Wave functions of the super Tonks-Girardeau gas and the trapped 1D hard sphere Bose gas

TL;DR

This paper demonstrates the exact equivalence of wave functions between the super Tonks-Girardeau gas and the 1D hard sphere Bose gas under certain conditions, providing analytical solutions and insights into their metastability.

Contribution

It establishes the precise relationship between sTG and HSB wave functions, including exact solutions for two particles and an approximation for many particles, clarifying their connection and metastability.

Findings

sTG and HSB wave functions are identical for two particles when certain conditions are met

Exact solutions for N=2 particles are expressed in terms of parabolic cylinder functions

The metastability of the sTG phase is explained by minimal overlap with collapsed states

Abstract

Recent theoretical and experimental results demonstrate a close connection between the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gas with hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ of the sTG gas, a highly excited gas-like state with nodes only at interparticle separations $|x_{j\ell}|=x_{node}\approx a_{1D}$. It is shown herein that when the coupling constant $g_B$ in the Lieb-Liniger interaction $g_B\delta(x_{j\ell})$ is negative and $|x_{12}|\ge x_{node}$, the sTG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the sTG and HSB wave functions is that the sTG wave function allows a small penetration into the region $|x_{12}|<x_{node}$, whereas for a HSB gas with hard sphere diameter $a_{h.s.}=x_{node}$, the HSB wave function vanishes when all $|x_{12}|<a_{h.s.}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The sTG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ge 2$, $x_{node}$ is given accurately by a simple parabola. The metastability of the sTG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.