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.

Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions

The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

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Graphical Models, Exponential Families, and Variational Inference

Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain region-based free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a "valid" or "maxent-normal" approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the "Bethe method", the "junction graph method", the "cluster variation method", and the "region graph method". Finally, we explain how to tell whether a region-based approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP.

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Constructing free-energy approximations and generalized belief propagation algorithms

Critical phenomena and renormalization-group theory

Diffusion in regular and disordered lattices

Physical, analytical, and numerical properties of the lattice Green's functions for the various lattices are described. Various methods of evaluating the Green's functions, which will be developed in the subsequent papers, are mentioned.

Lattice Green's Function. Introduction

The general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution functions in the formalism of the cluster variation method. The parameters or the Lagrange multipliers occurring in the distribution functions must be determined by the reducibility condition of the distribution functions or by the stationariness condition of the free energy.

General Structure of the Distribution Functions for the Heisenberg Model and the Ising Model

A recurrence relation, which gives the values of the lattice Green's function along the diagonal direction from a couple of the elliptic integrals of the first and second kind, is derived for the square lattice by an elementary partial integration. The values of the square lattice Green's function at an arbitrary site are then calculated in a successive way with the aid of the difference equation defining the function. Discussions are given of the application of this result to the calculation of the lattice Green's function of the tetragonal and body‐centered cubic lattices.

Useful Procedure for Computing the Lattice Green's Function‐Square, Tetragonal, and bcc Lattices

Formulas are provided which are convenient for the evaluation of the lattice Green's functions for the bcc, fcc, and rectangular lattices, at an arbitrary complex variable. The formulas involve the complete elliptic integral of the first kind with complex modulus; the integral has been found to be evaluated efficiently by the method of the arithmetic‐geometric mean, generalized for the case with complex modulus. The expansions of the lattice Green's functions around the singular points are given for the bcc and fcc lattices. These lattice Green's functions diverge at a variable. The singular points responsible for the divergences are found to form one‐dimensional lines.

Calculation of the Lattice Green's Function for the bcc, fcc, and Rectangular Lattices

The real and imaginary parts of the lattice Green's functions for the simple cubic (actually the tetragonal), body‐centered cubic, and face‐centered cubic lattices, at the variable from −∞ to +∞, are expressed as a sum of simple integrals of the complete elliptic integral of the first kind. The results of the numerical calculations obtained with the aid of the formulas are shown by graphs.

Tohru Morita

Papers

Lattice Green's Function. Introduction

General Structure of the Distribution Functions for the Heisenberg Model and the Ising Model

Useful Procedure for Computing the Lattice Green's Function‐Square, Tetragonal, and bcc Lattices

Calculation of the Lattice Green's Function for the bcc, fcc, and Rectangular Lattices

Lattice Green's Functions for the Cubic Lattices in Terms of the Complete Elliptic Integral