Springer Tracts in Modern Physics Volume 137

Transcription

1 Springer Tracts in Modern Physics Volume 137 Managing Editor: G. Hohler, Karlsruhe Editors: J. Kuhn, Karlsruhe Th. Muller, Karlsruhe R. D. Peccei, Los Angeles F. Steiner, Ulm J. Trumper, Garching P. Wolfle, Karlsruhe Honorary Editor: E. A. Niekisch, Julich Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Par is Santa Clara Singapore Tokyo

2 Springer - - Tracts in Modern Physics Volumes are listed at the end of ~lle book Covering reviews with emphasis on the fields of Elementary Particle Physics, Solid-State Physics, Complex Systems, and Fundamental Astrophysics Manuscripts for publication should be addressed to the editor mainly responsible for the field concerned: Gerhard Hohler lnstitut fur Theoretische Teilchenphysik Universitat Karlsruhe Postfach 6980 D Karlsruhe Germany Fax: +49 (7 21) Phone: +49 (7 21) hoehler@fphvax.physik.uni-kar1sruhe.de Johann Kuhn lnstitut fur Theoretische Teilchenphysik Universitat Karlsruhe Postfach 6y80 1) Karlsruhe Germany Fax: +49 (7 21) Phone: +49 (7 21) Ernail: johann.kuehn@physik.uni-karlsruhe.de Thomas Muller IEKP Fakultat fur Physik Universitat Karlsruhe Postfach ) Karlsruhe Joachim Triimper Max-Planck-lnstitut fur Extraterrestrische Physik Postfach 1603 D Garching Germany Fax: +49 (89) Phone: +49 (89) jtrumperfn~pe-garching.rnpg.de Peter Wolfle lnstitut fur Theorie der Kondensierten Materie Universitat Karlsruhe Postfach D Karlsruhe Germany Fax: t49 (7 21) 6y Phone: +49 (7 21) woelfle@tkm.physik.uni-karisr11he.de Fax:t49 (7 21) Phone: +qy (7 21) Ernail: mullerth@vxcern.cern.ch Roberto Peccei Department of Physics University of California, Los Angeles 405 Hilgard Avenue 1.0s Angeles, California USA Fax: +I Phone: +I robertop@college.ucla.edu Frank Steiner Abteilung fur Theoretische Physik Universitat Ulm Albert-Einstein-Allee 11 D Ulm Germany Fax: +49 (7 31) Phone: +49 (7 31) lo steiner@physik.uni-uim.de

3 Florian Gebhard The Mott Metal-Insulator Transition Models and Methods With 38 Figures Springer

4 Dr. Florian Gebhard Institul Max von Laue-Paul Langevin J. P. 158 F Crenoble Cedex 9 France florian@ill.fr Library of Congress Cataloging-in-Publication Data Gebhard. Florlan The mott metal-insulator transltlon. models and methods! Flor~an Gebhard. p. cm. -- (Sprlnger tracts In modern physlcs. ISSN : v Includes blbllographlcal references and Index. ISBN (Berlln. alk. paper) 1. Metal-Insulator transltlons. 2. Electron-electron Interacttons. 3. Condensed matter. 4. Many-body problem. 5. Mathematlcal physlcs. I. Tltle. 11. Serles: Springer tracts In modern physlcs ; 137. OCl.S797 no. 137 [QC176.8.M s--dc ' CIP Physics and Astronomy Classification Scheme (PACS): h, w, o5.7o.fh, a, 7i.i5.-m ISSN ISBN Springer-Verlag Berlin Heidelberg New York 'This work is subject to copyright. All rights are re~erved~whether 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 provisionsof 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. 0 Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even In the absence of aspecific statement,that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesett~ng: Camera-ready copy by the author uslng a Springer TEX macro-package SPIN: z I o - Printed on acid-free paper

