence at ro=O, a logarithmic divergence at OF=J, and a square-root cusp at «r=2J. The coefficients ~1""'~13' shown in the inset to Fig. 9-11 are clearly consistent with a linear growth rate (A.=l). The slope of the regression line detenriines the value of the parameter, CJ.lo=1.071J, of the Gaussian terminator. These ingredients are then put into our standard procedure for the reconstruction of the spectral density. The result is represented by the solid line in the main plot of Fig. 9-11. The outcome is very encouraging. The reconstructed spectral density reproduces all major features of the exact function at least qualitatively. Even more important is that our method has not produced any artificial features which may invite misinterpretation. The successful reconstruction of the spectral density CIlQ(ro) by means of the recursion method is an important test not only for the Gaussian terminator but for the recursion method itself, given the fact that this function exhibits many properties that are believed to be generic for spectral densities in quantum manybody dynamics [MUller and Shrock 1984, MUller 1987].

o. 0 -l-,....-,-.....,..............---r--r-'"'~.....,......'T""""""T---r-..,.-"........,......'T""""""T--r,.=;:=:;::=r==l 0.0

0.5

1.0

1.5

2.0

2.5

Figure 9-11: Spectral density CIl~X((O) for the spin autocorrelation function <8MB';> of the 10 s=1/2 XX model (7.33) with J=1 at T=O. The full line represents the result derived from the continued·fraction (9.3) (via (9.7) with £=0.001) terminated at level K=13 as described in the text. The dashed line represents the exact result first derived by Muller and Shrock [1984]. The inset shows the 13 known continued·fraction coefficients along with the regression line Ak = 0.574k [from Viswanath and MOller 1990].

Section 9-13

133

9-12 Infrared Singularities in Spectral Densities with Unbounded Support The spectral density reconstructed in this last example exhibits a number of singularities for which we may wish to determine the exponent values. Is it possible to predict these singularity exponents by an analysis similar to that employed in Secs. 9-4 and 9-7 for spectral densities with compact support? The answer is a qualified yes, at least for the singularity at c.o=O. Here we propose a procedure for the determination of the exponent value which characterizes the infrared singularity of a spectral density whose Ak-sequence has a linear growth rate (1..=1). Consider the model spectral density (5.24), (00)

o

=

21t1C% -oil 2 loYC% lae Cl\) r(aJ2 + 112)

(9.57)

whose only singularity is the one at c.o=O. It is characterized by the ~k-sequence (5.26): -

1

2

A2k - 1 = 2C%(2k-l +a),

-

~

1

2

= 2C%(2k)

.

(9.58)

The singularity exponent a determines the vertical displacement of the ~2k-1 from the line ~2k = oo'f/<. In most applications, there is no such easy-to-read signature of infrared singularities. The complication is clearly evident in the Ak-sequence shown in Fig. 9-11. Under these circumstances we can determine the value of a, for example, by calculating the average difference in vertical displacement of the coefficients ~ and the coefficients ~_I from the linear regression line that was derived for the entire sequence. For the Ak-sequence of Fig. 9-11 we thus obtain the exponent value

a = -O.5±0.4 .

(9.59)

That value is certainly consistent with the exact exponent, ex = -112, but the uncertainties are considerable. A larger number of known Ak'S will no doubt result in increased predictive power. In Chapters 10 and 11 we shall further explore ways to determine the values of these exponents in the context of spin dynamics applications. 6 9-13 Unbounded a-Terminator Once we have been able to determine at least a rough estimate for the value of the singularity exponent a by the method suggested in Sec. 9-12, we may attempt to sharpen some of the genuine features of the reconstructed spectral density by using expressions (9.57) and (9.58) as the building blocks of a new terminator, which we name unbounded a-terminator. For the case a=O it reduces to the Gaussian terminator. The associated model relaxation function can be expressed in terms of a Lommel function as follows [Gradshteyn and Ryzhik 1986]: ~he dk-sequence (9.58) was also reported to be relevant for the dynamics of a 3D electron gas [Hong and lee 1993].

134

Chapter 9

co(z)

22z-2a+I

C% (C% J

=- -

( 2z a-I - 2'-2 C%

S_a+I

J

.

(9.60)

In the numerical implementation of the reconstruction procedure, this function can be generated by summing up the continued-fraction representation as had already been suggested for the model relaxation functions (9.26) and (9.51). However, in the case of (9.60), which describes a spectral density with infinite bandwidth, convergence is much slower than for the cases with finite bandwidths. In such a situation, it is more efficient to determine the model relaxation function by a numerical evaluation of the Hilbert transform (9.4). Applications of the unbounded (X-terminator will be presented in Chapter 10.

9-14 Split-Gaussian Terminator The last terminator to be introduced in this Chapter pertains to spectral densities with unbounded support and a gap, which describe one of the most frequent occurrences in quantum many-body dynamics. This category of spectral densities is indicated by ~k-sequences of the type shown in the insets to Fig. 5-5. It is then convenient to use the split-Gaussian model spectral density (5.35), (1)0(00)

= 21tAa(ro)

+

2/i (l-A)EXlml-Q)e -(lrol-n)2/~

(9.61)

C%

and the modelli.k's derived via (3.33) from the frequency moments (5.36) as the ingredients of such a terminator. The model relaxation function must be computed from (9.61) via (9.4) by numerical integration. Applications of the split-Gaussian terminator are standard in quantum spin dynamics at zero temperature as we shall see in Chapter 11. For a preview, consider the T=O dynamic structure factor Szz(q,m) of the ID s= 1/2 Heisenberg antiferromagnet (4.35). This fun~tion has a q-dependent gap of size £L(q)=(1tl/2)lsinql as determined by the lowest branch of excitations. For q=TC!2, the recursion method applied to a system with N=16 spins yields 8 nearly size-independent ~k's as shown in the inset to Fig. 9-12. The pattern is clearly indicative of a gap and unbounded support. This is in contrast to what we have observed in Fig. 9-5 for the same quantity of the XX model (gap and bounded support). For the reconstruction of Szz(1tI2,m) in the Heisenberg case (Fig. 9-12), we have determined the parameters mo, {} of the model spectral density (9.61) with A=O by fitting the model li.k's to the ones shown in the inset and then following the standard procedure [Viswanath, Zhang, Stolze, and Muller 1994]. The reconstructed dynamic structure factor is shown in the main plot of Fig. 9-12. The exactly known gap £L(1tI2)=1tl/2 is fairly accurately reproduced. The result suggests that (i) the spectral weight is strongly peaked at £L(1tI2), (ii) most of the spectral weight (but not all!) is concentrated between £L(1tI2) and £u(1tI2)=1tl/..J2, all in agreement with predictions based on entirely different approaches [Muller, Thomas, Beck, and Bonner 1981].

Section 9-14

135

4.0.---------------r--r-----------.

q=1T/2

12

3.0 1\

<J~

rn 9" rn

N

N C"

8

4

V

::::::: 3

2.0

cr......... rn

o-1--..--.......--,----r-.---.,..--.-~ 4 8 o k

N N

1.0

J.

O.O-+---------r--=::::::~~~---r-~---===-~

0.0

1.0

eL

2.0

eu

3.0

Figure 9-12: Dynamic structure factor Sz)q,co) (normalized by <~~q» for fixed q=1tI2 of the 10 5=1/2 Heisenberg antiferromagnet at (4.35) T=O. The result was derived from the continued fraction (9.3) (via (9.7) with £=0.001) terminated at level K=6 with the splitGaussian terminator as described in the text. The inset shows the continued-fraction . coefficients l1 k vs k, k = 1,...8, for a system with N=16 spins.

10 TRANSPORT OF SPIN FLUCTUATIONS AT HIGH TEMPERATURE Quantitative studies of quantum or classical spin fluctuations expressed in terms of dynamic correlation functions of microscopic models for magnetic insulators are quite challenging under most circumstances. The spin dynamics of such models is, for the most part, governed by parameters or attributes from the following list: Symmetry of spin coupling Range of interaction Lattice dimensionality Competing interactions Randomness in coupling or site occupancy Long-range order Criticality Short-range order

In the low-temperature regime, there exists a seemingly unlimited variety of dynamical behavior, depending on the ordering that is present in the equilibrium state of the system. Specific types and degrees of magnetic ordering have their characteristic signature in the spectral densities. Some areas of that fascinating field of research will be explored in Chapter 11. In the high-temperature regime, by contrast, specifically at T=oo, the equilibrium state of the system is model-independent. It is a state of complete disorder, characterized by identically vanishing equal-time pair correlation functions, == 0 for i:t:j. That state is readily implemented in most calculational techniques. In spite of this simplification, the evaluation of dynamic correlation functions remllins a highly nontrivial task. However, their structure is expected to exhibit much less diversity than in the low-temperature regime and to depend primarily on the symmetry of the interaction Hamiltonian, i.e. on the presence or absence of conservation laws that have a bearing on the dynamical variable under investigation. It is not unreasonable to suppose that there exists what might be called generic dynamical behavior of, say, spin autocorrelation functions <S'j(t)~ at T=oo for systems with short-range interaction. Sections 10-1 to 10-13 report the study of such functions for the 10 s=1I2 XX and XXZ models as recently completed by Stolze, Viswanath and Muller [1992], and by Bohm et al. [1994]. In the remainder of this Chapter the main topic is spin diffusion in the classical Heisenberg model.

<SItS»

10-1 Generic High-Temperature Spin Dynamics What is the generic behavior predicted by the simplest phenomenological model compatible with the symmetry properties of a given interaction Hamiltonian? In the absence of any continuous rotational symmetry of H in spin space, the simplest

137

Section 10-1

phenomenological model for the long-time behavior would be exponential relaxation, 1

<sr(t)Sr>

- e -tit .

(10.1)

st,

If the interaction Hamiltonian conserves the total spin component S¥ = Ll we expect the same correlation function to exhibit a long-time tail, characterized by an algebraic decay law of the form <sr(t)Sr> - t-dJ2 ,

(10.2)

where d is the dimensionality of the lattice. In this situation, the dominant transport mechanism of spin fluctuations would be spin diffusion. The assumption is that for sufficiently long wavelengths and times, the time evolution of the dynamical variable SJ!(q,t)

=L

eiq""Sr(t)

(10.3)

I

is governed by a diffusion equation, (1004)

It must be said, however, that there exists as yet no rigorous theoretical derivation of this slow longitudinal spin motion from the rapid transverse spin motion, specified by the microscopic equations of motion for individual (classical or quantum) spins Si [Forster 1975, Fogedby and Young 1978]. The validity of the phenomenological equation (1004) implies that the correlation function of the variable (10.3) decays exponentially in time, <sJ!(q,t)SJ!(-q,O»

_ e-Dq2t ,

(10.5)

for small q and large t, and that the spin autocorrelation function decays algebraically as in (10.2). More details about the assumptions underlying expression (10.5) will be dicussed in Sec. 10-20 in the context of the classical Heisenberg model. In ID and 20 systems, this long-time tail, if indeed present, is in turn responsible for infrared divergences of the type

~J!(oo) _ (

00-

1/2

In(1/oo)

(10) (20)

(10.6)

in the corresponding spectral density. This quantity is directly amenable to experimental investigation, e.g. via NMR spin-lattice relaxation rates. Some experimental results will be discussed in Sec. 10-18.

I The prototype of this process is described, in the context of the recursion method, by the (n=1)-pole approximation of any dynamical system (see Chapter 8).

138

Chapter 10

10~2 10 s=112 XYZ Model on Semi-Immite Chain h is ironic but scarcely surprising that none of the exactly known functions <Slf(t)S'f> for quantum spin models is consistent with the expectations derived from the spin diffusion phenomenology. For classical spin systems too, most of the available simulation data exhibit significant deviations from predictions based on hydrodynamic assumptions (Secs. 10-14 to 10-23). Exact results for dynamic correlation functions of interacting quantum spin systems are scarce. Some such results have been reported in Chapter 7 for models with infinite-range interaction. With almost no exceptions, all other exact results pertain to the ID s=112 XY model. It is the special case lz=O of the ID s=1/2 XYZ model. The XYZ Hamiltonian for a semi-infinite chain reads

H XYZ

= -E {J?tS/:! /=0

+

lyS/S/:! + lzS/S/~d .

(10.7)

In the following the focus will be on the study of spin autocorrelation functions <Sl(t)S{>, Jl=X,y,z, at T=oo for this model. We shall review existing results and compare them with new results for bulk spins (/=00) and spins [=0,1,2,... at or near the boundary of the semi-infinite chain. Much attention will be given to the special case with uniaxial symmetry (Jx=ly=J), the XXZ model H xxz =

-E {J(stS/:! /=0

+ S/S/:!) +

l?/S/~!}

(10.8)

For each term alone, the dynamics can be analyzed exactly: the XX model (Jz=O) is equivalent to a system of noninteracting lattice fermions (Sec. 6-3), and the X model (l=O) is as trivial as the quantum harmonic oscillator (Sec. 6-1). For other parameter values, the T=oo dynamics of the XXZ model is quite complicated, and various interesting transitions between different types of dynamical behavior can be studied. For that purpose, the recursion method turns out to be an invaluable calculational tool. Although this Chapter focuses more on ph'ysical phenomena than on calculational techniques, due attention will be given to every new facet of the recursion method as it is introduced along the way. 10-3 Spin~1/2 XX Model: Neither Spin Diffusion nor Exponential Relaxation Let us begin our exploration of high-temperature quantum spin dynamics with the exactly solvable semi-infinite XX model H xx

= -lE

{sts /:! + s/s/:!} .

(10.9)

/=0

We are interested in spin autocorrelation functions <Slf(t)S'[> at T=oo and the associated spectral densities

Section 10-3

1111

_

4>0 (ro)[ =

f

-

dte

11 11 irot <S[ (t)S[ >

-00

_

, Il"""'x,y,z.

139

(l0.1O)

<SiSi>

For the bulk spin (1=00) these functions have been determined exactly many years ago. The results for Il=z, expressible in terms of a Bessel function and a complete elliptic integral, respectively,2 (lO.lla)

4>~(ro)oo = .?:-K(b -cJ14J 2 )a:1-cJI4J 2) , reJ

(1O.11b)

were first derived by Niemeijer [1967] and by Katsura, Horiguchi and Suzuki [1970]. In the fermion representation of the XX model, the evaluation of these quantities is straightforward e.g. in terms of a two-particle Green's function for noninteracting lattice fermions. Such a calculation was carried out in Sec. 6-3. Note that (lO.lla) decays more rapidly, _(1, than the diffusive long-time tail (10.2) with d= 1 does. Correspondingly, expression (l 0.11b) has only a logarithmic infrared divergence as opposed to the characteristic ro- 1I2-divergence of ID spin diffusion. The fluctuations of S;, in this model are obviously not governed by a diffusive process despite the conservation law Sf =const. The fluctuations of S/ also decay algebraically, _(112 with oscillations, rather than exponentially, -exp(-Dq2t), as would be expected for a diffusive process. A physical interpretation of this peculiar transport of spin fluctuations will be given in Sec. 10-11. The determination of the function <S;,(t)S;,> for that same model is far less straightforward. The exact result was first conjectured on the basis of a moment analysis for finite chains [Sur, Jasnow and Lowe 1975]. Rigorous derivations, based on the analysis of infinite Toplitz determinants, were reported within one year by Brandt and Jacoby [1976] and independently by Cape1 and Perk [1977]. The result is a pure Gaussian:

<S":(t)S":> = <S!,(t)S!,>

(l0.12a)

4>U«1» = o 00

(l0.12b)

2{it e -ofIJ2. J

It is remarkable that this result can be derived with little effort from the recursion

method in the spin representation [Florencio and Lee 1987] as reported in Sec. 6-4.

brhroughout this Chapter, the frequency moments of (10.10) as defined in (3.30) and the associated continued-fraction coefficients will be labelled accordingly, ~~(~ and !i'tt(~, respectively.

140

Chapter 10

The Gaussian decay of (l0.12a) is anomalous again. A nonnal relaxation process would be characterized by exponential decay at long times. The non-generic processes that govern the transport of spin fluctuations in this model are further indicated by the fact that all pair correlations <5'J(t)S[,>, 1*1', are identically zero. Whereas the free-particle nature of the excitation spectrum governing the correlation function <S;,(t)S;,> is readily recognizable by the bounded support of the spectral density (l0.11b). That same conclusion cannot be drawn from a mere inspection of the results (10.12). Spectral densities with unbounded support are typical for the dynamics of interacting degrees of freedom. In order to detect the free-particle nature of the XX model in the xx-autocorrelation function, it is necessary to study boundary effects.

10-4 Boundary Effects: Buildup of an Infrared Divergence The zz-autocorrelation function was determined in closed fonn for all sites on the semi-infinite chain [Gon~alves and Cruz 1980; Brandt and Stolze 1986]: <S/(t)S/> ::: 2.[Jo(Jt) -( _1)l+1 JZ(l+I)(Jt)]z .

4

(10.13)

In the bulk limit l~oo, only the first tenn in the square bracket survives, and the result (lO.lla) is recovered. The Fourier transfonn of the Bessel function In(Jt) is nonzero only on the interval [-J, 1]. The spectral density ~5Z(CO)l associated with (l0.13) is thus confined to the interval [-21,21]. The long-time asymptotic expansion (LTAE) of the function (10.13) was found to have the following general structure [Stolze, Viswanath, and Muller 1992]: <S/(t)S/> -

E- a;t-(2n+3) n=O

-

+ (eziJtE b;(it)-(n+3) + c.c}.

(l0.14)

n=O

The leading terms determine the dominant singularities ip the spectral density ~5Z(CO)l: quadratic cusp singularities at the endpoints (co = ±2J) and at ro=O, for all 1. The explicit result for 1=0, expressed in terms of complete elliptic integrals reads [Stolze, Viswanath, and Muller 1992]:

~~(co)o:::

128(1 +oY21)f(l +cJ/4J Z)E(21-CO) - CO K ( 21-CO]. (l0.15) 21+co J &1:21+CO

31CJ

L

The shape of the spectral density ~5Z(CO)l for selected values of 1 is shown in Fig. 10-1. The bulk limit is subtle: the quadratic cusps at the endpoints become steeper and steeper and, for l~oo, transfonn into the discontinuities displayed by (lO.llb). Likewise, the maximum at ro=O grows higher and narrower and, for l~oo, turns into a logarithmic divergence. In Chapter 9 we have already predicted some of these spectral properties in test applications of the recursion method. The quadratic cusp at the band edge of (l0.15) was determined in Sec. 9-4 from an extrapolation of the ~k-sequence (9.23) (see Fig. 9-3). The spectral density ~5Z(co)o itself was then reconstructed in Sec. 9-5 from the first 50 known continued-fraction coefficients Llf(O) by means of the ~-

Section 10-4

141

terminator (see Fig. 9-4). For spins near the boundary of the semi-infinite chain (~l), the recursion method yields ~k-sequences that tend to converge to the same value ~;,z(l) = J2, but their approach is alternating in character initially and then crosses over to uniform convergence. This is illustrated for the case 1=1 in Fig. 10-2 (inset). The associated ~k-sequence still converges to the same value ~=2 but much less uniformly than for the case of the boundary spin (1=0). Hence the ~-terminator (with ~=2 and 000=2) is still applicable for the reconstruction of the spectral density $ijZ(oo)l. The result of that calculation is shown in Fig. 10-2 (solid line) together with the exact result (dashed line) taken from Fig. 10-1. The two curves are almost indistinguishable. The emerging alternating character of the ~k-sequence for 1 ~ 1 reflects the buildup of the logarithmic divergence at ro=O in the bulk limit (l~oo). In Sec. 9-10 we have already analyzed that singularity by extrapolating the a ksequence (Fig. 9-1O(b)). Furthermore, we have reconstructed the spectral density (I0.11b) from the known ~k's by using the compact a-terminator (Fig. 9-IO(c)). 7.0-.-------------------------.

6.0

5.0

2.0 1.0 0.0 -¥--O:::::::""-r-----,----.,.----,...---.----r----.-----=::::~ 2.0 1.5 0.5 1.0 -2.0 -1.5 -1.0 -0.5 0.0

Figure 10-1: Spectral density $~Z(co), at T=oo of the semi-infinite 10 8=1/2 XX model (10.9) with ..1=1 as determined by the Fourier transform of expression (10.13). The four curves represent the cases /=0 (boundary spin), /=1,5 and /=00 (bulk spin) [from Stolze, Viswanath and MOller 1992].

142

Chapter 10

5.0...------------------------, 1.5-.------------,

4.0

-

...... '-'

~~

1.0


.... 3.0 ".......

3

0.5

'-"

N NO

t&

10

0

20

30

k

2.0

1.0

0.0+-------,------.,....------,-------"'-,,..--' 0.0 2.0 1.0

Figure 10-2: Spectral density cIl~Z((J))1 at T=oo of the semi-infinite 10 8=1/2 XX model (10.9) with .1=1. The solid line represents the result derived from the continued fraction (9.3) (with Z = E - ico, £=0.001) terminated at level K=30 as described in the text. The dashed line (hardly visible) represents the exact result obtained by Fourier transforming expression (10.13).

10-5 Boundary Effects: Crossover Between Growth Rates A.=O and A.=1 A general determinantal expression for the function <S[(t)S[> was derived by Stolze, Viswanath, and Muller [1992]. That derivation uses the Jordan-Wigner transformation (6.28) from spin-1I2 operators to Fermi operators and Wick's theorem, Le. precisely the same techniques that were used to derive expression (to. Ha) for <S](t)S]>. The more complex structure of <S[(t)S[> as compared to <S](t)S]> results from the fact that the spin operator Si turns into a product of Fermi operators involving all of the sites between and 1, whereas the fermion representation of S] involves only operators at site 1. The general structure of the function <S[(t)S[> turns out to be a sum of products of integer-order Bessel functions In(Jt) with n=0,1,...,2(1+1). Each term is the product of exactly 21+1 such functions. Explicit expressions for 1=0,1,2, corresponding to the first three sites of a semi-infinite chain, were evaluated by Stolze, Viswanath, and Muller [1992] as follows:

°

Section 10-5

x

x

<S2 (t)S2

>

1

143

2

= 4[(10 +J2)(10 -J4 ) + (J 1 +J3) ]

X{(10 +J 2)[(10 -J4)(10 +J6) + (11 -J5)2]

(lO.18)

+(J 1 +J3)[(J 1 +J3)(10 +J6) + (11 -J5)(12 +J4)] +(J2 +J4)[(J 1 +J3)(11 -J5) - (Jo -J4)(J2 +J4)]} Each Bessel function has the argument Jt. The spectral density <1>0(00)/ is then a multiple convolution of 21+1 functions with compact support on the interval [-J,1] and square-root singularities at the end points. Its spectral weight is thus restricted to the interval [-(21+1)J, (21+1)1], and nondivergent power-law singularities in O( (0)/ can again be inferred from the LTAB [Stolze, Viswanath, and Muller 1992]: /

<st(t)st> -

L

<0

Cmei(2m+I)JtUt)-'Ym

m=()

~

L

c:!j,(it)-n +

C.C.

(lO.19)

n=()

with leading exponents

'Y~

= 3/2

+ 1(1+2) + m(m+ 1) .

(10.20)

The number of m-terms in (10.19) increases with the distance of the spin from the boundary. Each m-term gives rise to a pair of (nondivergent) power-law singularities in <1>0(00)/ at frequencies oom = ±(2m+l)J. The associated singularity exponents, 'Y~l), increase monotonically with m and 1; the exponent for the endpoint singularity is 'Y~m) = 2z2+31+1I2. In the bulk limit, 1~00, the support of that spectral density is no longer bounded, and all singularities fade away completely. The result is the Gaussian function (10. 12b). In Fig. lO-3 we have plotted the exact results for 1=0,1,00. Convergence toward the bulk result is remarkably fast. To what extent would it have been possible to predict the spectral properties of the exact results (10.16-18) by means of the recursion method, i.e. by means of a general calculational technique that does not have to rely on the special circumstances which permit the exact solution by special techniques? Figure lO-4 shows computational data obtained by Stolze, Viswanath, and Muller [1992] for the ilk-sequences pertaining to the spectral densities <1>0(00)/. The dashed curves interpolate the values lIilf(l) plotted vs lIk for 1=1,2,3,4. The solid line represents the sequence ilf(oo) = J k/2 for the bulk spin. Not shown in Fig. lO-4 is the horizontal line which corresponds to the constant sequence ilf(O) = J2/4, for the

144

Chapter 10

boundary spin. The exact result (10.16) was, in fact, inferred by Sen, Mahanti and Cai [1991] from an application of the recursion method.

1=0

4.0

--

.....

3.0

:3

'-'

>< ><0 0&

2.0

1.0

0.0-+--....:y::~--,-----1--r--.,----r---+---,-----=;=------l

-2.0

-1.0

0.0

1.0

2.0

Figure 10-3: Spectral density «Il:(ro)/ at T=oo of the semi-infinite 1D s=1/2 XX model (10.9) with J=1. The three curves shown represent the cases /=0,1 as determined by the Fourier transform of expressions (10.16) and (10.17), respectively, and the case /=00 is given by the function (10.12b). The spectral densities for /=2 and /=00 coincide within line thickness [from Stolze, Viswanath, and MOller 1992].

All dashed curves in Fig. 10-4 start out superimposed on the solid line up to k=1, and then level off gradually toward a finite value, ~x(l) = J2(21+1)2/4. This is consistent with the exactly known band edge, ~l)=(21+1)J, of the function «I>Q(00)[. The uniform convergence of the ar(l) toward their limiting values a;,x(l) is consistent with the known fact that «I>Q(oo)[ does not have any infrared singularity. The exponents 13(l) of the endpoint singularities, - (~l) - oo)p, in the functions «I>Q(oo)[ can be determined directly from the ar(l) by extrapolation of the associated I3 ksequence (9.23). For 1=0 that analysis is trivial: ar(O) = const implies 113(0)1 = 112, which correctly describes the square-root cusp of «I>Q(oo)o (see Fig. 10-3). A simple extrapolation procedure a~plied to the sequences af(1) and ar(2) reproduces the exact exponent values 13(1 =-rll)-l = 11/2 and 13(2 = 'rl,2)... 1 = 29/2 to within one tenth of a percent and one percent, respectively. More substantial deviations from the exact values are found for 1>2, where fewer coefficients ar(l) are available for the analysis. There appears to be no practical means to extract from the ar(l) any

Section 10-6

145

quantitative information on the interior singularities known to exist in the spectral densities Q(ro)/ for 0<1<00. For given bandwidths and known endpoint singularity exponents, the spectral densities Q(ro)/ can be reconstructed very precisely by means of the explicitly known coefficients ilf(l) and the ~-terminator. 0.6--.----------------;,.---------,

.... 0.4 ,-... .-< .........

>< ><.::.::
"....... 0.2

.... .... ~

1=00

O.O-t----.--------r----,-------,-----t 0.0 0.1 0.2

Ilk Figure 10-4: Sequences 1/6.f(~ plotted vs 1/k for the spectral densities ct>~X(co)p 1=1,2,3,4, at T=oo of the semi-infinite 1D 8=1/2 XX model (10.9) with .1=1. The maximum value of kis 55,28,22,20, for the four cases, respectively. The solid line represents the sequence 6.f(oo) =k/2 for the bulk spin case. The arrows indicate the limiting values 1/6.~X(~ for 1=1,2,3,4 [from Stolze, Viswanath, and Muller 1992].

10·6 Spin·112 XXZ Model Consider first the bulk-spin autocorrelation function <S;'(t)S;'> at T=oo of the 10 s=1I2 XXZ model (l0.8). The nontrivial but exactly solvable case lz=O (XX model) is an ideal starting point for the analysis of the cases lz:t:O by means of a general method which can produce the exact result in the XX limit. The recursion method is such a method as we have demonstrated in Sec. 6-4. Its strength for the analysis

146

Chapter 10

of the function <S;'(t)S;'> derives from the fact that gradual deviations from the exactly solvable limit lz=O produce only gradual deviations from the linear sequence,

Ll~(oo)

=

(10.21)

!.-12k , 2

associated with the Gaussian spectral density (10.l2b). These gradual changes in the Llk-sequence in turn produce gradual changes in the spectral density c1>o(ro)oo' which can be analyzed quantitatively.

5.5 -r-----;:========================::::;--------i 1.2

4.5

.<

1.1

J =1

z

2.5

1.5 -+---------r-------r--------,------J 2.5 1.5 2.0 1.0

In(k) Figure 10-5: Log-log plot of the Ak-sequences for the bulk-spin autocorrelation function T=oo of the 10 5=1/2 XX model (..1=1, Jj=0) and XXX model (..I=Jj=1). The slope of the linear regression lines determines the growth rate A. of these Ak-sequences. The inset shows A. as a function of Jz (for ..1=1) as determined by this method from such Aksequences [from Bohm et al. 1994]. <~(~~> at

As lz increases from zero, we can identify two types of systematic deviations of the Llk's from the linear sequence (10.21): (i) a gradual increase in growth rate A.; (ii) gradually increasing alternating deviations of the Llk from the line k!'. Both effects are illustrated in Fig. 10-5. The main plot shows InLlk vs Ink for the two cases lz=O and lz=l. The former is the linear sequence (10.21), which has slope A.= 1. The latter has slope A... 1.22, and the deviations with predominantly alternating character from the regression line are clearly visible. The inset shows the variation

Section 10-8

147

of the growth rate A. with Jz between the XX and XXX models. What looks like a gradual increase of A. may, in fact, be a crossover between two fixed values of growth rate, one for free fermions (A.= 1) and another one for interacting fermions (1..... 1.2). Relevant for our analysis is the effective growth rate of the known finite ~k-sequence.

