# Topological Edge-State Manifestation of Interacting 2D Condensed Boson-Lattice Systems in a Harmonic Trap

## TL;DR

This paper demonstrates that interactions in a 2D bosonic lattice system under a harmonic trap can reveal topological edge states, with a clear band structure and increased edge modes, supported by analytical and numerical results.

## Contribution

It shows that weak interactions enable the observation of topological edge states in bosonic systems within harmonic traps, which is typically challenging in noninteracting systems.

## Key findings

- Edge states are clearly visible in the band structure with harmonic trapping.
- The number of edge states crossing the gap increases due to interactions.
- Analytical expressions for edge-state energies match numerical results.

## Abstract

In this Letter, it is shown that interactions can facilitate the emergence of topological edge states of quantum-degenerate bosonic systems in the presence of a harmonic potential. This effect is demonstrated with the concrete model of a hexagonal lattice populated by spin-one bosons under a synthetic gauge field. In fermionic or noninteracting systems, the presence of a harmonic trap can obscure the observation of edge states. For our system with weakly interacting bosons in the Thomas--Fermi regime, we can clearly see a topological band structure with a band gap traversed by edge states. We also find that the number of edge states crossing the gap is increased in the presence of a harmonic trap, and the edge modes experience an energy shift while traversing the first Brillouin zone which is related to the topological properties of the system. We find an analytical expression for the edge-state energies and our comparison with numerical computation shows excellent agreement.