A mathematical analysis of IPT-DMFT

TL;DR

This paper offers a rigorous mathematical analysis of the Dynamical Mean-Field Theory (DMFT), proving the existence of solutions for the equations in the context of the finite Hubbard model and exploring their properties.

Contribution

It provides a formal mathematical formulation of DMFT equations, proves the existence of solutions under certain conditions, and analyzes their properties, enhancing theoretical understanding.

Findings

Proves the existence of solutions for DMFT equations in the finite Hubbard model.

Establishes properties of the solutions within the mathematical framework.

Provides a self-contained mathematical formulation of DMFT and impurity models.

Abstract

We provide a mathematical analysis of the Dynamical Mean-Field Theory, a celebrated representative of a class of approximations in quantum mechanics known as embedding methods. We start by a pedagogical and self-contained mathematical formulation of the Dynamical Mean-Field Theory equations for the finite Hubbard model. After recalling the definition and properties of one-body time-ordered Green's functions and self-energies, and the mathematical structure of the Hubbard and Anderson impurity models, we describe a specific impurity solver, namely the Iterated Perturbation Theory solver, which can be conveniently formulated using Matsubara's Green's functions. Within this framework, we prove under certain assumptions that the Dynamical Mean-Field Theory equations admit a solution for any set of physical parameters. Moreover, we establish some properties of the solution(s).