5 To My Family

6 Preface Little do we reliably know about the Mott transition, and we are far from a complete understanding of the metal --insulator transition due to electronelectron interactions. Mott summarized his basic ideas on the subject in his wonderful book Metal--Insulator nansitions that first appeared in ). In his view, a Motk insulator displays a gap for charge-carrying excitations due to electron cowelations, whose importance is expressed by the presence of local magnetic moments regardless of whether or not they are ordered. Since the subject is far from being settled, different opinions on specific aspects of the Mott transition still persist. This book naturally embodies my own understanding of the phenomenon, inspired by the work of the late Sir Kevill Mott. The purpose of this book is twofold: first, to give a detailed presentation of the basic theoretical concopts for Mott insulators and, second, to test these ideas against the results from model calculations. For this purpose the Hubbard model and some of its derivatives are best suited. The Hubbard model describes a Mott transition with a mere minimum of tunable parameters, and various exact statements and even exact solutions exist in certain limiting cases. Exact solutions not only allow us to test our basic ideas, but also help to assess the quality of approxin~ate theories for correlated electron systems. As I describe in this book, the Mott transition is indeed a consequence of electron correlations, and Mott's ideas are found to be essentially correct. - not surprising for a physicist of his greatness. The Hubbard model serves as our paradigm for correlated clect,ron systems. This many-electron Hamiltonian is conceptually simple at first sight, but already poses a very difficult theoretical problem. Thus, w(: do not attempt to describe the phase diagram of real materials where the full band structure, banti degeneracies, lattice distortions, impurity scattering, and long-range Coulomb interactions must be taken seriously. Since these effects tend to obscure the physics of the Mott transition, and are not expected to change the overall picture qualitatively, a quantitative comparison to experiments is not the primary goal of this book. Numerical results for the Hubbard model have also been left out for the most part of the book, for good reasons: (i) I am not an expert in this vastly growing field which deserves a book of its own, (ii) numerical approaches have their own limitations (finite size

7 VIII Preface c~ff(?ct,s, niinus sign problem, analytical continuation, etc.), and in general it is very difficult to obtain sufficient energy resolution to allow for definitive statements on the Mott--Hubbard metal-insulator transition. Finally, I have left out t,hose analytical approaches to correlated electron systems which are not intcntied to describe the Mott transition. I apologize t80 the rriany workers on correlated electron systems who will find their cont,ributions inadequately considered. Despite these omissions, I hope that the book will be profitable for both researchers and graduate students working on correlated electron problems. 'I'he book cert,ainly requires a working knowledge of solid-state physics, and a basic familiarity with many-particle concepts would also prove helpful. In any case, I have tried to provide simple derivations, although I have deferred excessively detailed calculations to the appendices or primary literature. This project grew out of my Habilitation thesis at the Philipps University Marburg, and I am most grateful to Philippe Nozibres for the opportunity to turn it into 'a book during my stay at the Institute Laue-Langevin. I profited a lot from his opinions and from our discussions on the Mott transition. I am also greatly indebted to my dear friend David Logan who helped me to tie up niany loose ends, and to clarify my thoughts on insulators, charge gaps, magnetic moments, and the like. I also owe him my thanks for critical proofreading of substantial parts of the book, ;is did my friends Peter Thomas, Fabian EOler, and Kurt Schonhammer. Of course, the responsibility for errors and flaws rests entirely with me. My special thanks go to authors and publishers who granted nie pcrmission to reproduce their figures, in particular to Marcelo Rozenberg and Thomas Pruschke for their POSTSCRIPT originals. This book could not have been accomplished without the help and support of many people. and I am grateful to my family and friends for their constant support and encouragement. I appreciate the faithful collaboration with Dr. Kijlsch and the production team from Springer-Verlag, and thank them for their help with the layout, their patience, and their flexibility. Grenoble, January 1997

8 Contents 1. Metal-Insulator Transitions Classification of Metals and Insulators Definition of Metal and Insulator Classes of Insulators Gap Criterion for the Insulator Electrical Conductivity at Low Temperature Gap at Finite Temperature Types of Metal-Insulator Trarisitioris Quantum Phase Transition Thermodynamic Phase Transition Band, Peierls, and Anderson Insulators Band Insulators Peierls Insulators Anderson Insulators Mott Insulators: Basic Theoretical Concepts Electron-Electron Interaction in Metals Exchange and Correlations Magnetic Moments Slater Insulator Mott Insulator hlott-hubbard Insulator Mott-Heisenberg Insulator hlott Insulators: Some Experimental Observations Wigner Insulator Verwey Insulator hliilti-band Mott Insulator Charge-'llansfer Insulator Whither Theory? Hubbard Model Electronic Many-Particle Problem Hamilton Operator of Solid-state Physics Hamilton Operator of the Electronic Problern... 51

