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Physical Review Letters

Systematic 1/d Corrections to the Infinite-Dimensional Limit of Correlated Lattice Electron Models

Avraham Schiller, Kevin Ingersent

Published Jul 3, 1995• 71 citations
Publisher

Abstract

We present a self-consistent treatment of Hubbard-like lattice models at large, but finite, spatial dimensions $d$. This involves a systematic expansion in powers of $1/d$ about the limit $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$. The first-order corrections can be obtained by self-consistent solution of coupled one- and two-impurity models. We calculate the leading corrections to properties of the Falicov-Kimball model. The infinite-dimensional limit seems to describe this particular model remarkably well even for $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$.

DOI
10.1103/physrevlett.75.113

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Reviews of Modern Physics•Jan 1, 1996•6,900 citations
We present a self-consistent treatment of Hubbard-like lattice models at large, but finite, spatial dimensions ddd. This involves a systematic expansion in powers of 1/d1/d1/d about the limit d=\ensuremath∞d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}d=\ensuremath∞. The first-order corrections can be obtained by self-consistent solution of coupled one- and two-impurity models. We calculate the leading corrections to properties of the Falicov-Kimball model. The infinite-dimensional limit seems to describe this particular model remarkably well even for d=2d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2d=2.
In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.
The local approximation is applied to the spinless Falicov-Kimball model in one and two dimensions. The local approximation may be viewed as a zeroth-order expansion in 1/d that is exact in the weak-coupling limit. The local approximation rapidly becomes inaccurate as a function of interaction strength in one dimension, but is accurate in two dimensions. This result indicates that the 1/d expansion converges rapidly for the Falicov-Kimball model in d\ensuremath{\ge}2 and that exact solutions of many-body problems in infinite dimensions are quantitatively relevant for physical dimensions.
We discuss the infinite dimension limit of the Hubbard model by means of a perturbative expansion of the one-particle Green's function around the atomic limit. The diagrammatic structure is simplified in this limit, allowing a formal resummation that reproduces a previously proposed mapping to a single-site mean-field problem. The method provides a natural way of addressing this effective problem by means of a perturbative expansion in the local mean field. This gives the correct exact result for the Falicov-Kimball model, which is used as a starting point to study the Hubbard case. \textcopyright{} 1996 The American Physical Society.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').
We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.