Field Theories for Low-Dimensional Condensed Matter Systems: Spin Systems and Strongly Correlated Electrons

285 Pages·2000·7.854 MB·English

by  Giuseppe Morandi, Pasquale Sodano (auth.), Professor Giuseppe Morandi, Professor Pasquale Sodano, Professor Arturo Tagliacozzo, Professor Valerio Tognetti (eds.)

Springer Series in
SOLID-STATE SCIENCES 131
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Physics and Astronomy
http://www.springer.de/phys/
Springer Series in
SOLID-STATE SCIENCES
Series Editors:
M. Cardona P. Fulde K. von Klitzing R. Merlin H.-J. Queisser H. Stormer
The Springer Series in Solid-State Sciences consists of fundamental scientific
books prepared by leading researchers in the field. They strive to communica
te, in a systematic and comprehensive way, the basic principles as well as new
developments in theoretical and experimental solid-state physics.
126 Physical Properties of Quasicrystals
Editor: Z.M. Stadnik
127 Positron Annihilation in Semiconductors
Defect Studies
By R. Krause-Rehberg and H.S. Leipner
128 Magneto-Optics
Editors: S. Sugano and N. Kojima
129 Computational Materials Science
From Ab Initio to Monte Carlo Methods
By K. Ohno, K. Esfarjani, and Y. Kawazoe
130 Contact, Adhesion and Rupture of Elastic Solids
ByD. Maugis
131 Field Theories for Low-Dimensional Condensed Matter Systems
Spin Systems and Strongly Correlated Electrons
By G. Morandi, P. Sodano, A. Tagliacozzo, and V. Tognetti
Series homepage - http://www.springer.de/phys/books/sss/
Volumes 1-125 are listed at the end of the book.
Giuseppe Morandi Pasquale Sodano
Arturo Tagliacozzo Valerio Tognetti
(Eds.)
Field Theories
for low-Dimensional
Condensed Matter
Systems
Spin Systems
and Strongly Correlated Electrons
With 55 Figures
i
Springer
Professor Giuseppe Morandi Professor Pasquale Sodano
Physics Department, University of Bologna Physics Department, University of Perugia
Viale Berti Pichat, 6/2 Via A. Paseoli
40127 Bologna, Italy 06123 Perugia, Italy
E-mail: morandiObo. infn . it E-mail: sodanoOpg. infn. it
Professor Arturo Tagliacozzo Professor Valerio Tognetti
Physics Department Physics Department, University of Florenee
University of Naples "Federico II" Largo E. Fermi, 2
Monte S. Angelo, Via Cintia 50125 Florenee, Italy
80126 Naples, ltaly E-mail: tognettiOfi.infn.it
E-mail: arturoOna. infn. it
Series Editors:
Professor Dr., Dres. h. c. Manuel Cardona
Professor Dr., Dres. h. c. Peter Fulde*
Professor Dr., Dres. h. c. Klaus von Klitzing
Professor Dr., Dres. h. c. Hans-Joachim Queisser
Max-Planek-Institut rur Festkorperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany
* Max-Planek-Institut rur Physik komplexer Systeme, Nothnitzer Strasse 38
01187 Dresden, Germany
Professor Dr. Roberto Merlin
Department of Physics, 5000 East University, University of Michigan
Ann Arbor, MI 481°9-1120, USA
Professor Dr. Horst Stormer
Dept. Phys. and Dept. Appl. Physics, Columbia University, New York, NY 10023 and
Bell Labs., Lueent Teehnologies, Murray Hill, NJ 07974, USA
ISSN 0171-1873
ISBN 978-3-642-08650-2
Ubrary of Congress Cataloging-in-Publication Data
Field theories for low-dimensional condensed matter systems : spin systems and strongly correlated electrons
/ G. Morandi ... [et al.]. p.cm. -(Springer series in solid-state sciences; 131) IncJudes bibliographical references.
ISBN 978-3-642-08650-2 ISBN 978-3-662-04273-1 (eBook)
DOI 10.1007/978-3-662-04273-1
1. Condensed matter. 2. Field theory (Physics) I. Morandi, Giuseppe. II. Series.
QCI73.454.F542000 530.4'I-dc21 00-028515
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or
parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its
current version, and permission for use must always be obtained from Springer-VerIag Berlin HeideIberg GmbH.
Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 2000
Originally published by Springer-Verlag Berlin Heidelberg New York in 2000
Softcover reprint of the hardcover 18t edition 2000
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
Typesetting: Camera ready by the authors using a Springer TEX macro package
Cover concept: eStudio Calamar Steinen
Cover production: design & production GmbH, Heidelberg
Printed on acid-free paper SPIN 10753964 57/3141/di 5 4 3 2 1 o
Preface
This book is especially addressed to young researchers in theoretical physics
with a basic background in Field Theory and Condensed Matter Physics. The
topics were chosen so as to offer the largest possible overlap between the two
expertises, selecting a few key problems in Condensed Matter Theory which
have been recently revisited within a field-theoretic approach.
