Light-Matter Correlation Energy Functional of the Cavity-Coupled Two-Dimensional Electron Gas via Quantum Monte Carlo Simulations

We perform extensive simulations of the two-dimensional cavity-coupled electron gas in a modulating potential as a minimal model for cavity quantum materials. These simulations are enabled by a newly developed quantum-electrodynamical (QED) auxiliary-field quantum Monte Carlo method. We present a procedure to greatly reduce finite-size effects in such calculations. Based on our results, we show that a modified version of weak-coupling perturbation theory is remarkably accurate for a large parameter region. We further provide a simple parametrization of the light-matter correlation energy as a functional of the cavity parameters and the electronic density. These results provide a crucial step toward a numerical foundation for the development of the QED density functional theory, which was previously reliant on analytical approximations, to allow quantitative modeling of a wide range of systems with light-matter coupling.