Exact equivalence between one-dimensional Bose gases interacting via hard-sphere and zero-range potentials

TL;DR

This paper establishes an exact equivalence between one-dimensional hard-sphere Bose gases and a system with momentum-dependent zero-range interactions, enabling new analytical approaches to their ground state energies and related properties.

Contribution

It introduces the extended hard-sphere Bose gas model with a regular pseudopotential, facilitating momentum-space analysis and exact energy calculations for strongly interacting regimes.

Findings

Derived the ground state energy of Lieb-Liniger gas in the thermodynamic limit.

Calculated the energy of the super Tonks-Girardeau state with attractive interactions.

Applied Tan relations to analyze large-momentum behavior in the new framework.

Abstract

We prove the equivalence between the hard-sphere Bose gas and a system with momentum-dependent zero-range interactions in one spatial dimension, which we call extended hard-sphere Bose gas. The two-body interaction in the latter model has the advantage of being a regular pseudopotential. The most immediate consequence is the existence of its Fourier transform, permitting the formulation of the problem in momentum space, not possible with the original hard-core interaction. In addition, in the extended system, interactions are defined in terms of the scattering length, positive or negative, identified with the hard-sphere diameter only when it is positive. We are then able to obtain, directly in the thermodynamic limit, the ground state energy of the strongly repulsive Lieb-Liniger gas and, more importantly, the energy of the lowest-lying super Tonks-Girardeau gas state with finite, strongly attractive interactions, in perturbation theory from the novel extended hard-sphere Bose gas. Tan relations involving the large-momentum behavior of the Lieb-Liniger gas are also derived, and then applied to the super Tonks-Girardeau gas within our perturbative approach.