Unlike in classical spin dynamics, where diffusive long-time tails are readily detectable in simulation data and amenable to a direct analysis (Secs. 10-14 to 10-23), the most direct indicators of their presence in quantum spin dynamics are (at least in ID and 2D systems) infrared divergences in spectral densities. The continued-fraction analysis is an ideal instrument for the quantitative study of such singularities. We recall from Sec. 5-4 that alternating deviations such as observed in the ~k-sequence for Jz=J are the genetic code of an infrared divergence. Here it is expected to be of the type _00- 112 as implied by the spin-diffusive long-time tail _f 1l2. A quantitative analysis of that singularity will be presented in Sec. 10-9. It yields strong evidence for a transition between a relaxation process at Jz<J and a diffusion process at Jz=J in the fluctuations of S;'(t). 10-7 From Gaussian Decay to Exponential Decay Our first task should be to identify the relaxation process at Jz < J. We have already noted that the Gaussian decay of <S;,(t)S;'> for Jz=O is non-generic in that respect, attributable to the free-fermion nature of the XX model. The expectation is that a slight increase of Jz from zero causes a crossover in <S;,(t)S;'> from Gaussian decay to exponential decay. It would be impossible to detect such an effect directly in the ~k-sequence, because a transition from Gaussian decay to exponential decay at long times does not by itself have much impact on the shape of the spectral density. However, given the simplicity of the exact result (10.12a), the crossover might be discernible in a short-time expansion of <s;'(t)S;'>. This is indeed the case [B6hm et al. 1994]. In Fig. 10-6 we have plotted upper and lower bounds of the function In(<S;,(t)S;'»/ (Jt<S;'S;'» versus Jt for four different parameter values of the XXZ model near the XX limit. The dashed line with slope -1 represents the pure Gaussian (10. 12a). Pure exponential decay would be represented by a horizontal line with negative intercept. The results for Jz#J show strong indications that the decay is slower than Gaussian and consistent with exponential decay (convergence toward a negative constant). Power-law decay would imply convergence toward zero. Whether or not the observed exponential decay represents the true asymptotic behavior is, of course, beyond the reach of this type of analysis. 10-8 Analysis of ~k·Sequences with Growth Rates near A.=1 In Secs. 9-11 to 9-13 we have already discussed ways to analyze ~k-sequences with growth rate A.=1 for the double purpose of (i) reconstructing spectral densities with the Gaussian terminator or the unbounded a-terminator and (ii) estimating infrared exponents on the basis of the explicitly known functional dependence (5.26) of the model coefficients ~k on the singularity of the model spectral density (5.24).

148

Chapter 10

-1.6..,....;:-------------------------,

-1.8

-2.4 J /J=O

z

-2.6 1.6

1.8

2.0

2.2

2.4

2.6

Jt Figure 1~: Short-time expansion of the bulk-spin autocorrelation function Co(~ =

<S~(~~>/<S~S~> at T=oo of the 10 &=1/2 XXZ model (10.8) for three parameter values

JJJ = 0.02, 0.05, 0.1 (solid lines) near the exactly solvable XX case (dashed line). The data are plotted in a way suitable for visualizing the crossover from Gaussian decay (negative unit slope at zero intercept) to exponential decay (zero slope at negative intercept). Each result of the short-time expansion Is represented by two curves corresponding to an upper and a lower bound of the function. The bounds have been determined from 14 exact frequency moments ~oo) [from Bohm et al. 1994].

In applications to ak-sequences with M:1, as they occur in the situation now under consideration (see Fig. 10-5), such an analysis should be carried out on the basis of the more general model spectral density (5.27). This remains impractical as long as we lack closed-form expressions for the continued-fraction coefficients ak pe.rtaining to (5.27) as functions of the three parameters Olo, <X, A.. However for growth rates sufficiently close to 1..=1, we can approximate the (M:1)-problem with a (I..=l)-problem if we replace the ak-sequence by the rescaled sequence A* _

A

ilk - ilk

1/A.

(10.22)

and then proceed as outlined in Secs. 9-11 to 9-13. The main distortions in the reconstructed spectral density caused by this approximation are of two kinds: (i) a change in the large-ro decay law and (ii) a change in the frequency scale. Whereas the former has only a negligible impact on the shape of the spectral-weight

Section 10-9

149

distribution, the latter may warrant attention and lead to significant improvement upon proper adjustment. 10-9 From Exponential Relaxation to Diffusive Long-Time Tails Let us now present the results for the bulk-spin spectral density <1»0(00)00 of the XXZ model for oglJ~1 as reconstructed from the continued-fraction coefficients Llf( 00),... ,Llf4(00) (inferred from 14 exact moments Mik( 00» and a Gaussian terminator with its parameter determined from the slope of the Ll:-sequence [Bohm et al. 1994]. Figure 10-7 shows the reconstructed function <1»0(00)00 at ro0, for parameter values between JIJ = 0.6 and JIJ = 1.0. The six curves on the left illustrate how the pure Gaussian (l0.12b) (dashed line) evolves into a curve with some structure as JIJ increases from zero. The additional structure consists of (i) a central peak of increasing height and decreasing width, (ii) a shoulder of enhanced spectral weight at 00"1.5J. 4.0-r------------.,.....-------------,

3.0

-.8 3 2.0 >< ><

--

J =0.6

z

0

0&

1.0

0.0 -f-.-o!:=T---.,.--,.---r---+---.,.--..----r---,--.:::"""""I -1.0 0.0 2.0 -2.0 1.0

Figure 10-7: Spectral density 4>:(00)_ at T=- of the 10 8=1/2 XXZ model (10.8) with .1=1 as reconstructed from the continued-fraction coefficients 4f("::').....4f:(oo) and a Gaussian terminator. The calculation was carried out by the use of the 4 k-sequence in the role of the original 4 k-sequence and then as outlined in Sec. 9-11. The six curves plotted at 0><0 pertain to the values Jz = 0., 0.1 ..... 0.5 of the anisotropy parameter and the five curves plotted at 00>0 to values 0.6,.... 1.0 [from Bohm et al. 1994).

150

Chapter 10

The further development of the spectral density as JIJ approaches the XXX case is shown by the five curves on the right. The shoulder becomes more pronounced, and the strong peak at 00=0 signals the presence of an infrared divergence for Jz=J, in accordance with spin diffusion phenomenology. The curve for the XXX case is in qualitative agreement with previous results obtained from finite-chain calculations [Carboni and Richards 1969, Groen et al. 1980], and by a calculation which uses the first two frequency moments of the dynamic structure factor in conjunction with a two-parameter diffusivity [Tahir-Kheli and McFadden 1969]. Note that the infrared singularity in ~(ro).o which is strongly suggested by the curves for JIJ'" 1 in Fig. 10-7, is in no way artificially built into our approach. It is a structure resulting solely from the 14 known continued-fraction coefficients. The results shown in Fig. 10-7 are expected to be most accurate for small values of JIJ, where the growth rate is closest to A.=l (see Fig. 10-5, inset). As the growth rate increases toward 1.... 1.22, the curves are likely to become subject to some systematic errors as explained in Sec. 10-8. We have estimated the systematic error in frequency scale not to exceed 2% for the curves at ogIJ~0.5 and 12% for those at O.5gIJ~l. Once we have recognized the strong indications for the presence of an infrared singularity, we can investigate its nature more quantitatively by a direct analysis of the explicitly known continued-fraction coefficients. In Sec. 9-12 we have outlined and tested such a method for ~k-sequences with growth rate A.=l. In the present context, the analysis must be carried out for the associated ~;-sequence as defined in (10.22). The results of such an analysis are compiled in Fig. 10-8. The circles represent the mean exponent a as a function of JIJ ranging from the XX model (Jz=O) to the XXX model (Jz=J) and somewhat beyond. The error bars indicate the statistical uncertainty for each data point, which is due to the fact that the analysis is based on a finite number of known continued-fraction coefficients. On top of the statistical error, the data are likely to be subject to a systematic error whose potential impact increases with the deviation of the .growth rate from A.= 1. As Jz approaches zero, both types of uncertainties (statistical and systematic) become smaller and disappear. The data point a(O)=O is exact and describes the spectral density (lO.l2b), which has no infrared singularity. The dependence on Jp ofthe mean exponent values displayed in Fig. 10-8 is quite remarkable in spite of the limited overall accuracy. The data strongly indicate that the function a(JIJ) stays zero over some range of the anisotropy parameter. A vanishing exponent at small but nonzero JIJ is consistent with and thus reinforces the conclusion reached from the short-time analysis in Sec. 10-7 that the function <S;'(t)S;'> decays faster than a power-law, namely exponentially. While the data point at JIJ = 0.5 is still consistent with a=0, the mean avalues have already a strongly decreasing trend at this point. A minimum value is reached exactly at the symmetry point (Jz=J) of the XXX model - the only point for which the conservation law Sf LiSI const holds, and therefore the only point for with one expects a diffusive long-time tail in <S;'(t)S;'>. Upon further increase of JIJ, the data points rise again toward a=O as expected. The minimum exponent value, a = -0.37±0.12, obtained for the XXX case is only marginally consistent

=

=

Section 10-10

151

with the standard value, cx=-1I2, predicted by spin diffusion phenomenology. That discrepancy is likely attributable to the systematic error in our data. 0.1

0.0

-0.1

~

-0.2

-0.3

-0.4

-0.5 0.0

0.5

1.0

J

z

Figure 10-8: Infrared-singularity exponent <X vs anisotropy parameter Jz of the spectral density CI>~X«(J)_ at T=oo of the 10 5=1/2 XXZ (10.8) with ..1=1. The data points were obtained from the coefficients af(oo),...,af:(oo) by analyzing the a~-sequence as outlined in Sec. 9-12 [from B6hm et al. 1994].

10-10 Sustained Power-Law Decay The conservation law Sf LjS7 const for the spin fluctuations in z-direction holds over the entire parameter regime of the XXZ model (10.8). Consequently, the longtime behavior of the correlation function <S;,(t)S;,> or the low-frequency behavior of the spectral density ijZ( 00)00 is expected to be much less affected by the symmetry change of Hxxz at lz=1 than the functions <S;,(t)S;,> and <1>0(00)00 were. The verification of sustained power law decay in <S;,(t)S;,> at lz# as a contrast to the result presented in Secs. 10-7 and 10-9 will further support the case for quantum spin diffusion. Here we encounter a problem which prevents us from carrying through the calculation for parameter values near the exactly solvable XX limit (1/1=0). The breakdown is caused by a crossover in the growth rate of the relevant /1k-sequence. Figure 10-9 shows the sequences of /1iZ( 00) for four different parameter values.

=

=

152

Chapter 10

Between JjJ=o.6 and JjJ=1.0, the sequence of known coefficients has a welldefined growth rate somewhat higher than A.=1. For the XX model (Jz=O), on the other hand, growth rate A.=O is well known to be realized (see Sec. 10-4). The sequence for JjJ=O.1 has attributes of both regimes. It starts out with A.=o up to k-7 and then begins to grow with A.?1, thus causing a kink in tik vs k. It is impossible to analyze such sequences on the basis of a unique value of A, and, therefore, impossible to carry out the analysis described before without major modifications. 3

10.0

B.O

-8

.......... N

6.0

N~

<]

4.0

2.0

O. 0 +----.--.---...-...,...---r-.---,....-..----r-,.---r----,r-~___r___t o 6 9 12 15 3

k Figure 10-9: Continued-fraction coefficients ~f(oo) vs k for the bulk-spin autocorrelation function <S~(t)S~> at T:oo of the 1D 5=1/2 XXZ model with Jz = 0 (XX case), J z = 0.1,0.6, and J z = 1.0 (XXX case). The kink in the sequence for J z = 0.1 illustrates the crossover between growth rates A. 0 and A.
=

The bulk-spin spectral density cl»5Z(ro).. for four parameter values over the range 0.7 ~ JjJ ~ 1.0 as reconstructed from the known coefficients tifZ( 00)'00" tif~( 00) and a Gaussian terminator with its parameter inferred from the ti:-sequence is displayed in Fig. 10-10 (solid curves). Notice how the shape of the functions

31f the kink occurs at sufficiently large values of k, one might attempt to analyze that ~k­ sequence on the basis of growth rate A.=O. However, it is not a priori clear to what extent the physical phenomena of interest, here caused by the fermion interaction, are encoded in the ~kS before the kink. This question will be further discussed in Sec. 11-21.

Section 10-10

153

6Z(m)00 (Fig. 10-10) and <1>0(00)00 (Fig. 10-7), which start out identically, undergo different changes as the anisotropy parameter decreases from lll=1. While the function <1>0(00)00 gradually transforms into a pure Gaussian (dashed line in Fig. 10-7), the function 6Z(m)oo is supposed to approach the exact result (lO.lIb). The graph of that function has been added as dashed line to Fig. 10-10. The diminishing height of the central r:ak with decreasing 1p marks the weakening of the divergence from _00- 11 (diffusive) to -In(llm) (free fermions). Spectral weight removed from the central peak and from the high-frequency tail is transferred to the shoulder, which gradually transforms into a discontinuity at ro=2J. 4.0omm-------------------------, 0.1

0.0 -0.1

3.0

cs

3

N N

-0.3 -0.4

-.8 "'-"

-0.2

-0.5

2.0

-0.6

0

.eo

0.6

0.6

1.0

J

1.2

1.4

z

-----------_ J =0

1z

1.0

I I I

I I I

O.O+-----~--___,;__--_,_----.-..:~~---J

0.0

2.0

1.0

CJ

Figure 10-10: Spectral density cI>~Z(O»_ at T=oo for the bulk-spin of the semiinfinite 1D 5=1/2 XXZ model (10.8) with .1=1 as reconstructed from the continued-fraction coefficients AfZ(~), ...,Af:(oo) and a Gaussian terminator. The calculation was carried out by the use of the Ak-sequence as outlined in Sec. 9-11. The four solid curves represent the cases Jz = 0.7, 0.8, 0.9 and 1.0 (XXX model). The dashed curve is the exact result (10.11b) for Jz = 0 (XX model). In the inset we have plotted the infrared exponent a vs J~ The data points were obtained from the coefficients Af(oo) up to k=14 by analyzing the Ak-sequence as outlined in Sec. 9-12 [from Bohm et al. 1994].

The inset to Fig. 10-10 shows the results for the infrared singularity exponent a. over the parameter range 0.6 :5 lp :5 1.5. Within the statistical uncertainties indicated by error bars, the data points are consistent with a lzindependent exponent. The conservation law Sf = const, on which the diffusion process for the fluctuations of S7 hinges, holds for all values of the anisotropy

154

Chapter 10

parameter 1/1. These spin fluctuations are largely unaffected by the change in the symmetry at 1/1=1, in strong contrast to the observations made in Fig. 10-8 for the fluctuations of SI. The weak monotonic lz-dependence of the mean exponent values at lz ~ 0.8 and their deviation from the standard value a=-0.5 are probably attributable to the previously mentioned systematic errors, which we have not fully under control. However, the sloping tendency of the mean values toward the lowest values of lz is probably an artifact caused by the crossover between growth rates as discussed in the context of Fig. 10-9.

10-11 From Ballistic to Diffusive Transport of Spin Fluctuations The goal here is to gain further insight into the different transport mechanisms that govern the spread of local fluctuations of the conserved spin component Sf at lz=O (XX model) and at lz=1 (XXX model). We have already noted the different types of long-time tails in the bulk-spin autocorrelation function <S;,(t)S;,> (_r 1 versus _r Il2 ), which give rise to different infrared singularities in the spectral densities 8Z(oo).. (-In(1/oo) versus _00- 112). We have identified the strong divergence in the XXX case as the characteristic signature of spin diffusion and have attributed the weak divergence in the XX case to the free-fermion nature of that model. For the purpose of discriminating the underlying physical processes in the two cases, Bohm and Leschke [1992] introduced the spatial variance +00

(10.23) n=-oo

where C(n,t) denotes the (normalized) pair correlation function for the z-component of two spins n sites apart on an infinite chain. In the XX model, we know the exact result [Katsura, Horiguchi, and Suzuki 1970] C(n,t) = [J n(lt)f ,

(lz = 0)

(10.24)

in generalization of (1O.l1b). The resulting spatial variance4 (10.25) describes a ballistic spread of local spin fluctuations for arbitrarily long times as might have been expected for noninteracting ferrnions. Figure 10-11 depicts the function a2(t) (inset) and its first derivative (main plot). Here the purely ballistic behavior (10.25) is represented by the dashed lines. In a generic case, the ballistic regime is expected to be realized for very short times only. At longer times a crossover to diffusive behavior is expected to occur, the latter being characterized by a linearly increasing spatial variance a2(t). A crossover away from ballistic behavior is indeed borne out by the rigorous bounds for a2(t) of the XXX model calculated by Bohm and Leschke [1992] as 4-rhe function (10.25) coincides with a general rigorous upper bound for the spatial variance, valid for all cases of the XXZ model.

Section 10-12

155

shown in the two plots of Fig. 10-11 (solid lines). However, the time interval over which the bounds stay close together is just a bit too short for a clear picture of the long-time behavior in that case. From this perspective, it remains undecided whether the crossover leads to diffusive transport (zero slope in main plot) or some different type of transport (nonzero slope).

5.0 Jz/J=O / 4.0

3.0

2.0 /,

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

Jz/J=o / 10.0 C\l

b

5.0

""

1.0

/.

/

/

/

I

I

I

I

I

I

I

I

I

I

0.0 +"':::;:"""'-,.--.--,..---.--.--.,.;---1 0.0 2.0 4.0 6.0 6.0

Jt O.O+-----r------y----r----.------r-----,r------! 0.0 2.0 4.0 6.0

Jt Figure 10-11: Rigorous lower and upper bounds for the spatial variance
10-12 Boundary-Spin Spectral Densities The conclusions reached in Secs. 10-9 and 10-10 for the infrared singularities of the bulk-spin spectral densities o(ro)~ and 5Z(ro)~ in the semi-infinite XXZ chain (10.8) are further substantiated when we look at the results of the same analysis carried out for the boundary-spin spectral densities ~Il(ro)o' Jl=X,Z. For that calculation, Bohm et al. [1994] have computed 17 coefficients Ll~Il(O) (compared to 14 coefficients Ll~Il( 00) for the bulk case). 5 5Note that the A.-crossover now plagues the analysis of both cI>~Z(ro)o and cI>~X(ro)o for parameter values 0::; JjJ::; 0.6.

156

Chapter 10

6.0,,----------------------, 0.1

0.0

5.0

-0.1

t:S

4.0

o

~

-0.3 -0.4

..........

3 ~--

-0.2

3.0

o

.eo

-0.5 -0.6 -+-....-,--.---r-.......-....-,-+--r--+-....-,---.---r--l 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

J

2.0

z

1.0

O.O+------.,........-----r--------r--==~--==t

0.0

1.0

2.0

Figure 10-12: Spectral density <J)~X(c.o)o at T=oo for the boundary-spin of the semi-infinite 1D s=1/2 XXZ model (10.8) with ..1=1 as reconstructed from the continued-fraction coefficients

6f(0},...,6f;(0) and a Gaussian terminator. The calculation was carried out by the use of the 6 k-sequence as outlined in Sec. 9-11. The four solid curves represent the cases Jz = 1.0 (XXX model) and Jz = 0.6. Also shown is the exact result (10.26) for Jz = 0 (XX model). In the inset we have plotted the infrared-singularity exponent a vS.Jr The data points were obtained from the coefficients ~X(0), ... ,6f~(0) by analyzing the 6 k-sequence as outlined in Sec. 9-12 [from BOhm et a!. 1994].

The spectral densities fPQ( (0)0 for the two cases 1p = 0.6, 1.0 as reconstructed from the ~:-sequence and a Gaussian terminator are shown in Fig. 10-12 (solid lines). The curve for the XXX case (1/1=1) shows a pronounced peak at <0=0. That conspicuous enhancement of spectral weight has all but disappeared for 1/1=0.6, i.e. in the presence of anisotropy, where Sf is not conserved. This supports the interpretation that the central peak in the XXX result reflects a diffusive infrared divergence. As the anisotropy parameter drops below the value 1/1=0.6, the shape of the function fPQ(oo)o must approach that of the dashed line, which represents the exact result for 1/1=0 [Sen 1991], (10.26)

157

Section 10-12

and is the Fourier transfonn of (10.16).6 While the ~;-analysis breaks down for small values of lp, the way the function <1»0(0))0 develops between 1/1=1.0 and 0.6 can be extrapolated fairly smoothly toward the dashed line. We have calculated the infrared exponent a of the boundary-spin spectral density
6.0 I

