F-electron spectral function of the Falicov-Kimball model and the Wiener-Hopf sum equation approach

TL;DR

This paper introduces a new real-time approach to compute the f-electron spectral function in the Falicov-Kimball model, enabling analysis at very low temperatures using Toeplitz determinants and Wiener-Hopf methods.

Contribution

It presents an alternative real-time representation and exact analytic formulas for the spectral function using Wiener-Hopf and Szeg"o's theorem, improving low-temperature calculations.

Findings

Derived exact asymptotic formulas for Green's functions.

Validated formulas against matrix extrapolation results.

Extended analysis to cases with different winding behaviors.

Abstract

We derive an alternative representation for the $f$-electron spectral function of the Falicov-Kimball model from the original one proposed by Brandt and Urbanek. In the new representation, all calculations are restricted to the real time axis, allowing us to go to arbitrarily low temperatures. The general formula for the retarded Green's function involves two determinants of continuous matrix operators that have the Toeplitz form. By employing the Wiener-Hopf sum equation approach and Szeg\"o's theorem, we can derive exact analytic formulas for the large-time limits of the Green's function; we illustrate this for cases when the logarithm of characteristic function (which defines the continuous Toeplitz matrix) does and does not wind around the origin. We show how accurate these asymptotic formulas are to the exact solutions found from extrapolating matrix calculations to the zero discretization size limit.