Luttinger liquids with multiple Fermi edges: Generalized Fisher-Hartwig conjecture and numerical analysis of Toeplitz determinants

TL;DR

This paper numerically investigates Toeplitz determinants with Fisher-Hartwig singularities in many-body physics, confirming an extended FH conjecture and applying it to analyze Fermi edge singularities and tunneling in Luttinger liquids with complex energy distributions.

Contribution

It provides numerical validation of the extended Fisher-Hartwig conjecture for Toeplitz determinants with multiple singularities and applies this to problems involving Fermi edges and Luttinger liquids.

Findings

Numerical results agree with the extended FH conjecture including all branches.

Power-law singularities occur at multiple edges due to summation over FH branches.

Application to Luttinger liquids reveals complex edge behaviors in energy distributions.

Abstract

It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.