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

TL;DR

This work obtains an exact solution for two $δ$-interacting bosons on a ring in the presence of a $δ$-barrier, restricted to the spatially odd subspace. Using the Asymmetric Bethe Ansatz, the authors map the physical problem to an auxiliary Bethe-Ansatz-solvable system with a nonlocal interaction, derive Bethe Ansatz Equations for the rapidities $k_1$ and $k_2$, and construct explicit wavefunctions with energy $E=rac{ ext{hbar}^2}{2m}(k_1^2+k_2^2)$. They benchmark a $1/g$ expansion around the hard-core limit, finding that the leading correction matches expectations while the $1/g^2$ term is positive, indicating limitations of the pseudopotential approach beyond first order. An intriguing regime occurs when the barrier is traded for a δ-well with equal magnitude to the particle-particle interaction ($g_B=-g$), yielding a non-interacting spectrum while the eigenstates retain strong-interaction features. The results illuminate the interplay between integrability, barrier-induced reflections, and finite-size effects on quantum spectra, and lay groundwork for exploring level statistics and quantum chaos in related models.

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 $δ$-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 $δ$-barrier is converted to a $δ$-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.

Figures (2)

• Figure 1: Geometry of the problem. According to the Hamiltonian \ref{['H_C2_tilde']}, two bosons in a box of side length $L$ with periodic boundary conditions for particle 1 (short-dashed blue) and particle 2 (short-dashed green) interact with a barrier (solid magenta) at the origin. The particles also interact between themselves via a $\delta$-potential peaked at their point of contact (solid red). Additionally, an empirically irrelevant non-local $\delta$-potential is peaked at loci where the particles have equal-magnitude, opposite-sign coordinates (dashed red). This system represents an instance of Gaudin’s generalized kaleidoscope gaudin1983_book_english (Sec. 5.2), solvable via the Bethe Ansatz. The underlying reflection group is the symmetry of a square tiling of a plane, $\tilde{\mathcal{C}}_{2}$, with generating mirrors at $x_{1}=x_{2}$, $x_{1}=0$, and $x_{2}=\frac{L}{2}$. The empirically relevant Hamiltonian \ref{['H']} omits the unphysical interaction at $x_{1}=-x_{2}$ (dashed red) and is not generally solvable with the Bethe Ansatz. However, following the Asymmetric Bethe Ansatz recipe jackson2024_062, the eigenstates of the physical problem that are odd under point inversion about the barrier can be obtained, as they are shared with the unphysical problem. For bosonic particles, the states that are odd under point inversion are also odd with respect to reflection about the $x_{1}=-x_{2}$ mirror featuring a node there as the result.
• Figure 2: Finding the spatially odd eigenstates for two $\delta$-interacting bosons interacting with a $\delta$-barrier, subject to periodic boundary conditions. (a) Solution manifolds for the Bethe Ansatz Equation \ref{['BAE1']} (blue) and \ref{['BAE2']} (green). Their intersections provide one with the allowed pairs of rapidities, $(k_{1},\,k_{2})$. The red dot marks the ground state. The parameters are $g=4.\times \hbar^2/mL$ and $g_{\text{B}}=g/\sqrt{2}$. (b) The ground state wavefunction. It's rapidities are $k_{1} = 2.00 \times 1/L$ and $k_{2} = 7.27\times 1/L$, leading to the ground state energy of $56.7 \times \hbar^2/mL^2$. The wavefunction is normalized to unity.