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HUBBARD OPERATORS IN THE THEORY OF SDTRONGLY CORRELATED ELECTRONS This page intentionally left blank S.G. Ovchinnikov V.V. VaI'kov L.V. Kirensky Institute of Physics, Siberian Branch of the Russian Academy of Sciences and Krasnoyarsk State Technical University, Russia Imperial College Press Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. The authors and publisher would like to thank the American Physical Society and Springer-Verlag for permission to reproduce selected materials in this book. HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS Copyright © 2004 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 1-86094-430-2 Typeset by Stallion Press Email: [email protected] Printed in Singapore. June28,2004 11:4 WSPC/BookTrimSizefor9inx6in FM Contents Preface ix PART 1 Chapter 1. Hubbard Model as a Simplest Model of Strong Electron Correlations 1 1.1 Hamiltonian of the Hubbard Model and its Symmetry . . . 1 1.2 Time Inversion . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Electron Hole Symmetry . . . . . . . . . . . . . . . . . . . . 3 1.4 Pseudospin Symmetry for Half-Filled Case . . . . . . . . . . 4 1.5 The Band Limit . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 The Atomic Limit, Hubbard 1 Solution . . . . . . . . . . . 6 1.7 The Hubbard Model in One and Infinite Dimension Cases . 9 1.8 The t–J Model . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 The Hubbard Model in X-Operator Representation . . . . . 14 1.10Study of the Electronic Structure by the Angle Resolved Photoemission Spectroscopy . . . . . . . . . . . . . . . . . . 23 Chapter 2. Multielectron Models in X-Operator Representation 27 2.1 The Isotropic Heisenberg Model . . . . . . . . . . . . . . . . 27 2.2 The Heisenberg Model with Single-Ion Anisotropy . . . . . 29 2.3 The Atomic Representation for s–d(f) Exchange Model . . 30 2.4 The Periodic Anderson Model . . . . . . . . . . . . . . . . . 33 2.5 The Multiband Model of Transition Metal Oxides. . . . . . 37 v June28,2004 11:4 WSPC/BookTrimSizefor9inx6in FM vi Hubbard Operators in the Theory of Strongly Correlated Electrons Chapter 3. General Approach to the Quasiparticle Description of Strongly Correlated Systems 41 3.1 The Definition of Fermi- and Bose-Type Quasiparticles . . . 41 3.2 The Exact Intracell Local Green’s Function . . . . . . . . . 44 3.3 The Generalized Tight-Binding Method for Quasiparticle Band Structure Calculations. . . . . . . . . . . . . . . . . . 46 3.4 The Concentration Dependence of the Quasiparticle Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Ab Initio Approach to the Quasiparticle Band Structure of Strongly Correlated Electron Systems . . . . . 56 3.6 The Generalized Tight-Binding Method and the Exact Lehmann Representation. . . . . . . . . . . . . . . . . . . . 59 Chapter 4. Unitary Transformation Method in Atomic Representation 63 4.1 Unitary Operators and Transformation Laws in Atomic Representation . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Diagonalization of Two-Level and Quasi-Two-Level Forms . 67 4.3 The Diagonalization of Three-Level Forms . . . . . . . . . . 69 4.4 The Diagonalization of the N Level Hamiltonian . . . . . . 76 Chapter 5. Diagram Technique in the X-Operators Representation 77 5.1 Green’s Function in the X-Operators Representation . . . . 78 5.2 Wick’s Theorem for the Hubbard Operators . . . . . . . . . 82 5.3 Averaging the Diagonal X-Operators . . . . . . . . . . . . . 86 5.4 External and Internal Operators: Simplest Vertexes . . . . . 90 5.5 A Hierarchy of Interaction Lines and a Topological Continuity Principles . . . . . . . . . . . . . . . 97 5.6 Calculation of the Sign for an Arbitrary Diagram . . . . . . 99 5.7 Diagrams with Ovals . . . . . . . . . . . . . . . . . . . . . . 104 5.8 General Rules for Arbitrary Order Diagram . . . . . . . . . 112 5.9 The Larkin Equation . . . . . . . . . . . . . . . . . . . . . . 115 5.10The Self-Energy and the Strength Operator . . . . . . . . . 123 5.11Low Temperature Properties of Anisotropic Ferromagnet. . 129 5.12Effect of the Three-Site Correlated Hopping on the Magnetic Mechanism of dx2−y2-Superconductivity in the t–J∗ Model . . . . . . . . . . . . . . . . . . . . . . . 