Towards analytical approaches to the dynamical-cluster approximation

TL;DR

This paper proposes simplified schemes for the dynamical cluster approximation's self-consistency condition, demonstrating their effectiveness in approximating thermodynamic and dynamical properties of the Hubbard model with potential for analytic solutions.

Contribution

Introduces simplified schemes for the dynamical cluster approximation's self-consistency, validated numerically, enabling more analytic approaches to Hubbard and similar models.

Findings

Simplified schemes closely match full self-consistent results.

Thermodynamic properties are practically indistinguishable from full schemes.

Effective approximation of dynamical properties using analytic continuation.

Abstract

I introduce several simplified schemes for the approximation of the self-consistency condition of the dynamical cluster approximation. The applicability of the schemes is tested numerically using the fluctuation-exchange approximation as a cluster solver for the Hubbard model. Thermodynamic properties are found to be practically indistinguishable from those computed using the full self-consistent scheme in all cases where the non-interacting partial density of states is replaced by simplified analytic forms with matching 1st and 2nd moments. Green functions are also compared and found to be in close agreement, and the density of states computed using Pad\'{e} approximant analytic continuation shows that dynamical properties can also be approximated effectively. Extensions to two-particle properties and multiple bands are discussed. Simplified approaches to the dynamical cluster approximation should lead to new analytic solutions of the Hubbard and other models.