A mathematical analysis of the discretized IPT-DMFT equations

Abstract

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 \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 ${\left(i\ddot{I}{‰}_n\right)}_{n\in |[0,N_{\ddot{I}}‰]|}$, where $\ddot{I}{‰}_n=(2n+1)\ddot{I}€/{\hat{I}}^2$ is the $n$-th Matsubara frequency and $N_{\ddot{I}}‰\in \mathbb{N}$. We first prove the existence of solutions to the discretized equations in some parameter range depending on $N_{\ddot{I}}‰$. 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 $(N_{\ddot{I}}‰+1)$ equations with $(N_{\ddot{I}}‰+1)$ variables. We provide a complete characterization of the solutions for $N_{\ddot{I}}‰=0$ and some results for $N_{\ddot{I}}‰=1$ in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.