0.0

I

5.0 4.0 ..........0

--3

N N 0 >&

I

\ \ \ \

\

\

tS

,

-0.5

3.0 0.6

0.6

0.9

J

2.0

1.0

0.7

--- J =0

z

-

J =0.6 z - - J =1.0

z

---

~~~

"""........ ............

1.0

1.1

1.2

z

............

....

-

"..

---::...::.~

0.0 -+------,--------,------~----=.::::3:IiiIIIo_I 0.0 0.5 1.0 1.5 2.0

Figure 10.13: Spectral density «ll~Z(o»o at T=oo for the boundary-spin of the semi-infinite 10 5=1/2 XXZ model (10.8) with .1=1 as reconstructed from the continued-fraction coefficients dfZ(O),... ,df~(O) and the a-terminator. The calculation was carried out by the use of the d~­ sequence as outlined in Sec. 9-11 for the two cases Jz = 1.0 (XXX model) and Jz = 0.6. Also shown is the exact result (10.15) for Jz = 0 (XX model). In the inset we have plotted the infrared-singularity exponent a vs Jr The data points were obtained from the coefficients ~Z(O), ... ,df~(O) by analyzing the d~-sequence as outlined in Sec. 9-12 [from Bohm et al. 1994].

6 Recall from Sec. 10-5 that the function (10.26) is characterized by a constant sequence of continued-fraction coefficients, d'!:(0)=.}/4.

158

Chapter 10

Note the strongly contrasting Jz-dependence of the singularity exponent pertaining to the spectral density aZ( ro)o as shown in the inset to Fig. 10-13. Here the data points indicate the presence of an infrared divergence over the entire parameter range shown. However, a much stronger Jz-dependence of the mean values of a. (attributable to systematic errors) is indicated than was the case for the corresponding bulk-spin result (Fig. 10-10). In view of the fact that infrared divergences are very likely to be real in the function aZ(ro)o for all values of J;>O, we have treated them as such for its reconstruction from the known coefficients aiZ(O). Two of the curves in the main plot of Fig. 10-13 represent the function aZ(ro)o for JP=1.0 and 0.6 as reconstructed from the 17 known coefficients af(O) and the unbounded a.-terminator (Sec. 9-13), which has a built-in infrared divergence. 7 As the parameter drops from the higher to the lower value, the central peak in the spectral density weakens considerably, and the shoulder at ro..J present for Jp=1 all but disappears. The short-dashed line represents the exact result (10.15) for Jp=O. This limiting case is perfectly in line with the change of shape between J/J=1.0 and 0.6 of the reconstructed spectral density.

10-13 Spectral Signature of Quantum Spin Diffusion in Dimensions d=I,2,3 The continued-fraction analysis of infrared singularities in spin autocorrelation functions at T=oo as reported in Secs. 10-9 and 10-10 for the s=1I2 XXZ model (10.27) with nearest-neighbor coupling on a linear chain (d=I), is readily extended to the cases of a square lattice (d=2) and a simple cubic lattice (d=3) [Bohm et al. 1994].8 The method of estimating the infrared singularity exponents in spectral densities !r(ro) used in this context is a refined version of that used previously, which takes the varying growth rates A. of the sequences of continued-fraction coefficients a~ more adequately into account. It proceeds as follows: We choose a model spectral density with (i) a variable overall frequency unit roo' a power-law infrared singularity with variable exponent a., (iii) a high-frequency decay law of the type (5.23) with variable growth rate A.. The simplest function which meets these requirements and has the correct normalization is the model spectral density (5.27),

7The ~nbounded a-terminator has two parameters: 000 is determined by the average slope of t:. k vs k and a by our best estimate of the singUlarity exponent.

8All the results presented in this Section are for infinite lattices. No boundary effects are considered here. For that reason, labels which refer to lattice sites are suppressed.

Section 10-13 IX

21t/~

n

159

00 exp( -100/0\)1 21;1..) 0\) 1.(1 +a)] 2

(10.28)

The associated model continued-fraction coefficients ~k can be calculated via (3.33) from the exact frequency moments (5.28). For all A. values relevant in this application (IS~2), a singularity of growing strength in (10.28) causes an alternating pattern of growing amplitude in Ak• However, only for the case A.=1 is the relation between exponent and alternation known analytically (see Eq. (5.26)).

><

><~


30.0

><

><~ <]

... diagonal

80

40.0

Go£)

site

80

Jz/J= 1.0

40 20 0

d=2 0

5

20.0

10

k

A

15

J z /J=0.3

10.0 d=1 0.0 +----.---r----.---r----,--r----,,..---r--,..---.---.,....--.---.,....---,----l 6 9 12 15 o 3

k Figure 10-14: Continued-fraction coefficents a~x,. .. ,a~: (circles connected by solid lines) of the xx site-spin autocorrelation function at T=oo for two cases of the XXZ chain. The sequences for JjJ=0.3 and 1.0 have growth rates A.=1.03 and A.=1.18, respectively. The model ak-sequences which provide the best fit as described in the text are shown as dashed lines. The inset shows the x plotted vs K" for the site-spin (open circles) and the (diagonal) chain spin (full circles) of the 5=1/2 XXX model at T=oo on the square lattice. Here A.=1.54 is the growth rate of the site-spin data [from B6hm et al. 1994].

aZ

The known coefficients I1f of the spectral density ~o(oo) for a site-spin component ~ of the XXZ chain at J p= 1 and Jp=O.3 are displayed vs k in the main plot of Fig. 10-14 (circles connected by solid lines). Note the different growth rates and degrees of alternation. In order to estimate the exponent value a that ltH gives rise to the observed amount of alternation in the data sequence Lil'l,...,Lig.ll, we

160

Chapter 10

determine a matching model sequence liJ,...,liK obtained from (10.28) by numerically minimizing the mean-square deviation K

L

k=kmin

(~~-~i

(10.29)

with respect to the undetermined parameters A., a, 0>0. The lower cutoff kmin was found to be necessary because the first few continued-fraction coefficients tend to deviate significantly from the asymptotic behavior described by the model coefficients ~k' We have set ~in=3 for all sequences analyzed here. Two optimized model ~k-sequences are displayed as dashed lines in Fig. 10-14 (main plot) along with the data sets to which they have been fitted. We know the exact moments ~ up to K=14,7,6 in d=I,2,3 space dimensions, respectively [Bohm and Leschke 1992, Morita 1971, Stolze and Brandt 1989, Bohm et al. 1994]. The growth rates A. range between 1.0 and 1.3 for d=l and up to 2.0 for d=2, 3. 9 In order to set the stage for a proper interpretation of our results, consider the Green's function of the d-dimensional diffusion equation (with Dirichlet boundary conditions at infinity), G(r,t)

(10.30) (41tDt)dl2

where D is the diffusion constant. This function describes the density of a diffusing (globally conserved) quantity initially concentrated in a point at the origin. The density autocorrelation function of that quantity exhibits the characteristic diffusive long-time tail: G(O,t) DC (dl2. Integration of the d-dimensional Green's function over one spatial coordinate yields the (d-l)-dimensional Green's function. Physically, this describes the diffusive spreading of a line-like distribution. A plane-like initial distribution is obtained by integrating over two dimensions. The spectral density of G(O,t) has infrared singularities of the form _loor Il2 , -lnlool, _1001 112 in d=I,2,3, respectively. Our method of analysis is sensitive mainly to the strongest of these three singularities, _lror Il2 • That singularity is expected for site-spin autocorrelation functions in d= 1 as well as for autocorrelation functions of chain-like and plane-like spin aggregates in d=2 and 3, respectively. In more general terms, the idea behind our method of analysis may be stated as follows: Associated with any d-dimensional diffusion process on ad-dimensional lattice is a one-dimensional diffusion process on the same lattice, described in terms of aggregate dynamical variables. Therefore, the alternating pattern in the ~~-data, which is a direct and sensitive indicator of one-dimensional diffusion processes, is at the same time an indirect but equally sensitive indicator of d-dimensional diffusion processes for d> 1. The J!J-dependence of the infrared exponents axx and azz for the autocorrelation functions of a site spin on a lattice of dimension d=I,2,3 is displayed in Fig. 9Growth rates ~ are known to cause some mathematical problems (see Sec. 5-4), but these problems do not jeopardize the exponent analysis carried out here.

Section 10-13

161

to-15. We see at one glance that our indicator detects fairly reliably where site-spin autocorrelation functions describe one-dimensional diffusion, namely for the d=1 lattice only. The exponent axx varies strongly with JjJ as expected and assumes a minimum value at the symmetry point JjJ=I, consistent with d=1 spin diffusion. The exponent av;' by contrast, stays near that value over the entire anisotropy range shown, thus reflecting sustained diffusive behavior. 1O The broad nature of the minimum in a xx is attributable to the fact that the true long-time behavior is only nebulously encoded in the first few continued-fraction coefficients. I I 0.25.---..:_.::------------------------, ........

d=l,2,3 lattices

............................

-'-.-.

-.-.

0.00J....--------

--------

::t ~

(d=2, site) --=_:-:_::-:_~-~---=

:::t -0.25 ...

"-

-0.50

"- .....

........

----- ---

---

zz -0.75-t----,-x-x--,---.,.-----,--....----,--r--_-,-_---,_ _-; 0.5 0.7 1.5 0.9 1.3 1.1

Figure 10-15: JjJ-dependence of the infrared exponents a xx (solid lines) and a zz (dashed lines) for the site-spin spectral densities at T=oo of the 5=1/2 XXZ model in lattice dimensions d=1,2,3 [from Bohm et al. 1994].

IOFor the d=1 model the analysis of the exponent a zz becomes inapplicable in the parameter range O~ JjJ ~0.6 because of a crossover in growth rate which is related to the free-fermion nature of that system at JjJ=O (see Sec. 10-10). IIA similar JjJ-dependence of the infrared exponents has been obtained for the d=1 XXZ models with 5=1 and 5=00 (classical spins), but with considerably stronger deviations from the expected dominant patterns in both the ~~I1-sequences and the ~11' For these systems we know K=9 and K=8 exact moments, respectively [Bohm and leschke 1993].

162

Chapter 10

The data for the site-spin exponents <Xxx and <Xzz in lattice dimensions d=2,3 lie significantly above the d= 1 data. In d=2, site-spin diffusion is characterized by the logarithmic divergence in the spectral density. That weak divergence causes a shallow minimum in <Xxx at JjJ=l and a sustained negative <Xzz of much smaller magnitude than in d=1. The characteristic - lc.ol1/2 cusp singularity of d=3 site-spin diffusion is unlikely to be detectable by our analysis because of terms in the spectral density that are regular at 0.>=0. Our data for <Xxx and <Xzz' which are nonnegative except near the margins, indeed do not bear any signature of the diffusive cusp singularity. Figures 10-16 and 10-17 summarize our numerical evidence for quantum spin diffusion in lattice dimension d = 2, 3. Consider first the square lattice (Fig. 10-16). The two uppermost curves, which we have already discussed in the context of Fig. 10-15, reflect the weak logarithmic divergence in tr(co) associated with the two-dimensional site-spin diffusion. The two pairs of curves underneath bear the signature of the much stronger CO- 1I2 divergence in tr(co) associated with the onedimensional diffusion of chain-like spin aggregates consisting of entire rows or diagonals of spins on the lattice. The exponent <Xxx now has a much deeper minimum at JjJ=l, and <Xzz stays strongly negative over the entire parameter range. 0.25.,-------------------------, square laltice xx

zz

0.00 sile tS

::t ::t -0.25

-0.50

row

--

---- - - -

-0.75 -t=.=..::;;;;..-,.-------T----....---,---:-....---,-----,.-----.------,---i 0.5 0.7 0.9 1.5 1.1 1.3

Figure 10-16: JjJ.dependence of the infrared exponents <Xxx (solid lines) and <Xzz (dashed lines) for three types of spectral densities at T=oo of the 5=1/2 XXZ model on the square lattice: site spin, chain spins in (1 O)-direction (row), and (1 1)-direction (diagonal) [from Bohm et al. 1994].

Section 10-13

163

0.25.----0-:_=-------------------------, - simple cubic lattice

------------

0.00 -1-----------------

site

~

:t :t-0.25

row

-0.50

--

--

---------------

--

----- zz --- ---...--_--r__.-_--.-__ ,---_--r-===--,::.;;x'-x_-.-_-i ----plane

-0.75 -t-_--,._ _ 0.5 0.7

0.9

1.1

1.3

1.5

Figure 10-17: JjJ-dependence of the infrared exponents a xx (solid lines) and a zz (dashed lines) for three types of spectral densities at T=oo of the 5=1/2 XXZ model on the simple cubic lattice: site spin, chain spin in (1 0 O)-direction (row), and (1 0 O)-plane spin [from Bohm et al. 1994].

The difference between the results for the two types of chain spins in Fig. 10-16 is attributable to the fact that the A~II used in our analysis to gain information on the isotropic long-time dynamics are strongly influenced by the anisotropic short-time dynamics. The deviations between the two sets of curves are nonnegligible but sufficiently small to make our approach meaningful. The different strengths of infrared divergence in the site-spin and chain-spin spectral densities at JjJ=l is already detectable in the corresponding continued-fraction coefficients Ak as displayed in the inset to Fig. 10-14 by their different amplitudes of even-odd alternation. On the simple cubic lattice (Fig. 10-17), we investigate one-, two-, and three-dimensional diffusion processes. The two curves at the top represent the infrared exponents for the site-spin spectral densities, which are, as already stated in the context of Fig. 10-15, largely insensitive to the _00 112 cusp singularity associated with three-dimensional spin diffusion. The two curves in the middle of Fig. 10-17 resemble those at the top of Fig. 10-16 insofar as both sets reflect the weak divergence of two-dimensional spin diffusion, Le. chain-spin diffusion on the cubic lattice and site-spin diffusion on the square lattice, respectively. Likewise, the two curves at the bottom of Fig. 10-17, which describe the infrared exponents of

164

Chapter 10

lattice-plane spin spectral densities for the d=3 lattice, exhibit the same characteristic signature of one-dimensional spin diffusion as did the exponents of chain-spin spectral densities for the d=2 lattice (Fig. 10-16) and those of the site-spin spectral densities for the d=l lattice (Fig. 10-15). With this method we can thus identify quantum spin diffusion processes in lattice dimensions d=I,2,3 and discriminate between diffusion processes of different dimensionality on a given (tk.2) lattice. 10-14 Spin Diffusion in the Classical Heisenberg Model Classical spin dynamics at infinite temperature is an ideal field of operation for computer simulations. All equal time correlations are identically zero in such systems. Initial configurations are chosen completely at random. Nevertheless, interesting temporal and spatial correlations evolve in time - correlations whose exact nature has eluded a full understanding to this day. The primary focus of most T=oo spin dynamics studies has been the identification and characterization of the spin diffusion process (see Sec. 10-1). The prototype model of such studies is the classical Heisenberg model (XXX model), H

= -lE S(Sj'

(10.31)



for 3-component vectors (6.48) and with a uniform coupling extending over all nearest-neighbor pairs on a lattice to be specified. During the first wave of simulation studies probing T=oo spin dynamics, which started around 1970, the available computational power was quite limited [Steinet, Villain and Windsor 1976]. However enthusiastically the new calculational tool was employed, not much new insight into the spin diffusion process was gained. At that time, the results of spin diffusion phenomenology were not seriously questioned. For the most part, simulation results were interpreted in the straightjacket of hydrodynamic assumptions and used for the sole purpose of determining the values of the transport coefficient (diffusion constant). This was also the time when high-quality quasi-ID magnetic compounds became available in abundance for experimental investigation. It triggered a flurry of experimental studies of ID spin diffusion by means of NMR, ESR and neutron scattering. 10-15 Is Classical Spin Diffusion Anomalous? In the late seventies classical spin diffusion was considered fully understood with no promising avenue for further investigations. As a research topic it remained dormant until it was revisited unsuspectingly (as a distraction from a study of chaos in spin clusters) by one of the present authors [Miiller 1988]. His simulation study focused on spin autocorrelation functions at T=oo of the classical XXX model (10.31) in dimensions d=l (linear chain), d=2 (square lattice), and d=3 (simple cubic lattice). The recipe for such a calculation is simple and familiar: Choose the f(h set of initial conditions (S(k)} at random and integrate the equations of motion

Section 10-15

dS;

_

dt

=

aH -s ·x_ I

as;

, i=l,.··,N

165

(10.32)

numerically on a time interval [0, tmaxl. The spin autocorrelation function of the canonical ensemble at T=oo is then simulated by a combination of site and ensemble average, Co(t)

K

N

K k=1 N

;=1

= -.!.. L -.!.. L S~k)(t)'S~k)

(10.33)

,

Le. an average over all N lattice sites 12 and an average over K randomly chosen initial spin configurations. For finite K, Co(t) is a fluctuating quantity, and for finite N, that function is subject to finite-size effects. The goal then is to choose both N and K sufficiently large to eliminate systematical errors and to minimize statistical uncertainties on the time interval over which the function Co(t) is analyzed. -2.0

-- ---

o d=1 N=50

...........,.....

-2.5

...........

'-_,

-3.0

....,

...........

'--,

'- "

0

d=2 N=20*20

6

d=3 N=8*8*8

" '-,

-,

-'-

..........

U

o

-

.......... l::

-3.5

-4.0

-4.5

-5.0 -t---r-----,--,----,:=,-......,....-.-------r---,--....-----t 30. 40; 50. 4. 5. 6. 7. 8.9.10. 20. 2. 3.

Jt Figure 10-18: Log-log plot of the spin autocorrelation functions Co(t) ;: <S,{t)'Sj> at T:oo for the classical Heisenberg model on a linear chain of N=50 spins, a square lattice of N=20x20 spins and a simple cubic lattice of N=8x8x8 spins. The three sets of data represent averages over K = 22792, 4563, and 4041 random initial conditions, respectively. The dashed lines represent the slopes -d/2, predicted by spin diffusion phenomenology. The simulation was carried out with a 4th-order Runge-Kutta integration with fixed time step ..klt=0.025 [from Muller 1988].

12rhe use of periodic boundary conditions makes all lattice sites equivalent.

166

Chapter 10

The main results of that simulation are displayed in Fig. 10-18. It shows a log-log plot of the spin autocorrelation functions for the cases d=I,2,3 over time intervals of different sizes. The data are in support of the spin-diffusion picture on a qualitative level: uniform power-law decay with a characteristic exponent that increases with increasing dimensionality. Comparison with the slopes of the standard fdl2 prediction (represented by the dashed lines in Fig. 10-18) suggests the presence of an anomaly. The characteristic exponent ad' as obtained from the slope of the linear regression line for the data shown, exceeds the value d/2 in all three dimensionalities: (10.34) The case in support of anomalous spin diffusion is strongest (amounting to some 20%) in the d= 1 system. But the last word on that has not yet been spoken.

,

1.000

....

-0.1

,....., ,..., .... ><- -0.3 ><0

0.500

U ...... ..5

.-.... ......

-0.5

><......... 0.100 ><0 U

3.

9.

12.

15.

18.

Jt

0.050

6.

0.010

6.

o o

d=3 N=8*8*8 d=2 N=20*20

d-1 N-50

0.005+---,---,--,..---,..---,..---...---...---...---...---,----'1 O. 1. 2. 9. 10.11. 3. 4. 5. 6. 7. 8.

Jt/Vd 3<SMSf>

Figure 10-19: Logarithm of the spin autocorrelation function ~(~ == at T=oo for the classical XX model on a linear chain of N=50 spins,·a square lattice of N=20x20 spins and a simple cubic lattice of N=8x8x8 spins. The three sets of data represent averages over K = 37621, 5039, and 3392 random initial conditions, respectively. In the inset we have replotted the same data in the mode previously used in Fig. 10-6 for the same correlation function of the quantum XXZ model. The simulation was carried out with a 4th-order RungeKutta integration with fixed time step "kjt=0.025.

Section 10-17

167

10-16 Exponential Decay Versus Long-Time Tails Let us pause and juxtapose the diffusive long-time tails depicted in Fig. 10-18 with simulation data obtained for the spin autocorrelation function <Sj(t)S? at T=co for the classical XX model, (10.35) in dimensions d= 1,2,3. In Fig. 10-19 we have plotted the logarithm of that function versus time. The lattice sizes are the same as in Fig. 10-18. We have rescaled the time axis by d l /2, which makes the three curves start out with the same curvature. In the absence of a conservation law for the variable ST==LjS!, this correlation function is expected to describe a standard relaxation process, which is characterized by exponential decay. This type of behavior is indeed suggested by all three sets of data. For easy comparison with the corresponding short-time results for the quantum XXZ model displayed in Fig. 10-6, we have plotted in the inset to Fig. 10-19 the quantity In(3<Sj(t)S?)/Jt vs Jt.

10-17 Anomalous Exponent or Non-Asymptotic Effect? Now we return to the XXX model and the unusual long-time tails displayed in Fig. 10-18. Gerling and Landau [1989] undertook a more extensive simulation for the d=l case and evaluated the spin autocorrelation function Co(t) == <Sj(t)"Sj> over a much longer time interval (up to Jt=2oo). They also evaluated the energy autocorrelation function Eo(t) == <Sj(t)"Sj+l(t)Sj'Sj+l> over the same range. Local energy fluctuations are expected to exhibit diffusive behavior, too. The results are shown in Fig. 10-20. The solid lines have slope -112 and the dashed line -0.58. Several interesting observations have been made. The energy correlation function reaches aslzmptotic behavior very quickly and displays a standard diffusive long-time tail, _(12. The data for the spin autocorrelation function are manifestly consistent with an anomalous long-time tail, _rO. 58 (dashed line).n However, the data for In[Co(t)] in Fig. 10-20 exhibit a slope with a significant trend toward decreasing magnitude for increasing t. That trend is not readily perceptible in the log-log plot itself because of the statistical fluctuations. Gerling and Landau [1990] calculated the slope of the data over successive time intervals in the log-log graph and plotted that slope vs (Jty1l2 as shown in the inset to Fig. to-20. From the unmistakable trend toward smaller magnitude they concluded that the data points in the inset can be extrapolated to an asymptotic slope of
a.

13The same conclusion can be drawn for the data shown in Fig. 1 of Muller [1989].

141n Fig. 2 of Gerling and Landau [1990]. a linear extrapolation was suggested for the data reproduced in Fig. 10-20.

Figure 10-20: Log-log plot of the spin autocorrelation function Co(t) E <S~t)'Sj> and the energy autocorrelation function Eo(t) E <S~t)'Si+1S;,Si+1> at T=oo for the classical 10 Heisenberg model with N=20,OOO spins. The number of configurations averaged over was K=400 (spin) and K=310 (energy). The simulation was carried out with a vectorizable predictor-corrector method. The solid lines have slope of magnitude a=O.5 and the dashed line a=O.58. The data points in the inset show the results of least-square fits to simple power laws using data for Co(t) over intervals of different sizes. The statistical uncertainty of each data point is indicated by an error bar.The horizontal coordinate of each data point is at the midpoint of the interval used [adapted from Gerling and Landau 1989, 1990].

10-18 Experimental Evidence for Anomalous Spin Diffusion A search of the literature for suitable quasi-ID magnetic insulators in which the occurrence of anomalous spin diffusion might be investigated experimentally revealed that the phenomenon had, in fact, already been observed in at least three different NMR proton spin-lattice relaxation studies [Hone, Scherer, and Borsa 1974; Borsa and Mali 1974; Boucher et al. 1976] on the s=512 Heisenberg antiferromagnet (CH3)4NMnCI3 [tetramethyl amonium manganese trichloride (TMMC)]. However, at the time those studies were undertaken, the possibility of anomalous spin diffusion was not taken into consideration as a possibility in the data analysis. The measurements discussed here were all taken at room temperature (kBTIJ"23), where the correlation length is sufficiently small to permit direct

Section ID-18

169

comparison with T=oo simulation results. The relaxation rate lITI as determined in these experiments is expected to contain a frequency-independent contribution of unknown strength and a contribution which is proportional to the spectral density ct»!ill(ro), evaluated at the nuclear Larmor frequency, which is very low in electronic units. Here cI»ti1l(ro) is dominated by the infrared divergence associated with the diffusive long-time tail, _ra, in <S~(t)SIi>. This yields an expression of the form

lITI

= Proa - I +Q

(10-36)

for the relaxation rate, with three variable parameters P,Q,a [Borsa and Mali 1974].15

4.0.--------------------------, 3.---,----r------.---r-----,

3.5 2

....

o

3.0 o

o

O'---::'-_---'_ _- ' - _ - - - l . . _ - - . J o 0.1 0.2 0.3 0.4 0.5 11;112 (MHflll)

1.5 1.0 a=O.57

0.5-;-----,------,,----....-----,..-------.---' 0.0 0.1 0.2 0.3 0.4 0.5 a-I CA)

Figure 10-21: Frequency-dependence of the NMR proton spin-lattice relaxation rate 1/T1 at room temperature as measured on the quasi-1D Heisenberg antiferromagnet TMMC by Hone, Scherer and Borsa [1974]. The inset shows the data as originally pUblished. One set of data (open cirlces) was used to determine the three parameters P, Q, a of (10-36) by nonlinear regression analysis, yielding the exponent value a = 0.57±O.11. We have replotted that set of data values versus OJa.·1 along with the regression line.

15 Note that for a spin system described by a Hamiltonian H = Hint - hL,Sf with [Hinl'L,Sf] = 0, the dependence on the magnetic field h of the spectral density ~(OJ) is

equal to its ro-dependence, at T = 00 (under appropriate scaling).

170

Chapter 10

5.0 ~ (..c·,)

4.5 3

4.0

T,

TMMC

CIIH

T.300·K

2

3.5 _ 3.0

Eo-<

"'"

H" "1 ( KO- \<1) 1.5

1.0

0.5

2.5 2.0 1.5 a=O.60

1.0 0.5 0.0

0.5

1.0 U)

1.5

a-I

Figure 10-22: Effective frequency-dependence of the NMR spin-lattice relaxation rate 1/T1 at room temperature as measured on the quasi-1 D Heisenberg antiferromagnet TMMC by Borsa and Mali [1974]. The inset shows the data as originally pubished. The data were used to determine the three parameter p. Q. a of (10.36) by non-linear regression analysis, yielding the exponent value a = 0.60±0.06. We have replotted those data vs roa - 1 including the regression line.

Specific sets of experimental data from the three aforementioned NMR studies were reanalyzed by fitting them to expression (10.36) with unconstrained infrared exponent a [Miiller 1988]. The frequency-dependence of these data as originally published are reproduced as insets to Figs. 10-21 to 10-23. For the data reanalysis, two independent methods of nonlinear regression were employed in order to determine the parameters P,Q,a i.n (10.36). In the main plot of Figs. 10-21 to 10-23, we show those sets of experimental data redisplayed vs roa -1 or vs ro along with the regression line. For the characteristic exponent a, both methods consistently yield the following values for a best fit of the experimental data: a(exp) = 0.57 ± 0.11 [Hone, Scherer and Borsa 1974], a(exp) = 0.60 ± 0.06 [Borsa and Mali 1974], a(exp) = 0.62 ± 0.05 [Boucher et al. 1976]. All three sets of experimental data confirm the anomalous nature of 10 spin diffusion. In two out of three cases the standard value a(SD) = 0.5 is clearly ruled out.

Section 10-19

4.0

171

misec· 11

1 )

3.5 0

3

..................

3.0 ....-t

2.5

E-o

-........ ....-t

2.0

5

10

15

H(kG) a=O.62

1.5 1.0 0.5

O.

5.

10.

15.

20.

CJ

Figure 10-23: Effective frequency-dependence of the NMR spin-lattice relaxation rate 1/T1 at room temperature as measured on the quasi-1D Heisenberg antiferromagnet TMMC by Boucher et al. [1976]. The inset shows the data as originally pUblished. Two sets of data (open and solid circles, from measurements performed at different places) were used to determine the three parameter P, Q, a of (10-36) by non-linear regression analysis, yielding the exponent value a = 0.62±O.05. We have replotted those data versus Q) along with the regression line.

10-19 Two Kinds of Computational Errors Before proceeding with the main story line, we pause for a brief discussion of the computational errors that must be kept under control in these simulations [Liu et al. 1991]. Consider the 2N-dimensional phase space of the classical XXX model with N spins. It is foliated by lower-dimensional invariant surfaces. Their dimensionality depends on the number of existing analytic invariants: the total energy H, the total spin ST' and perhaps some less obvious ones. In a nonintegrable many-body system the dimensionality of the invariant surfaces is believed to be not much lower than 2N. The motion of any individual phase point is confined to an invariant surface, but the flow of phase points on an invariant surface is chaotic. 16

16rhe exception is the case where the invariant surfaces have dimensionality N. Then they have the topology of N-tori and the phase flow confined to a torus is regular.

172

Chapter 10

The numerical integration of the equations of motion (10.32) is subject to two different types of computational error: D Deviations Ar(t) tangential to the invariant surface: the (chaotic) trajectories of two nearby phase points tend to separate exponentially in t, implying an exponential error propagation in the numerical integration. It causes a loss of M significant digits per time interval Mato on average. D Deviations ap(t) perpendicular to the invariant surface: a computational trajectory also tends to stray off the invariant surface on which it starts out, but at a much slower rate, namely linearly in t. The associated loss of M significant digits occurs on a time interval 1oM-I ato(P) on average. 100 10-1 10-·

