Mass-imbalanced atoms in a hard-wall trap: an exactly solvable model with emergent $D_{2n}$ symmetry

Yanxia Liu, Fan Qi, Yunbo Zhang, Shu Chen

Published 2019-03-20Version 1

We show that a system consisting of two interacting particles with unequal masses in a hard-wall box can be exactly solved by using Bethe ansatz, when the mass ratio takes some specific values. The ansatz is based on finite superpositions of plane waves associated with a dihedral group $D_{2n}$, which enforces the momentums after a series of scattering and reflection processes to fulfill the $D_{2n}$ symmetry. Starting from a two-body elastic collision model in a hard-wall box, we demonstrate how a finite momentum distribution is related to the $D_{2n}$ symmetry and gives the condition of permitted mass ratios corresponding to classical superintegrable points. For a quantum system with mass ratio $3$, we obtain exact eigenenergies and eigenstates by solving Bethe ansatz equations for arbitrary interaction strength. A many-body excited state of the system is found to be independent of the interaction strength, i.e. the wave function looks exactly the same for non-interacting two particles or in the hard-core limit.