Exactly Solvable Models of Strongly Correlated Electrons

EXACTLY SOLVABLE MODELS OF STRONGLY CORRELATED ELECTRONS

This page is intentionally left blank

Advanced Series in Mathematical Physics Vol. 18

EXACTLY SOLVABLE MODELS OF STRONGLY CORRELATED ELECTRONS

Editors

Vladimir E. Korepin Fabian H. L. Epler Institute for Theoretical Physics State University of New York at Stony Brook

V f e World Scientific wb

Singapore • New Jersey -L • London • Hong Kong Sinqapore-

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

The editors and publisher are grateful to the authors concerned and to publishers of the following journals for their permission to reproduce the articles found in this volume: American Institute of Physics (/. Maths. Phys. and Sov. Phys. JETP); American Physical Society (Phys. Rev. B and Phys. Rev. Letts.); Elsevier Science Publishers (Nucl. Phys. B and Phys. Lett.); Institute of Physics Publishing (J. Phys. A, J. Phys. C and J. Phys: Cond. Mat.); Physical Society of Japan (J. Phys. Soc. Jpn.); Plenum Publishing Corp. (/. Stat. Phys.); Progress of Theoretical Physics (Prog. Theor. Phys.) and Springer-Verlag (Z. Phys. B)

EXACTLY SOLVABLE MODELS OF STRONGLY CORRELATED ELECTRONS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. 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., 27 Congress Street, Salem, MA 01970, USA.

ISBN 981-02-1534-7

Printed in Singapore.

V

PREFACE

In recent years there has been a strong increase of interest in strongly correlated electronic systems in the Condensed Matter community. T h e reason for this development is t h a t certain phenomena like high-T c superconductivity cannot seem to be explained in the framework of weak interactions, that is mean-field theory and perturbation around it. There are mainly two different approaches in dealing with strong correlations: numerical studies and exact analytical solutions. Analytical solutions provide a complete and unambiguous picture of the dynamics of the models under consideration. T h e analytical approach is mainly limited to one spatial dimension, but there are indications that two-dimensional systems may share certain features with their one-dimensional analogs (see e.g. P. W . Anderson in Ref. 1; for a modified Hubbard model it was recently shown t h a t the ground state structure is identical in one and two dimensions 2 ). This volume is devoted to exact solutions of models of strongly correlated electrons in one spatial dimension by means of the Bethe Ansatz. T h e first such exactly solved model is the non-relativistic continuum model of electrons with local interaction, solved by C. N. Yang in 1967 3 (see also Ref. 4). In the solution of the model the nested Bethe Ansatz, which is the basis for all exact solutions of electronic models in one dimension, was discovered. T h e most important model of strongly correlated electrons, and the central topic of this volume, is the Hubbard model. T h e first chapter is devoted to its exact solution in one space and one time dimension, which started with E. H. Lieb's and F . Y. Wu's work of 1968. T h e chapter starts with reprints dealing with the construction of eigenstates of the Hubbard hamiltonian and the determination of the ground state and excitation spectrum at zero temperature. T h e next few topics are the study of magnetic properties in an external magnetic field, the thermodynamics of the model, and the asymptotic behaviour of correlation functions. T h e chapter closes with reprints on transport properties of the model. The second chapter is devoted to the t-J model at t h e supersymmetric point J = ±2r. T h e t-J model first appeared as an effective model describing the strong coupling limit of the Hubbard model for H oo the Bethe Ansatz wave functions take an especially simple form (see also [repr.1.5]). This fact allows for direct studies of asymptotic properties of correlation functions. M. Ogata, T. Sugiyama and H. S h i b a 8 6 - 8 8 performed numerical studies of momentum distribution and spin-correlation functions in the limit U —• oo. An analytic derivation of the asymptotics for spin-spin correlators in the limit U —• oo at zero magnetic field and quarter filling was obtained by A. Parola and S. Sorella 89 . 86