---'

---

lO-a 10-4

10-0 10-Eo-<

<]

10-7 10-10-' 10-10 10-11 10-12

0.5 20

10- la 10-14

40

60

80

Jl 0

10

20

30

40

60

50

70

80

Jt Figure 10-24: Computational error ~(t) tangential to global invariants (main plot) and computational error ~(t) perpendicular to global invariants (inset). The integration of the equation of motion (10-32) for the 10 Heisenberg model with N=250 spins was performed by RK4 with fixed time step: ..k:lt=0.05, 0.025, 0.01, respectively, for the three lines. For better visual effect, the curves in the inset have different vertical scales. The horizontal dashed line in the main plot indicates the value of the spin autocorrelation function Co(t) at Jt=80 [from Liu et al. 1991].

For the purpose of illustration, Liu et al. [1991] have computed the timedependence of the following quantities representative of the two types of error: N

Ar(t)

] 1/2

= T(t/2) [ ~tr I!~Z [Sr(t) -Sr(O)f

(10.37)

Section 10-20

L\p(t) = T(tl2) IH(t) - H(O) I ,

173

(10.38)

where t(tI2) symbolizes the property that the spin motion is reversed at time tl2 in the numerical integration. The two quantities (10.37) and (10.38) as evaluated by means of a 4 th -order Runge-Kutta integration with fixed time step are plotted in Fig. 10-24 for given initial conditions, the former with an exponential scale on the vertical axis (main plot) and the latter with a linear vertical scale (inset). The three curves in each plot represent data from integrations with different time steps dt. The results clearly demonstrate the exponential and linear error propagation in the two quantities Ll.y.(t) and L\p(t), respectively. In both quantities, the rate of error propagation increases with increasing time step dt. The function Ll.y.(t) is observed to level off when its size reaches 0(1). The value of the spin autocorrelation function Co(t) at 1t=80 is about 0.023 (indicated by the dashed line in Fig. 10-24). The computational error Ll.y.(t) is expected to become non-negligible when it has grown to a size comparable to Co(t). This is the case at 1t<80 at least for the two larger time steps used. The function L\p(t), by contrast, stays orders of magnitude below the value of Co(t) at 1t=80 for all three time steps used. It is, of course, the computational error Ll.y.(t) that is likely to cause systematic errors in the simulation data unless it is kept under strict control. The computational effort to keep Ll.y.(t) small increases exponentially in t. It tends to become very costly for 1t';?loo no matter which integration algorithm is employed.

10-20 q-Dependent Correlation FunctioQ Our conclusions from the simulation data presented thus far may be summarized as follows [Liu et al. 1991]: (i) The anomalous character of the diffusive long-time tail persists out to the largest value of t for which a quantitative analysis of the slope (in the log-log representation) yields reliable results. (ii) The slope 1/2) or of the standard type (a= 112) has eluded a conclusive answer. The answer to (iii) requires a credible extrapolation of the varying slope loo with sufficient accuracy at an acceptable low level of statistical fluctuations seemed prohibitive in 1991. The unresolved issue was about to go dormant again, when it was reinvigorated by a new simulation study of De Alcantara Bonfim and Reiter [1992] of q-dependent spin correlation functions. Such a study is expected to yield more detailed information about the transport mechanism of spin fluctuations than can be gained from the long-time tails of spin autocorrelation functions alone.

17Preliminary attempts at that task have been reported in Fig. 2 of Gerling and Landau [1990] and Fig. 2 of Liu et al. [1991].

174

Chapter 10

Consider the normalized q-dependent correlation function Co(q,t) == <S(q,t)·S( -q,O» <S(q,O)'S( -q,O»

(10.39)

for the classical ID Heisenberg model with uniform exchange. It plays the role of the (normalized) fluctuation function (3.21) in the jargon used throughout this book. This can be expressed in terms of a memory function as the solution of (3.39a): t

~Co(q,t) + fdtlf,(q,t-tl)Co(q,tl) = 0

at

0

(10.40)

.

In the framework of the projection operator formalism (Sec. 3-7), the memory function !(q,t) can be interpreted as the fluctuation function of a new dynamical system, specified by the old Liouvillian sandwiched between a pair of projection operators. For small q it is well approximated by the quantity f,(q,t) ... q 2<j(q,t)j( -q,O» Co(q,O)

,

(10.41)

which is in essence the current-correlation function of the original dynamical system. 18 The q-dependent spin current is defined by the expression j(q,t) =

L eiqlJ[S I(t) xSI+I(t)]

.

(10.42)

1

Unlike the total spin S(q=O,t), the total spin currentj(q=O,t) is not conserved. The time-decay of the memory function (10.41) is therefore expected to be governed by exponential relaxation on a microscopic time scale. The fluctuation function (10.39), by contrast, decays on a time scale that grows arbitrarily long as q~O. It can thus be argued that for sufficiently small q, the function Co(q,t) does not vary much over the time interval where f,(q,t) is significantly di.fferent from zero. For memory functions with very short memory it might then be justified to replace t' by t in Co(q,t') of Eq. (10.40). That amounts to using a memory function with no memory: (10.43) where D(q)

= fdt l <j(q,t)j( -q,O» = D[1 +O(q 2)]

.

(10.44)

o

To leading order in q, Eq. (10.40) with (10.43) reduces to the diffusion equation

I~he time dependence of the exact memory function is modified (for ~) by a projection operator.

Section 10-21

a

2

175

(10.45)

at Co(q,t) = -Dq Co(q,t) .

We recognize expression (10.43) to be the memory function of a fluctuation function in the I-pole approximation as discussed in Chapter 8. Consider the Laplace transform of (10.39) in the continued-fraction representation, co(q.z) == jdte-ztCo(q,t)

o

= _----:1-.,....~ ~1(q)

z+--,---

(10.46)

~(q)

z+-z+ ... with coefficients that can be determined recursively as described in Chapter 3. The dominant long-time asymptotic behavior of Co(q,t) is then governed by the singularity in co(q,z) closest to the imaginary z-axis. The approximation (10.43) now hinges on the assumption that this dominant singularity is a pole at z=-Dl for small q and that it remains isolated for q-?O. With

~1(q) = ~J2(1-cosq) 3

.. 'l:..J 2q 2 ,

3

(10.47)

the implication of this assumption is that (10.44) can be replaced by the I-pole relaxation function D = 'l:..J 2 lim lim 1 3 q~O z~O 1 + ~(q) z+ ...

(10.48)

without alteration of the dominant singularity. Expression (10.48) is the Laplace transform of (10.5). However, the validity of the assumption leading to this form is by no means guaranteed. It had been questioned before [Forster 1975, Roldan, McCoy and Perk 1986], but substantial counter-evidence has remained elusive until recently.

10-21 Power Law Long-Time Tail with Logarithmic Correction Having been alerted by the puzzling simulation results for the spin autocorrelation function, De Alcantara Bonfim and Reiter [1992] focused their simulation study on the direct evaluation of the q-dependent spin correlation function (10.39) and the current correlation function (10.41). The latter function was observed to decay algebraically ,_f 1, instead of exponentially. This implies a logarithmically divergent diffusivity (10.44) and thus invalidates the hydrodynamic assumption that leads to the diffusion equation (10.45) and its solution (10.5). The singularity in the transport coefficient affects not only the t-dependence but also the q-dependence in the decay law (10.5) of the fluctuation function (10.39). From their simulation data, Bonfim and Reiter infer that the decay (10.5) should be modified as follows:

176

Chapter 10

Co(q,t) - exp[ -q 2.12ltln(lt)] .

(10.49)

This has far reaching consequences for the diffusive long-time tail of the spin autocorrelation function: Co(t) - [ltln(.Qt)]-« ,

(10.50)

with D.=l and a=0.472. This result is at variance with the standard result, _(ltr 1l2 , in a way that sheds a great deal of light onto the unresolved issues from previous simulation studies. More about that in Sec. 10-22. Bohm, Gerling, and Leschke [1993] were quick to point out that the asymptotic form (10.49) is in contradiction to the non-negativity of <Sj(t)"Sj+n>' for which strong numerical evidence exists. Based on their analysis of the spatial variance defined in (10.23), they proposed an alternative asymptotic expression, (10.51) which also implies an asymptotic form (10.5) for the spin autocorrelation function, but with a different exponent, a=O.5.

10-22 Effective Exponent The proposed decay law (10.50) for the spin autocorrelation function implies that the slope of Co(t) in a log-log plot can be described by an effective exponent,

a.

= a[1 +_1_] .

In(Ot)

(10.52)

The asymptotic decay law (10.49) proposed by Bonfim and Reiter predicts a=O.472, and the alternative law (10.51) proposed by Bohm, Gerling and Leschke entails a=O.5. There is no compelling reason for setting D.=J in the logarithmic correction as done in both references. Since neither of the two conclusions was primarily based on the analysis of spin autocorrelation functions, the logical next step is to use the best available simulation data for Co(t) as a discriminant between expressions (10.50) with a=O.472 and a=O.5, respectively. To that end, we have carried out a new simulation for a system of 1024 spins with periodic boundary conditions [Srivastava et al. 1994]. We have employed CM-5 machines with various numbers of processors programmed in Connection Machine Fortran for up to 4096 parallel time integrations. For the integration over the time interval 0::; It::; 102.2, we have used a 4th-order Runge-Kutta method with fixed time step Idt=O.OO5. In this massively data-parallel programming mode previously unattained statistics can be reached with no undue effort. For the intended analysis, we have determined the average slope a. of the simulation data in a log-log representation over a time interval of length ltav as follows: each data point of a.(t) is calculated by linear regression from Nav consecutive data points (lnJt,lnCo(t» spaced at ll:!t = 0.2 and assigned the It-value

Section 10-22

177

at the midpoint of the interval of length Jtav = Na,J!:J.t. Figure 10-25 shows the slope function a. plotted versus lIJt for three different sizes of Jtav' This representation enhances the visibility of the subtle features in the long-time tail, but it also magnifies the statistical fluctuations. The latter are kept under control by adjusting Jtav' 19

0.60

0.00

(a)

0.01

0.03

0.02

0.04

0.05

l/Jt Figure 10-25: Slope function a(~ for the 10 classical Heisenberg model [(10.54) with uniform exchange Ji,i+1=J] as determined from the slope of Co(~ in a log-log plot. The data for Co(~ represent an average over 404484 randomly chosen initial conditions and over the 1024 sites of the lattice. Each data point a is determined by linear regression from Nav consecutive data points (InJt, InCo(~) spaced at JM=0.2 and plotted vs 1/Jt at the midpoint of the interval of length Jta..rNa."JM. The simulation data are represented by the circles. The asymptotic form (10.50) subjected to the same procedure yields the dashed lines for a=0.5 and the solid lines are for a=0.478. The three plots correspond to different sizes of averaging time interval: (a) Jta..r30. (b) Jta..r20, (c) Jta..r10. [from Srivastava et al. 1994]

19The price to be paid in exchange of a smooth slope function is a systematic deviation from the true slope at a given value of Jt for all functions except pure power laws. For a given set of simulation data, if one increases JtaV' one gaines smoothness and along with it the ability to extrapolate. Conversely, if one decreases JtaV' the systematic deviations go down, but the statistical fluctuations grow more intense. The three plots in Fig. 10.25 are intended to illustrate that the systematic deviations are negligible except (in some cases) for short times.

178

Chapter 10

In order to facilitate a direct comparison of our simulation results with the proposed functional form (10.52) for the effective exponent aCt), we have subjected the asymptotic expression (10.50) to the same exponent analysis as the data. The resulting slope function aCt) still depends on the parameters ex. and n. Minimizing the relative rms deviation between the two slope functions aCt), namely the one representing the simulation data and the one representing the averaged exponent over the interval 5+Jta j2 ~ Jt ~ 102.2-Jta j2, yields parameter values in the range

a(t)

ex. = 0.478±0.OOl

, O/J = 2.30±0.02 ,

(10.53)

for the three values of averaging intervals Jtav used. The solid lines represent a vs lilt for the optimal parameter values. The agreement with the simulation data is quite satisfactory. If we perform the fit for fixed a.=0.5, we obtain the optimal value QlJ=9.70±0.05 for the other parameter, and the result, represented by the dashed lines, is in clear disagreement with the simulation data. We have repeated the analysis with moreof the (evidently non-asymptotic) data at small times omitted (up to l5+Jta j2) and found a decreasing trend of the optimal exponent value. Now it is in the range a.=0.472±O.OO2, in even better agreement with the value proposed by Bonfim and Reiter. Nevertheless, the problems attached to this scenario, as pointed out by Bohm, Gerling, and Leschke, cannot be dismissed. The inevitable consequence is that the true asymptotic behavior is even more subtle. In this context, we should also mention a recent study by Lovesey and Balcar [1994], which reports new results based on a coupled-mode approximation. That calculation starts from the exact equation (10.40) and uses a particular approximate expression for the memory function f.(q,t) in terms of the correlation function Co(q,t). The analysis of the resulting (closed) equation yields an anomalous long-time tail for the spin autocorrelation function, in the form of power laws with different characteristic exponents at intermediate times, _(213, and asymptotically, _r2l5 • To what extent these findings are consistent with the simulation data of Fig. 10-25 remains to be seen.

10-23 Effect of Exchange Inhomogeneities How typical is the occurrence of anomalous long-time tails in 10 classical spin systems with isotropic exchange? It had already been noted that the anomaly disappears in the presence of uniaxial single-site anisotropy [De Alcantara Bonfim and Reiter 1992]. The question is what happens if we modify the spin coupling without altering the 0(3) rotational symmetry in spin space, for example, by reducing or removing the translational symmetry along the chain. In order to investigate that question, we have carried out simulations of comparable extent on three further variants of the classical Heisenberg model

H

= - L Ji ,i+l S (Si+l

.

(10.54)

i

In addition to the model (i) with uniform exchange, Ji,i+l=J, dis~ussed previously, we consider the model (ii) with alternating exchange, Ji,i+l=(-l)'J, and two models

Section 10-23

179

0.30"""1":""-------------------,

0.20

alternating

0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03

0.02-+------.,----......--.,-~-,..._ ......--.__;_' 10 20 30 40 50 60 708090

Jt Figure 10-26: Log-log plot of the spin autocorrelation function Co(f)=<S/{trSj> for the 1D classical Heisenberg model (10.54) with four types (i)-(iv) of isotropic exchange as explained in the text. The simulations were carried out by means of an RK4-integration with fixed time step Jdt=0.005 on systems of 1024 spins with periodic boundary conditions. Each of the four curves represents data averaged over more that 400000 random initial conditions. In the two random-exchange models, the exchange constants Ji,i+1 were randomly picked for each initial configuration. The solid lines have slope of magnitude a=0.5.

with random exchange: model (iii) has Ji,i+1=±l with equal probabilities and model (iv) has IJi i+11 ~ {jJ with a rectangular probability distribution. 20 The long-time tails of Co(t) for the four variants are shown in the log-lo~ plot of Fig. 10-26. Significant deviations from the phenomenological result, _(1 2 (solid lines), can be

20rhe width 2-J3Jof the J;,i+1-distribution has been chosen to match the initial curvature Co(f) with those of the other three models.

180

Chapter 10

discerned in all cases for times up to Jt..40 and in three out of four cases up to considerably larger times. 21

0.54

(a) o o

0 0 00

0.52

ltS

0.50

CP 0

o

0.52

(~

0.50

0.48

0.48 0.46

0.46 0.00

0.00 0.01 0.02 0.03 0.04 0.05

0.01

0.02

0.03

0.04

0.05

l/Jt Figure 10-27: Slope function n(1) for the classical Heisenberg model (10.54) with altemating exchange J,.i+1=(-1)'J, produced by the same method as that of Fig. 10-25. The number of integrations with randomly chosen initial conditions was 479232 for this case [from Srivastava et al. 1994].

What looks like subtle differences in the log-log representation of the longtime tails as shown in Fig. 10-26 translates into much more dramatic differences in the associated slope functions
21MOller [1989] had already compared three of the four long-time tails and found (ii) and (iii) similar but distinct from (i). The superior statistics of the new data presented in Fig. 10-26 reveals subtle but significant differences between (ii) and (iii) as well.

Section 10-23

(a)

0.54-

·~~o

0.520.50 0.54

0.48

.~

0.520.50 ....

0.46 0.44 0.42 0.40 0.00

181

00

0.48 0.460.440.420.400.00 I

0.01

0.01

0.02

0.03

I

0.04 I

0.02

0.03

0.05 I

0.04

0.05

l/Jt Figure 10-28: Slope function a(~ for the classical Heisenberg model (10.54) with random exchange Ji,i+1=±J, produced by the same method as that of Fig. 10-25. The number of integrations with randomly chosen initial conditions was 409600 for this case. For each initial configuration, the exchange constants were randomly chosen as well [from Srivastava.et al. 1994]

The slope function a of the random-exchange model (iii) is shown in Fig. 10-28. Unlike in the previous two cases, it has an increasing trend for increasing t up to a"0.53 at the tail end of the data, where it seems to level off. While a limiting value of (XSD=O.5 cannot be ruled out, the data do not show any tendency to extrapolate to that value. Changing the distribution of random exchange constants from (iii) to (iv) produces a quite different slope function as can be observed in Fig. 10-29. It starts out at a much smaller value for shorter t and reaches a.. O.4 at the tail end of the data (Jt=102.2). In some way this slope function looks like the mirror image of that for model (i). It may very well extrapolate to (XsD=0.5 or there abouts by some logarithmic law - a modification of (l0.52). But any such law would have to be motivated by an investigation of q-dependent correlation functions as was done by Bonfim and Reiter for model (i). For the random-exchange model (iii), the anomaly, if it indeed exists, is much weaker than in models (i) and (iv) and will therefore be much harder to identify and analyze in q-dependent correlation functions.

182

Chapter 10

0.52 0.50

0.52

(a)

0.50 0.48

·0.48

(c)

0.46 0.44

0.46

0.42 0.40

0.44

les

(b)

0.38 0.36

0.42

0.34

0.40

0.01

I

0.02

I

0.03

0.04

0.05

0.38 0.36 0.34 0.00

0.01

0.02

0.03

0.04

0.05

l/Jt Figure 10-29: Slope function ii(~ for the classical Heisenberg model (10.54) with random exchange IJi,H.11~-..f3J, produced by the same method as that of Fig. 10-25. The number of integrations with randomly chosen initial conditions was 424960 for this case. For each initial configuration. the exchange constants were randomly chosen as well [from Srivastava et al. 1994].

11 QUANTUM SPIN DYNAMICS AT ZERO TEMPERATURE One-dimensional quantum spin systems with short-range interaction are strongly fluctuating many-body systems under most circumstances. Any spontaneous magnetic long-range order (LRO) that might exist in the ground state is destabilized by thermal fluctuations at all nonzero temperatures no matter how small. Even at T=O the order parameter is, in general, considerably reduced by correlated quantum fluctuations. It is not unusual that the zero-point motion prevents the onset of magnetic ordering entirely and gives rise to a ground state that is either critical or magnetically disordered. Quantum spin chains have surprised us more than once with new and unexpected Tc=O critical behavior and T=O phase transitions as functions of certain parameters (exchange or single-site anisotropy, magnetic field etc). In some cases, the zero-temperature phase transitions of quantum spin chains map onto phase transitions of exactly solvable 20 classical statistical models, where the role of quantum fluctuations is taken over by thermal fluctuations. Perhaps the most fascinating aspect of quantum spin chains is the impact of the T=O phase changes on the dynamical properties. Inevitably, the questions of primary interest from an experimentalist's point of view - dynamically relevant dispersions and line shapes - are extremely difficult to handle technically. Even the most simple 10 quantum spin models reveal a degree of complexity in their dynamical properties that is virtually impossible to describe in terms of single-mode fluctuations such as result from standard approximation schemes. Those rare examples of ID quantum spin models for which nontrivial dynamical properties can be determined rigorously provide ample evidence for the inadequacy of the singlemode picture. At the same time, these exact results provide challenging tests for general calculational techniques including the recursion method and demand a higher level of sophistication in the interpretation of dynamics experiments on quasi-ID magnetic materials than was previously found acceptable.

11·1 10 s=1/2 XY Model with Magnetic Field The Hamiltonian of the 10 s=1I2 XY model in the presence of a uniform magnetic field h in z-direction reads H XY =

-E {(1+y)StS/:l /

+

(1-y)S/S/~l + hS/l .

(11.1)

We have already encountered the XX model ("(=0) in previous applications of the recursion method. A brief discussion of the magnetic ordering in the ground state of H XY in the parameter space spanned by the exchange anisotropy 'Y and the magnetic field h (see Fig. 11-1) will help us understand the dynamical properties for different parameter values. At nonzero exchange anisotropy (O
184

Chapter 11

1

Figure 11·1: Parameter space (h, y) of the 10 8=1/2 XY model (11.1). The shaded area denotes the region in which the ground-state properties of this model map onto the thermodynamic properties of the 20 Ising model with nearest neighbor couplings J1, J2 on a rectangular lattice. The circular line h=~ corresrgnds to T=O. and the dashed line h=hc corresponds to T=Tc- The dot-dashed line h=(i2-1) 12 maps onto the temperature axis ofthe square-lattice Ising model J1=J2 [adapted from Barouch and McCoy 1971; Suzuki 1971].

Mx = <st> = V'Y 112/2(1 +y)(1 -h 2)118

(11.2)

•

As h is raised across the c:ritical value he= 1, the system undergoes a continuous transition to a phase with Mx O. In the isotropic case (y.=O), however, every point on the line OSh~e is a critical point. For parameter values in the shaded region of Fig. 11-1, the ground state properties of the 1D quantum spin model (11.1) are related to the thermodynamic properties of the 2D Ising model on a rectangular lattice,

=

HI

= -11L

O"n,mO"n+l,m

m,n

L O"n,mO"n,m+l

-12

(11.3)

'

n,m

where the coupling constants 11, 12 and the temperature T are determined by the parameters 'Yand h [Suzuki 1971]: tanh(2J1/kB1)

= 'Y,

tanh(2J2IkB1)

= hNlh

.

(11.4)

In this mapping, the special field hN = (1_y.)1I2 of the quantum spin chain (circular line) corresponds to T=O in the 2D Ising model, whereas the critical field he=1 (dashed line) maps onto the critical temperature Te determined by the equation (11.5)

Section 11-1

185

The square-lattice case (11=12) of the Ising model maps onto the line h = (1-2-1)112 in the parameter space of the XY model (shown dot-dashed in Fig. 11-1). The equal-time two-spin correlation functions of the two models HI and H XY are related to one another as follows: --

..

2 --< 1ry Smxs x I> - 2 l-r <SmYS Y1> . 1 m 1 m

(11.6)

In the limit Im-m'l~oo, this equation establishes the equivalence between the result (11.2) for the order parameter of the ID s= 1/2 XY model and the Onsager-Yang result for the spontaneous magnetization of the 2D !sing model [Yang 1952]:

lry -2

-2

= 2_M x = (I -cosech(2JlkB1)cosech(2JikB1) }1/4.

M

1

(11.7)

Suzuki's mapping also establishes the equivalence between the logarithmic divergence of the susceptibility Xzz(T=O,h) =dM/dh at h=hc in the ID quantum spin model H XY and the logarithmic divergence of the specific heat at T=Tc in the 2D Ising model. The magnetization M" which is not the order parameter of the T=O phase transition in the quantum spin chain, thus plays a role analogous to the entropy in the 2D Ising model and the quantity hXzz a role analogous to the specific heat. The divergence of the 2D Ising model susceptibility has its quantum counterpart in the susceptibility Xxx(T=O,h) - Ih-h/ 7/4 of the 10 XY model. In Secs. 11-2 to 11-12, we shall report studies of the T=O dynamics of HXY for two contrasting situations: (i) a situation in which the model has a fully ordered ferromagnetic ground state - a state with no correlated quantum fluctuations; (ii) a situation in which the model has a critical ground state - a state with no magnetic LRO and very strong quantum fluctuations. Situation (i) is realized along the circular line h = hN = (1_y)I/2, 0~-r-:;1, in Fig. 11-1. Here the system has a simple product ground state, characterized by the saturated ferromagnetic order parameter

M = <S I> = !..(b1 1(lry) 2

(11.8)

, 0, V(1-r)/(1ry) ) ,

as will be further discussed in Sec. 11-2. Situation (ii) is realized (not exclusively) at the following two points (marked by asterisks in Fig. 11-1): o h=y=1. This is the transverse Ising (TI) model at the critical field, with equal-time two-spin correlation functions decaying algebraically as follows [Barouch and McCoy 1971]: x <S1Xs I+n>

-

n

-1/4

Y , <S1YSI+n>

-

n

-9/4

' - M- , 2 , <S1'sI+n>

-

n

-2

.

(11.9)

o h=y=O. This is the XX model in zero field. Its correlation functions have different power-law long-distance asymptotic properties [McCoy 1968]: (11.10)

186

Chapter 11

11-2 Product Ground State of the ID Spin-s XYZ Model with Magnetic Field Here we are concerned with situation (i) as specified above, but instead of (11.1) we consider the more general spin-s XYZ model, N

H XYZ

= -L [JxStSI:l 1=1

+

J;lSI~1

+ Jf/SI:l + hS/] .

(11.11)

The search for a ground state with saturated magnetization leads to the following question: Under what circumstances does H XYZ with J~~;::.o, even N, and periodic boundary conditions have a ground state wave function of the form N

(11.12a)

IG> = ®1i),1> , 1=1 1i),1>

= UI(i))ls>1 =

E[

f

(2s)! ]1I2[cos i)1]s+m sin i)1]s-mlm>I' (11.12b) m=-s (s+m)!(s-m)! 2 2

L

where Uf...i)) describes a unitary transformation representing a rotation of the spin direction at site 1 by an angle i) away from the z-axis in the xz-plane? IG> is a state of maximum ferromagnetic order, M

= = (ssini),

0, scosi)) ,

(11.13)

with no correlated fluctuations (see Fig. 11-2). An alternative formulation of this question, which turns out to be useful for the analysis of the dynamics, is the following: Under what circumstances does the Hamiltonian

HXYZ

= U- 1H xyz U,

N

U = ®UI(i))

(11.14)

1=1

have a ground-state wave function of the simple form N

IG>

= U- 1IG> = ®ls>1 = lii···i>

(11.15)

1=1

with all spins aligned paralled to the z-axis? For given values of the exchange constants J~y'?J;::.o, the XYZ model (11.11) has indeed a ground state of the form (11.12) [Kurmann, Thomas and Muller 1982] with ,

(11.16)

= hN = 2sJ(Jy -Jz)(Jx -Jz)

(11.17)

cosi)

= J(Jy -Jz)/(Jx -Jz)

provided the magnetic field assumes the value h

The ground-state energy is (11.18)

Section 11-3

187

x

.. z

y Figure 11·2: Schematic illustration of the product ground state (11.12) of the spin-s XYZ ferromagnet HXYZ with Jx ~ Jy ~ Jz ~ 0 in a magnetic field of magnitude hN in z-direction [from Muller 1987].

HXYZ

and the transformed Hamiltonian

whose ground-state wave function is

10>=ltt...tt>, reads [Muller and Shrock 1985]:

HXYZ

N

= -

E [Jy<StSI~1 +S/SI~I) 1=1

+

(Jx-Jy+Jz)S/SI~1 + 2s(Jy-Jz)S/

v

(11.19)

x +SIxSz x x 1+1 -s(SI+1 +SI )}] . + (Jx-Jy)(Jy-Jz) {SI zSI+I

The circular line in Fig. 11-1 represents a special case of this situation. 1

11·3 Conditions for the Existence of Linear Spin Waves Once we have established the conditions for the realization of the simple product ground state 10>, we may ask: Under what additional conditions are the ferromagnetic spin-wave states Iq> = S;IO>,

S; = N-

1I2

N

E e-iqISI-

(11.20)

1=1

also eigenstates of HXyz ? Their existence guarantees that the T=O dynamic structure factors SIlIl(q,<.o) as defined in (1.1) have single-mode spectra. The condition is that the second term on the right-hand side of the following equation vanishes:

I For antiferromagnetic coupling (J ::::;O), HXYZ has a spin-flop product ground state if

h has a certain strength.

I1

188

Chapter 11

(11.21)

+ If(J

2: V

x-

J)(J y

y-

(l+e-iq)N-1I2~e-iqIS-S-IG-> z LJ 1 1+1 '

J)

I

where ~w(q) = 2s(Jx -Jlosq)

(11.22)

is the dispersion predicted by linear spin-wave theory. This implies that one of the following three conditions (in addition to h=hN ) must be satisfied [Muller and Shrock 1985]: o Jx J y or J y Jz; arbitrary q and s. q = n; arbitrary Jx' Jy' J z' and s. o s ~ 00; arbitrary Jx' J y' J z' and q. Only under anyone of these additional conditions do the T=O dynamic structure factors of HXYZ have zero linewidth. In all other cases, they have a nontrivial structure in spite of the very simple product ground state. However, Muller and Shrock [1985] proved that, if the product ground state IG> is realized, the following linear relations between the dynamic structure factors SII,iq,ro), Il=x,y,z of H XYZ at T=O are satisfied:

