THE HUBBARD MODEL Recent Results SERIES ON ADVANCES IN STATlSllCAL MECHANICS Editor-in-Chief: M.Rasetti Published Vol. 1: IntegrableSystflm~ in,StatisticalMeqha~~ ...' .'. .: editedbyG. M. D'tiritino,A.·;~ntor.tiantfM~'RasBtti Vol. 2: Modern Met~ods inEquilibriumStatisticalMechanics byM. Rasetti Vol.3: ProgressinStatistical.Mechanlcs editedbyC. K. Hu Vol.4: ThermodynamicsofComplexSystems byL. Sertorio Vol.5: PottsModelandRelatedProblemsinStatisticalMechanics byP. Martin Vol. 6: NewProblems, MethodsandTechniquesinQuantumFieldTheory andStatisticalMechanics editedbyM. Rasetfi SeriesonAdvances in Statistical Mechanics- Vol. 7 It. r World Scientific Singapore•NewJersey•London •HongKong ' PublisMdby WorldScientificPublishingCo.Pte•.Ltd. POBox128,FarrerRoad,Singapore9128 USAoffICe: SuiteIB,1060MainStreet,RiverEdge,NJ07661 UKoIMe: 73LyntonMead,Toueridge,LondonN208DH THE HUBBARD MODEL -RECENT RESULTS Copyrighte1991byWorldScientificPublishingCo.Pte.Ltd. AllrighUreserved.Thisboole,orpartstMreof,mtlytlDtbereproducedillanyform orbyanymeans,electronicormechanical,includingphotocopyillg,rect>rdillgora,.., iIIfoT'l'lttltionstorageandretr~WJl systemnowbaownortobe i1&Yented, without writtenpermissi01lfromtMPub/isMr. ISBN 981-02-0623-2 ISBN 981-02-0624-0(Pbk) mw PrintedinSingaporeby PrintenandBindenPte.Ltd. PREFACE The Hubbard model, an interactingfermion model independentlyproposed in 1963 by Gutzwiller, Hubbard and Kanamori, has played over the years an extremely relevant role in condensed matter physics. Originally designed to describe in approximate way the effects induced by the competition between delocalization and correlations for electrons forming a narrow d-band in transition metals, the model has then been shown to be indeed the ideal tool to describe the relevant collective features of these materials, namely itinerant magnetism and metal-insulator transition. More recently the model has been the object of a tremendous revival of interest, due to the discovery ofheavy fermions and high-T superconductors, as c the mostpromisingmodeltoaccountforstrongcorrelations. Itis infact believed, for instance, thatjustthe interplaybetween strongantiferromagneticcorrelations and Mott transition in proximityofhalf-filling ofthe band plays a crucial role in generating a superconducting phase in the new ceramic superconductor of the YBaCuO family. r4> Allofthis has stimulatedtheachievementofawealthofnewsignificantresults concerning the properties ofthe Hubbard model, ranging from exact or rigorous statistical mechanics results (either on it or on related models), to novel approximate approaches to its relevant features in different physical regimes, to numerical Quantum Monte Carlo and exact diagonalization studies. This book presents a collection ofrecent interesting papers dealing with some ofthose results, which should be ofinterest for all scientists working in the field. v This page is intentionally left blank CONTENTS Preface v The Phase Diagram ofthe Hubbard Model M W Long Charge and Spin Fluctuations in the Hubbard Model: Fermi Liquid Properties at Low Temperatures M Lavagna 21 The t-J and Frustrated Heisenberg Models: A Status Report on Numerical Studies E. Dagotto 43 The Hubbard-Stratonovich Transformation and the Hubbard Model S. Sore/fa 73 S04 Symmetry in a Hubbard Model C. N Yang andS. C. Zhang 113 Mott Transition in an Exactly Solvable K.S.S.H. Model A. Montorsi andM Rasetti 121 The Hubbard Model: From Small to Large U D. Baeriswy/ and W von der Linden 135 First Order Metal-Insulator Transition in an Alloy Analogy Approach to the Hubbard Model M Corrias 151 Renormalization Group Approach to Quasi-One-Dimensional Conductors C. Bourbonnais and L. G. Caron 169 vii This page is intentionally left blank THE PHASE DIAGRAM OF THE HUBBARD MODEL M. W. LONG School of Physics, Bath University, Claverton Down, Bath, BA2 7AY, UK A brief review of the types of ideas which have been used to investigate the phase diagram of the Hubbard model is presented. No clear solution emerges although the competitions at work are established. Hartree-Fock, Gutzwiller's projection, Nagaoka's theorem, Kanamori's paramagnetism, Lieb's exact results and charge-spin separation are the main concepts. 1. Introduction The Hubbard model1 is probably the simplest model which can be used to study the many-body aspects of the electronic properties of condensed matter. All the complexities of atomic physics and the corresponding multi-band description of condensed matter have been stripped away, until all that remains is a stark competition between chemical bonding and Coulomb repulsion; the two dominant direct forces which decide the behaviour of mobile electrons in condensed matter. The model is usually written down in the form H = -t^clc^* Ujn n (1) j n n where c] {c ) are creation (annihilation) operators which create (annihilate) an G io electron of spin a(=t,l) on a site /, in terms of which n = c] c is an electron i(T a i(J number operator and simply counts the number of electrons of spin a on site /. Attention is normally restricted to hybridisation between neighbouring atoms, denoted by <//'>. The first term corresponds to chemical bonding and is known as 'hopping'. This term is a single particle interaction and hops an electron from one atom to a nearest neighbour atom with hopping matrix element t. This contribution favours itinerant electrons. The second term corresponds to the Coulomb repulsion between two electrons and is therefore a two particle interaction. The long range contribution is assumed to be screened and only the interaction when both the electrons are on the same atom is retained, yielding an additional energy of U when the atom is doubly occupied. PACS Nos.: 71.10, 75.10.L, 75.10.J. 1