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arXiv:2508.17371 · SciPost Physics Core

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

Maxim Olshanii, Mathias Albert, Gianni Aupetit-Diallo, Patrizia Vignolo, Steven Glenn Jackson

Published Nov 12, 2025• 2 citations
arXivHTMLPDF

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 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 1 / g 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 δ -barrier is converted to a \delta 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.

DOI
10.21468/scipostphyscore.8.4.083

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We consider two particular one-dimensional quantum many-body systems with local interactions related to the root system CN. Both models describe identical particles moving on the half-line with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as delta-prime interaction. We show that the Bethe ansatz solution of the delta-interaction model is consistent even for the general case where the particles are distinguishable, whereas for the delta-prime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta- and the fermionic delta-prime model, and we elaborate on the physical interpretations of these models.
We consider molecules made of two one-dimensional short-range-interacting bosonic atoms. We show that in the process of scattering of these molecules off a narrow barrier, odd incident waves produce no unbound atoms, even when the incident energy exceeds the dissociation threshold . We demonstrate that such a feature originates from the fact that our system shares a part of its spectrum, namely its odd eigenstates, with a Bethe Ansatz integrable model, a δ -interaction Hamiltonian with local and non-local terms, associated with a reflection group C 2 —a full symmetry of a square. This partial integrability induces prohibition of chemical reactions, allowing us to make predictions about scattering processes. We suggest several experimental implementations of the effect. We also propose to use the monomer production as an alternative read-out channel in an atom interferometer: unlike in the standard interferometric schemes, no spatial separation of the output channels will be required. Our approach offers an example of use of integrable problems beyond the realm of fully integrable models.
The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.
one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.
We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).