The presentation of the material is aimed not only at providing the reader
with an overview of this exciting frontier area of modern theoretical physics,
but also at elucidating most of the tools needed for a technical comprehen
sion of the many papers appearing in current issues of physics journals and,
hopefully, to enable the reader to tackle research problems in this area of
physics. This makes the material a live creature: while not pretending it to
be exhaustive, it is tutorial enough to be useful to young researchers as a
starting point in anyone of the topics covered in the book.
Each chapter in this book has its own standpoint as guideline and intro
duces the reader to the different techniques used in dealing with the many
theoretical aspects of today's Condensed Matter Physics. The material of
each chapter is rather self-contained, except for Chap. 3, where it is required
some familiarity with Conformal Field Theory. An introduction to this sub
ject, as well as to most of the material presented in this book, may be found
in the books quoted in the Introduction. Chapter 6 is mostly sketchy and
well reflects the outburst of creative activity in the young field of Mesoscopic
Physics.
After a short introduction, the topics are organized into six chapters,
each inspired by the mini-courses delivered at the "Workshop with Learning"
on "Field Theories for Low-Dimensional Condensed Matter Systems: Spin
Systems and Strongly Correlated Electrons" held in Chia - Laguna (Italy) in
the fall of 1997. Lecturers were (in the order of the chapters appearing in the
book) H.J. Schulz, R.B. Laughlin, L Affleck, A. Auerbach, G. Semenoff and
B.L. Altshuler. This stimulating workshop was promoted by the Laboratory
"Forum of Condensed Matter Theory" of the Italian Istituto N azionale di
Fisica della Materia (LN.F.M.) and greatly benefited from the close interplay
with the Fifth Chia Meeting on "Common Trends in Condensed Matter and
High Energy Physics" organized by Profs. A. Barone of the University of
Naples and A. Devoto of the University of Cagliari.
VI Preface
This book would not be out without the moral support of many friends
and the financial support of the Laboratory "Forum of Condensed Matter
Theory" of the Istituto Nazionale di Fisica della Materia (LN.F.M.), of the
Gruppo Nazionale di Struttura della Materia (G.N.S.M.) and the Istituto
Nazionale di Fisica Nucleare (LN.F.N.).
We greatly thank all the lecturers, senior researchers and students who
attended the Chia Workshop; their enthusiasm, interest and skills were de
termining at all the stages of this enterprise; in particular, we are grateful to
R.B. Laughlin who advised us in the early stages of the project and strongly
encouraged us to converge towards the final form of this book. We warmly
thank A. Devoto for his invaluable help and, most importantly, for keeping
alive in Chia -Laguna the charming tradition of periodical international meet
ings on the interplay between Field Theory and Condensed Matter Physics;
we feel that, without this long standing tradition, this project for a book
could not have taken its shape.
Special thanks are due to Giovanna Savoldi of LN.F.M. (Unita di Genova)
and Isabella Panico of LN.F.N. (Sezione di Perugia), who collaborated with
great skill and enthusiasm in gathering the first materials for the book. We
thank also Guido Celentano who greatly helped in preparing the final version
of the book.
Most sadly, shortly after completing his contribution, Prof. H.J. Schulz
passed away. We mourn him as a great person and an outstanding scientist
and we feel that his death is an invaluable loss for the whole scientific com
munity. We like to dedicate this volume to his memory.
Naples, March 2000 Giuseppe Momndi
Pasquale Sodano
Arturo Tagliacozzo
Valerio Tognetti
Contents
1 Introduction
Giuseppe Morandi, Pasquale Sodano, Arturo Tagliacozzo,
and Valerio Tognetti ............................................ 1
References ............................................ . . . . . . . . . 5
2 Fermi Liquids and Luttinger Liquids
Heinz J. Schulz, Gianaurelio Cuniberti, and Pierbiagio Pieri 9
2.1 Introduction................................................ 9
2.2 Fermi Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 The Fermi Gas ........................................ 10
2.2.2 Landau's Theory of Fermi Liquids. . . . . . . . . . . . . . . . . . . . . . .. 12
2.2.3 Microscopic Basis of Landau's Theory. . . . . . . . . . . . . . . . . . .. 17
2.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
2.3 Renormalization Group for Interacting Fermions ................ 22
2.3.1 One Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.3.2 Two and Three Dimensions ............................. 28
2.4 Bosonization and the Luttinger Liquid . . . . . . . . . . . . . . . . . . . . . . . .. 32
2.4.1 Spinless Model: Representation of Excitations ............. 32
2.4.2 Model with Spin: the Concept of the Luttinger Liquid. . . . .. 39
2.4.3 Summary............................................. 47
2.5 Applications..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
2.5.1 Transport............................................. 48
2.5.2 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
2.5.3 The Spin-1/2 Chain as a Luttinger Liquid. . . . . . . . . . . . . . . .. 59
2.6 Spin Ladders and Coupled Luttinger Liquids. . . . . . . . . . . . . . . . . .. 69
2.6.1 Coupled Spin Chains .................................. 69
2.6.2 Two Coupled Luttinger Liquids. . . . . . . . . . . . . . . . . . . . . . . . .. 71
2.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
Appendix: When to Bosonize in Peace. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
