LANDAU FERMI-LIQUID THEORY Concepts and Applications
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LANDAU FERMI-LIQUID THEORY Concepts and Applications GORDON BAYM University of Illinois at Urbana-Champaign and CHRISTOPHER PETHICK Nordita, Copenhagen and University of Illinois at Urbana-Champaign WI LEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank LANDAU FERMI-LIQUID THEORY Concepts and Applications GORDON BAYM University of Illinois at Urbana-Champaign and CHRISTOPHER PETHICK Nordita, Copenhagen and University of Illinois at Urbana-Champaign WI LEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free oferrors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available I)om the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Ribliothek lists this publication in the Deutsche Nationalbibliogratie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 199 I by John Wiley & Sons, Inc. 0 2004 WII,T:Y-Vc'I I Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those oftranslation into other languages). No part or this book may be reproduced in any form ~ nor transmitted or translated into machinc language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Printing Strauss CimbH, Miirlenbach Bookbinding Litgcs & Dopf Buchbinderei Gmbl I, I leppenheim ISBN-13: 978-0-47 1-824 1 8-3 ISBN-10: 0-47 1-824 18-6 This Page Intentionally Left Blank CONTENTS Preface vii 1 Landau Fermi-Liquid Theory and Law Temperature Properties of Normal Liquid %e I Introduction, 1 1.1 Static Properties, 2 1.1.1 Quasiparticles, 2 1.1.2 Quasiparticle Energy and Interactions, 4 1.1.3 Equilibrium Properties, 10 1.2 Nonequilibrium Properties, 16 1.2.1 Quasiparticle Energies and Interaction, 16 1.2.2 The Kinetic Equation, 18 1.2.3 The Conservation Laws, 21 1.2.4 Transport Coefficients, 25 1.3 Collective Effects, 45 1.3.1 Sound in Fermi Liquids, 45 1.3.2 Spin Waves and Related Phenomena, 56 1.3.3 Response Functions, Inequalities, and Form Factors, 69 1.4 Scattering of Quasiparticles and Finite Temperature Effects, 76 1.4.1 Landau Parameters and Scattering Amplitudes, 76 1.4.2 The Low Temperature Transport Coefficients of Liquid 3He; Theory and Experiment, 85 1.4.3 Finite Temperature Transport Coefficients, 86 1.4.4 Finite Temperature Contributions to the Specific Heat and Magnetic Susceptibility, 95 1.5 Concluding Remarks, 113 Appendix A: Some Useful Fermi Integrals, 114 Appendix B: Properties of Q,,, 116 V vi CONTENTS Appendix C Fermi Liquid Parameters for Liquid 3He, 117 2 Low Temperature Properties of Dilute Solutions of 'He in Superhid 4He 123 Introduction, 123 2.1 Elementary Excitations of Dilute Solutions, 125 2.2 Properties of one 3He Atom in 4He at T = 0, 128 2.2.1 Volume Occupied by 3He, 128 2.2.2 3He Effective Mass, 130 2.3 Interactions of the 3He at Very Low Temperature, 131 2.3.1 'He Landau Parameters, 131 2.3.2 Low Temperature Properties of Dilute Solutions, 134 2.3.3 Phenomenological Effective Interaction, 138 2.3.4 Microscopic Approaches to the Effective Interaction, 142 2.4 Interaction Between the 3He and 4He, 146 2.4.1 Effectives of Superfluid Flow on 'He Quasiparticles, 147 2.4.2 Interaction of 3He Quasiparticles with Long-Wavelength Phonons, 149 2.4.3 Scattering of Phonons by 'He Quasiparticles, 155 2.4.4 First Sound in Dilute Solutions, 158 2.4.5 Second Sound, 167 2.4.6 Transport Properties, 168 3 Further Developments 177 Introduction, 177 3.1 Liquid 3He, 177 3.1.1 Quasiparticle Spectrum and Thermodynamic Properties, 177 3.1.2 Measurements of Transport Properties, 178 3.1.3 Density and Spin Fluctuations, 179 3.1.4 Calculations of Scattering Amplitudes, 180 3.1.5 Superfluid 3He and the Landau Theory of Fermi Liquids, 181 3.2 Dilute Solutions of 'He in Superfluid 4He, 183 3.2.1 Equilibrium and Transport Properties, 183 3.2.2 Higher-Momentum Excitations, 184 3.3 Spin-Polarized Systems, 185 3.3.1 Spin-Polarized 'He, 185 3.3.2 Dilute Solutions of 3He in 4He, 186 3.3.3 Other Systems, 187 3.4 Nuclear Applications, 187 3.4.1 Particles and Quasiparticles, 188 3.4.2 Quasiparticle Interactions in Nuclei and Nuclear Matter, 189 3.5 Electrons in Metals, 191 INDEX 201 Landau’s theory of Fermi liquids, advanced in 1956, stands out as one of the high points of modern theoretical physics, a theory whose profun- dity goes beyond mere phenomenology. Although strongly interacting many-particle systems are very difficult to describe in general, Landau saw, with great insight, that at low temperature an exact simplicity arises, enabling one to describe the properties of extended systems in terms