Solving the Bethe-Salpeter Equation in Real Frequencies at Finite Temperature

TL;DR

This paper introduces a method to solve the Bethe-Salpeter equation directly on the real frequency axis at finite temperature, avoiding the ill-conditioned analytic continuation process, and demonstrates its effectiveness on the electron gas model.

Contribution

It presents a novel approach combining diagrammatic Monte Carlo and series resummation to solve the Bethe-Salpeter equation directly in real frequencies at finite temperature.

Findings

Accurately solves the Bethe-Salpeter equation on the real axis.

Shows how multiple scattering renormalizes Landau damping.

Demonstrates method on the homogeneous electron gas model.

Abstract

Self-consistent Hartree-Fock approximation combined with solutions of the Bethe-Salpeter equation offers a powerful tool for studies of strong correlation effects arising in condensed matter models, nuclear physics, quantum field theories, and real materials. The standard finite-temperature approach would be to first solve the problem in the Matsubara representation and then apply numerical analytic continuation to the real-frequency axis to link theoretical results with experimental probes, but this ill-conditioned procedure often distorts important spectral features even for very accurate imaginary-frequency data. We demonstrate that the ladder-type finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the 3-point vertex function and, ultimately, system's polarization can be accurately solved directly on the real frequency axis using the diagrammatic Monte Carlo technique and series resummation. We illustrate the method by applying it to the homogeneous electron gas model and demonstrate how multiple scattering events renormalize Landau damping.