Exact Solution for Two $\delta$-Interacting Bosons on a Ring in the Presence of a $\delta$-Barrier: Asymmetric Bethe Ansatz for Spatially Odd States

TL;DR

This paper applies an asymmetric Bethe Ansatz to exactly solve a two-boson system on a ring with a delta barrier, revealing insights into odd states and benchmarking expansion methods, with notable spectral phenomena.

Contribution

It introduces an exact solution for odd spatial states of two bosons with delta interactions on a ring with a barrier, using the asymmetric Bethe Ansatz, and explores spectral properties.

Findings

Benchmarking of the 1/g expansion method.

Spectral equivalence when delta barrier becomes a well.

Features of strongly interacting systems in eigenstates.

Abstract

In this article, we apply the recently proposed Asymmetric Bethe Ansatz method to the problem of two one-dimensional, short-range-interacting bosons on a ring in the presence of a $\delta$-function barrier. Only half of the Hilbert space--namely, the two-body states that are odd under point inversion about the position of the barrier--is accessible to this method. The other half is presumably non-integrable. We consider benchmarking the recently proposed $1/g$ expansion about the hard-core boson point [A. G. Volosniev, D. V. Fedorov, A. S. Jensen, M. Valiente, N. T. Zinner, Nature Communications 5, 5300 (2014)] as one application of our results. Additionally, we find that when the $\delta$-barrier is converted to a $\delta$-well with strength equal to that of the particle-particle interaction, the system exhibits the spectrum of its non-interacting counterpart while its eigenstates display features of a strongly interacting system. We discuss this phenomenon in the "Summary and Future Research" section of our paper.