Another method for the calculation of correlators at zero temperature is the so-called Luttinger-liquid approach invented by F . D. M. H a l d a n e 9 0 - 9 4 , which is essentially based on the fact that the Hubbard model, as well as many other critical one-dimensional models, are certain realizations of the Gaussian model 9 5 . By generalizing Haldane's method H. J. Schulz 9 6 ' 9 7 [repr.1.18] and independently Y. Ren and P. W . Anderson 9 8 obtained the expressions for the critical exponents of the charge and spin correlators in zero magnetic field. The singularities of the electron momentum distribution function (which are connected to the long-distance asymptotics of correlation functions by Fourier transform) were studied by M. Brecht, J. Voigt and H. B u t t n e r " in the framework of Luttinger liquid theory. An extension of F. D. M. Haldane's description to the case of finite magnetic fields is provided by the Landau-Luttinger-liquid approach of J. M. P. Carmelo, P. Horsch and A. A. Ovchinnikov 5 3 ' 1 0 0 . In Ref. 100 the critical exponents were obtained in the framework of a generalized Landau-liquid theory, confirming the results of H. Frahm and V.E. Korepin 1 0 1 ' 1 0 2 (see below). T h e concept of a LandauLuttinger liquid was introduced by J. M. P. Carmelo and A. A. Ovchinnikov in [repr.1.19] 53 , where the spectral properties and asymptotic behaviour of correlation functions of the one-dimensional Hubbard model were reformulated in terms of interacting charge and spin pseudoparticles. The modern version of the Luttinger-liquid approach is related to conformal quantum field theory 1 0 3 . It was shown in Refs. 104-106 that the critical exponents can be extracted from the finite size corrections to the spectrum of the hamiltonian. In Bethe Ansatz solvable models it is possible to evaluate the finite size corrections

7 exactly 1 0 7 n 0 . T h e corrections for the low-lying states in the repulsive half-filled Hubbard model were obtained by F . Woynarovich and H. P. Eckle 1 1 1 [repr.1.20]. The finite size corrections and critical exponents in the attractive Hubbard model were determined by N. M. Bogoliubov and V. E. Korepin 1 1 2 ' 1 1 3 [repr.1.21]. Unlike in the repulsive case there is only one branch of gapless excitations in the attractive case (note that the expression for the correlation length £ [describing the exponential decay of the correlator of Fermi fields] given in the paragraph between formulas (7) and (8) is incorrect). Expressions for the finite size corrections in a general solvable model were given A. Izergin, V. E. Korepin and N. Yu. Reshetikhin in Ref. 114. Using these results F . Woynarovich treated the case of the non-half-filled Hubbard model 1 1 5 [repr.1.22]. T h e derivation of the critical exponents from the finite-size corrections in the repulsive case for arbitrary values of the external magnetic field, the density, and the coupling U was performed by H. Frahm and V. E. Korepin 1 0 1 ' 1 0 2 [repr.1.23]. They were the first to evaluate the critical exponents for finite magnetic fields. They found t h a t the conformal dimensions depend strongly on the external magnetic field and the other system parameters like density and coupling constant. In Ref. 102 an expression that determines the conformal dimensions for any (gapless) model solvable by a nested Bethe Ansatz was obtained*. They also investigated the singularities of the electron momentum distribution function. T h e case of zero magnetic field was also studied by N. Kawakami and S. K. Yang 1 1 7 . Logarithmic amplitude corrections to the asymptotical behaviour of the correlators were calculated by T. Giamarchi and H. J. Schulz 1 1 8 . Other methods for determining the critical exponents, based on perturbation theory and a renormalization group analysis, can be found in Refs. 119-121. The transport properties of the one-dimensional Hubbard model were studied by B. Sutherland and B. S. Shastry 1 2 2 , M. Fowler and N. Yu 1 2 3 , J. M. P. Carmelo, P. Horsch and D. Campbell 1 2 4 , and by M. J. Martins and R. M. F y e 1 2 5 . In [repr.1.24] B. S. Shastry and B. Sutherland showed that the boundary energy of a fermionic lattice system with twisted boundary conditions can be identified with t h e inverse effective charge- and spin-carrying masses (or stiffnesses). They evaluated the chargeand spin-stiffnesses for the Hubbard model and showed that these quantities can be used as a probe for a Mott-Hubbard transition. T h e twisting of boundary conditions can be viewed as the effect of a transverse ("two-dimensional") magnetic field.

*An application of this formula to other models can be found in [repr.11.5] and Ref. 116

8 N. Yu and M. Fowler studied the persistent current in Hubbard rings as a function of the flux through the ring and the coupling U [repr.1.25]. They found t h a t the persistent current is a periodic function of the flux and showed t h a t the ground state is always a spin-singlet. J. M. P. Carmelo, P. Horsch and D. K. Campbell investigated the transport properties of the Hubbard model in the framework of the Landau-Luttinger-liquid approach. They evaluated the charge and spin dynamical form factors for arbitrary momentum in a magnetic field124. M. J. Martins and R. M. Fye evaluated the charge and spin stiffnesses for the repulsive and attractive Hubbard models with toroidal boundary conditions.