=

o

=

Sxx(q,ro) = Syy(q,ro)cos 2i} + 4n 2s 2o(ro)o(q)sin 2i} 2 Szz(q,ro) = Syy(q,ro)sin 2i} + 4n 2s o(ro)O(q)cos 2i}

(11.23)

with i} from (11.16).

11-4 Resonances with Intrinsic Width The only nontrivial case for which the structure of the functions SIIII(q,ro) have been determined exactly, pertains to the s=1I2 XY model (Jx=I+y, J y=I-Y, Jz=O) at h = h~ (1-r)ll2 [Taylor and Muller 1983]: Szz(q,ro) =n 2 1 -y o(ro)o (q)

lry

+

(11.24) 2 y2 V4(1 -y2)cos (q/2) _(ro-2)2 e[4(1 -y2)cos2(q/2) _(ro-2)2]

1-y2 [ro-2sin 2(q/2)]2 +y2 sin 2q

and is plotted in Fig. 11-3. 2 It is nonzero for values of (q,ro) within the range of the two-fermion spectrum

brhe functions Sxx(q,co) and Syy(q,co), which are not readily evaluated in the ferrnion representation, can then be inferred from the result (11.24) and the relations (11.23).

Section 11-5

100-21 <

189

(11.25)

2h~os(qI2)

and is strikingly different from the B-function predicted by linear spin-wave theory,

s~SW)(q,oo)

=

rr:y B(OO-~w(q»

1ry

(11.26)

.

In fact, the spin-wave dispersion (11.22) differs considerably from the peak frequency of the exact result (11.24) except at q=rt.

TT

3'

$N Ul

N

4 3

2

ot:.-l~==;====;::::==---r---L.J

3

2

o

Figure 11-3: Dynamic structure factor Szz
11-5 Finite and Inrmite Bandwidths For the analysis of generic cases of H XYZ at h=hN (s> 1/2 XY or XYZ with arbitrary s), we employ the recursion method in the Hamiltonian representation [Viswanath, Stolze and Muller 1994]. The recursion algorithm for the orthogonal expansion of the wave function 1'P~(t» = ~(-t)IG> produces (after some intermediate steps) a sequence of continued-fraction coefficients il,(q), il~lI(q),... for the relaxation

190

Chapter 11

function cl)1l(q,z) in the general form (3.25), which is the Laplace transform of the syrnrnetrized correlation function 9kS~(t)S~?/<SI!;~? The dynamic structure factor Slliq,oo) is the Fourier transform of ~(t)S~? and can be obtained directly from the continued fraction as follows (for T=O): Slliq,oo) = 4<S:S~?EXoo)lim9t[crill(q,E -;(0)] .

(11.27)

E-tO

The simple dependence of the product wave function IG> on the size of the system offers the advantage that we can compute a significant number of size-independent coefficients Ll~Il(q).3 In Fig. 11-4 we have plotted the sequences Ll~Il(q=O) for four different applications of the recursion method. Each one of the four qualitatively different patterns displayed by these sequences bears the signature of a characteristic property of the associated S ll(q=O,oo). In panels (a) and (b), the Ll2k_1 and the Ll2k tend to converge to different (finite) values Llio) and Llie), respectively. If Llio) > Llie) as in (a), the implication for the dynamic structure factor is that all its spectral weight is confined to the interval OOmin ~ 00 ~ OOmax with oomin , OOmax determined as in (5.19). If Ll5.,°) < Ll5.,e) as indicated in (b) for k > 5, the dynamic structure factor has a o(oo)-contribution in addition to the continuous part (see Sec. 5-3). These are precisely the properties of the known functions Syy(O,oo) and Su(O,OO) for the s=1/2 XY model as inferred from (11.24) and (11.23). In panels (c) and (d) the two subsequences Ll2k and ~k-l grow roughly linearly with k to infinity but with different average slopes. The linear growth of a Llk-sequence implies that the associated dynamic structure factor has unbounded support and that the spectral weight tapers off by a Gaussian decay law, -exp(-(02), at high frequencies (see Sec. 5-4). If the Ll2k_1 grow more steeply than the Ll2k as in (c), it can be concluded that the dynamic structure factor has a gap at O<<.o 5, it signals the presence of an additional o(oo)-contribution in the dynamic structure factor. Our observations indicate that patterns (c) and (d) are generic for Syy(O,oo) and Su(O,OO), Su(O,OO), respectively, of the s-spin XYZ model. The exception is the s=1/2 XY case, where patterns (a) and (b) obtain. For q i; 0, the Llk-sequences of all three functions SIlIl(q,oo), /l=X,y,z are the same in consequence of (11.23). For q = 0, by contrast, the additional 0(00)contributions in S.o;(O,OO) and Su(O,OO) lead to a pattern reversal from (a) to (b) or from (c) to (d), with two characteristic properties: (i) it leaves the sum of successive pairs of coefficients, Ll~~_I(O) + Ll~(O), invariant; (ii) the factor by which

3Sincethe functions 8~0,oo) and 8z)O,oo) contain (for finite N) a strongly N-dependent contribution at 00=0, which represents the (extensive) order parameter N<M>, the corresponding coefficients Lir(O), IJ.=x,z generated by the recursion method are not automatically N-independent. However, they can be made N-independent by a simple rescaling and then describe 8 (0,00) of the infinite system. For this calculation 1111 we have converted the Lif(O} into frequency moments, multiplied those by <~s'!.rj>/<SZ;S:rj> (for q=0) and then converted them back to continued-fraction coefficients.

Section 11-6

191

the first coefficient changes determines the weight of the o( ro)-contribution: .1f(O)/.1f(O) cos 2t'}, .1fZ(O)/.11'Y(O) sin2t'}.

=

=

(a)

,.....

(b)

4

,.....

0

4

0

'-'"

'-'"

>. >.

N N ~2

~2 <]

<]

s=1/2 Q

,.....

S

>>.~ <]

0

5

k

10

s=1/2 0

15

150

,.....

S

100

N

0

5

10

15

k 10

15

k

150 100

N~

<]

50 5

k 10

50

15

5

Figure 11-4: Continued-fraction coefficients aV(o) and (rescaled) af(O) as obtained from the recursion method for the determination of the T=O dynamic structure factors S~~(q=O,ro), Jl=Y,z of the 10 spin-s XV model (J,?1+y, J 1-y, Jz=O, h=hN) with r=314 and the spin quantum number 5 as specified in each of the four panels [from Viswanath. Stolze, and Muller 1994].

r

11-6 Limitations of Single-Mode Picture The reconstruction of the dynamic structure factor Syy(O,ro) from the known coefficients .1f(O), ..., .1P(O), such as shown for distinct cases in panels (a) and (c) of Fig. 11-4, proceeds according to the well-tested methods discussed in Chapter 9. For case (a) we choose the gap terminator G2 (Sec. 9-6), and for the case (c) the split-Gaussian terminator (Sec. 9-14). In order to demonstrate the degree of accuracy of our method, we first reconstruct the function SY,Y(O,ro) for the s=112 XY model with "(=3/4 from the .1P'(O) of panel (a) and compare It with the exact expression determined in Sec. 11-4. The two results are plotted in the inset to Fig. 11-5. The coefficients .1P'(O) of panel (c) pertain to spin quantum number s= 1. The reconstructed dynamic structure factor Syy
192

Chapter 11

result Syy(O,co) = 21tO(c0-4sy), which is exact in the classical limit s=oo. We conclude that quantum effects are very significant. They produce nontrivial line shapes and move the peak positions by as much as a factor 2.1 relative to the spin-wave prediction. Nevertheless, convergence toward the classical result of the quantum results for increasing s is indicated. 300.0

3.0

3.

I I I I I I I s=ool I I

reconstr. - - exact

2.0

0

..---

200.0

len

~

s=1/2

1.0

3

I

0

s=3/2

~

Icn

>. >.

0.0

100.0

0.0

2.0

(,,)/2s

s=1/2 0.0 0.5

4.0

s=l

(x 10)

I I I I I I I I I I I

I

1.0

1.5

c.>/2s Figure 11-5: Normalized dynamic structure factor Syy:!! SvJ.q, ro)/<srP-r? (for q=O) at T=O of the 10 XV model (Jr1+y, Jy=1-Y, J:t=0' h=~) With "(=314. The three curves in the main plot represent the results for 8=1/2, 1, 3/2 as obtained from the recursion method combined with the continued-fraction analysis outlined in the text. For better visibility, we have expanded the vertical scale by a factor of 10 for the spin-112 curve. The number of continued·fraction coefficients used is K=16. The vertical dashed line represents the classical spin-wave result for the same function. The inset shows again the spin-1/2 result (solid line) on different scales, now in comparison with the exact expression derived from (11.24) via (11.23) (dashed line) [from Viswanath, Stolze, and MOller 1994].

These results expose the limitations of spin-wave theory in quantum spin dynamics very clearly. No matter how favorable the circumstances for the application of a harmonic analysis or single-mode approximation are, the generic structure functions SJliq,co) for quantum spin systems at T=O deviates considerably from the results prodUced on that basis, especially for small spin quantum numbers: The spectral weight is distributed over bands of infinite width (unbounded support) and is dominated by lines with nonzero intrinsic width at frequencies that differ significantly from the spin-wave dispersion. From a different perspective, the

Section 11-7

193

quantum mechanical line broadening may be viewed as evidence for zero-point spin-wave damping. Since these quantum effects cannot be attributed to the strongly fluctuating nature of typical ID phenomena, there is no reason to assume that they are less important in 2D and 3D magnetic systems. All the conclusions reached in this study for the spin-s XYZ ferromagnet (JJl~) can be translated into similar conclusions for the same model with antiferromagnetic coupling (JJl<5.0). That model has a spin-flop ground-state at a particular strength of the magnetic field [Kurmann, Thomas, and Muller 1982]. In that case, the antiferromagnetic order parameter associated with the spin-flop product ground state causes a pattern reversal at q=1t in Sxx(q,ro) and at q=O in Szz(q,ro).

11-7 Spin Dynamics at Tc=O Critical Point: Exact Results for the Transverse Ising Model and the XX Model Consider the time-dependent two-spin correlation functions Sn(t) ==

4<S1~(t)SI:n>'

~ =X,y.z

(11.28)

for the two cases h=r=l and h=r=O of the ID s=1I2 XY model (11.1) at T=O, which represent the transverse Ising (TI) model at the critical field and the XX model in zero field, respectively. These two cases are realizations of situation (ii) as specified in Sec. 11-1. The correlation function Zn(t) for the TI and XX models can be evaluated in terms of a 2-particle Green's function for free lattice fermions [Niemeijer 1967, Katsura, Horiguchi and Suzuki 1970] and expressed in terms of Bessel and Weber functions [Tommet and Huber 1975; Gon~alves and Cruz 1980]: Zn(t)n

=~ 1t2

+ [J2n(2t) +tE2n(2t)f

(11.29)

- [J 2n _1(2t) +tE2n _1(2t)][J 2n +1(2t) +tE2n +1(2t)],

(11.30)

Zn(t)XX = [J n(t) +lE n(t)]2 .

The correlation functions Xn(t) and Yn(t), by contrast, have a much more complicated structure and are expressible in terms of infinite block Toplitz determinants [McCoy, Barouch, and Abraham 1971]. For the two cases of interest here we have the expression [McCoy, Perk, and Shrock 1983]

[Xn(t)n]4 = lim

°0

0_ 1

al

ao

aN

aN-I

a_N a_N + I

N~oo

ao

(11.31a)

194

Chapter 11

fde e _'116 ~tanh(~Sine)

e2in6-2itsin6[1 +tanh(~Sine)]]

1 21t

=_

a I

with

21t

0

-

e -2in9+2itSin9[1-tanh(~sine)]

~=lIkBT and the relations

tanh(~sine)

[Lajzerwoicz and Pfeuty 1975, Capel and Perk 1977]

d2

= -_Xn(t)n 2

Yn(t)n

dt

[x

(11.32)

'

J

(n even)

;(tI2)n

Xit)xx

= Yn(t)xx

( 11.31b)

=

1

(11.33)

X~ (t12)nX ~ (t12)n 2

(n odd)

2

which are valid for arbitrary temperatures. McCoy, Perk, and Shrock [1983] showed that expression (11.31) for T=O is equivalent to the more manageable form

Xn(t)n = XiO)n ex{

t ttr. T

1 2

-2

2t

+

1 C'J

l )] ('t I

'

(11.34)

where Xn(O)n is the known equal-time correlation function and C'Jn(z) satisfies the nonlinear ordinary differential equation (ODE) (11.35) for specific initial conditions. The analysis of this solution has revealed a degree of complexity in the structure of time-dependent correlation functions not previously encountered in exactly solvable quantum spin systems. 11-8 Long-Time Asymptotic Expansions The enormous difference in structural complexity between the functions Zn(t) as given by the closed-form results (11.29-30) and the functions Xn(t) and Yn(t) resulting from expression (11.34) is strikingly manifest in their long-time asymptotic expansions (LTAEs). For a comparison of LTAE structures we report here results for n=O (autocorrelations) [McCoy, Perk, and Shrock 1983; Muller and Shrock 1984]. The LTAEs determine the singularity structure of the spectral densities, which, in turn yields valuable insight into the structure of the dynamically relevant excitation spectra. LTAEs for the explicit expressions (11.29) and (11.30) can be inferred to all· orders from standard mathematical references: 2

o n -~2 - ~ L.J

Z (t)

1t

m~

T(TI) Z,m

(l1.36a)

Section 11-8

•

(z)

T ::;:) = 1tml2-2e-2Imt(_2it)--am z

with

L a~z,m\-2it)-n 00

195

(I 1.36b)

n~

r4z) = 2, afz) = 312, c4z) = 2, and rational coefficients a~z,m); 2

Zo(t)XX -

L

Tz~X)

(l1.37a)

m~

(l1.37b) with ~bz) = 2, ~fz) = 3/2, ~iz) = 1, and rational coefficients b~z,m). This is to be contrasted with the much more complex LTAEs that have been derived from the solution of (11.35): 00

X (t)

o

TI

- A(it)-1I4~ T(TI) LJ x,m '

(l1.38a)

m~

(xl

T ::;:) = (21t)-m12e -2imt( -2it) -lXm x

L a~x,m)( -2it)-n 00

(l1.38b)

n~

with a~x,m) = m212, rational coefficients an(x,m), and the overall multiplicative constant

A = 21112e3~/(-1)

= 0.64500248... ;

(11.39)

00

Xo(t) XX - A 2V£. '2(it)-1I2~ T(XX) LJ x,m '

(11.40a)

m~

(xl

00

T(XX) = (21t)-m12e-imt(-itff3m~ b(x,m)(_it)-n x,m

L...J n

(l1.40b)

n~

with ~~x) = m2/4 for even m, ~~x) = (m 2+1)/4 for odd m, and rational coefficients

b~x,m).

A difference in complexity of similar magnitude has been noted in Secs. 10-4 and 10-5 for the LTAEs of the T=oo spin autocorrelation functions <Sj(t)S}> and <Si(t)S/> of the semi-infinite XX model. In <Sj(t)S}>, the number of m-terms was invariably small, similar to (11.37) at T=O. In <S/(t)S/>, on the other hand, the number of m-terms was found to increase with the distance of SI from the boundary of the semi-infinite chain. This is in line with expression (11.40), which pertains to the bulk spin and has an infinite number of m-terms. Each m-term contains an oscillatory function, exp( -immt), which produces a singularity at frequency mm in the spectral density or structure function.

196

Chapter 11

11-9 Structure Functions and their Singularities The structure functions

Jdteirot<S!(t)S!>

+00

SIlIl(ro) ==

(11.41)

associated with the T=O spin autocorrelation functions of the TI and XX models were studied in great detail by Muller and Shrock [1984]. The Fourier transform of expressions (11.29-30) for Zo(t)TI and Zo(t)xx can be evaluated in terms of elliptic integrals as follows: SZZ(ro>n

= ~O(ro) 1t

2

+

~[K(k) -E(k)]EXro)EX4-ro)

+

1t

V(ro)EXro)EX2-ro) ,

V(ro) .: -(F(Yl,k) - F(Y2,k) - E(Yl,k) + E(Y2,k)

1t + tanu[cos 2u-(<.tY4)2]1I2 - cotu[sin 2u-(<.tY4)2]1I2} ,

where Yl

(11.42)

= arcsin[sinu/k], Y2 = arcsin[cosu/k], k = [1_(0014)2]112, sin(2u) = 0012;

SZZ(ro)xx =

~F[arcsin({..!.[1-(1-ot)1I2]}1I2Ik),[1-(<.tY2)2]1I2]EXro)EX1-ro) 1t

+

2

(11.43)

~K{[l _(<.tY2)2]1I2) EXro-1)EX2-ro) 1t

with k=[l-(00I2)2]l/2. These two functions are plotted in Figs. 11-6 and 11-7. They both have compact support and three singularities corresponding to the three mterms in the LTAEs (11.36) and (11.37): SZZ(ro)¥;) -

SZZ(ro)~

~EXro) 21t

, -2.(2-ro)1I2EX2-ro) , ..!.(4-ro)EX4-ro) , (11.44) 1t 4

- 2ro EXro) , -25/21t-1(1-ro)1I2EX1-ro) , EX2-ro). 1t

(11.45)

No closed-form expressions are known for the structure functions SU(ro) and r(ro). The curves for the TI and XX cases shown in Figs. 11-6 and 11-7 are the result of high-precision numerical calculation combined with an exact analysis of the singularity structure. The function SU(ro)TI' for example, was calculated by Fourier transforming the function Xo(t)TI as determined by integrating (11.35) up to t=4O, where the solution (11.34) matches the explicitly known terms of the LTAE (11.38) to within an error of 10-6 . The characteristic properties of these functions, which are likely to reflect many generic features of quantum many-body dynamics may be summarized as follows [Muller and Shrock 1984]: The structure functions SU(ro>n, r(ro)TI ro2SU(ro)TI and SU(ro)xx r(ro)xx have unbounded support, in contrast to the functions SZZ(ro>n and SZZ(ro)xx, which have compact support.

o

=

=

Section 11-9

I

I I

1.0

I

I

/

I

I

I

I

/

/

I /

I I I I

I I I

,, , , , . .'. \

~~=xx

--

yy - - -

zz -.-

\

\

\ \

,,~

~.,,\

\

,"." ".

.

"" .....' ....... ."......

01£.._ _---1.

197

,....... ..... ~

---I..._ _-=:I::::==-~:t:::::=_._l

o

2

w

3

4

11-6: Frequency-dependent spin autocorrelation functions (structure functions) ~ ~=x.y.zfor the 1D 5=1/2 transverse Ising model at the critical field and at T=O [from MOller and Shrock 1984]. Fi~ure

«(Oh,.

o

The function SXX(ro)n has an infinite set of singularities at frequencies ro=2m, m=0,1,2,...., one for each m-tenn in the LTAE (11.38):

SXX(ro)~)

-

~mr(3/4 -m 2/2)lro-2mlv~) 2

x {.!.[1-(-1)m]EX2m-ro) + 2-

2

o

1I2

(11.46) EXro-2m)}

with exponents v~n) = m212-3/4 and known amplitudes Am' These power law singularities are one-sided for even m and two-sided for odd m. Only the first two singularities (m=O,I) are divergent: _ro- 3/48(ro) at ro"O and -I
198

Chapter 11

2.0

e-~= XX,

I

...........

......

1.0

0.5

1.0

...... ...... ......

yy

zz

-- -... --

1.5

2.0

w Fi!ure 11-7: Frequency-dependent spin autocorrelation functions (structure functions) fI. (ro)xx' ~=x,y,z for the 10 8=1/2 XX model at zero field and at T=O [from MOller and

Shrock 1984].

S xx(oo)(m) _ xx

o

~Bmr(1I2 -m 2/4)(ro-m)V~X)EXro-m)

1-~Bm{[(m2-1)/4]!}-llro-mIVm =

(even m) (11.47)

(XX)

Inlro-ml

(odd m)

=

with exponents v~XX) (m 2-2)14 for even m, v~XX) (m2-1)14 for odd m, and known amplitudes Bm' The singularities visible in Fig. 11-7 are the two divergences -m- I12 8(00) at 00"0, -lnlro-1I at 00"1, and the cusp -(ro-2) 1I28(ro-2) at 00..2. The uniform spacings between singularities in SU(oo>n and SU(OO)xx can be understood in terms of the dynamically relevant excitation spectrum of these functions, which (in the fermion representation) consists of sets of mparticle excitations with energies m

m

(11.48)

for arbitrarily large m, where oor) =

2cos(k/2), oo~XX) = cosk

(11.49)

Section 11-10

o

are the one-particle energies for the two models, respectively. The m-particle densities of states have van Hove singularities at exactly the frequencies c.o=2m (TI) or c.o=m (XX), m=O,1,2,... This illustrates the fact that the functions S=(OO)TI and S=(OO)xx couple to the m-particle spectrum for arbitrarily large m in contrast to the functions SU(OO)TI and SU(OO)xx' which couple only to the 2-particle excitations, characterized by just 3 van Hove singularities [McCoy, Barouch, and Abraham 1971; Muller and Shrock 1984].4 On the intervals between the singularities, the structure functions of both the TI and XX models are convex functions of 00: 2 -d- SI!I!(00)

dor

o

o

199

~>

0 ,Il-x,y,z. -

(11.50)

They have no smooth maxima; all maxima occur at points of nonanalyticity. The Luttinger model represents a continuum version of the ID s=112 XXZ model, which contains the isotropic XX model as a special case, namely that of free Fermi fields. The calculation of S=(OO)xx for the Luttinger model [Luther and Peschel 1975] yields the correct exponents for the leading and next leading singularities at c.o=O. Understandably, the continuum analysis cannot account for the singularities at 00>0, which are intrinsic features of the discrete system. 5 The structure functions S=(OO)xx and sYY(OO)xx can be expressed, like SU(OO)xx, in terms of 2-particle Green's functions, but (unlike SZZ) for interacting fermions. This representation is obtained by means of a switch of coordinates in spin space preceding the Jordan-Wigner transformation. The coupling of the functions s= and sYY to the m-particle excitations for arbitrarily large m can then be understood as being caused by the infinite hierarchy of m-particle Green's functions generated in the equation of motion for the 2-particle Green's function by the interaction term of the fermion Harniltonian. 6

11-10 Structure Functions Reconstructed by Continued-Fraction Analysis In Sec. 9-9 we have already reconstructed the structure function SU(OO)xx or, more precisely, the associated spectral density from the first 15 continued-fraction

~he structure of the 2-particle spectrum and its role in ,szz(ro) will be further discussed in Sec. 11-11. 5The Luttinger model will be further discussed in Sec. 11-19 in the context of the 10 5=1/2 XXZ model.

~he relevant 2-particle Green's function has already been determined in Sec. 6-3 for the noninteracting case and will be further analyzed for a case with interaction (Hartree-Fock approximation), in the context of the 10 5=1/2 XXZ model (Sec. 11-20).

200

Chapter 11

coefficients and a tenninator with matching bandwidth and infrared singularity. In Table 11-1 we have listed the !::J.k's for the three structure functions SU(ro>n, S>'Y(ro)TI' and SU(ro)xx which we detennined from an exact moment expansion for a system of infinite size. A smaller number of !::J./s is readily reproduced with reasonable accuracy by means of the recursion method in the Hamiltonian representation. 7 ,8 All three !::J.k-sequences are consistent with linear growth (A.=l). The function SU(ro)xx has been reconstructed in Sec. 9-11 from 13 !::J.k's as a first application of the Gaussian terminator. The same analysis carried out for the structure functions SU(ro)TI and S>'Y(ro)TI yields the short-dashed lines in Figs. 11-8(a) and 11-8(b), respectively, which are to be compared with the exact results, here reproduced as long-dashed lines. k

!::J.(AX,TI)

!::J.(yy,TI)

!::J.(AX,XX)

1

1.00000oo

4.0807604

0.7026425

2

3.0807593

1.8726733

1.4222254

3

2.4805557

6.9274231

1.7805298

4

6.3195583

3.9917053

2.5736554

5

4.3756690

7.5153267

2.7468790

6

7.1313039

7.7418177

3.9361211

7

8.1587912

9.3998502

3.6102501

8

8.9829565

8.7562723

5.1165807

9

9.1618756

11.8240840

4.7705698

10

11.4202618

11.1498487

5.8053100

11

11.5490359

14.7374968

6.1941504

12

14.2726272

12.1189133

6.4730222

13

12.8601515

k

k

k

7.6809316

Table 11·1: Continued-fraction coefficients pertaining to the spin autocorrelation functions (structure functions) ,soc(ro)n. ,sYY(roh" and ,soc(ro)xx = ,sYY(ro)xx at T=O as inferred from a moment expansion, which is based on a short-time expansion of the function O'o(~ of (11.35). This analysis does not require the evaluation of ground-state expectation values.

7At T=O and for ro>O the structure function and the (normalized) spectral density differ only by a constant factor: ~(ro) = 2<~~>~(ro)e(ro) with <~~>=1/4 for s=1/2. 8For the function ,soc(ro)xx. the recurrence relations (4.5) applied to the N=18 ground state yield (after conversion) 8 Ak'S that are size-independent to within 1 part in 106•

Section 11-10

4.0

12.0 9.0

3.0

..Ill
e::

--

•

6.0

••

3.0

.........

3 >< >
•

15.0

(a)

201

2.0

0.0

0

2

4

• 6

8

10

12

k

1.0

o.o-j£----..-------,r----...".---~::=.-..=~

0.0

1.0

2.0

3.0

4.0

5.0

Figure 11-8: Continued-fraction reconstruction of the structure functions (a) SCX(roh, and (b) ,sw(roh, for the 1D 5=1/2 transverse Ising model at the critical field and at T=O. The longdashed lines represent the exact results from Fig. 11-6. The short-dashed curves result from reconstructions with the Gaussian terminator and the solid lines from reconstructions with the a-terminator as described in the text. The insets show the continued-fraction coefficients from Table 11-1 that have been used for the reconstructions. Also shown are the linear regression lines, the slope of which determines the parameter roo of either terminator used.

202

Chapter 11

In evaluating the degree of success in this application of the reconstruction procedure, we must keep in mind that the Gaussian terminator is completely unbiased with respect to the spectral-weight distribution anywhere except at high frequencies. The function SU(oo)TI has a strong divergence at 00=0 and a weaker divergence at 00=2 as its only discernible structures. They are represented in the reconstructed function by a tall peak at ro=O and a much lower peak at 00-2. In the function sY>'(00)n, the divergence at ro=O is suppressed by a factor 002 , while the divergence at 00=2 is enhanced by a factor of 4 in amplitude, all with respect to the function SU(oo)TI' Both changes are reasonably well represented by the reconstructed structure function: the peak at ro=O has disappeared and the one at 00-2 has grown much stronger. These results are further fine-tuned if we replace the Gaussian terminator by the unbounded a-terminator (Sec. 9-13), which produces a power-law infrared singularity in the reconstructed structure function. The patterns of the Llk-sequences shown in the insets to Fig. 11-8 suggest a negative exponent for SU(oo>n and a positive exponent for SYY(oo)TI' We might as well use the exact exponents a=-3/4 and a=5/4 (Sec. 11-9) in the unbounded a-terminator. The reconstruction on that basis produces the solid lines shown in Fig. 11-8. They bring a substantial improvement over the Gaussian reconstruction. In both cases artificial wiggles have now largely disappeared. The second peak of SU(oo)TI has moved closer to the correct position, and the curve for SYY(oo>n starts out at zero as it should.

11-11 Dynamic Structure Factors Szz(q,oo)TI and Szz(q,oo)xx: Two-Particle Spectrum In the fermion representation of the TI and XX models, the ground state is characterized by a half-filled band of noninteracting lattice fermions with oneparticle spectra (11.49). The 2-particle excitations then form a continuum consisting of two partly overlapping sheets with energies E2(k,q)TI

= 2Icos(k/2-q/4)1

+ 2Icos(k/2+q/4)1',

E2(k,q)xx = Icos(k-ql2)1 + Icos(k+q/2)1.

(11.51a) (11.51b)

as functions of wave number q and parameter k (0~1t). The two sheets have upper boundaries EU(q)TI

= 41 sin(q/4) I ,

Eu(q)xx

= 21 sin(ql2) I ,

(11.52)

EU(q)TI

= 41 cos(q/4) I ,

Eu(q)xx

= 2Icos(q/2) I ,

(11.53)

respectively, and a common lower boundary, EL(q)TI = 2Isin(q/2)I , EL(q)xx = Isinql , as shown on the right of Fig. 11-9.

(11.54)

Section 11-11

203

w

-+ 4[2]

,...-

2'77' [4'77']

o

'77' [2'77']

0[0]

[0]

L.-..

~

-.--

0 [0]

D2 (w) Figure 11-9: Two-particle excitation spectrum of the TI model and XX model. It consist of two partly overlapping continua. The curve on the left represents the density of the twoparticle states. Labels withlWithout brackets pertain to the XXlTI model [from Muller and Shrock 1984].

The dynamic structure factors Sliq,ro) for the two models, derived via 2particle Green's functions for free lattice fermions (Sec. 6-3), have all their spectral weight confined to that region in (q,ro)-space: 9 Szz(q,ro)TI

= 4o(q)o(ro)

V16cos2(q/4)-cJ

+ ....:....-_~-.:..._ _ 8(4Icos(q/4)I-ro)8(ro-2Isin(q/2)1)

8cos 2(q/4) 2

+ Vt6sin (q/4)

-cJ 8(4Isin(q/4)I-ro)8(ro-2Isin(q/2)1)

8sin 2(q/4)

-;::===2===== 8(21 sin(q/2) I-ro) 8(ro -I sinql) . V4sin 2(q/2)

-cJ

(11.55)

,

(11.56)

9Additional selection rules suppress contributions to (11.56) from one of the two sheets of excitations. The function Szz(q,ro)xx was used in Sec. 9-6 to test the gap terminator G2.

204

Chapter 11

Consider the density of 2-particle excitations,

fo f

21t

D 2(ro) ==

1t

dq dkO (ro-E 2(k,q» ,

(11.57)

0

as shown on the left of Fig. 11-9. It has van Hove singularities at 0>=0,2,4 for the

TI model and at 0>=0,1,2 for the XX model. The singularities (11.44) of the structure function SU(ro)TI are the combined effect of the van Hove singularities in D 2(ro)TI and the singular behavior of the matrix elements at the boundaries of the 2-particle continuum. In the XX case, these matrix elements are independent of k,q, and the structure function (11.43) differs from the 2-particle density of states (11.57) only by a constant factor: SZZ(ro)xx

= _1-D2(ro)xx 41t

.

(11.58)

The singularities (11.47) in SU(ro)TI at 0>=0,2,4 and those in SU(ro)xx at 0>=0,1,2 as reported in Sec. 11-9 are likely to be at least in part an effect of 2-particle excitations. The nonanalyticities in these structure functions at higher integer-valued frequencies must then be attributable to the effects of matrix elements and densities of m-particle states (11.48) with m>2.

11-12 Dynamic Structure Factor Sxiq,ro)xx: Continued-Fraction Analysis The fact remains that the lower boundary EL(q)XX of the 2-particle spectrum coincides with the threshold of the entire spectrum which is dynamically relevant for the function Sxx(q,ro)xx. That function thus exhibits the same q-dependent gap as the exact expression (11.56) of Szz(q,ro)xx, but, unlike the former, the latter has no upper boundary in its spectral-weight distribution. In principle, the dynamic structure factor Sxx(q,ro)xx can be calculated on the same level of rigor as the structure function SU(ro)xx was determined by Muller and Shrock [1984] (Sec. 11-9), but such a calculation is exceedingly cumbersome, and nobody has carried it out as of yet. Under these circumstances, the recursion method presents itself as a welcome alternative. In Sec. 11-10 we have already demonstrated that the continued-fraction analysis captures the dominant structures of the structure function SU(ro)xx. Results of comparable accuracy are expected for the dynamic structure factor Sxx(q,Ol)XX' even though the reconstruction is based, in this case, on a smaller number of continued-fraction coefficients li.k, extracted from the finite-size ground-state wave function by means of the recursion method in the Hamiltonian representation. In Fig. 11-10 we show such results for two q-values: (a) q=1t and (b) q=71t18. In both cases, the six li.k's shown in the insets have been used to determine the parameters 000, n of the split-Gaussian model spectral density (9.61) with A=O (no

0.0+--4-.---~~==:::::;:===:::::::=-;c==--~

0.0

1.0

2.0

Figure 11-10: Dynamic structure factor Sxxf,q,ro) for fixed wave numbers (a) q=1t and (b) q=7rrJ8 of the 10 s=1/2 XX model at T=O. The results were derived from the continuedfraction (9.3) (via (11.27) with £=0.001) terminated at level K=6 with the split Gaussian terminator as described in the text. The insets show the nearly size-independent continuedfraction coefficients Al""'~ for system with N=16 spins (full circles). The open circles represent the AkS of the model relaxation function. The circles on the ro-axis mark the upper and lower boundaries of the 2-particle spectrum.

206

Chapter 11