9 X (:ontents 2.2 Band Structure Calculations Independent Electrons Hartree Theory Local Density Approximation Iniprovements to the Local Density Approximation Derivation of the Hubbard Model Stx:ond Quantization Hubbard Model Hubbard Model and Real Materials Polymers Transition Metals and Their Oxides Fullerenes High-Temperature Superconductors Heavy Fermion Materials Liquid Helium Relevance of the Hubbard Model Model Properties Model Parameters Model Variants Symmetries Basic Properties of the Hubbard Model Magnetism Approximate Methods Hartre* F'ock Theory Method and Exact Statements Slater's Theory of the Antiferromagnet Stoner's Theory of the Ferromagnet Simplified Hartree-Fock Phase Diagram Advantages and Problems Improvements Local Derisity Approximation Site Occupation Functional Theory Site Occupation Local Density Approximation Xa Local Density Approximation LSDA with Self-Interaction Corrections Green-Function Decoupling Method Atomic Limit and Hubbard Bands Hubbard-I Approximation Advantages and Problenis Irriprovements Alloy-Analogy Approximat. ion Hubbard-I11 Approximation Advantages and Problems Irriprovements

10 Contents XI 3.4 Variational Wave Functions Gutzwiller-Correlated Wave Functions Variational Characterization of an Insulator Exact Statements Gutzwiller Wave Function in One Dimension Variational Wave Functions in Infinite Dimensions Brinkman-Rice Transition Expansion Around the Limit of Infinite Dimensions Variational Phase Diagram in Infinite Dimensions Extensions and Improvements Advantages and Problems Slave Boson Approach Slave Bosons and Functional Integrals Kotliar-Ruckenstein Slave Bosons Kotliar-Ruckenstein Approximation Advantages and Problems Barnes-Coleman Slave Bosons and Approximation Advantages and Problems One-Dimensional Hubbard Models Hubbard Model with Nearest-Neighbor Hopping Two-Electron Problem for Two Lattice Sites General Two-Electron Problen~ in One Dimensiori Bethe Ansatz for the N-Electrorl Problem Ground-State Energy at Half Band-Filling Metal-Insulator Transition at Half Band-Filling Drude Weight Below Half Band-Filling Metal-Insulator I'ransition Approaching Half Filling Hubbard Model with Long-Range Hopping Basic Properties General Two-Electron Problem in One Dimension Effective Hamiltonian for the N-Electron Problern Ground-State Energy and Metal-Insulator Transition Thermodynamic Properties Hubbard Model with Variable-Range Hopping Basic Properties General Two-Electron Problem in One Dimension S Matrix and Generalized Lieb-Wu Equations Ground-State Energy at Half Band-Filling Metal-Insulator Transition at Half Band-Filling Mott- Hubbard Transition in One Dimension g-ology Hamiltonian Perfect Nesting in One Dimension Long-Range Hopping Comparison with Basic Theoretical Concepts

11 XI1 Contents 5. Hubbard Model in Infinite Dimensions Limit of Infinite Dimensions Spin Models in Infinite Dimensions Non-Interacting Itinerant Electrons Generalized Mean-Field Approaches Mean-Field Spin Models Random Dispersion Approximat ion Simplifications in Infinite Dimensions Position-Space Collapse of Diagrams Irrelevance of Momentum Conservation Effective Models with Single-Site Interaction Analytical Results in Infinite Dimensions Perturbation Theory for Small Interactions Perturbation Theory for Large Interactions Variational Wave Functions Falicov-Kimb,a 11 Model Spinless Fermions Approximate Solutions of the Hubbard Model Model Specifications Constraints on Approximate Treatments Quantum Monte-Carlo Calculations Exact Diagonalization Studies Iterated Perturbation Theory Non-Crossing Approximation Local Moments in Hubbard's Approximations Local-Moment Approach Metal-Insulator Transitions Mott-Heisenberg Insulator Correlated Metal Mott-Hubbard Insulator Mott-Hubbard Transition Comparison with Basic Theoretical Concepts Further Models with Hubbard Interaction Degenerate and Extended Hubbard Models Band-Degenerate Hubbard Model in One Dimension Extended Hubbard Model Generalized Hubbard Models Models with Conserved Double Occupancies Bond-Charge Interactions and Harris-Lange Model Harris-Lange Model in One Dimension Montorsi-Rasetti Model Supersymmetric Hubbard Model Models with Commuting Operators Hubbard Model with Infinite-Range Hopping

12 Contents XI Stoner Model with Long-Range Hopping Landau-Hubbard Model Conclusions Appendix A.l Calculations for Hartree-Fock Theory A.l.l Diagonalization of the Hartree-Fock Hamiltonian A.1.2 Ground-State Properties for Half Band-Filling A.2 Green Functions and Hubbard-I Approximation A.2.1 Green Functions A.2.2 Hubbard-I Approximation A.3 Generalized Lieb-Wu Integral Equations A.3.1 Derivation of the Integral Equations A.3.2 Solution for Half-Filled Bands A.3.3 Quasi-Particle Dispersions A.3.4 Limit of Long-Range Hopping References Index