2025/05/27 by Éric Cancès, Andreas Kirsch, Cancès, E. +3
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2505.21287
openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
In a previous contribution (E. Cancès, 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 \acrshortipt-\acrshortdmft 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 (iωn)n ∈ |[0,Nω]|, where ωn=(2n+1)π/ β is the n-th Matsubara frequency and Nω∈ \mathbb N. We first prove the existence of solutions to the discretized equations in some parameter range depending on Nω. 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 (Nω+1) equations with (Nω+1) variables. We provide a complete characterization of the solutions for Nω=0 and some results for Nω=1 in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.