9 VOLUME 20, NUMBER 25

PHYSICAL REVIEW

LETTERS

17 JUNE 1968

ABSENCE OF MOTT TRANSITION IN AN EXACT SOLUTION OF THE SHORT-RANGE, ONE-BAND MODEL IN ONE DIMENSION Elliott H. Lieb* and F. Y. Wu Department of Physics, Northeastern University, Boston, Massachusetts (Received 22 April 1968) The short-range, one-band model for electron correlations in a narrow energy band is solved exactly in the one-dimensional case. The ground-state energy, wave function, and the chemical potentials are obtained, and it is found that the ground state exhibits no conductor-insulator transition as the correlation strength is increased. The correlation effect of electrons in a partially filled energy band has been a subject of interest for many years. 1 " 4 A realistic model which takes this correlation into consideration, and which is hopefully amenable to mathematical treatment, is the short-range, one-band model considered by a number of authors. 2 " 5 In this model, one pictures the electrons in a narrow energy band hopping between the Wannier states of neighboring lattice sites, with a repulsive interaction energy between two electrons of opposite spins occupying the same lattice site. The central problems of interest have been (a) the possible existence of a "Mott transition" between conducting and insulating states as the strength of the interaction is increased, and (b) the magnetic nature (ferromagnetic or antiferromagnetic) of the ground state. While previous treatments of this model have always been approximate, we have succeeded in solving the model exactly in the one-dimensional case. Our exact result shows that the Mott transition does occur in the ground state of the one-dimensional model. Furthermore, a general theorem of Lieb and Mattis 6 on one-dimensional systems tells us that the ground state is necessarily antiferromagnetic. It may be argued that the absence of a Mott transition in one dimension is irrelevant for the study of real three-dimensional systems because of the folkloristic dictum that there are never any phase transitions in one dimension with short-range interactions. In actual fact, the dictum is only true for nonzero temperature; the ground state is another matter. Generally speaking, when a Hamiltonian is considered to be a function of some parameter, U (which in our case is the electron-electron repulsion), singularities with respect to U usually do appear in the ground-state wave function, energy, polarizability, etc., even in one dimension. A good example of this is the one-.dimensional Heisenberg chain (to which the present model is very close) which, when considered as a function of the anisotropy parameter, does have two singularities

in the ground state and, presumably, no singularities for nonzero temperatures. 7 ) 8 Consider a crystal (one-, two-, or three-dimensional) of Na lattice sites with a total of N ^2Na electrons. We suppose that the electrons can hop between the Wannier states of neighboring lattice sites, ajid that each site is capable of accommodating two electrons of opposite spins, with an interaction energy U>0. The Hamiltonian to consider is then 2 " 5 (1) W> oi where Cj ff t,c,- 0 a r e , respectively, the creation and annihilation operators for an electron of spin a in the Wannier state at the ith lattice site, and the sum

is restricted to nearest-neighbor sites. First of all, it can be shown that the energy spectrum of H is invariant under the replacement of T by - 7 \ 9 Therefore, for simplicity we shall take, in appropriate units, T = - l . Since the numbers JW of down-spin electrons and M' of upspin electrons a r e good quantum numbers (M +M' = N), we may designate the ground-state energy of H by E(M,M'; U). It is then easy to derive the following relations [by considering holes instead of particles in (1)]: E(M,M';U)=-(N

-M-M')V

+ E{N -M,N a ' a =MU+E(M,N =M'U+E(N

~M';U) ' -M';-U)

-M,M';-U).

(2)

Without loss of generality, therefore, we may take S z{(N-2M)>0 z

and N^N a

(less than half-filled band). 1445

10 PHYSICAL REVIEW

VOLUME 20, NUMBER 25

It can similarly be shown that the maximum energy G(Af,M'; [/) is related to the ground-state energy by G(M,M';U)=M'U-E{N

-M,M';U).

£ ftev---,x. + s,---,xN)

^mn Y

(7)

* s a n operator defined by

°b ran sih£ -sin£ + ziV n m sinfe -sin* n m pab sink -sinfe + jz'i/ n m

(8)

where, for j = i + 1,

Qk=Q'k for all

i=ls=±l + U £ 6(x.-x)f( Xl -

Pi=m=P'j,

--x^

t