Mean-Field Theory

Prosenjit Haldar

In this chapter, we present a bird’s-eye view of dynamical mean field theory (DMFT) and its extensions to highly correlated systems. We first demonstrate the elementary introduction to DMFT on a single site with detailed derivations and then a solution of the simple interacting models: the Hubbard model and the Falicov–Kimball model on infinite dimensional Bethe lattice using DMFT. Thereafter, we focus on two kinds of non-local extensions of single-site DMFT - (a) cluster dynamical mean field theory (CDMFT) on real space and (b) dynamical cluster approximation (DCA) on momentum space. Finally, we conclude the chapter with the application of DMFT combining with density functional theory (DFT) in the real-material systems (LDA+DMFT).

Nikolai Borisovich Melnikov, Boris Il'ich Reser

Dominika Zgid, Garnet Kin-Lic Chan

We investigate the dynamical mean-field theory (DMFT) from a quantum chemical perspective. Dynamical mean-field theory offers a formalism to extend quantum chemical methods for finite systems to infinite periodic problems within a local correlation approximation. In addition, quantum chemical techniques can be used to construct new ab initio Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-field theory to connect to quantum chemical language. Next, we describe an implementation of dynamical mean-field theory where we start from an ab initio Hartree-Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the configuration interaction hierarchy in DMFT as an approximate solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions.

Takashi Mori

Equilibrium properties of long-range interacting systems on lattices are investigated. There was a conjecture by Cannas et al. [Phys. Rev. B 61, 11521 (2000)] that the mean-field theory is exact for spin systems with nonadditive long-range interactions. This is called "exactness of the mean-field theory." We show that the exactness of the mean-field theory holds for systems on a lattice with nonadditive two-body long-range interactions in the canonical ensemble with unfixed order parameters. We also show that in a canonical ensemble with fixed order parameters, exactness of the mean-field theory does not hold in one parameter region, which we call the "non-mean-field region." In the non-mean-field region, an inhomogeneous configuration appears, in contrast to the uniform configuration in the region where the mean-field theory holds. This inhomogeneous configuration is not the one given by the standard phase separation. Therefore, the mean-field picture is not adequate to describe these states. We discuss phase transitions between the mean-field region and the non-mean-field region. Exactness of the mean-field theory in spin glasses is also discussed.