# Mass-imbalanced atoms in a hard-wall trap: an exactly solvable model with emergent $D_{6}$ symmetry

## TL;DR

This paper presents an exactly solvable model of two interacting particles with specific mass ratios in a hard-wall trap, revealing an emergent $D_{6}$ symmetry and providing explicit eigenstates and energies.

## Contribution

It introduces a Bethe-type ansatz based on dihedral group symmetry to solve for eigenstates of a mass-imbalanced two-particle system in a hard-wall trap.

## Key findings

- Exact eigenenergies and eigenstates for mass ratio 3 obtained.
- Many-body excited states are interaction-independent.
- The model links momentum distribution to $D_{2n}$ symmetry.

## Abstract

We show that a system consisting of two interacting particles with mass ratio $3$ or $1/3$ in a hard-wall box can be exactly solved by using Bethe-type ansatz. The ansatz is based on a finite superposition of plane waves associated with a dihedral group $D_{6}$, which enforces the momentums after a series of scattering and reflection processes to fulfill the $D_{6}$ 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 for permitted mass ratios. For a quantum system with mass ratio $3$, we obtain exact eigenenergies and eigenstates by solving Bethe-type-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.