Cluster dynamical mean-field theories: Causality and classical limit

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, periodized cluster 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’HooftVeltman cutting equations are generalized and derived for nontranslation 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 in general and the pair scheme in particular are not. Constraints on further extension of these schemes are discussed.