central peak).lO For case (a) the best fit yields zero gap (0=0) and for case (b) nonzero gap. The model Ak's (shown as open circles) thus follow, for case (a), a single line with linear growth and, for case (b), two subsequences that grow with different slopes. The reconstructed function Sa(1t,m)xx is strongly peaked at c.o=O. Unlike the function sa(m)xx (Fig. 9-11), it shows no structure at c.o= 1. The 2particle spectrum (Fig. 11-9) indeed indicates that the enhancements of spectral weight at 00.. 1 in SXX(m)xx comes from excitations at q .. rrJ2. Most of the spectral weight in Sxx(1t,m)xx is concentrated within the range of the 2-particle spectrum (~=:;;2) but not entirely. The alternating pattern of the Ak-sequence shown in the inset to Fig. 11-10(a) indicates the presence of a strong infrared divergence in Sxx(1t,m)xx' That feature is conspicuously present in the reconstructed dynamic structure factor shown in the main plot. 11 An estimate of the infrared singularity exponent from the coefficients AI"'" As yields the value -l.l±O.4, roughly consistent with the exact value -3/2 (Sec. 11-22). This singularity is much stronger than the one, _00- 112, previously estimated (Sec. 9-12) and rigorously identified (Sec. 11-9) in the structure function SXX(m)xx' The continued-fraction analysis thus leads to the correct conclusion that most of the spectral weight at small m in SXX(m)xx comes from q=1t and not from q=O, where low-lying excitations exist as well. This conclusion is also corroborated by our observation that the Ak-sequence for Sxx(O,m)xx suggests zero gap with a positive infrared exponent. The spectral-weight in Sxx(O,m)xx must, in fact, go to zero for 00---70 in order to be consistent with the nondivergent direct susceptibility Xxx(q=O) [Muller and Shrock 1984]. The curve shown in Fig. 11-10(b) for the reconstructed Sxx(7rrJ8,m)xx tells us that again most of the spectral weight, but not all of it, is concentrated within the range of the 2-particle spectrum (0.38=:;;00=:;;1.96). The size of the dynamically relevant gap obtained by the continued-fraction analysis is in very good agreement with the lowest branch £L(q)XX of excited states. The observed line shape strongly suggests the presence of a divergence at the spectral. threshold, which is in agreement with field-theoretic predictions [Luther and Peschel 1975]. A more complete discussion of the q-dependence of the function Sxx(q,oo)xx will follow later in the context of the ID s=1I2 XXZ model (Sec. 11-24).

11-13 iD s=1I2 XYZ Model: Ground State and Excitation Spectrum The spin-1I2 XYZ chain, specified by the Hamiltonian N

HXYZ = -:L {JxStSI:1 1=1

+

J;/SI~I

+

J?/SI~d

'

(11.59)

IOrhe minimization criterion was the same as that used in Sec. 10-13 for a different model spectral density. lIThe Gaussian terminator, which has been used for that reconstruction, produces a tall peak at co=O, but no singUlarity.

Section 11-13

207

is an extraordinary model with numerous connections in solid state physics, statistical mechanics and field theory. Its ground-state properties are related to the thermodynamic properties of important classical 20 models such as the Ising model, the KDP and F six-vertex models, the Baxter eight-vertex model, the q-state Potts model, the planar spin model and others. They have been used to describe a large variety of quasi-2D phenomena including ferro- and antiferromagnetism, ferro- and antiferroelectricity, adsorbed monolayers etc. In the continuum limit, the XYZ chain is related to various field theories in (I +1) space-time dimensions such as the Thirring model, the quantum sine-Gordon model, and the Luttinger model. It is also related to the 10 Hubbard model for itinerant electrons in the limit of strong on-site repulsion. Last but not least, the XYZ chain is directly relevant in the context of experimental studies of a large and rapidly growing number of real quasi-ID magnetic insulator compounds. Without loss of generality we can restrict the values of the exchange constants Jfl to the domain

Jx :::;; J y

:::;;

-IJzl .

(11.60)

All physically distinct cases are represented in the interior or on the edges of the triangle shown in Fig. 11-11. The two most prominent special cases are the XY model (Jz=O), whose T=O dynamics has been the main subject of Chapter 11 up to this point, and the XXZ model (Jx = Jy )' which will be the focus of our attention for much of the remainder.

HEISENBERG FERROMAGNET

HEISENBERG ANTIFERROMAGNET

-I

ISING MODEL

Figure 11-11: Two-dimensional parameter space of the 10 8=1/2 XYZ model (11.59). The expressions for the ground-state energy and the excitation spectrum of HXYZ model as given in the text are for parameter values Jx :s;; Jy' :s;; -IJzl corresponding to points in the interior or on the edges of the triangle shown. The lines (i)-(iii) around the edge represent the XXZ model and the center line (iv) the XV model.

208

Chapter 11

The ground-state energy per site, EG = EdN, of H XYZ in the limit N-7 00 was derived by Baxter [1972] from the largest eigenvalue of the eight-vertex transfer operator, Le from his own solution of that 2D model. The result is expressible in the form of a series,

EG = Jx [.!.+_1t_sn(2~,I)E 4

=

Kt (I)

n=1

2

Sinh [(:-A)n]tanh(A,n)], smh(2'tn)

(11.61)

=

where A 1ttjK.'(I), 't 1tK(I)/K.'(I), and the parameters ~ and I are determined from Baxter's parametrization of the XYZ model, covering all cases of the triangle shown in Fig. 11-11: Jx : J y

:

Jz

=1

: dn(2~,1) : cn(2~,1) .

(11.62)

The relation between the two models was used by Johnson, Krinsky, and McCoy [1973] for the determination of a set of low-lying excited XYZ excitations from a band of next-largest eigenvalues of the eight-vertex transfer matrix. The excitation spectrum thus found consists, in general, of a continuum of free states and a set of branches (one-parameter continua) of bound states. The free-state spectrum for wave numbers -1t~1t and parameter values ~1t reads

£(k,q) • A,[

(11.63)

with amplitude

Ac

==

.!.IJxISn(2~,I)Kt (m)/K t (I) 2

,

(11.64)

and where m is determined from 1tK(m)/K.'(m) = A. The bound states make their appearance in the spectrum of low-lying excitations only if the parameter /J = 1ttjK(1) exceeds the value 1tI2. This is the case for JjIJ) > 0 in Fig. 11-11, Le. for the XYZ ferromagnet. The number of bound-state branches is then equal to the integer part of JJ!(1t-/J), denoted [JJ!(1t-/J)]. The dispersion of the nth branch is

r

2

2]

2

En(q) == An~ sin 2(q/2) +ancos (q/2) sin 2(q/2) +bncos2(q/2)

for n=1,2,3,...,[JJ!(1t-/J)] and wave numbers parameters an' b n are given by

-1t~1t.

(11.65)

The amplitude An and the

Section 11-13

209

1.5

1.0

0.5

OL...-

....I-

o

-..J

TT

TT/2 q

Figure 11-12: Spectrum of Iow-lying excitations of the 10 5=1/2 XYZ model. For this generic case shown (Jx = -1, Jy = -0.732, Jz = 0.666), the spectrum consists of a two-parameter continuum (11.63) and four branches of bound states (11.65), all specified by parameters ~=1.873, 1=0.913, A.=3.580, t=4.476, p.=2.513.

A =IJIK1(m) n x K I (I)

sn(2K(I)-2~,1)

y)' sn(n,m

a =msn(y m) n

n"

b =sn(y m) n

n"

(11.66)

where Yn = nt..'K'(m)/Tt and t..' = t-t... A generic case of this excitation spectrum is shown in Fig. 11-12. For small wave numbers, the bound states are generally located below the continuum. The energy differences between the bound states and the lower edge of the continuum for q--70 can be interpreted in terms of binding energies of different types of (bosonic or fermionic) fundamental particles of quantum field theories related to the XYZ model (quantum sine-Gordon model, massive Thirring model). 12 The spectrum at larger energies and wave numbers are, however, intrinsic properties of the discrete system. In Table 11-2 we present simplified expressions for the relations

12-rhese relationships have been described, for example, in the work of Coleman [1975], Luther [1976], Bergknoff and Thacker [1979], Dashen, Hasslacher, and Neuveu [1975].

210

Chapter 11

between Baxter's parameters and the exchange constants Jx ' Jy' Jz on the lines (i) (iv) of Fig. 11-11. (i)

J x = -J , J y = -aJ , J z = aJ ;

=1,

1

O~ex~ 1

~ ~oo ; ",~oo , 't ~oo , ~

= 1t

.!.n""

sech",' = ex ' n Y = 2 Ac = 0 , m = 1

tanh",' An = J _ _ , an = bn = tanhY n tanhY n (ii)

Jx =Jy = -J ,Jz =!iJ; -l~A~l 1=0 , cos2~ = -A ; '" = 0 , 't = 0 , ~ = 2~ _ n1t(1t -~) Yn 2~ A c

= J1t sin~

An =

J1tsin~

.

~lOYn

(Hi)

=0

. , an = 0 , b n = SlOY n (for O
Jx = -J , J y = J z = aJ ; O~ex~ 1 1 = 1 , sech2~ = ex ; '" = 2~ , 't ~oo A c

(iv)

m

2~'

= J~K' (m) 1t

, m from 1tK(m) K/(m)

~ = 0

='"

Jy =-J(1 -y) , J z = 0 ; O~"( ~ 1 2 r:::"( 1 1 1t J(,("() 1t 1 = _V_T , ~ = -K(l) , '" = _'t = _ _ , ~ = _ K'("() 2 1+"( 2 2 Ac = J , m = "( Jx

= -J(l +"()

,

,

Table 11·2: The values of Baxter's parameters I, ~, )." t, 11 and expressions for the amplitudes Ac' An and the parameters rn, an' bn for the special cases (i)-(iv) of the 108=1/2 XYZ model (11.59) as indicated in Fig. 11-11.

Section 11-14

211

11-14 ID s=I/2 XXZ Model: Criticality and Long-Range Order For parameter values on the lines (i)-(iii) in Fig. 11-11, the XYZ model (11.59) has uniaxial symmetry (XXZ model). It has become standard to express the XXZ Hamiltonian in terms of a single anisotropy parameter ~: N

H

= -JL {StSI:1 1=1

+

S/S/~1 + M/S /: 1} , -oo<~
(11.67)

The ground-state energy per site, EG(~)' in the limit N~oo as determined exactly by means of the Bethe ansatz 13 is plotted in Fig. 11-13.

-

ORDER PARAMETER Mz

----- EXCITATION GAP G/4J 1/2

...... ......

.... -.. -

-2.0

... ...

.......

-1.0

....

.. . .

...1.0

5/3 2.0

6

GROUND -STATE ENERGY

-0.5

Figure 11-13: Ground-state energy per spin EG, excitation gap G(~). and order parameter Mi~) of the 1D 5=1/2 XXZ model (11.67). All quantities are singular at the symmetry points ~=±1. In the regime -1~~<1, there is no LRO, and the spectrum is gapless. For A>1, the order parameter is the magnetization <~> and for ~<-1 the staggered magnetization (-1)/<~>. In the regime ~>1, the energy gap switches from being determined by continuum states (Gd to being determined by bound states (GB) at ~=5/3.

13Bethe [1931] proposed a representation for the eigenfunctions of the 5=1/2 Heisenberg model (1~1=1) which enabled him to classify all 2 N eigenstates in terms of a discrete set of quantum numbers, and to give a prescription for calculating the eigenfunctions, energies, and wave numbers in terms of the solutions of a set of coupled nonlinear equations. Bethe's ansatz was later generalized by Orbach [1958]. Des Cloizeaux and Gaudin [1966], and Yang and Yang [1966] to the XXZ model in its entire parameter range.

212

Chapter 11

The function EG(6.) is singular at 6. = ±1, due to symmetry changes in the ground state. They represent T=O phase transitions driven by quantum fluctuations. Also shown in Fig. 11-13 is the amount of magnetic LRO as described by the order parameter

-

M z = lim 1<S/S/:n>1

2

n--+ oo

(11.68)

•

For A> 1 the ground state is a ferromagnetic doublet with all spins aligned parallel to the z-axis. The order parameter is the magnetization and assumes its maximum value: Mz = 112. At 6.=1 the ground state changes into a singlet. Here Mz drops to zero discontinuously. Throughout the regime, -1~<1, the XXZ model has a critical ground state. At 6.=-1, the ground state changes back into a doublet, and antiferromagnetic LRO in z-direction develops gradually. Here the order parameter (staggered magnetization) was first determined by Baxter [1973] in the context of an exact calculation for the F model, a six-vertex model related to the XXZ chain: 00

Mz = V21tlxLexp[-(n-1I2)21t212x],

coshx

= -6..

(11.69)

n=l

Mz goes to zero exponentially in the Heisenberg limit (6.=-1) and, unlike in the ferromagnetic case, reaches saturation only asymptotically for 6.~-oo (Ising limit).

11·15 Excitation Spedrwn of the ID s=1I2 XXZ Ferromagnet: Spin Waves and Bound States

In each subspace with quantum number Sf = l:tS] the eigenfunctions of H xxz can be constructed as linear combinations of simple product states with reversed spins at r = NI2-Sf lattice sites. The only state with r=O, is one component of the twofold degenerate ground state in the regime A> 1; it has energy EO) = EG = -NJA/4. The N eigenstates with r=1 are simple traveling-wave states with wave numbers q = 21tmIN,m = 0,1 ,.. .N-1 and energy ofl)(q) = J(6. -cosq)

(11.70)

relative to that of the ground state. These are ferromagnetic magnon states [Bloch 1930] corresponding to small-amplitude normal-mode solutions of the classical XXZ model. From a naive linear-spin-wave point of view one might hope to generate traveling-wave solutions with 1'> 1 as simple superpositions of one-magnon states. For r=2 that would produce N(N+1)/2 two-magnon states. However, apart from representing an overcomp1ete set, which arises from the s= 1I2-specific property SiSili>/ = 0, these states are non-orthogonal and subject to scattering. The magnon interaction gives rise to qUalitively new features in the r=2 spectrum [Bethe 1931, Orbach 1959, Thompson 1972]. The N(N-l)/2 eigenstates with r=2 belong to one of two distinct classes: to the continuum of 2-magnon scattering states with excitation energies,

Section 11-15

cJ;)(k,q)

= 2J(ti-coskcosi) 2

, 0 ~ k ~

1t ,

213

(11.71)

or to the discrete branch of 2-magnon bound states with dispersion (11.72)

(cl

4.0

.,

.,

'3- 2.0

'3- 2.0

Ol..---------.------.J Tf TfI2 o q

TfI2

Tf

q

Figure 11-14: Spectrum of low-lying excitations of the 10 5=1/2 XXZ model (11.67) in the regime &1. The two plots represent (a) the uniaxial ferromagnet ~=1.5 and (b) the (isotropic) Heisenberg ferromagnet ~=1. The spectrum shown consists of the branch (J)~1)(q) of ferromagnetic magnons, the continuum of 2-magnon scattering states, and several branches of magnon bound states £n(q). For ~=1.5, the spin-wave branch and the boundstate branches with n ~ 3 lie completely outside the continuum, whereas the bound-state branches with n ~ 4 penetrate the continuum at small wave numbers. For ~=1 , on the other hand, the spin-wave branch is completely inside the continuum, whereas all bound-state branches are outside.

How do these one- and two-magnon states relate to the excitation spectrum of the XYZ model found by Johnson, Krinsky, and McCoy [1973]? For H xxz with ti:>1, i.e. for H xyz on line (i) in the parameter space depicted in Fig. 11-11, the continuum states (11.63) are absent, and the discrete branches of bound states (11.65) become infinite in number. Their excitation energies can be rewritten in the form

214

Chapter 11

E (q) n

=

J~tanhA/

sinh(nA/)

/ / -1 [COSh(nA ) -cosq], COShA =~

(11.73)

Thus the one-magnon states (11.70) can be identified with the first branch (n=l) of (11.73) and the two-magnon bound states (11.72) with the branch n=2. Correspondingly, we can interpret the remaining branches of (11.73) as n-magnon bound states. This excitation spectrum is plotted in Fig. 11-14(a). There is an energy gap between the ground state and the lowest excitations. Note that the gap GB of the bound states (11.72) has a different dependence on the anisotropy parameter ~ than the gap Gs of the scattering states (11.71): (11.74) Both gaps are plotted in Fig. 11-13. The kink at ~=5/3 of the true gap gives rise to crossover phenomena in the low-temperature thermodynamic properties of the XXZ model [Johnson and Bonner 1980]. In the case of isotropic ferromagnetic exchange (~=1), the excitation spectrum is gapless as shown in Fig. 11-14(b). Expression (11.73) for the bound-state branches turns into En(q)

J = -(1 -cosq), n

n =1,2,3,...

(~=1)

.

(11.75)

11-16 Excitation Spectrum of the 10 s=1I2 XXZ Antiferromagnet: Spinons and Solitons In the regime -1~<1, the excitation spectrum of the XXZ model has a quite different structure. The ground state is a singlet Sf = 0 state (r=NI2). The spectral threshold is gapless throughout this regime. Two typical cases are illustrated in Fig. 11-15. The continuum states (11.63) form two partly overlappipg sheets C+ and cbounded by sine-like branches, EL(q)

1tJsinJl . =_ _ I smq I , 2Jl

cOSJl

= -~

qI. _ 1tJsinJl, . q I - ( ) _ 1tJsinJl, cos_ u (q ) - - - - sm_ , Eu q Jl 2 Jl 2

E

(11.76a)

(11.76b)

For -1 <6<0 only the continuum states are present. As ~ increases from zero, the bound state branches En(q) with n=1,2,3,... progressively emerge from the top of continuum c+ at Jl=1tI( 1+ l/n). Their dispersion is described by expression (11.65) with parameters specified in entry (ii) of Table 11-2. In the limit ~=1, the two continua disappear completely, leaving behind an infinite set of discrete branches with energies (11.75). Historically, Des Cloizeaux and Pearson [1962] were the first to calculate the energies of the lowest branch of excitations for the Heisenberg antiferromagnet (~=-1):

Section 11-16

(a)

215

(b)

3.0

1.5

2.0

o.

1.0

0.0 "--

o

~-----..ll

TT

1T12

1T

TT/2

q

q

Figure 11-15: Spectrum of low-lying excitations of the 1D s=1/2 XXZ model (11.67) in the regime -1 ~<1. The two plots represent (a) the (isotropic) Heisenberg antiferromagnet A=-1 and (b) the ferromagnet with planar anisotropy A=O.8. The spectrum shown consists of the continuum states (11.76) and, in the ferromagnetic case, three discrete branches of bound states (11.65) [from Beck and MOller 1982].

£DP(q)

= 2:Jlsinql 2

.

(11.77)

Since this result is very similar to the dispersion of classical antiferromagnetic spinwaves [Anderson 1952], £sw(q)

= 2sJlsinql

,

(11.78)

these states were commonly identified as antiferromagnetic magnons, and the discrepancy in prefactor was attributed to quantum effects. The fact is, however, that (11.77) marks the lower boundary of continuum C+ of triplet states, all of which must be treated on equal footing [Yamada 1969; Muller, Beck, and Bonner 1979; Faddeev and Takhtajan 1981]. In field-theoretic approaches, these states are known as unbound spinon pairs, and in the context of the lattice-fermion representation (Sec. 6-3), they are particle-hole excitations (for A=O). Finally, the excitation spectrum for the regime A<-1 of H xxz was first analyzed by Johnson and McCoy [1972] in the framework of Bethe's ansatz. Their result, which coincides with (11.63) for A c' m as specified in entry (Hi) of Table 11-2, again consists of two partly overlapping continua C+ and C-, bounded by the three branches (11.79a)

216

Chapter 11

2.---

1T

q Figure 11-16: Spectrum of low-lying excitations for the case m=1/3 (a=-1.322) of the 10 8=1/2 XXZ model (11.67) in the antiferromagnetic regime with uniaxial anisotropy. The spectrum consists of two partly overlapping continua with boundaries (11.79) [from Mohan and MOller 1983].

2 . 2q (11.79b) _+m sm _ , cos2q

2

2

with qc = (1-m)/(1+m). This excitation spectrum, shown in Fig. 11-16 for 1:1=-1.322, is separated from the two-fold degenerate ground state by a gap of magnitude [Des Cloizeaux and Gaudin 1966] G(I:1)

= 2Acm =

xl I sinhA

A

i:

n=-oo

sech[ (2n + l)x

2A

2

],

COShA =-1:1 .

The l:1-dependence of the energy gap is shown in Fig. 11-13. As vanishes exponentially, G - 4xJexp[ -x 212J -2(1:1 + 1) ] ,

1:1~-I,

(11.80) the gap (11.81)

and the continuum boundaries (11.79) become sine-like. In the Ising limit, 1:1-+-00 the continuum collapses into a single dispersionless branch. Ishimura and Shiba [1980] proposed an illuminating interpretation of this excitation spectrum for 1:1 « -1 in terms of mobile antiferromagnetic domain walls (solitons) on the basis of a perturbation calculation about the Ising antiferromagnet.

Section 11-17

217

11-17 Equal.Time Correlation Functions As noted previously, the ground state of the 10 s=112 XXZ model (11.67) is critical in the planar regime -1~<1. implying that the equal-time two-spin correlation functions are expected to decay algebraically for large distances:

<s I Zsl+n z> _

(-l)nn -TIt

(I 1.82a) (11.82b)

Such behavior was indeed found by exact calculations for the XX case (il=O) as discussed previously (Sec. 11-1). The exact exponent values over the entire critical phase were first obtained by Luther and Peschel [1975]. Their derivation was based on the rigorous mapping between the XXZ model at T=O and the Baxter eightvertex model at T=Tc . The latter has a line of critical points that can be parametrized by the XXZ parameter il, and the correlation function exponents T\z' T\x of (11.82) can be inferred from the exactly known electric critical exponents of the eightvertex model. 14 The calculation relies on the Luttinger model, an exactly solvable Fermi field theory, as an intermediate step. On the level of the Luttinger model, the correlation function exponents, T\z' T\x of the XXZ model and the electric critical exponents of the the eight-vertex model can both be expressed in terms of a single parameter 8. The il-dependence of T\z' T\x can then be derived from the known ildependence of the electric critical exponents:

T\z 8

= 1/8, T\x = 8 .

= 112

- (ll1t)arcsinil .

(11.83a) (11.83b)

If the il-dependence of the exponents T\z. T\x is determined from an explicit evaluation of Luttinger correlation functions, it agrees with the exact result (11.83) only to O(il). This reflects the approximate nature of the mapping between the Luttinger model and the XXZ model (or eight-vertex model). The discrepancy is caused by the neglect of urnklapp processes in the Luttinger model. Black and Emery [1981] and Den Nijs [1981] later demonstrated that the exact exponents result from the continuum model with the urnklapp terms restored. A different route to deriving the exponents T\z' T\x was pursued by Haldane [1980]. He pointed out that in the framework of the Luttinger model, the parameter 8 can be expressed in terms of the low-energy spectral properties of that model. If the same formal relationship holds for the XXZ model, it yields, effectively, the correlation function exponents T\z. T\x in terms of such spectral properties as can be determined by the Bethe ansatz.

14-yhe eight-vertex model has two physically distinct interpretations. (i) as a model for 20 (anti-)ferroelectricity or (ii) as a 20 Ising model with 2-spin and 4-spin coupling. Consequently the critical properties of the relevant quantities are described by different sets of critical exponents - (i) the electric exponents and (iii) the magnetic exponents. It is the former set that plays a role in the present context.

218

Chapter 11

HBAFM £\ = -1

XX £\=0

HBFM £\ = 1-

<S]S7+n>

-n-1

-n-2

_ n-oo

<S/S/+n>

-n-1

-n-112

-n-0

Table 11-3: Long-distance asymptotic decay of the equal-time two-spin correlation functions

(11.82) for three special cases of the 10 5=1/2 XXZ model (11.67) at T=O. The exponents 'IlZ' 'Il x are continuous functions of the anisotropy parameter as given in (11.83). The limiting values for the Heisenberg ferromagnet (a=1) reflect the exact results <~~+ri> = (1/4)~n.o. <S'fS'f+ri> =1/4.

The different types of power-law decay exhibited by the correlation functions (11.83) at the midpoint (£\=0) and at the two endpoints (£\=±l) of the critical regime in the XXZ model are summarized in Table 11-3. At £\=-1 we have equal exponents for both correlation functions, in accordance with the full rotational symmetry of the ground state. At the other end of the critical line (£\~ 1-), the ground state breaks the symmetry of the Hamiltonian. It is a state with a saturated magnetization pointing in a direction perpendicular to the z-axis and without correlated fluctuations. These properties are reflected in the limiting algebraic decay cited in Table 11-3 for £\=C

11-18 Dynamic Correlation Functions Sections 11-18 to 11-25 are devoted to the study of dynamic structure factors SIlIl(q,ro) and frequency-dependent spin autocorrelation functions (structure functions) SIlIl(ro) of the ID s=1I2 XXZ model (11.67) at T=O and in the critical regime -1~<1. We are particularly interested in dynamically relevant dispersions, in detailed line shapes, and in infrared singularities where applicable. In a methodological context, the goal is to demonstrate that the recursion method is capable of providing significant and reliable answers to these questions. The flexibility and robustness of the recursion method stands out particularly in comparison with other calculational techniques used for the same purpose. o An exact analysis of dynamic correlation functions is restricted to the XX case (£\=0). We have gained a fairly deep insight into the dynamics of that particular model (Secs. 11-7 to 11-9). However there is little hope for extending a rigorous analysis to £\ '# O. o An analysis of dynamic correlation functions in the framework of the continuum approximation (Luttinger model) yields useful results over the entire critical regime, as will be discussed in Sec. 11-19, but their significance is limited to small frequencies, which occur only near the zone boundary (q=n:) and near the zone center (q=O). When the result~ are

Section 11-19

219

intended to be used for the interpretation of experimental data from, say, inelastic neutron scattering, this restriction is a serious handicap. We can remove this restriction if we insist on carrying out the calculation for lattice fermions instead of Fermi fields. This results in different types of restrictions. The Green's function approach presented in Sec. 6-3 for free lattice fermions (the case A=O of the XXZ model) can be extended to the case of interacting fermions in the framework of a perturbation calculation. The results of a Hartree-Fock calculation for the dynamic structure factor Szz(q,ro), to be presented in Sec. 11-20, are no longer restricted to small pockets in (q,ro)-space. However, unlike the continuum results, they are only significant for small values of the coupling constant, IAI « 1. The recursion method is not subject to any of these restrictions. It can be carried out in the original spin representation of the XXZ model. Its scope is not restricted to special regions in (q,ro)-space, nor is the range of the anisotropy parameter restricted to the critical regime. It is a truly universal tool, and produces, as we shall see in Secs. 11-21 to 11-25, valuable new results for the T=O dynamics of Hxxz .

o

o

11-19 Continuum Approximation (Luttinger Model)

The Luttinger model is an exactly solvable Fermi field theory in (1 + 1) space-time dimensions. It represents the continuum limit of the ID s=1I2 XXZ model for IAl
t = -J~ L.J cosk aka k

L\J~ t t L.J cosq akak+qaklak' _q' - -

N

k

(11.84)

kk1q

under the assumption that the fermion interaction leaves the Fermi surface intact. On the left of Fig. 11-17 we show an illustration of the ground state for the noninteracting case (A=O) - a half-filled band of free fermions. In the continuum limit, the cosine-like one-particle spectrum of the lattice model is replaced by two infinite dispersionless branches of Fermi fields, (11.85) as illustrated on the right of Fig. 11-17. The slopes of the two branches are determined by the Fermi velocity of the lattice model. The Luttinger Hamiltonian reads HL

t t 2J~ = J~ L.J k[a 1,ka l,k -a 2,ka 2,k] +T L.J k

kk1q

t

t

v(q)al,ka 2,k+qa2,k'a2,k' -q ,

(11.86)

where J sets the energy scale and L=Nb the length scale. In the XXZ context the coupling strength is q-independent, v(q)=A. The interaction term in (11.84) represents three different types of processes involving the excitations in the vicinity

220

Chapter 11

of the Fermi surface: (i) small-k processes, (ii) backward scattering processes, (iii) umklapp processes [Black and Emery 1981; Den Nijs 1981]. Only the processes (i) and (ii) are represented in HL .

(0)

(b)

cv=Jcosk

w=±Jk

--4---+----,I----k

-----A----k

CONTINUUM LIMIT

Figure 11·17: One-particle spectrum of the noninteracting lattice fermions described by (11.84) with d=O (left) and its continuum version, described by (11.86) with v(q) ;: 0 (right). The half-filled band turns into an infinite Dirac sea. The two Fermi velocities stay invariant but the Fermi momenta are shifted to the origin [adapted from Den Nijs 1981].

The Luttinger model was first solved by Mattis and Lieb [1965]. They rewrote H L as an expression that is bilinear in boson-like operators, and determined the unitary transformation which brings that expression into diagonal form. In order to calculate the dynamic spin correlation functions of interest in the context of the XXZ model, it is necessary to find expressions for the continuum spin operators SZ(x,t), SX(x,t), in terms of the same boson operators. For that purpose, Luther and Peschel [1975] constructed a continuum version of the Jordan-Wigner transformation, in which the resulting Fermi-density operators are identified with the boson operators of the Luttinger model. The expectation values ~(x,t)~(O,O», ~=x,z, can then be evaluated for the (diagonalized) boson Hamiltonian. The resulting expressions for the asymptotic region tx2-~rl » 1 are the following: 1

26L, + cos(1tx)(x -cLt) 2

2 2

(l1.87a)

Section 11-19

X 2 +C 2t 2

<s X(X,t)S X(O,O»-

2

L-=-_ _ +

1+ ElL +_1_ (X

222)

-cLt

221

_ ElL

COS(1tX)(X 2 -c t 2) 2, L

(11.87b)

22ElL

with exponent parameter (11.88) and renormalized Fermi velocity

cL

2d = 1(1--) 1t

(11.89)

.

A somewhat different approach toward the same goal was pursued by Fogedby [1978], who calculated the two-spin correlation functions <S"'(x,t)~(O,O», Il=x,z for the Luttinger model directly in the continuum fermion representation. The results are essentially the same as those obtained by Luther and Peschel via bosonization.1 5 The validity of these continuum results for the dynamic structure factors SI!I!(q,c.o) is restricted to small excitation energies (00« 1), which occur, throughout the critical regime, near q ..O and q ..1t (see Fig. 11-15). The Fourier transform of (11.87) yields the following expressions for the dynamic structure factors at 00<<1 [Fogedby 1978, Muller et al. 1981]: (11.90a)

Szz(q,c.o) - cOO(c.o-cLq) ,

(11.90b) for small q, and (11.91a)

2

ElL _-I

Sxx(q,oo) - EXoo-cLQ)(of -cLQ 2) 2

(11.91b)

for small Q == 1t-q. The exponent parameter eL and the Fermi velocity CL as obtained in the continuum calculation are consistent with the exact results (11.83b) and (11.76a) to O(d2) and O(d), respectively.

