# Effective spin physics in two-dimensional cavity QED arrays

## TL;DR

This paper explores the complex behavior of light-matter interactions in two-dimensional cavity arrays, deriving effective spin models and analyzing phase transitions and frustration effects through analytical and numerical methods.

## Contribution

It introduces an effective spin Hamiltonian for 2D cavity QED arrays and provides comprehensive analysis using mean-field and quantum Monte Carlo simulations.

## Key findings

- Identification of phase diagram and transition points.
- Characterization of frustrated spin interactions.
- Quantitative analysis of correlation properties.

## Abstract

We investigate a strongly correlated system of light and matter in two-dimensional cavity arrays. We formulate a Jaynes-Cummings Hamiltonian for two-level atoms coupled to cavity modes and driven by an external laser field which reduces to an effective spin Hamiltonian in the dispersive regime. In one dimension we provide exact analytical solution. In two dimensions, we perform mean-field study and large scale quantum Monte Carlo simulations of both the Jaynes-Cummings and the effective spin models. We discuss the phase diagram and the parameter regime which gives rise to frustrated interactions between the spins. We provide quantitative description of the phase transitions and correlation properties featured by the system and we discuss graph-theoretical properties of the ground states in terms of graph colorings using P\'{o}lya's enumeration theorem.