Cluster Dynamical Mean Field Theories

TL;DR

This paper analyzes the properties of Cluster Dynamical Mean Field Theories, focusing on their semiclassical limits, causality, and introduces a translation invariant formulation of PCDMFT, with implications for classical and quantum models.

Contribution

It presents a translation invariant formulation of PCDMFT and analyzes its causality, contrasting it with other cluster schemes, and generalizes cutting equations for non-invariant systems.

Findings

PCDMFT is shown to be causal.

Nested cluster schemes are generally not causal.

The semiclassical limit relates to classical Ising models.

Abstract

Cluster Dynamical Mean Field Theories are analyzed in terms of their semiclassical limit and their causality properties, and a translation invariant formulation of the cellular dynamical mean field theory, PCDMFT, is presented. The semiclassical limit of the cluster methods is analyzed by applying them to the Falikov-Kimball model in the limit of infinite Hubbard interaction U where they map to different classical cluster schemes for the Ising model. Furthermore the Cutkosky-t'Hooft-Veltman cutting equations are generalized and derived for non translation invariant systems using the Schwinger-Keldysh formalism. This provides a general setting to discuss causality properties of cluster methods. To illustrate the method, we prove that PCDMFT is causal while the nested cluster schemes (NCS) in general and the pair scheme in particular are not. Constraints on further extension of these schemes are discussed.