15The results (11.87), (11.90), and (11.91) pertain to the XXZ Hamiltonian as defined in (11.107) below with JjJ -d for use in later Sections. This substitution has no effect on <SZ(x,~SZ(O,O» and S;aC.q,ro). Its effect on <,sx(x,~,sx(O,O» is an overall multiplicative factor cos(7tx) and its effect on Sxx
=

=

222

Chapter 11

In the context of the two-particle spectrum shown in Fig. 11-15, the results (11.90a) and (11.91a) reflect the contributions of continuum C+ to Szz(q,ro), whereas expressions (11.90b) and (11.91b) are attributable to C- and C, respectively, in Sxx(q,ro). Naturally, no information on SIlIl(q,ro) at the upper continuum boundaries or on the spectral weight of the bound state branches or any other higher lying excitations can be obtained from the continuum approach. Moreover, the results (11.87) do not always represent the leading terms in the LTAB of the relevant XXZ correlation functions. In some cases, the leading terms are known to be oscillatory in time. 16

11-20 Hartree-Fock Approximation The starting point here is again the XXZ Hamiltonian (11.67) in the lattice fermion representation, (11.92a) Eo(k) '" -Jcosk,

V(q) '" -2Mcosq .

(11.92b)

We consider the two-particle Green's function Gjk,q,t;,) ==

E «a:ak+Q;akt,ak' -q»~

(11.93)

k'

as defined in Sec. 3-8. It determines the dynamic structure factor Szz(q,ro). In Sec. 6-3 we have worked out the exact solution for the noninteracting case (~=O). The equation of motion (6.31) for the system with interaction couples to higher-order Green's functions of the type <
.

(11.94)

This puts the exact solution out of reach for this approach. However, the function G_(k,q,t;,) and the hierarchy of equations it generates via (6.31) may be approximated by artificial closure at some level. On the lowest level (Hartree-Fock approximation), the higher-order Green's functions (11.94) are expressed in terms of the 2particle Green's function (11.93). Applied to Hxxz with I~I « 1, that analysis yields valuable insight into the structure of Szz(q,ro) despite obvious limitations [Todani and Kawasaki 1973; Schneider, Stoll, and Glaus 1982; Beck and Muller 1982]. The equation of motion collapses into Gjk,q,t;,)

= Go(k,q,t;,)[l

+

N-1E {V(q) - V(k-k/)}Gjk,q,t;,)] ,

(11.95)

k'

16For the isotropic Heisenberg antiferromagnet (~=-1) we must have SxJ.,q,ro) = SuC.q,ro). In (11.91) this symmetry is indeed realized for 9l =1, but in (11.90) the relationship is more subtle [Muller et aI.1981].

Section 11-20

223

where n k - ql2 - nk+q12 = -::----: -:-"~----::::--:~

Go(k,q,~)

is the Green's function of the noninteracting system with ECk)

= eo(k)

(11.96)

~ +e(k-q /2) -e(k+q/ 2)

+ N

nk

= [el3£(k)+lr l and

-lE [V(O) - V(k-e )]ne k'

.

(11.97)

At T=O, the renormalized I-particle spectrum (11.97) becomes ECk) = _JI cosk, JI = J(1 -2!:V7C) . The solution of (11.95) is straightforward:

(11.98)

(11.99a) with

]

[~2

E

Ko(q,~) _ - -1 Go(k,q,~) , W(q,~) _ - 2Ll - Jcosq . (11.99b) N k 4J I sin 2q/2 The T=O dissipation function X~(q,ro) is then inferred from that solution as in (6.35b), 11

XZZ (q,ro)

'Cl ZZ= - hm....> [<<Sq ;S-q»Q)l-i£]

£-+0

,

(1 I.lOO)

and the T=O dynamic structure factor from (2.23) : Szz(q,o»

= 2e(ro)Xzz11 (q,ro)

.

. (II.lOl )

The salient features of the Hartree-Fock result for the dynamic structur e factor

Szz(q,ro) may be summarized as follows [Beck and Muller 1982]:

o

The spectrum of 2-particle excitations o:>,.(k,q)

o

= le(k-q/2)1

+ le(k+q/2)J ,

(11.102)

generated from the renormaIized I-particle spectrum (11.98) agrees with the exact low-lying excitations identified in Fig. 11-15 to O(Ll). For .1>0 the Hartree-Fock result (11.99) has a pole at energy (1 I.l03) above and outside the continuum of 2-particle excitations (I I.l02). This resonance can be identified as representing the first branch of bound states with energies el(q) as discussed in the context of Fig. 11-15. In fact, the Hartree-Fock result (1 I.103) agrees with the exact expression (11.65) for n=I to O(~).

224

Chapter 11

3.0

..-.. 3

2.0

0' .......,

N

U)

N

1.0

0.0 +-------r-.. . . . . . - -----'"---,-------,,.-.-----'-'--.L.-T'------' 0.0 2.0 1.0

Figure 11-18: Dynamic structure factor Sdq,ro) as a function of frequency at fixed wave number q=47t15 for the 1D 5=1/2 XXZ model (11.67) in the critical regime. The two solid

curves represent the Hartree-Fock results for .i =±O.1. The dashed line represents the exact result (6.38) [adapted from Beck and MOller 1982].

o

The spectral weight of these bound states in the dynamic structure factor

Szz(q,ro) as inferred from (11.99) is given by the following expression: 17

S~B)(q,oi) = 41t~sin!O(ro-C%(q)) . 2

o

o

(11.104)

By construction, the Hartree-Fock approximation neglects the contributions to Szz(q,ro) from m-ferrnion excitations with m>2. Such contributions, while manifestly absent in the exact result (6.38) for ~=O, cannot be ignored at ~;t{). They are readily taken into account by the recursion method as will be discussed later. In Fig. 11-18 we have plotted the dynamic structure factor Szz(q,ro) as a function of ro at fixed q=41ti5 for the case ~=O (exact expression) and for ~=±O.1 (Hartree-Fock results). From the latter we expect a reasonably accurate picture of the impact which an attractive (~O) or repulsive (~
17For .i
Section 11-21

o

225

fermion interaction has on the spectral-weight distribution of Szz(q,ro) within the 2-particle spectrum in the weak-coupling regime (IAI « 1). The discontinuity in Szz(q,ro) present for A=O at the spectral threshold EL(q) Jsinq changes into a round shoulder (for A=O.1) or into a tall and narrow peak (for A=-O.1) at the renormalized lower boundary O\.(q) = J'sinq of the 2-particle continuum. The Hartree-Fock singularity at O\.(q) is, in fact, logarithmic in nature. If it is interpreted as the logarithmic correction to a power-law singularity of the form

=

for

ro ?

roL(q), then the exponent

~~F = -AITt

(11.106)

is found to agrees to 0(.:1) with the exact exponent ~iA). The latter can be inferred from the exact mapping between H xxz at -1~ A <1 and the critical eight-vertex model as will be discussed further in Sec. 11-22.

11-21 Weak-Coupling and Strong-Coupling Regimes The Hartree-Fock analysis of the dynamic structure factor Szz(q,ro) in the lattice fermion representation is an elementary weak-coupling approach. It is based on the assumption that the ground state is essentially that of a free-particle system. The interaction is only summarily taken into account. The dynamically relevant excitation spectrum remains confined to a band of finite width. The effects of the fermion interaction at this level include renormalized 2-particle excitation energies, modified line shapes, and the formation of bound states. Some of these results are known to be accurate to leading order in the coupling constant (Sec. 11-20). The continuum analysis of the same dynamical problem on the level of the Luttinger model (Sec. 11-19) is also a weak-coupling approach albeit superior to the Hartree-Fock analysis in some respect. Unlike Hartree-Fock, it produces genuine power-law infrared singularities for the critical fluctuations with exponents that are accurate to second order in the coupling constant. However, like Hartree-Fock, it fails to produce and describe the transitions at the endpoints of the critical phase (see Fig. 11-13). Within the continuum analysis, this deficiency can be overcome by incorporating the urnklapp terms into the calculation. That turns it into a strongcoupling approach. A strong-coupling approach to the lattice-fermion problem associated with the ID XXZ model would distinguish itself from, say, the Hartree-Fock picture by producing dynamically relevant excitations that are truly collective in nature. Such excitations would necessarily involve arbitrarily many fermions e.g. in the twoparticle Green's function. The associated spectral density would then have unbounded support, in general. One particularly useful feature of the recursion method is that it can be employed for the same purpose and with comparable success in both the weak-

226

Chapter 11

coupling and strong-coupling regimes of nontrivial problems in quantum many-body dynamics. This will be demonstrated here in the context of the ID s=1I2 XXZ model (11.67). In order to facilitate comparison with results for the 2D s=1I2 and ID s=1 XXZ antiferromagnets in later Sections, we rewrite the Hamiltonian in the form

H xxz =

L

{J(StSi:l

+

i

S/Si~l)

+

J..s/Si:tl

(11.107)

with J>O and periodic boundary conditions.

4.0

3.0

2.0 \; 1.0

Jz=O.l

',----_>-----------..-------1 -------.-----.-------------.-----

O.O-t----,----,---.----r----,---..-----l 5 . o 1 2 3 4 7 6

k Figure 11-19: Continued-fraction coefficients li.k vs k for the dynamic structure factor Szz(1t,ro) at T=O of the 10 s=1/2 XXZ model (11.107) with Jt=0 (free fermions), JjJ=0.1 (weak-coupling regime) and JjJ=1.0 (strong-coupling regime) as derived from the groundstate wave function of a chain with N=16 spins.

How do we recognize and distinguish the weak-coupling and strong-coupling regimes in the framework of the recursion method? The relevant information is encoded in the ~k-sequences pertaining to the dynamic quantities of interest. In Fig. 11-19 we have plotted ~k vs k for the functions Szz<"lt,ro) of the s=1I2 XXZ chain at three different values of the anisotropy parameter Jp (coupling constant). At J!J = 0, the ~k-sequence has growth rate A.=O, reflecting the fact that the dynamically relevant spectrum is the finite band of 2-particle excitations. For J!J 'I:: 0 the fermion interaction causes the growth rate to switch from A.=O to ).,,1 or larger. This

Section 11-22

227

dramatic change signals the dynamical relevance of many-body collective modes, which form a band of infinite width. For sufficiently strong coupling, the new growth rate is manifest in the entire ~k-sequence. That is the case for J/J=1.0 in the context of Fig. 11-19. For very weak coupling, however, the ~k-sequence starts out with growth rate A.=O as can be observed for J/J=O.1. For larger k this sequence crosses over to growth rate b 1. The interaction affects all continued-fraction coefficients ~k' but for weak coupling the impact is much more subtle on the first Kw coefficients than on the rest. The number Kw decreases with increasing coupling strength. When it reaches zero we are in the strong-coupling regime, and the dynamical quantity is governed by the full many-body spectrum. Under these circumstances the continued-fraction analysis must be carried out on the basis of the actual growth rate (A.:? 1). If, on the other hand, Kw is sufficiently large, we may proceed with just those coefficients ~k which are still consistent with growth rate A.=O and carry out the continued-fraction analysis on that basis. This is what we call a weak-coupling approach within the framework of the recursion method.

11-22 Infrared Singularities in SIlIl(1t,ro) and SIlIl(ro) The critical nature of the ground state in the planar regime -1<J/l-5,1 of H xxz manifests itself not only in terms of algebraically decaying equal-time correlation functions <SlfSlf+n> or, equivalently, power-law singularities in the corresponding integrated intensities <SI!~?, but also in terms of power-law infrared singularities in certain dynamic quantities, specifically the dynamic structure factors SIlIl(1t,ro) and the structure functions SIlIl(ro). Both types of singularities describe different aspects of the same critical fluctuations and are characterized by interrelated sets of critical exponents. For the model at hand, these exponents are continuous functions of J/J. The explicit results for the exponents T\x' T\z of the equal-time correlation functions have already been quoted in (11.83). The infrared singularities are (11.108) with exponents

a z =APz +1 = 1/8-1

'

a x =AP x +1 =8-1

(11.109)

and the parameter 8 from (11.83b), which is equivalent to

8 = 1-(1/1t)arctan(J/J) .

(11.110)

The relation between the static exponents, T\1l' and the dynamic exponents, is established by the Lorentz invariance of the Fermi field theory which was shown to describe the long-distance and long-time properties of the XXZ chain [Luther and Peschel 1975, Fogedby 1978]: all' ~Il'

(11.111)

228

Chapter 11

Two attempts at detennining the J!J-dependence of infrared exponents approximately have already been reported. The Luttinger model reproduces the parameter e correctly to second order in the coupling constant (Sec. 11-19), and the HartreeFock approximation yields the exponent ~z correctly to first order (Sec. 11-20). An exact detennination of the exponents describing the dynamical aspects of the critical ground state of a quantum many-body system is, in general, out of reach. Therefore, the availability of a general method by which exponents of infrared singularities in spectral densities can be estimated with reasonable accuracy is an asset of considerable value. The recursion method is such a method. In the following we use it, first in a weak-coupling approach and then in a strong-coupling approach, to detennine the exponents ~, ~I! of the s= 1/2 XXZ chain. A different general method, based on Luck's formula, was previously employed by Schulz and Ziman [1986] with considerable success for the detennination of the exponents Tlx' Tl z from finite-size data for the energies of the ground state and two specific lowlying excitations. In our approach, by contrast, the infrared exponents are derived from the finite-size ground-state wave function [Viswanath et al. 1994]. 0.10..,.---~---------------------,

0.003

0

,......

0.06

0.002

eN IQ. 0.001

--::::::

0.008

0.02

0.000 +----,...---.,...-"T"""--..-----l 0.2 0.4 0.0

0.004

Ilk

-0.02

IQ.N

0.000 -0.004

-0.06

o o

-0.008 +-"""""-"-""'-'--"""-""""'-'-"'" -0.004 0.000 . 0.004

Jz -0.08

-0.04

0.00

0.04

0.08

Jz Figure 11-20: Jz·dependence of the infrared exponent ~z pertaining to the dynamic structure factor Szz<1t,Cll) at T=O of the 10 5=1/2 XXZ model (11.107) with J=1. In the main plot the solid line represents the exact result (11.109) and the circles the results of our weakcoupling analysis based on the near1y size-independent continued-fraction coefficients ~ ..,As computed for a system of N=18 spins. The lower inset is a blowup of the main plot near JrO. The upper inset illustrates the extrapolation procedure used for the exponent sequences (9.37) at Jr -0.003 (top) and Jr -0.OO1 (bottom) as described in the text.

Section 11-22

In the weak-coupling regime, the infrared exponent

~z

229

in the function

Szz(1t,ro) can be extracted from the emerging alternating pattern in the .1k-sequence

caused by the onset of the fermion interaction. This effect is conspicuously present in the .1k-sequence for J/J=O.1 displayed in Fig. 11-19. The method was introduced in Sec. 9-7. It involves the extrapolation of the sequence of exponents (9.37), which we have renamed ~~k) in the context of the present application. Two such ~~k)_ sequences, plotted vs lIk, are shown in the upper inset to Fig. 11-20. For the extrapolation we use a very simple scheme, based on a linear extension of the trends indicated by the ~~k) for even k and odd k, separately. These extensions, marked by dashed lines, extrapolate, for k~oo, to roughly the same exponent value. We take the average of the two intersection points with the vertical axis as the data point (open circle) for ~z in the main plot. These data points display a smooth dependence on J/J over the interval of anisotropy shown. Also shown in the main plot is the exact J/J-dependence of the exponent ~z (solid line) as given in (11.109). Note that the line through the data points has the correct slope near the free-fermion limit (1/J=O). The lower inset shows a blowup of that region. The conclusion is that the weak-coupling continuedfraction analysis of the infrared exponent ~z is accurate to at least leading order in J/J, like the result (11.88) produced by the continuum analysis and the result (11.106) inferred from the Hartree-Fock approximation. Now we turn to the exponent analysis in the strong-coupling regime of the ID s=1I2 XXZ antiferromagnet. For the critical fluctuations perpendicular to the symmetry axis (z), the strong-coupling regime coincides with the entire (antiferromagnetic) critical regime, 09/J'5.1, whereas for the fluctuations in z-direction it is limited to O.5'5,1/J'5.1. Here, the determination of the infrared exponents is based on the alternating deviations of the .1k's from average linear growth. The method has been introduced in Sec. 9-12. The results of the strong-coupling exponent analysis are summarized in Fig. 11-21 (circles and squares). For comparison, the exact J/Jdependence of the exponents aJ.l' ~Il' Jl=x,z, given by (11.109), is shown by four solid lines. This type of analysis is subject to considerable statistical and systematic uncertainty, as we have pointed out previously. Starting at J/J=I, the strong-coupling analysis correctly predicts an exponent ax=az near zero and an exponent ~x=~z very close to -I. For decreasing values of J/J, the increasing trend of a z' ~z and the decreasing trend ofax' ~x are both correctly reproduced by our estimates. For completeness we have added (at the center left) the data points for ~z obtained from the weak-coupling analysis discussed previously (Fig. 11-20). The results for the exponents ax' a z are evidently less accurate than those for ~X' ~z' The reason for this is the more complex structure of the structure functions S'll!(ro) as compared to the dynamic structure factors SI!I!(q,ro). As will be seen in Sec. 11-25, the former have a singularity at ro=Ed7tl2) of strength comparable to that at 0>=0. This second singularity is more likely to interfere with the simple pattern of the model ~k-sequence (9.58), on which our exponent estimates are 'based. More accurate exponent values can be obtained if more .1k 's are available for the analysis. This expectation is supported by the following result: For J/J=O

230

Chapter 11

we can compute up to 13 completely size-independent !:ik's from the exactly known frequency moments (listed in Table 11-1). When we use all of them in our exponent analysis, we obtain a data point for (Xx at Iz=O which is right on target. 1.5.-------------------------,

1.0 -o-l

~

Q)

~

o

0.5

p..

><

Q)

....s..>.

0.0

-o-l

aj

'3tl.O

-0.5

~

'00 -1.0 -1.5

(Jz=1/e-2 cxx=e-l

. ""

_-r

,,"

G--::-a:.-. .

.....

.....

.."

"El-_. -'9 __ ' 9 ......

11-_- ..... - - . - - - . -

"

o Pz

-----

px =e-2 0.0

0.2

• Px 0.4

0.6

0.8

1.0

Figure 11-21: JjJ-dependence of the infrared-singularity exponents ~, 131l' J.L=x,z for ,5f1J1(CIl) and SJ1J1(X,CIl) of the 10 8=1/2 XXZ model at T=O. The circles and sguares represent the results derived, via strong-coupling analysis as explained in the text, from a number of nearly size-independent AkS computed for systems with up to N=18 spins. The exact results (11.109) are shown as solid lines. Also shown (on the left of the 13z"branch) are some weakcoupling results [adapted from Viswanath et al. 1994].

11-23 Reconstruction of Su:(7t,co) (Weak-Coupling Analysis) We recall from Sec. 6-3 the exact expression (6.38) for the dynamic structure factor Su:(q,co) of the case Iz=O (XX model). For q=1t that function is characterized by the particularly simple !:ik-sequence, !:it 2J2, ~ !:i3 12. This sequence is plotted in both insets to Fig. 11-22 (full circles). Also shown there are six nearly size-independent !:ik's for (a) two cases with weak, repulsive fermion interaction (lz>O) and (b) one case with weak, attractive coupling (lz
=

= =... =

Section 11-23

/

2.0

3,0

__B- __


.... _- •.... _-

1.0

•..._-.

t

.f

0.0+---'--2r--"3--'.--5--I

o

1.0

(a)

./

e

2.0 \,

231

8 ..../ . /

1//

,I

i.

"------ _.._~_···~----------":,·,i" \ \

....... Jz=o Jz=O.1 --- Jz=0.2

I \

0.0i-----r----,----........-----J----.l-.J.....----J 0.0

1.0

2.0

(b)

1.0

2.0

Figure 11·22: Dynamic structure factor Szz(7t,ro) at T=O ofthe 10 5=1/2 XXZ model (11.107) with (a) antiferromagnetic coupling (.1=1, Jj!.O) and (b) ferromagnetic coupling (.1=1, J~O). The short-dashed curve in each plot represents the exact result (6.38) for the XX case (JrO). The remaining curves are the results of a weak-coupling reconstruction as described in the text. The inset shows the ~k-sequences for the same cases. The coefficients shown are exactly size-independent for JrO and nearly size-independent for Jro. The latter have been extracted from the ground-state wave function of a system with #.=16 spins.

232

Chapter 11

Two effects of the fennion interaction are readily discernible in the I1k sequences for Jz:#.). Both can be analyzed quantitatively and the information used for the optimization of the termination function: o From the average value of the coefficients 112=...=115 we determine a J/Jdependent bandwidth (Sec. 5-2). It turns out to agree very accurately (for IJ/J1 < 0.1) with the exactly known upper boundary (l1.76b) of the spinon continuum. o The alternating deviations of the coefficients 112=...=115 from that average value detennine the J/J-dependent infrared exponent ~z' which we have analyzed in the previous Section. For the weak-coupling continued-fraction reconstruction of Szz(1t,ro) we should like to employ a tennination method which takes into account both pieces of (implicit) information. The logical choice is the compact a-terminator (Sec. 9-10), whose two parameters are a bandwidth (<00) and an infrared exponent (a). The implementation of this terminator at level K=6 yields the curves labelled J/J = 0.1, 0.2, and -0.1 in Fig. 11-22. In the antiferromagnetic case (J?O), we observe an increase in bandwidth and the emergence of an infrared divergence. These are the effects previously identified in the I1k-sequences and then incorporated in the termination function. Both must be regarded as genuine consequences of a weak, repulsive fermion interaction. Near the band edge the compact a~tenninator is unbiased with respect to any structures that emerge from the weak-coupling reconstruction. Its model spectral density (9.49) is flat and drops to zero discontinuously. What we can observe in the curves of Fig. 11-22(a) is a softening of the square-root divergence at the band edge, which, in the free-fermion case, is wholly attributable to a density-of-state effect within the 2-particle spectrum. At Jp = 0.2, the spectral enhancement near the band edge has all but disappeared. That trend combined with a strengthening of the infrared divergence connects perfectly with the strongcoupling result for J/J=1 to be presented in Sec. 11-24. In the ferromagnetic case (lz
18 1n the reconstructed relaxation function (9.3), that contribution is represented by an isolated pole. For z = e-iro with e = 0.0001, it has a small but visibly nonzero width.

Section 11-23

a l = 2a +r, a 2 = a 3 = ... = a .

233

(11.112)

With r cc -JjJ and a-J2 cc JjJ, it describes the most dominant pattern of the weakcoupling ak-sequences displayed in the insets to Fig. 11-22. For r
'Vo(ro) =

1t

In +r

2 4(a +f)

[8(ro-rop) + 8(ro+rop)]

(11.113)

with 2a+r fA I r2 rop = _ _ = 2ya(1 + - - + ...) . 8 a2 a +r

v

(11.114)

This simple model demonstrates that the presence of a bound state in the weakcoupling k-sequences of Fig. 11-22 depends principally on the displacement of the first coefficient with respect to the nearly constant sequence of the remaining ones. The results of Fig. 11-22, obtained from a weak-coupling continued-fraction analysis, should be compared with those of Fig. 11-19, derived from the HartreeFock analysis of the 2-particle Green's function. Although the curves pertain to somewhat different wave numbers, it is evident that the results are remarkably consistent with one another. Note that in both the Hartree-Fock result (11.103) and the continued-fraction result (11.114), the gap between the bound state and the edge of the continuum grows cc (JjJ)2, in agreement, to leading order, with the exact results (11.65) and (11.76b). Likewise, the spectral weight of the bound state increases linearly with JjJ in both the Hartree-Fock result (11.104) and the continued-fraction result (11.113). It is appropriate to mention, in this context, an explicit expression for the dynamic structure factor Szz(q,ro) of the ID s=1I2 XXZ antiferromagnet in the critical regime (05JjJ-:;'I) that was proposed on the basis of numerical finite-chain results and sum rule arguments [Miiller et aI. 1981, Muller 1982]:

a

Szz(q,ro)

= Az

EXro-EL(q) )8(E u (q) -ro)

[at _E~(q)]~/2 [E~(q) -at] 1/2 -~,!2

(11.115)

with ~z from (11.109) and a JjJ-dependent factor Az. It was designed to provide a reasonably accurate description of the spectral-weight distribution within the spinon continuum. Expression (11.115) connects smoothly with the exact result (6.38) for Jz=O and exhibits the correct singularity at q"'1t over the entire critical regime. Moreover, the line shapes produced by the weak-coupling results presented in this Section and in Sec. 11-20 agree well with the result (11.115) for I1j1l « I. That expression for JjJ 1 was recently found to provide a near perfect match with the data of an inelastic neutron scattering experiment by Nagler et al. [1991] on the quasi-ID compound KCuF3.

=

234

Chapter 11

11-24 Reconstruction of Sxx(q,ro) (Strong-Coupling Analysis) There exists no weak-coupling regime for the dynamic structure factor Sxx(q,ro) in the antiferromagnetic critical phase, ogp5.1, as explained in Sec. 11-21. For the strong-coupling reconstruction of that function, we use K=6 nearly size-independent Ak'S extracted from the ground-state wave function of a system with N=16 spins. The Ak'S at 0
=

l~he Gaussian terminator is also used for the reconstruction of Sxx
JjJ'" 0,

Section 11-24

235

n A (a)

\~

80

60

3:er [IJ

/

4.0

~><

><

2.0

/

/

/

/

:::----

/

t-----;:' '-

"--../ \

j

/

0.0 0.0

'---

3.0

20

10

/ /0

/

/

/

/

/

/

q

(b)

60

'3

'"

~>< [IJ

4.0

>< 2.0

(c)

10

2.0

3.0

w 8.0

8.0

'3

'"

~><

4.0

en ><

q 2.0

2.0

10

3.0

0

w

Figure 11·23: Dynamic structure factor Sdq,ro) at T=O for fixed q=mrI8. 11=0,1 •...8 of the 10 5=1/2 XXZ antiferromagnet (11.107) with .1=1 and (a) Jz=1, (b) Jz=0.5. and (c)Jz=0' The results were derived from the reconstructed relaxation function (9.3) with nearly sizeindependent continued·fraction coefficients ~1 ,... ,<\, and a termination function as described in the text. The circles in the (q,ro)-plane indicate the boundaries (11.76) of the twoparameter continuum of dominant excitations [from Viswanath et al. 1994].

236

Chapter 11

(with upper boundary EU(q» of the two-parameter continuum. This effect is best visible at small q. The integrated intensity <S~? is nonzero for q=O. There is no longer a conservation law which prohibits that from happening. The line shapes strongly suggest that the spectral-weight distribution is divergent at the lower continuum boundary EL(q) throughout the planar regime. This confirms the conclusions of an early finite-chain study of XXZ dynamics [Muller et al. 1981]. The mild enhancement of spectral weight at the continuum boundary EU(q), most conspicuous in Fig. 11-23(c), is attributable to a divergent density of states at the upper edge of the two-parameter continuum. We conclude this Section with a remark on the T=O transition from the critical phase at JjJ5.1 to the antiferromagnetically ordered phase at JjJ> 1 which takes place at the symmetry point (d=-1 in Fig. 11-13). We have already mentioned that the analysis of this transition is not within reach of weak-coupling approaches. The Hartree-Fock approximation does not produce any transition. While the Luttinger model does, in fact, have an instability at some critical coupling strength, nothing significant about the transition in the XXZ chain can be leamed from it unless the umklapp terms are included in the analysis, which makes it a strong-coupling theory. Likewise, the weak-coupling continued-fraction analysis of the dynamic structure factor Szz(q,ro) (Sec. 11-23) does not yield any clues about the onset of antiferromagnetic LRO. That transition takes place well inside the strong-coupling regime as defined in Sec. 11-21 for the continued-fraction analysis. It leaves its unmistakable signature in the dk-sequences of the dynamic structure factors SJ.lIl(q,ro). We recall that the dk-sequence of Su(1t,ro) at OgP5.1 is of the type shown in Fig. 5-4 (left), which indicates zero gap and an infrared divergence and reflects the presence of critical fluctuations. This is in strong contrast to the dk-sequence of, say, Sxx<71t18,ro), which is of the type shown in Fig. 5-5 (left) and produces a gap equal to the spectral threshold (l1.76a). These contrasting properties have been illustrated in Fig. 11-10 for the case Jz=O. The same patterns are observable in the function Szz(q,ro) for the same two q-values, provided we stay in the strong-coupling regime (JjJ?;O.5) and in the critical phase (JjJ5.1). As we increase the anisotropy parameter beyond the symmetry point, a dramatic change in pattern occurs in the dk-sequences of these functions. The new patterns are shown in Fig. 11-24 for the case JjJ=3.0. The two sequences shown in the inset pertain to go;(q,ro) at q=71t18 and 1t. They are almost identical, both indicating a spectral gap. The spectral threshold for that parameter value does indeed not vary much between the two wave numbers. A quite different picture emerges for the function Szz(q,ro) at the same two q-values, as is shown in the main plot. The dk-sequence for q=71t18 is of the same type as those in the inset and thus suggests a spectral gap for the zz-fluctuations too. For q=1t, however, we observe the pattern reversal which is characteristic of a lone spectral line at ro=O separated by a spectral gap from a continuum of dynamically relevant excitations. In our context, that spectral line is, of course, the signature of antiferromagnetic LRO. Recall that we have identified the presence of ferromagnetic LRO in the spin-s XYZ model (Sec. 11-3) by the same type of pattern reversal occurring at q=O (Fig. 11-4).

Section 11-25

30.0

Jl-=X

80.0

237

Jl-=Z

60.0

25.0 40.0

20.0

20.0

::t

::t~


0.0

15.0

0

10.0 5.0 0.0

0

2

4

8

6

