Exact spectral function of the Tonks-Girardeau gas at finite temperature

TL;DR

This paper derives exact determinant formulas for the spectral function of the Tonks-Girardeau gas at finite temperature, applicable to various potentials and statistics, enabling efficient numerical analysis of both equilibrium and nonequilibrium states.

Contribution

It provides a novel, exact determinant representation for the spectral function of the Tonks-Girardeau gas applicable to arbitrary potentials, temperature, and particle statistics, including nonequilibrium scenarios.

Findings

Spectral function of a trapped system has only two singular lines.

The formulas are efficient for numerical implementation.

Results apply to both lattice and continuum models.

Abstract

We report on the derivation of determinant representations for the Green's functions and spectral function of the trapped Tonks-Girardeau gas on the lattice and in the continuum. Our results are valid for any type of statistics of the constituent particles, at zero and finite temperature and arbitrary confining potentials, including nonequilibrium scenarios induced by sudden changes of the external potential. In addition, they are also extremely efficient and easy to implement numerically with the main computational effort being represented by the calculation of partial overlaps of the dynamically evolved single particle wavefunctions. In the lattice case we show that the spectral function of a system with a strong harmonic potential presents only two singular lines compared with three singular lines in the case of a homogeneous system.