138 June28,2004 11:4 WSPC/BookTrimSizefor9inx6in FM Contents vii PART 2 Chapter 6. Spectral Properties of Anisotropic Metallic Ferromagnets 149 6.1 The Hamiltonian of the Anisotropic s−f Magnet in Atomic Representation . . . . . . . . . . . . . . . . . . . . . 150 6.2 The Green Function and Dispersion Equation . . . . . . . . 154 6.3 The Ferromagnetic Metal with Single Axis Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.4 The Metallic Ferromagnets with the SA of Cubic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 166 Chapter 7. Peculiarities of the de Haas-van Alphen Effect in Strongly Correlated Systems with Magnetic Polaron States 175 7.1 The Hamiltonian of a Strong Correlated Narrow Band Antiferromagnet . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2 The Hamiltonian in the Non-colinear Phase: Unitary Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.3 Construction of the Atomic Representation Basis . . . . . . 179 7.4 The Hamiltonian of the Narrow Band Antiferromagnet in the Non-colinear Phase in Atomic Representation . . . . . . . . . . . . . . . . . . . . . 183 7.5 The Green’s Functions and Dispersion Relations . . . . . . 184 7.6 The Spectrum of Holes Near the Spin-flip Transition . . . . 187 7.7 The Peculiarities of the de Haas-van Alphen Effect in Antiferromagnetic Semimetal with Magnetic Polaron States . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.8 The Temperature Quantum Oscillations Caused by Non-Fermi Liquid Effects . . . . . . . . . . . . . . . . . . 191 Chapter 8. The Electronic Structure of Copper Oxides in the Multiorbital p–d Model 195 8.1 The Exact Diagonalization of the CuO4 Cluster . . . . . . . 196 8.2 The Construction of the Wannier Functions and X-Representation of the Multiband p–d Model . . . . . . . 200 June28,2004 11:4 WSPC/BookTrimSizefor9inx6in FM viii Hubbard Operators in the Theory of Strongly Correlated Electrons 8.3 The Evolution of the Band Structure and ARPES Data with Doping from the Undoped Antiferromagnetic Insulator to Optimal Doped and Overdoped Metal . . . . . 210 8.4 Comparison of the Electronic Structure and Superconductivity of Cuprates and Ruthenates . . . . . . . 218 References 231 Index 239 June28,2004 11:4 WSPC/BookTrimSizefor9inx6in FM Preface Systems with strong electron correlations (SEC) have been known for a long time, since Mott (1949) explained the controversy in the insulating propertiesofNiOandFe3O4,andtheruleofone-electronbandfillingwhich results in the metal states of these oxides. The SEC effects determine the localization of electrons, thermodynamic and kinetic properties of a large group of d-, f-metals and their compounds. While a conventional single- electron band theory was successfully used to describe normal metals and alloys, its application to the SEC systems often yields incorrect results. That is why the problem of SEC is one of the essential problems of the modern condensed matter physics. Recent discovery of new SEC systems such as the copper oxide-based high temperature superconductors and manganese oxides with colossal magnetoresistivity has raised the interest in the problem of SEC. Inten- sive studies all over the world have given insights to the ground state and elementary excitations in SEC systems. Very important results have been obtainedbynumericallyexactcomputationslikethequantumMonte-Carlo (QMC) and the exact diagonalization methods in the Hubbard model, t– J and p–d models for finite clusters (see e.g. reviews by Dagotto (1994); Kampf(1994);Brenig(1995);Ovchinnikov(1997)).Someprogressachieved fortheinfinitesystemsisduetothedevelopmentofdiagramtechniquesfor the Hubbard operators (Zaitsev 1975, 1976; Izyumov et al 1994), projec- tionoperatorsmethod(Fulde1991)andtheself-consistenttreatmentofthe Hubbard model by a fluctuation exchange (FLEX) approach (Bickers et al 1989)andthedynamicalmean-fieldtheoryininfinitedimensions(Metzner and Vollhardt 1989; Georges et al 1996). Analysis of the Mendeleev periodic table shows that SEC-determined localization of electrons is typical for the 4f-electrons of the rare-earth ix