A mathematical analysis of the discretized IPT-DMFT equations

TL;DR

This paper rigorously analyzes the discretized IPT-DMFT equations, proving existence and uniqueness of solutions, characterizing solutions in specific cases, and demonstrating numerical simulations on the Hubbard dimer.

Contribution

It provides the first mathematical proof of existence and uniqueness for discretized IPT-DMFT equations and characterizes solutions in symmetric cases.

Findings

Existence of solutions for discretized equations within certain parameter ranges.

Uniqueness of solutions in a smaller parameter range.

Complete solution characterization for the case N_omega=0 and partial results for N_omega=1.

Abstract

In a previous contribution (E. Canc\`es, A. Kirsch and S. Perrin--Roussel, arXiv:2406.03384), we have proven the existence of a solution to the Dynamical Mean-Field Theory (DMFT) equations under the Iterated Perturbation Theory (IPT-DMFT) approximation. In view of numerical simulations, these equations need to be discretized. In this article, we are interested in a discretization of the \acrshort{ipt}-\acrshort{dmft} functional equations, based on the restriction of the hybridization function and local self-energy to a finite number of points in the upper half-plane $\left(i\omega_n\right)_{n \in |[0,N_\omega]|}$, where $\omega_n=(2n+1)\pi / \beta$ is the $n$-th Matsubara frequency and $N_\omega \in \mathbb N$. We first prove the existence of solutions to the discretized equations in some parameter range depending on $N_\omega$. We then prove uniqueness for a smaller range of parameters. We also study more in depth the case of bipartite systems exhibiting particle-hole symmetry. In this case, the discretized IPT-DMFT equations have purely imaginary solutions, which can be obtained by solving a real algebraic system of $(N_\omega+1)$ equations with $(N_\omega+1)$ variables. We provide a complete characterization of the solutions for $N_\omega=0$ and some results for $N_\omega=1$ in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.