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A mathematical analysis of the discretized IPT-DMFT equations

E. Cancès, A. Kirsch, S. Perrin--Roussel

TL;DR

This paper provides a rigorous treatment of discretized IPT-DMFT equations by restricting the hybridization function and local self-energy to a finite Matsubara grid. It establishes existence of solutions for small dimensionless parameters and proves uniqueness in a tighter regime, with special attention to bipartite systems at half filling where purely imaginary solutions arise; in this case the problem reduces to real algebraic equations and, for the Hubbard dimer, to tractable scalar or low-degree polynomial systems. The authors also perform numerical experiments on the Hubbard dimer, including fixes to discretization and interpolation procedures, revealing a conductor-to-insulator transition and highlighting challenges in Nevanlinna-Pick interpolation and in preserving Pick-function structure under discretization. Overall, the work provides a mathematically grounded framework for analyzing discretized DMFT solvers and offers insight into parameter regimes and interpolation issues that affect numerical IPT-DMFT implementations.

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

A mathematical analysis of the discretized IPT-DMFT equations

TL;DR

This paper provides a rigorous treatment of discretized IPT-DMFT equations by restricting the hybridization function and local self-energy to a finite Matsubara grid. It establishes existence of solutions for small dimensionless parameters and proves uniqueness in a tighter regime, with special attention to bipartite systems at half filling where purely imaginary solutions arise; in this case the problem reduces to real algebraic equations and, for the Hubbard dimer, to tractable scalar or low-degree polynomial systems. The authors also perform numerical experiments on the Hubbard dimer, including fixes to discretization and interpolation procedures, revealing a conductor-to-insulator transition and highlighting challenges in Nevanlinna-Pick interpolation and in preserving Pick-function structure under discretization. Overall, the work provides a mathematically grounded framework for analyzing discretized DMFT solvers and offers insight into parameter regimes and interpolation issues that affect numerical IPT-DMFT implementations.

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 , where is the -th Matsubara frequency and . We first prove the existence of solutions to the discretized equations in some parameter range depending on . 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 equations with variables. We provide a complete characterization of the solutions for and some results for in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.

Paper Structure

This paper contains 18 sections, 7 theorems, 106 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Matsubara frequency discretization of IPT-DMFT equations
  3. Derivation of the IPT-DMFT equations
  4. Matsubara frequency (MaF) discretization
  5. Dimensionless formulation
  6. Existence and uniqueness results
  7. Existence of solutions
  8. Uniqueness
  9. The case of bipartite systems
  10. Particle-hole symmetry
  11. Algebraic structure of MaF-discretized IPT-DMFT for generic bipartite systems
  12. Hubbard dimer
  13. Low-temperature regime for $N_{\omega}=0$
  14. Purely imaginary solutions for the Hubbard dimer
  15. Numerical simulations of the Hubbard dimer
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $N_{\omega} \in \mathbb{N}^*$ and There exists $t_{N_{\omega}} > 0$ depending only on $N_{\omega}$ such that for all $0 \le t \le \frac{t_{N_{\omega}}}{\sqrt{\mathrm{deg}(\mathcal{G}_{{H}})}}$, the MaF-discretized ipt-dmft equations eq:NondimensionalBU--eq:NondimensionalIPT have a solution where $\mathfrak{D}_t^{N_{\omega}}$ is given by

Figures (7)

  • Figure 1: Left: Pariser--Parr--Pople model of benzene C$_6$H$_6$. Right: supercell model $(\mathbb Z/6\mathbb Z)^2$.
  • Figure 2: The set $\left(i\omega_n\right)_{n \in [\![ {0},{N_{\omega}}]\!]}$ of points in $\mathbb{C}_+$ used for the discretization of $\Delta$ and $\Sigma$ with $N_{\omega}=4$
  • Figure 3: Number of admissible solutions $(x_0,x_1) \in (0,1]^2$ to \ref{['eq:system_dimer_Nomega=1_0']}-\ref{['eq:system_dimer_Nomega=1_1']} in the range of parameters $(a,b) \in [0,10] \times [0,25]$. This gives the number of purely imaginary solutions to the dimensionless MaF-discretized IPT-DMFT equations \ref{['eq:NondimensionalBU']}--\ref{['eq:NondimensionalIPT']} for the Hubbard dimer in the range of parameters $(t,u) \in [0,\sqrt{10}\pi] \times [0,5\pi^2]$.
  • Figure 4: Density $\rho$ of the spectral function $A$ obtained by analytic continuation using Pade approximants, for different values of the on-site repulsion $U_{{}}$. Other parameters are fixed as in \ref{['eq:ParametersMottTransition']}.
  • Figure 5: Linear convergence of the fixed-point algorithm. The residual $\Vert {\Delta^{(n+1)}-\Delta^{(n)}} \Vert_2$ is plotted in log scale, as a function of the iteration $n \in [\![ {0},{N_{\mathrm{iter}}}]\!]$. Left and right sides differ in the initial guess $\Delta^{(0)}$.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Definition 2.1: MaF-discretized IPT-DMFT equations
  • Remark 2.2: comment on the set of admissible solutions
  • Definition 2.3: dimensionless MaF-discretized ipt-dmft equations
  • Remark 2.4: comparison with other dimensionless versions of the IPT-DMFT equations
  • Theorem 3.1: Existence of solution to the discretized ipt-dmft equations.
  • Remark 3.2: on the non-interacting setting $u=0$
  • proof : Proof of Theorem \ref{['thm:GlobalExistenceMFDiscretized']}
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 10 more