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Jose Gonzalez Miguel A. Martin-Delgado
German Sierra Angeles H.Vozmediano
Quantunl Electron Liquids
and High-T
c
Superconductivity
Springer
Authors
JoseGonzalez
InstitutodeEstructuradelaMateria
CSIC,Serrano123
E-28oo6 Madrid,Spain
MiguelA.Martfn-Delgado
DepartamentodeFfsicaTe6ricaI
FacultaddeCienciasFfsicas
UniversidadComphitensedeMadrid
E-28040 Madrid,Spain
GermanSierra
InstitutodeMatematicasyFfsicaFundamental
CSIC,Serrano123
E-28006 Madrid,Spain
Angeles H.Vozmediano
DepartamentodeMatematicas
UniversidadCarlosIIIdeMadrid
E-28913Leganes (Madrid),Spain
Cataloging-in-Publicationdataappliedfor.
..
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Quantum electron liquids and high-t superconductivity/ J.
Gonzalez ... - Berlin; Heidelberg; New York: Barcelona;
Budapest ; Hong Kong; London ; Milan ; Paris ; Tokyo :
Springer, 1995
(Lecture notes in physics: N.s. M, Monographs; Vol. 38)
ISBN 3-540-60503-7
NE: Gonzalez, Jose; Lecture notes in physics I M
ISBN3-540-60503-7Springer-Verlag Berlin Heidelberg·NewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe
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©Springer-VerlagBerlinHeidelberg1995
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Preface
This book originated from a course given at the Univcrsidad Aut6noma of
Madrid in the Springof1994 and in the Universidad ComplutenseofMadridin
1995. The goal of these courses is to give the non-specialist an introduction to
someold and new ideas in thefield ofstrongly correlated systems, in particular
the problems posed by the high-1~ superconducting materials. As theoretical
physicists, our starting viewpoint to address the problem of strongly correlat
ed ferlnion systems and related issues of modern condensed matter physics·is
the renormalization group approach applied both to quantU111 field theory and
statistical physics. In recent years this has become not only a powerful tool for
retrieving the essential physics of interacting systems but also a link between
theoretical physics and modern condensed matter physics. Furthermore, once
we have this common background for dealing with apparently different prob
lems, we discuss more specific topics and even phenomenological aspects ofthe
field. In doing so we have tried to make the exposition clear and simple, with
out entering into technical details but focusing ill the fundamental physics of
thephenomenaunder study. Therefore,veexpect that ourexperiencell1ayhave
some value to other people entering this fascinating field.
We have divided these notes into three parts and each part into chapters,
which correspond roughly to one or two lectures.
Part I, Chaps. 1-2 (A.H.V.), reviews the essentials of the Landau Fermi
liquid theory and the modern approach of the renormalization group methods
as applied to fermionic systems.
Part II, Chaps. 3-5 (J.G.), discusses the 1d electron systems and the Lut
tinger liquid concept using different techniques: the renormalization group ap
proach, bosonization, and the correspondence between exactly solvable lattice
models and continuumfield theory.
Part III, Chaps. 6-11 (l\1.A.M.-D. and G.S.), introduces the basic phe
nomenology of the high-T compounds and the different theoretical nlodels to
c
explain their behaviour: Hubbard, t-J, Heisenberg. A modern review of the
real-space renormalization group method is also given.
VI
We would like to express our gratitude to all the people who have helped
us through discussions but especially to J.L. Alonso, L. Brey, J.G. Esteve, J.
Ferrer, G. Gomez-Santos, F. Guinea, F. Jimenez, and C. Tejedor.
M.A.M.-D. wishes to thank··Artemio Gonzalez-Lopez for many computer
hints in preparing part III ofthe manuscript and for sharing with us his access
tothe Alphamachine Ciruelowith which someofthecomputationsinthis book
were carried out.
One of us (MAHV) wants to thank Xenia de la Ossa for her help with the
figures and the hospitality of the Institute for Advanced Study of Princeton
where her part ofthe manuscript was completed.