References .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
VIII Contents
3 Quantum Number Fractionalization in Antiferromagnets
Robert B. Laughlin, Domenico Giuliano, Raffaele Caracciolo,
and Olivia L. White ............................................ 83
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
3.2 Haldane-Shastry Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
3.3 Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.4 Spinons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
3.5 Annihilation Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
3.6 Spin Current ............................................... 102
3.7 Supersymmetry ............................................. 104
3.8 Holons ....... '. ............................................. 106
Appendix A: Fourier Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Appendix B: Problems ........................................... 112
References ..................................................... 115
4 Conformal Field Theory Approach
to Quantum Impurity Problems
Ian Affleck ..................................................... 117
4.1 Old Results on the Kondo Problem ............................ 117
4.1.1 Reduction to 1D ....................................... 118
4.1.2 Perturbative RG ....................................... 119
4.1.3 Local Fermi Liquid Theory .............................. 120
4.2 Some Basic Conformal Field Theory/ Luttinger Liquid Techniques . 124
4.3 CFT Approach to the Kondo Problem ......................... 132
4.3.1 Current Algebra Formulation and the Infrared Fixed Point .. 132
4.3.2 BCFT and the Multi-Channel Kondo Problem ............. 136
References ..................................................... 141
5 Quantum Magnetism Approaches
to Strongly Correlated Electrons
Assa Auerbach, Federico Berruto, and Luca Capriotti ............... 143
5.1 Deriving the Effective Hamiltonian ............................ 143
5.1.1 Two-Site Hubbard Model ............................... 144
5.1.2 Renormalization to Low Energy Subspace ................. 145
5.1.3 From Hubbard to t - J and Heisenberg Models ............ 147
5.1.4 The Negative-U Hubbard Model ......................... 149
5.2 Quantum Magnetism ........................................ 150
5.2.1 Spin Coherent States .................................. 150
5.2.2 Spin Path Integral ..................................... 151
5.2.3 Spin Wave Theory ..................................... 153
5.2.4 Continuum Theory for Anisotropic Models ................ 155
5.2.5 Anisotropic Quantum Rotators .......................... 157
5.2.6 Haldane's Mapping .................................... 159
5.2.7 Spin Liquid States ..................................... 160
Contents IX
5.3 Pseudospins and Superconductivity ............................ 161
5.3.1 Charge Density Wave to Superconductivity ................ 162
5.3.2 Order of Transition from Rotator Theory ................. 163
5.3.3 SO(5) Rotator Theory and High-Tc Superconductors ...... 165
5.3.4 Vortex Dynamics in Superfluids ......................... 167
References ..................................................... 170
6 Introduction to Some Common Topics in Gauge Theory
and Spin Systems
Gianluca Grignani and Gordon W. Semenoff ....................... 171
6.1 Overview ................................................... 171
6.2 What Is a Gauge Theory? .................................... 173
6.2.1 Electrodynamics ....................................... 174
6.2.2 Geometry ............................................. 180
6.2.3 Schrodinger Equation for a Charged Particle ............... 183
6.2.4 Bohm-Aharonov Phase ................................. 185
6.2.5 Classical Yang-Mills Theory ............................. 186
6.3 Relativistic Fermions ........................................ 191
6.3.1 1+1 Dimensions ....................................... 195
6.3.2 2+1 Dimensions ....................................... 198
6.3.3 3+1 Dimensions ....................................... 200
6.4 Quantized Yang-Mills Theory ................................. 202
6.4.1 Canonical Quantization of Electrodynamics ............... 207
6.4.2 Path Integrals ........................................ 209
6.5 Quantum Spins and Strong Coupling in Gauge Theory ........... 211
6.5.1 Overview of the Problem ................................ 211
6.6 Staggered Fermions .......................................... 216
6.7 Strong Coupling Lattice Gauge Theory ......................... 220
6.7.1 Hamiltonian Formulation of Lattice Gauge Theory ......... 220
6.7.2 Strong Coupling ....................................... 222
6.8 Results on Strong Coupling ................................... 228
References ..................................................... 231
7 Quantum Chaos and Transport in Mesoscopic Systems
Boris L. Altshuler, Cristiano Biagini, and Marco Patriarca ........... 235
7.1 The Dimensionless Conductance 9 ............................. 235
7.2 Quantum Chaos ............................................ 236
7.2.1 Beyond Random Matrix Theory ......................... 241
7.2.2 From Disordered to Chaotic Systems ..................... 244
7.3 Quantum Dots .............................................. 249
7.3.1 Spectroscopy .......................................... 249
7.3.2 Orthodox Theory ...................................... 251
7.3.3 Od Hamiltonian ........................................ 254
7.4 Quasiparticle Lifetime in a Finite System ....................... 257
X Contents
7.4.1 Transition Region ...................................... 261
References ..................................................... 267
Index ......................................................... 271