Simulating the spectral gap with polariton graphs

TL;DR

This paper demonstrates that polariton graphs can simulate the spectral gap of the XY model by relating the energy spectrum to polariton condensation, validated through numerical simulations on a hexagonal lattice.

Contribution

It establishes a connection between the XY Hamiltonian's energy spectrum and polariton condensation, enabling spectral gap simulation with polariton graphs.

Findings

Lower energy states are accurately reproduced by simulations.

The relationship between the XY spectrum and polariton particles is elucidated.

The approach is validated on a hexagonal lattice.

Abstract

We recently proposed polariton graphs as a novel platform for solving hard optimization problems that can be mapped into the $XY$ model. Here, we elucidate a relationship between the energy spectrum of the $XY$ Hamiltonian and the total number of condensed polariton particles. Using as a test-bed the hexagonal unit lattice we show that the lower energy states of the $XY$ Hamiltonian are faithfully reproduced by mean-field numerical simulations utilising the Ginzburg--Landau equation coupled to an exciton reservoir. Our study paves the way to simulating the spectral gap of the XY model using polariton graphs.