Exact solution for two $\delta$-interacting bosons on a ring in the presence of a $\delta$-barrier: Asymmetric Bethe Ansatz for spatially odd states

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> expansion about the hard-core boson point [D. Sen, Int. J. Mod. Phys. A 14, 1789 (1999); A. G. Volosniev et al., Nat. Commun. 5, 5300 (2014)] as one application of our results. Additionally, we find that when the \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> -barrier is converted to a \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> -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.