# Symmetry-enriched Bose-Einstein condensates in spin-orbit coupled bilayer system

## TL;DR

This paper explores how symmetry properties in a spin-orbit coupled bilayer system influence Bose-Einstein condensation, revealing novel phases and the role of symmetry in degeneracy and condensate competition.

## Contribution

It demonstrates how symmetry operators' commutation or anti-commutation affects band degeneracy and condensate phases in spin-orbit coupled BECs, introducing new stripe phases and crossover phenomena.

## Key findings

- Degeneracy depends on symmetry operator relations.
- Stripe phases emerge from condensate competition.
- Crossover involves excited band condensation.

## Abstract

We consider the fate of Bose-Einstein condensation (BEC) with time-reversal symmetry and inversion symmetry in a spin-orbit coupled bilayer system. When these two symmetry operators commute, all the single particle bands are exactly two-fold degenerate in the momentum space. The scattering in the two-fold degenerate rings can relax the spin-momentum locking effect resulting from spin-orbit coupling, thus we can realize the spin polarized plane wave phase even when the inter-particle interaction dominates. When these two operators anti-commute, the lowest two bands may have the same minimal energy, which have totally different spin structures. As a result, the competition between different condensates in these two energetically degenerate rings can give rise to interesting stripe phases with atoms condensed at two or four colinear momenta. We find that the crossover between these two cases is accompanied by the excited band condensation when the interference energy can overcome the increased single particle energy in the excited band. This effect is not based on strong interaction, thus can be realized even with moderate interaction strength.