Bogoliubov-like mode in the Tonks-Girardeau Gas

TL;DR

This paper develops a 1D boson-fermion duality framework in path-integral form, revealing a Bogoliubov-like collective mode in the Tonks-Girardeau gas with phonon dispersion.

Contribution

It introduces a novel 1D duality approach that allows analytical derivation of collective excitations in strongly interacting bosonic systems.

Findings

Derivation of the Bogoliubov-like phonon dispersion in the Tonks-Girardeau regime

Establishment of a 1D boson-fermion duality analogous to 2D Chern-Simons theory

Analytical expressions for long-wavelength collective modes

Abstract

We reformulate 1D boson-fermion duality in path-integral terms. The result is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The theory is consistent and enables, using standard resummation techniques, to obtain the long-wave-length asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: $\omega(q)=q \sqrt{\bar\rho U(q)/m}$, where $\bar\rho$ is the bosons density and $U(q)$ is a Fourier component of the two-body potential.