Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions Antoine Georges Laboratoire de Physique The ´ orique de l’Ecole Normale Supe ´ rieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Gabriel Kotliar Serin Physics Laboratory, Rutgers University, Piscataway, New Jersey 08854 Werner Krauth and Marcelo J. Rozenberg Laboratoire de Physique Statistique de l’Ecole Normale Supe ´ rieure, 24, rue Lhomond, 75231 Paris Cedex 05, France 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. CONTENTS I. Introduction 14 II. The Local Impurity Self-Consistent Approximation: An Overview 17 A. Dynamical mean-field equations 17 B. Physical content and connection with impurity models 19 C. The limit of infinite dimensions 20 III. Derivations of the Dynamical Mean-Field Equations 21 A. The cavity method 21 B. Local nature of perturbation theory in infinite dimensions 23 C. Derivation based on an expansion around the atomic limit 25 D. Effective medium interpretation 26 IV. Response Functions and Transport 27 A. General formalism 27 B. Frequency-dependent conductivity, thermopower and Hall effect 30 V. Phases with Long-Range Order 31 A. Ferromagnetic long-range order 31 B. Antiferromagnetic long-range order 31 C. Superconductivity and pairing 32 VI. Methods of Solution 33 A. Numerical solutions 34 1. Quantum Monte Carlo method 34 a. Introduction: A heuristic derivation 22 b. The Hirsch-Fye algorithm: Rigorous derivation 23 c. Implementation of the Hirsch-Fye algorithm 24 d. The LISA-QMC algorithm and a practical example 26 e. Relationship with other QMC algorithms 26 26 2. Exact diagonalization method 40 3. Comparison of exact diagonalization and Monte Carlo methods 43 4. Spectral densities and real frequency quantities: Comparison of various methods 44 5. Numerical calculation of susceptibilities and vertex functions 46 B. Analytic methods 48 1. Exact methods at low energy 48 2. The iterated perturbation theory approximation 50 3. Slave boson methods and the noncrossing approximation 51 4. Equations of motion decoupling schemes 53 C. The projective self-consistent technique 54 VII. The Hubbard Model and the Mott Transition 59 A. Early approaches to the Mott transition 59 B. Models and self-consistent equations 61 C. Existence of a Mott transition at half-filling 62 1. Metallic phase 62 2. Insulating phase 64 D. Phase diagram and thermodynamics 65 1. Paramagnetic phases 65 2. Thermodynamics 67 3. Antiferromagnetic phases 69 E. The zero-temperature metal-insulator transition 70 F. On the T=0 instability of the insulating solution 71 G. Response functions close to the Mott-Hubbard transition 72 13 Reviews of Modern Physics, Vol. 68, No. 1, January 1996 0034-6861/96/68(1)/13(113)/$23.95 © 1996 The American Physical Society