J. Gonzalez
M.A. l\1artin-Delgado
l\fadrid
G. Sierra
August 1995
A.H. Vozmediano
Contents
I
1 Fermi Liquid in D ~ 2 . . . . . . . . . . 3
1.1 Introduction . 3
1.2 l'he Landau Fernli Liquid Theory 3
1.2.1 The ~1aguitudesofInterest 4
1.2.2 l'he Landau IIypothesis ... 5
1.2.3 Review of the Fermi Gas . .. . 5
1.2.4 The Concept of Quasiparticles. The Phonon Analogy 7
1.2.5 1"'he l'heory of the Effective Mass . 9
1.3 Method ofSecond Quantization and Green's Functions . 11
1.3.1 Similarities to and Differences from Quantum Field The-
ory. The Dirac Versus the Fermi Sea . 11
1.3.2 Second Quantization and the Field Operators 12
1.3.3 The One-Particle Green's Function ..... 14
1.3.4 General Properties of the Green's Function. 16
1.3.5 Analytic Properties of the Green's Function 17
1.3.6 Physical ~feaningof G(p,w). The Spectrum 19
1.3.7 The Momentum Particle Distribution . . . . 21
1.3.8 Computing the Green's Function. Wick Theorem and
~eynmanDiagrams . . . . . . . . . . . . . . . . 22
1.3.9 The Four-Point Function. Bosonic Excitations. 25
1.3.10 11arginal Fermi Liquids .. 28
References . . . . . . . . . . . . . . . . . . . . . . 29
2 Effective Actions and the Renormalizatioll Group 31
2.1 Introduction.................. 31
2.2 The Renormalization Group in Quantum Field Theory 32
2.2.1 The Physical Origin of the Divergences . . .. 32
2.2.2 The Expression ofthe Divergences in QFT . . .. 33
2.2.3 Renorulalization. Substracting Infinities ...... 34
2.2.4 Renormalizatioll Prescriptions. TheRenormalizationGroup 38
2.2.5 Uses of the Renormalizatioll Group in QFT. Fixed Points 40
VIII
2.2.6 Effective Field Theory. The Classification of Operators. 41
2.3 The Renormalization Group in Statistical Physics . . . . . . .. 45
2.3.1 Renormalization Group Analysis of Critical Phenomena. 46
2.4 Renormalization Group Analysis of the Fermi Liquid .. . . .. 50
2.4.1 The Gaussian Model ....-. . . . . . . . . . . . . . .. 51
2.4.2 The Fermi Liquid as Fixed Point of the Renormalization
Group . . . . . . . . . . . . 56
2.4.3 Comments on Fine Points 60
2.5 Non-Fermi Liquids 62
References . . . . . . . . . . . . . . . . . 66
II
3 Electronic Systems in d=1 . . . . . . . . . . . . . . 71
3.1 Introduction . 71
3.2 Perturbation Theory. Renormalization Group 72
3.2.1 Interactions....... 73
3.2.2 Quantum Corrections . . . . 76
3.2.3 Renormalization Group ... 79
3.2.4 Ground-State Properties .. 82
References . . . . . . . . . . . . . 86
4 Bosonization. Luttinger Liquid 87
4.1 Luttinger Model. Bosonization . 87
4.1.1 Bosonic Excitations. 87
4.1.2 Bosonization . 91
4.1.3 Interacting Theory . . . 93
4.2 Charge-Spin Separation. Luttinger Liquid 97
4.2.1 Charge-Spin Separationin a Simple Case. . 97
4.2.2 Boson Representation of Fermion Operators 99
4.2.3 Electron Green Function . . . . . . . . . . . 102
4.2.4 Intuitive Picture of Charge-Spin Separation 105
References . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Correspondence from Discrete to Continuum Models 109
5.1 Introduction . 109
5.2 The Harmonic Chain . . . . . . 110
5.2.1 Continuum Limit .... 110
5.2.2 Correlation Functions. . 113
5.2.3 Massive Interactions 115
5.3 The Hubbard Model .. . . . . 116
5.3.1 Weak Coupling ..... 117
5.3.2 Large-U Limit. Correlation Functions 121
References . . . . . . . . . . . . . . . . . . . . .. 124
IX
III
6 From the Cuprate Compounds to the Hubbard Model 127
6.1 Phenomenology ofthe Cuprate Compounds .... 127