of a dilute collection of elementary excitations. These elementary excitations, or “quasiparticles,” are not merely the motion of single particles, but represent the motion of many particles of the system simultaneously. As Landau realized, one can keep track of the thermodynamic and transport properties, that is, do the bookkeeping on conserved quantities and their currents, in terms of the elementary excitations; the detailed proof of this picture requires extensive use of field theory, including conservation laws, associated Ward identities, and renormalization theory. At the same time that the Landau theory has deep conceptual roots, it is a useful and practical tool for accounting for the low temperature properties of Fermi liquids in terms of a rather small number of parameters. Beyond being a vital part of a condensed matter physicist’s culture, the theory is a necessary skill for the practitioner. The major emphasis in this book is on the practical development and application of the theory, rather than on the description of its microscopic derivation. The first chapter delineates the basic theory of the thermody- namic and nonequilibrium properties of Fermi liquids, with application to liquid 3He in its normal state. The second chapter discusses the important and tractable extension of the Fermi liquid theory to dilute solutions of 3He in superfluid 4He, an experimentally rich system where, in addition to Fermi excitations, boson degrees of freedom-of the 4He-play a role. viii PREFACE The basis of this book, two review articles on the Landau Fermi-liquid theory and the application to dilute solutions of 3He in superfluid 4He, were originally written as part of a collection of articles in the books, The Physics of Liquid and Solid Helium, edited by K. H. Bennemann and J. B. Ketterson (Wiley, Part 1, 1976, and Part 2, 1978). In retrospect, because of the general utility of the Landau Fermi-liquid theory, it appeared to be valuable to make them more widely available in the present format. Despite the passing of over a decade, the two review articles remain timely, having been written to emphasize the fundamental ideas of the theory. Experiments on normal ’He in the intervening years have contin- ued to confirm the general theory, and provide information on its exten- sion to higher temperatures. In addition, the theory has found further application in other areas, such as nuclear and neutron star matter, superfluid ’He, and modern problems in superconductivity. To avoid resetting type for the entire book, we have chosen to give a brief summary of the developments in the physics of Fermi liquids since the writing of the original text in an added final Chapter 3, and at the same time have made minor modifications to the introductory sections of the two original chap- ters. Citations in Chapter 3 references in earlier chapters are given in the format “[Chap. 1: 46, 1021.’’ We would like to take this opportunity to thank our colleagues in Urbana and Copenhagen for the many interesting and provocative conver- sations over the years that have helped to shape this monograph. The supportive environment of Nordita, and the continued funding by the U.S. National Science Foundation of our research in low temperature physics, have both contributed greatly to our writing of the present volume. We dedicate this book to the memory of our friend and colleague, John Wheatley, whose virtuosity in extracting the physics of Fermi liquids, through both his imaginative experiments and his challenging and probing interactions with theorists, made the field for us vital and exciting. GORDONBAYM CHRISTOPHERPETHICK Urbana, Illinois Covenhagen, Denmurk 1 LANDAU FERMI-LIQUID THEORY AND LOW TEMPERATURE PROPERTIES OF NORMAL LIQUID 3He INTRODUCTION Landau advanced his theory of normal Fermi liquids [l,21 in 1956, and by the early 1960s, experiments indicated that the predictions of the Landau theory were satisfied, at least qualitatively, by liquid 3He. During the past three decades, precision measurements of the properties of liquid 3 He have been carried out at temperatures below 100 mK, where the Landau theory enables one to calculate. More recent theoretical and experimental attention has turned to properties at higher temperatures, and phenomena that vary rapidly in space and time. We shall focus our attention here chiefly on theoretical deveIopments and related experi- ments. A detailed account of experimental work on Fermi liquid proper- ties of 3He is given by Wheatley in his review article [3].