Integrability and weak diffraction in a two-particle Bose–Hubbard model

Abstract

A recently introduced one-dimensional two-particle Bose–Hubbard model with a single impurity (Zhang et al 2012 Phys. Rev. Lett. 109 116405) is studied on finite lattices. The model possesses a discrete reflection symmetry and we demonstrate that all eigenstates odd under this symmetry can be obtained with a generalized Bethe ansatz if periodic boundary conditions~(BC) are imposed. Furthermore, we provide numerical evidence that this holds true for open BC as well. The model exhibits backscattering at the impurity site—which usually destroys integrability—yet there exists an integrable subspace. We investigate the non-integrable even sector numerically and find a class of states which have almost the Bethe ansatz form. These weakly diffractive states correspond to a weak violation of the non-local Yang–Baxter relation which is satisfied in the odd sector. We introduce a method based on the Prony algorithm to check whether a numerically obtained wave function has the Bethe form or not, and if so, to extract its parameters. This technique is applicable to a wide variety of other lattice models.