6.1.1 Lattice Structure ofthe Cuprates . . . . . . 127
6.1.2 Basic Features of the Cuprates Phase Diagrams 129
6.1.3 Normal State Properties ofthe Oxide Superconductors . 138
6.2 Hubbard ModelDescription of the Cuprate Compounds. 143
6.2.1 One-Band Hubbard Model. . 146
6.2.2 Three-Band Hubbard Model. 147
References . . . . . . . . . . . . . . . . . . . . 149
7 The Mott Transition and the Hubbard Model 151
7.1 Mott Theory ofthe Metal-Insulator Transition. 151
7.2 The Hubbard Approximation . 156
7.2.1 Hubbard Approximation . . . . . . . . . 158
7.2.2 Hubbard Parameters . . . . . . . . . . . 159
7.2.3 Solvable Limits of the Hubbard Model 160
7.2.4 Hubbard's Results ..... 163
7.2.5 Weak Coupling Approach . 165
7.2.6 Strong Coupling Approach. . 166
References . . . . . . . . . . . . . . . . . 173
8 Strong Coupling Limit and Some Exact Results 175
8.1 The Strong Coupling Limit. . . . . . . . . . 175
8.2 Exact Results for the Hubbard Model and its Strong Coupling
Limit Relatives 182
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9 Resonating Valence Bond States and High-T Superconductivity 191
c
9.1 Resonating Valence Bond States (RVBS) . . . . . . . . . . . 191
9.2 Anderson's RVB Ground States in High-T Superconductors 199
c
9.3 Excitation Spectrum of RVB States . . . . . . . . . . . . . . 204
9.4 Other Applications of RSV States: The Majumdar-Ghosh and
AKLT Constructions 208
References . . . . . . . . . . . 212
10 The Hubbard Model at D = 1 213
10.1 The Bethe Ansatz. . . . 213
10.1.1 XXZ MODEL. 1and 2 Magnon Sectors. 214
10.1.2 Physical Meaning of A21/ A12 • • • • • • • 217
10.1.3 b-Function Many-Body System. 1 and 2 Particle Solutions219
10.1.4 XXZ Model. Multi-magnon Solutions . . . . . . . . . . . 222
10.1.5 b-Function Many-Body System. Multiparticle Solutions. 223
10.2 Bethe Ansatz for the Hubbard Model 231
10.2.1 Bethe Ansatz and Eigenstates . . . . . . . . . . . . . . . 232
x
10.2.2 The Eigenvalue Condition . . . . . . . . . . . . . . . .. 235
10.2.3 Continuity Conditions . . . . . . . . . . . . . . . . . . . 236
10.2.4 Compatibility of Eigenvalue and Continuity Conditions:
The Yang-Baxter Equation . 237
10.2.5 Periodic Boundary Conditions . . . . . 238
10.2.6 Nested Bethe Ansatz . . . . . . . . . 239
10.2.7 Ground State ofthe Hubbard Model 241
10.3 Physical Consequences of Lieb-Wu's Equations 241
10.3.1 Excitation Spectrum 243
References . . . . . . . . . . . . . . . . . . . . . . . . 245
11 New and Old Real-Space Renormalization Group Methods for Quantum
Lattice Hamiltonians 247
11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 247
11.2 Foundations of the Real-Space Renormalization Group 248
11.3 Block Methods (BRG) . . . . . . . . . . .. . . 251
11.3.1 Ising Model in a Transverse Field (ITF) . . . . . 252
11.4 Ising Model in a Transverse Field (ITF) 257
11.4.1 Correlation Length Exponent v. . 261
11.4.2 Dynamical Exponent z . 262
11.4.3 Magnetic Exponent f3 . . . . . 262
11.4.4 Gap Exponent s. . . . . . . . 263
11.5 Antiferromagnetic Heisenberg Model 263
11.6 Quantum Groups and the Block Renormalization Group Method 269
11.7 Density Matrix Renormalization Group Methods: Introduction. 275
11.8 The Role of Boundary Conditions: The CBC Method 277
11.9 Density Matrix Renormalization Group Foundations. . 281
11.9.1 Density Matrix Algorithm 286
11.10DMRG Study of the ITF Model 288
11.10.1Variational DMRG . . . 289
11.10.2Fokker-Planck DMRG 294
References . . . . . . . . . . . . . . . 299