k Figure 11-24: Continued-fraction coefficients Iik vs k for the dynamic structure factor Szz(q,ro) (main plot) and Sxx(q,m) (inset) at q=7rc/8 and q=1t of the 10 5=1/2 XXZ antiferromagnet (11.107) with JjJ = 3.0 at T=O. These coefficients, which are nearly sizeindependent, have been derived from the ground-state wave function for a system with N=16 spins.

11-25 Strong-Coupling Reconstruction of SU(ro) The frequency-dependent spin autocorrelation function (structure function) was defined in (11.41) and is related to the dynamic structure factor via the integral (11.116) Subjected to its own continued-fraction analysis, the function SXX(ro) provides further insight into the dynamics of the XXZ chain. Starting from the same finite-size ground-state wave function for a ring of N=18 spins, we have determined K=7 nearly size-independent tik's. Here we are dealing with sequences of the same type as previously observed in Sxx(1t,ro) (no gap!). In an effort to be as unbiased as possible with respect to the spectral-weight distribution emerging from the reconstruction process, we have employed the Gaussian terminator (Sec. 9-11). The results of this reconstruction are depicted in Fig. 11-25. The six curves on the left pertain to J!J 0., 0.1,...,0.5 and the five curves on the right to Jp 0.6,...,1.0.

=

=

238

Chapter 11 2.0.,..-~~~~--------rr.-------------,

J =0 z

..

O. 0 -+--~-r----,.----.--.---+---r-~r----"---,---'-=-' 3.0 -3.0 2.0 -1.0 0.0 1.0 -2.0

Figure 11·25: Spin autocorrelation function ,s"X(ro) at T=O of the 10 s=1/2 XXZ antiferromagnet (11.107) with ..1=1 and Jz=0, 0.1, ..., 1.0. The results are derived from the reconstructed relaxation function (9.3) with seven nearly size-independent continued-fraction coefficients d k and a termination function chosen as described in the teXt. The tick marks on the auxiliary horizontal axes inside the frame indicate the range of values of EL(7tl2) for the Jz" values selected on either side of the central line [from Viswanath et al. 1994].

The reconstructed SU(ro) evolves rather smoothly with 'varying J/J. None of the observed structures are artificially imposed by our choice of termination function, nor are they artificats by nature. An exact result exists only at J/J = 0 (Sec. 11-9). The enhancement of spectral weight at low frequencies signals the presence of an infrared divergence with variable strength. It is strongest at J/J = o and weakens monotonically with increasing J/J. The exact result for J/J = 0 has a square-root infrared divergence. The J/J-dependence of that singularity has already been investigated by a continued-fraction analysis in Sec. 11-22. Now consider the peak at roIJ? 1. Its position moves monotonically to h.igher frequencies as J/J increases from 0 to 1. This peak echoes a divergent smgularity of some kind at ro =EL(1CI2). It is a consequence of the singularity in Sxx(q,ro) along the spectral threshold Edq). The peak position in the continuedfraction result is indeed very close to ro = EL(1CI2) throughout the parameter range as indicated by the marks on the auxiliary axes. The variable strength of the peak at roIJ? 1 might indicate that the associated singularity is also characterized by a

Section 11-26

239

lll-dependent exponent like the one at 0>=0. 20 If that is so, then the results of Fig. 11-25 indicate that with lz increasing from zero, the singularity gains strength at least initially (up to lp .. 0.5). Since the singularity at ro=O is rigorously known to lose strength with increasing lz' that observation would imply that the divergence at CJ.)::£L(q) in the dynamic structure factor Sxx(q,ro) is not characterized by one and the same exponent for different values of q. For III = 0, the function SXX(ro) is known to have a further detectable singularity - a square-root cusp at 0>=2J. In the curves of Fig. 11-25 there is indeed a hint of structure at or near the upper continuum boundary EU(1t) = Eu(O). There is very little spectral weight beyond that frequency, which further underlines the importance of the two-parameter continuum bounded by the branches (11.76) for the T=O dynamics of the 10 s=1/2 XXZ model.

11-26 2D s=1I2 XXZ Antiferromagnet Here and in the next Section we present some results for T=O dynamic structure factors of the s= 1/2 XXZ antiferromagnet, H xxz

=L

[l(StS/

+

S/S/) + l?/S/] ,

(11.117)



with nearest-neighbor interaction on the square lattice. In order to set the stage for the interpretation of these results in comparison with those presented previously for the same model on the linear chain, we show in Fig. 11-26 schematic representations of the T=O phase diagrams (order parameter and spectral gaps) for the two cases. The ID phase diagram was already discussed in Sec. 11-14. In 2D the consensus is that the XXZ model is antiferromagnetically ordered for all values III ~ 0 [Manousakis 1991]. Consequently there must be a transition, logically at the symme~ry point III = 1, involving a 90° rotation of the order parameter (from Mz :;:. 0 to M1. :;:. 0). In order for that to happen, the spectral gap tiE must go to zero as III approaches unity!rom above. At lll:S; 1, the spectral gap stays zero because the order parameter, M1.' now breaks a continuous symmetry of H xxz . Since the staggered magnetization does not commute with the Hamiltonian, its magnitude is reduced from the saturation value, and the ground state contains a certain amount of correlated quantum fluctuations, except in the Ising limit III ~ 00. As the anisotropy parameter decreases toward III = 1, the quantum fluctuations gain strength and cause an increasing amount of spin reduction. In the ID system, they make the LRO disappear completely at the symmetry point. Moreover, the planar anisotropy does not weaken them sufficiently to allow an inplane staggered magnetization to establish itself for 1p < 1. In the 2D system the quantum fluctuations lead to only a partial spin reduction anywhere on the lll-axis. The magnitude of the order parameter is expected to have a minimum for isotropic exchange coupling [Bames, Kotchan, and Swanson 1989].

20rhe exact result for JjJ = 0 has a logarithmic divergence at ro/J = 1.

240

Chapter 11

(a) linear chain

.....

----

criticality

o

........

_---~E

(b) square lattice

id z

I

1,.--

I

o

-------_.

........ ----~E

k

Figure 11-26: Phase diagrams of the 10 and 20 5=1/2 XXZ antiferromagnets in a schematic representation: JjJ-dependence of the staggered magnetization Mz or Ml. (solid lines) and of the spectral gap !J.E between the ground state and the lowest branch of excitations at q=rc or q=(7t,7t) respectively (dashed lines) [from Viswanath et al. 1994].

It is interesting to compare the impact of zero-point quantum fluctuations on the antiferromagnetic ordering with the impact of thermal fluctuations at small nonzero temperature. In the planar regime (J!J S 1), 2D LRO is stable against the former (T=O) but, according to the Merrnin-Wagner theorem, not against the latter (T>O), whereas ID LRO is nonexixtent, even at T=O. In the uniaxial regime (J!J ~ 1), the ID and 2D staggered magnetizations both survive the zero-point motion, but only the order parameter of the 2D system can withstand some amount of thermal fluctuations. The different types of zero-temperature phase transition take place at J!J = 1 in the 1D and the 2D system, and the different degrees of ordering at Jp S 1- are reflected in the dynamical properties by contrasting signatures. Widespread interest in 2D quantum spin models was no doubt kindled by the excitement about the oxide high-Tc superconductors. The electronic properties of the Cu02-planes in the undoped parent compound La 2CuO4 are describable, in some approximation, by the prototype model for a Mott insulator - the 2D Hubbard model. For very strong on-site repulsion, it turns into the 2D s=1I2 Heisenberg antiferromagnet [Manousakis 1991, Bames 1991]. By embedding the latter in the more general XXZ model (tU I?}, we gain a parameter, J!J, which controls the direction of the magnetic ordering in the ground state. This enables us to identify and interpret finite-size effects even though our numerical analysis is limited to a single system size.

Section 11-27

241

In a classical description, the T=O phase change at J/J = 1 consists of a simple spin-flop transition between two Neel states representing saturated antiferromagnetic LRO parallel and perpendicular to the z-axis. The two classical ground states are further distinguished by their degree of degeneracy. The one for J/J> 1, which breaks a discrete symmetry of H xxz' is twofold degenerate, whereas the degeneracy of the other (for J/J < 1), which breaks a continuous symmetry of H xxz' is proportional to the lattice size. Both types of degeneracy are removed in the presence of quantum fluctuations, and all finite-size ground-state properties change smoothly across the transition point J/J 1. In finite systems, therefore, the characteristic signature of phase transitions must be searched in the dynamically relevant excitation spectrum. The most direct access to that spectrum is provided by the recursion method as we have already amply demonstrated for the ID case.

=

11-27 Dynamic Structure Factors Sxx(1t,1t,ro) and Szz(1t,1t,ro) The numerical results presented here for the T=O dynamics of the 2D s=1I2 XXZ model are all derived from the ground-state wave function of a 4x4 lattice with periodic boundary conditions. The question is then not how to avoid finite-size effects but how to identify them and how to estimate their impact. For the reconstruction of the functions SlIiq,ro), Il=X,Z over the parameter range Og/J~ 2, we have used the first six /),/ s and a termination function selected according the criteria established previously. It is quite clear that not all six /),k's are nearly sizeindependent. We have indirect evidence that the number of nearly size-independent /),k's varies with J/J and is (not unexpectedly) lowest near the transition point. However, for our reconstruction procedure, we need at least 5 or 6 /),k's for the identification of the pattern, which determines the type of termination function to be used. Inevitably, they are strongly influenced by finite-size effects, at least in some instances. Nevertheless, observing the variation with J/J of the patterns in the /),k-sequences yields important clues about changes in the structure of the ground state, specifically their nature and their location in parameter space. Naturally, we wish to scrutinize the continued-fraction coefficients pertaining to SlIiq,ro), 11 =X,Z at the wave vector q = (1t,1t) associated with antiferromagnetic ordering. The /),k-sequence for Szz(1t,1t,ro) exhibits the characteristic pattern of Fig. 5-5 (left) at small Jp, indicative of a spectral gap. With increasing Jp, that pattern is stable at first, then, at J/J .. 0.9, begins to change its character to the type shown in Fig. 5-5 (right), which reflects the emergence of a 8--function central peak as caused by antiferromagnetic LRO in z-direction. That metamorphosis is completed at J/J .. 1.3. A similar change in pattern can be observed in the /),k-sequence of Sx/1t,1t,ro), but in opposite direction. These changes of pattern together with the observation that no such changes occur in the /),k-sequences for other wave vectors, clearly single out the phase diagram sketched in Fig. 11-26(b) from other possible scenarios. For comparison recall that the patterns in ID were of the type shown in Fig. 5-5 (right) for Szz(1t,ro) and (left) for Sxx(1t,ro) at Jp > 1, where antiferromagnetic LRO is established, but of the type shown in Fig. 5-4 (left) for both functions in the critical phase Jp ~ 1.

242

Chapter 11

~

r

(a)

~I\ ~ ~~ '\.

...L--'"

\

ArT-.

\~'\ \ \ \ >1\ \ \. '-. .i\\'\."'-

30

3' t='

i: en

V

>< >< 10

-

-----

'-

/

V\ "'-

"'-

I

/

I

/

I

"'-

1.0

I 1

11.0

y\ \. ' -

'\ 0.0

'\.

"-

......

1

'-

'-

V\ \.. 1\\. \.. ' -----

20

o·

\

\.

2.0

I

I

Jz

/

0.0

3.0

"\

n

(b)

30

1 .5

// r

1/

I

/

/

/

l::

i:

N N

I

en 10

.,/ / .,/

./ ./ ./ ./

/

I 0.0

/

------

/

1.0

I

. 11.0

./

~ 20

o

/

I

2.0

I

I

\-..

"-

3.0

I

I Jz

/

10 .0

Figure 11-27: Dynamic structure factors SI1l1(q.ro) at T=O for fixed q=(1t,1t) and (a) IJ.=X. (b) J.1=z of the 2D 5=1/2 XXZ model with ..1=1 and Jz=0. 0.1 •...., 1.6. The results are derived from the continued-fraction coefficients 6 1 ,.... Aa and a matching termination function as described in the text. The circles in the (JZ'ro)-plane mark the linear spin-wave frequency at q=(1t,1t) for that model in the Neel state, specifically the frequencies for modes involving spin fluctuations (a) perpendicluar and (b) parallel to the z-axis. For greater clarity, the curves at Jz=1.2, 1.4, and 1.5 have been omitted from (b) [from Viswanath et al. 1994].

Section 11-27

243

The reconstructed functions SIJl.I(7t,7t,ro) themselves are displayed in Fig. 11-27 over a broad range of lll-values around the transition region. A general observation is that the spectral-weight distribution is considerably more localized than in SJ.li7t,ro) of the 10 system. In the planar regime, Sxx(7t,7t,ro) is dominated by a narrow central peak, as is Szz(7t,7t,ro) in the uniaxial regime. These central peaks represent the antiferromagnetic order parameter in the two phases, respectively. The nonzero width is entirely due to our use of the Gaussian terminator for the reconstruction procedure. 21 The fluctuations transverse to the order parameter are reflected in Sxx(7t,7t,ro) at lll> 1 and in Szz(7t,7t,ro) at III < 1, in the form of well-defined peaks with nearly symmetric line shapes. The position of these peaks is in fair agreement with the prediction of linear spin-wave theory. However, let us keep in mind that linear spin waves are not exact eigenstates of the XXZ antiferromagnet and that the ground state is subject to correlated quantum fluctuations, which cause a partial reduction of the order parameter. All this will surely result in an intrinsic broadening of the spin-wave peak such as is evident in Fig. 11-27. Note that the spin-wave peaks as obtained from our reconstruction procedure do not represent individual poles of a truncated continued fraction. On approach to the transition point III = 1 from either side, the spin wave peaks at q = (7t,7t) become soft as expected, but the gap in the reconstructed functions SJ.lJ.I(7t,7t,ro) reaches zero only somewhat beyond the transition point - a clear finite-size effect. Here the coefficients A1,... ,A6 as extracted from the 4x4 system are more strongly size-dependent than for other parameter values. In Fig. 11-28 we have plotted the peak positions of the reconstructed SJ.li7t,7t,ro) as functions of III and the corresponding linear spin-wave frequencies. Also plotted are the exact spectral gaps between the ground state (k = 0, Sf = 0) and the lowest state at k = (7t,7t) relevant for in-plane fluctuations (Sf = ±1) or ·out-of-plane fluctuations (Sf = 0), and corresponding quantum Monte Carlo data of Barnes, Kotchan, and Swanson [1984] for lattice sizes 6x6 and 8x8. In the transition region, the peak positions from Fig. 11-27 agree quite well with the spectral gaps of the 4x4 lattice, which confirms the strong size-dependence of some of the coefficients A1,... ,AK used for the reconstruction. However, unlike the 4x4 excitation gap, which stays nonzero throughout the parameter range shown in Fig. 11-28, the spin-wave peak of the reconstructed SJ.lJ.I(7t,7t,ro) turns into a true central peak at the edge of the transition region. Here our input data A1,...,AK extracted from the 4x4 ground state are considerably less size-dependent than the finite-size spectral gap. The linear spin-wave frequency of the transverse mode at the ordering wave vector goes to zero as IlIl - 11 1/2 near the Heisenberg point in both 10 (q =7t) and 20 (q = (7t,7t». However, it is rigorously known in 10 that the lowest out-of-plane excitation at q=7t stays soft in the planar regime, and the lowest in-plane excitation rises by an exponential law in the uniaxial regime (see Fig.

21 When we use the 3-parameter terminator (9.61) instead, the central peak becomes a true o-function, but fitting six data points with three parameters introduces potential biases whose adverse effects may well outweigh the benefit of having a o-function central peak.

244

Chapter 11

11-26). It is not clear how accurate the spin-wave prediction near f If = 1 is in the 2D case.

o 6

6x6 8x8

•

4x4

2.0

1.0

, "

•

•

•

I

"

€I

•

•

•

0.0 -r----,.----r-=--=-----i---=::....--,------r---.J 0.4 0.6 0.8 1.2 1.4

Figure 11-28: Dependence on the anisotropy parameter JjJ of the spectral gap between the ground state and the lowest transverse excitation at wave-vector q=(1t,1t) for the 20 5=1/2 XXZ model. The solid lines represent the spin-wave peak of the dynamic structure factors displayed in Fig. 11-27 (Szz for JjJ'$ 1 and Sxx for JjJ ~ 1). The full circles denote the exact spectral gaps of the 4x4 lattice. The open squares and triangles are spectral gaps for 6x6 and 8x8 lattices, respectively, as quoted from a quantum Monte Carlo study [Bames, Kotchan, and Swanson 1989]. The linear spin-wave predictions are shown as dashed lines [from Viswanath et al. 1994].

11-28 ID Spin-I Heisenberg Antiferromagnet with Uniaxial Exchange and Single-Site Anisotropy Spin chains with quantum number s= 1 have been the object of sustained intense study ever since Haldane [1983] first predicted the existence of a non-magnetic phase with an excitation gap for the isotropic Heisenberg antiferromagnet. A broad consensus has now been reached on the main features and many intricate details of the T=O phase diagrams pertaining to several model Harniltonians involving spin-l chains. The prototype among them is the Heisenberg antiferromagnet with exchange and single-site anisotropy,

Section 11-29 N

H

= L [l(StSI:1 1=1

+

S/S/~I)

+

l~/S/~1

+

D(S/)2] .

245

(11.118)

Within the domain 0 :5: l!l :5: 2, 0 :5: DIJ :5: 2 for the two anisotropy parameters, a total of four distinct T=O phases have been identified: (i) A Neel phase (N) with antiferromagnetic LRO in z-direction is realized for sufficiently large l!l. (ii) A singlet phase (S) characterized by a non-degenerate ground state and an excitation gap exists for large DIJ. (iii) A critical phase (C) with algebraically decaying in-plane correlation function <SiSi+n> is present for sufficiently small l!l and DJl. (iv) Contiguous to these three phases is the phase (H) - the one first predicted by Haldane. Like (S), it is characterized by a non-magnetic singlet ground state with an excitation gap. The ground state is critical on the boundaries of (H). The (CIH)-transition is of the Kosterlitz-Thouless type and the (NIH)-transition is of the 2D-Ising variety. Both transitions are specified by a single set of critical exponents for the entire phase boundary except at the endpoints. The critical exponents of the (SIH)-transition, by contrast, vary continuously along the phase boundary [Botet, Jullien, and Kolb 1983; Glaus and Schneider 1984; Schulz and Ziman 1986; Schulz 1986; Bonner 1987]. What is the characteristic signature of the four phases and the impact of the three transitions on the T=O dynamic sturcture factors SJlJl(q,ro)? Our first goal here is to investigate this important question with an application of the recursion method. The second goal will be a study of line shapes of SJlJl(q,ro) for an experimentally relevant situation.

11-29 Dynamically Relevant Excitation Gaps Our continued-fraction analysis uses the K=6 nearly size-independent continuedfraction coefficients A'f(q),...., A,tJl(q) that can be derived from the ground-state wave function of (11.118) with N=12 spins for the detection of dynamically relevant excitation gaps and for the determination of the spectral-weight distribution in SJlJl(q,ro). In order to map out the rather complex T=O phase diagram of this model in the given parameter range, we choose an operator (in our case ~ with q=1t and Il=x,z) which we know or expect to represent the critical fluctuations along a specified phase boundary. The method of determining the dynamically relevant excitation gap for that operator involves the matching of the nearly size-independent coefficients A'tJl(q),...., A,tJl(q) with the first K coefficients K 1(Q),..., K~Q) of a suitable model dynamic structure factor that has a gap of size Q. 22 Since the matching criterion is not unique we cannot expect to obtain from this analysis accurate values for the gap sizes. Nevertheless, it proves to be a reliable indicator for the identification of phase boundaries and the dynamical variables that are subject to critical fluctuations.

221n this application, we use the split-Gaussian model spectral density (9.61) with A=O.

Figure 11·29: Dynamically relevant excitation gaps for operators S; (circles) and ~ (squares) along three lines in the parameter space of the Hamiltonian (11.118). The gaps are derived from K=6 nearly size-independent continued-fraction coefficients ~'ft(1t). k=1, .... K computed by means of the recursion method from the N=12 ground-state wave function. The vertical arrows indicate the locations, where the three lines in parameter space cross phase boundaries [from Zhang et al. 1994].

We have carried out the gap analysis along four lines in (JIJ, DIJ)-space. In Fig. 11-29(a) we have plotted "the JIJ-dependence for fixed DIJ=O of the excitation gaps which are dynamically relevant for the operators S~ (circles) and Si (squares). At JIJ=O, our analysis yields a S~-gap very close to zero, and it remains near' zero over some range of JIJ-values, reflecting the extended critical phase (C). The large Si-gap in that region indicates that the out-of-plane fluctuations are not critical. Even past the transition to the Haldane phase, marked (CIH) in Fig. 11-29(a), the S~-gap opens only very slowly, which is typical near a KosterlitzThouless type phase boundary. This is in marked contrast to the more rapid closing of the Si-gap at the other end of the Haldane phase. The (HIN)-transition marks the onset of antiferromagnetic LRO in z-direction. The large S~-gap near the phase boundary confirms Haldanes's prediction that the in-plane fluctuations have no part in that transitions. For the second line in parameter space (Fig. 11-29(b», we keep the exchange parameter fixed at JIJ = I and vary the single-site parameter DIJ. At DIJ = 0, which is equivalent to the midpoint of line (a), the Si-gap starts out nonzero and grows with increasing DIJ, which reflects the fact that low-frequency out-of-

Section 11-29

247

plane fluctuations are more and more suppressed as the easy-plane anisotropy becomes stronger. The S~-gap, by contrast, has a decreasing trend. Starting from the same value at DIJ=O, it reaches a minimum near zero at DIJ -1.0 and then grows again. The minimum marks the (H/S)-transition between two non-magnetic phases, where only the staggered in-plane fluctuations become critical. Now we shift the line 0 ~ DJ ~ 2 in parameter space from Jp = 1 (Fig. 11-29(b)) to JIJ = 1.5 (Fig. 11-29(c)). At DJJ =0 the system is in the Neel phase. Here the S~-gap is large, reflecting the transverse spin-waves of a uniaxial antiferromagnet, whereas the Si-gap is effectively zero, reflecting the twofold degeneracy of the ground state associated with antiferromagnetic ordering. As DIJ increases from zero, the Si-gap stays near zero up to the (NIH)-phase boundary at DIJ .. 0.5, where it starts to increase very rapidly with no further change in course. The S~-gap decreases from some large value as DIJ increases from zero. It goes through a minimum near zero at DIJ - 1.7, marking, as in Fig. 11-29(b), the (H/S)transition, now shifted to a higher value of single-site anisotropy in agreement with the broadly accepted picture. 1.5--,----------------------,------,

1.0

0.5

o. 0 ~~~:::_--=~~~*fZ::::e:::£l=£l=:ljl=a:a=&=lEl=!jl::::a:::£l:=£l:::~ -3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

D/J

s:

Figure 11-30: Dynamically relevant excitation gaps for operators (circles) and ~ (squares) along the line JjJ = 0, -3 s; D/J s; 0, in the parameter space of Hamiltonian (11.118). The gaps are derived from K=6 nearly size-independent continued-fraction coefficients ~r(1t), k=1, ..., K computed by means of the recursion method from the N=12 ground-state wave function. The vertical line indicates the phase boundary between two different critical phases.

248

Chapter 11

The significance of the results displayed in Fig. 11-29 is that they have been derived entirely from the ground-state wave function of a chain with just N=12 spins. It is remarkable that the spectral signatures of a rather complex phase diagram are so clearly encoded in that quantity and that this information is so easily retrievable. As a further demonstration of the versatility of this type of gap analysis for the stated purpose, we have carried out the same analysis along a fourth line, 0 ~ DU ~ -3 at fixed J!J = 0, in the parameter space of the Hamiltonian (11.118). It represents an XX model with easy-axis anisotropy. For DJJ near zero, the system is in the critical phase (C) discussed previously. As the easy-axis anisotropy gains strength, it was predicted that (C) becomes unstable and that a transition to a different critical phase (C2) takes place - a phase which represents the critical ground-state of an effective spin-1I2 chain [Schulz and Ziman 1986]. This transition is reflected in our simple dynamical gap analysis as shown in Fig. 11-30 for DU decreasing from zero, the S~-gap stays near zero as far as the (C)-phase lasts, Le down to DU .. -2.3, where it starts to increase. The Si-gap starts out large and becomes effectively zero at the same DU-value. It stays small beyond that point.

11·30 Dispersion and Line Shapes The same type of gap analysis carried out for the q-dependence of the operator ~ yields the dispersion of the lowest branch of excitations which is dynamically relevant for SII (q,ro). Toward our second goal we reconstruct the dynamic structure factor itself from the coefficients Ll']II(q), ...., Llkll(q) and the split-Gaussian termination function. Here we limit the discussion to a couple of specific issues: What is the main difference in line shape between the functions SIIII(q,ro) of the s=l Heisenberg antiferromagnet and its s=1I2 counterpart, and what is the impact of a small easy-plane single-site anisotropy on the peak positions and line shapes in the s=1 case? Figure 11-31(a) shows the reconstructed (normalized). dynamic structure factor SIIII(q,ro) == SlIiq,ro)J<$qSli.q> at q=1t/6 and q=51t16 for the ID s=1 Heisenberg antiferromagnet. For both wave numbers the function consists of a single peak with nonzero intrinsic linewidth. This is in strong contrast to the results obtained from truncated continued fraction [Haas, Riera, and Dagotto 1993] or equivalent procedures [Golinelli, Jolicoeur, and Lacaze 1993], which are sums of O-functions, often broadened into Lorentzians for graphical representation. We note two main differences between the results of Fig. 11-31(a) and the corresponding results of the s=1I2 model (Fig. 11-23(a»: (a) the peak positions at wave numbers q and 1t-q are the same for s=1I2 but differ substantially for s=1. The symmetric dispersion of the s=1I2 chain is exactly known and the asymmetric dispersion of the s= 1 chain has been well established by extensive numerical diagonalizations [Haas et al. 1993, Golinelli et al. 1993]. (b) The linewidth is known to increase monotonically with increasing q in the s=1I2 case [Muller et al. 1981], but in Fig. 11-31(a) the s=1 result at small q has considerably larger linewidth than the one at q near 1t. Our s=1 results are consistent with the findings of Haas et al. and Golinelli et al. that the spectral weight in SIIII(q,ro) is dominated

Section 11-30

249

by a single o-function at q near 1t but less so at small q. The opposite trends of line broadening in the s= 1/2 and s= I chains may be understood by interpreting the observable excitations as composites of different kinds of elementary states. The presence of some easy-plane anisotropy, DU = 0.18, alters the functions SIlIl(1tI6,ro) and SIl/51t16,ro) as shown in Fig. 11-31(b). In-plane (xx) and out-of-plane (zz) fluctuations are now represented by separate peaks. At small q, the in-plane peak has moved up and the out-of-plane peak down. At Small1t-q, they have moved in opposite directions. The anisotropy causes an increase in in-plane linewidths. The peak positions of the reconstructed functions SIl/51t16,ro) shown in Fig. 11-31 are in good agreement with the energies of the lowest-lying dynamically relevant excitation of a chain with N=12 spins. The latter, quoted from Golinelli et al. [1993], are indicated by arrows in Fig. 11-31. However, for q=1tI6 the peaks lie significantly higher than the lowest N=12 excitation.

(a)

D/J=O

-.. 3

- - xx=zz

J z /J=l

-!.

&

. . . . .: t 10.0 :t

ICIJ

0.0

q=rr/6

(b)

xx zz

J z /J=l D/J=0.18

-.. 3

&

. . . . .: t 10.0 :t

len

q=rr/6

0.0 --l----~~=~===::::::::.._...:::::::::=:j~ .........;:::===_l 3.0 2.0 1.0 0.0

(,)/J Figure 11-31: Normalized dynamic structure factors SI'I'(q,Ol)/<~~q> at T=O of (a) the 10 5=1 Heisenberg antiferromagnet and (b) the same model with an additional easy-plane single-site anisotropy D/J = 0.18. We show results for Il=x,z and q=rtl6, 51t/6. They have been derived from K=6 continued-fraction coefficients ,i~I'(q). k=1, ... ,K extracted from the N=12 ground-state wave function combined with the same continued-fraction analysis as used in Sec. 11-27. The arrow near the peak of each curve indicates the energy of the lowest excitation with nonzero matrix element in SI'I'(q,Ol) of the N=12 chain [from Zhang et al. 1994].

250

Chapter 11

=

=

The significance of the point (Jp 1, D/J 0.18) in the parameter space of (11.118) is its physical realization by the quasi-ID magnetic compound Ni(C2HgN2)IV0 2C104 (NENP). A recent inelastic neutron scattering study on that compound [Ma et al. 1992] observes well defined resonances for 0.3 S q S n, including q = 57t16, where our result predicts a peak with small but nonzero linewidth. At smaller wave numbers, including q = 7tl6, the resonance in the experiment has disappeared in a broad background intensity. Here our analysis predicts a broad signal, which, when multiplied by the very small integrated intensity, becomes indeed undetectable for all practical purposes.

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