# Exact functionals for correlated electron-photon systems

## TL;DR

This paper constructs exact density-to-potential maps for correlated electron-photon systems, revealing strong exchange-correlation effects and the limitations of mean-field approximations in the strong coupling regime, with implications for understanding coupled matter-photon phenomena.

## Contribution

It introduces exact density-to-potential maps for electron-photon systems and generalizes the intra-system steepening concept to coupled fermion-boson systems, providing new insights into strong coupling effects.

## Key findings

- Intra-system steepening indicates strong exchange-correlation effects.
- Mean-field approximation fails in the strong coupling regime.
- Large polarizability observed due to ultra-strong electron-photon coupling.

## Abstract

For certain correlated electron-photon systems we construct the exact density-to-potential maps, which are the basic ingredients of a density-functional reformulation of coupled matter-photon problems. We do so for numerically exactly solvable models consisting of up to four fermionic sites coupled to a single photon mode. We show that the recently introduced concept of the intra-system steepening (T.Dimitrov et al., 18, 083004 NJP (2016)) can be generalized to coupled fermion-boson systems and that the intra-system steepening indicates strong exchange-correlation (xc) effects due to the coupling between electrons and photons. The reliability of the mean-field approximation to the electron-photon interaction is investigated and its failure in the strong coupling regime analyzed. We highlight how the intra-system steepening of the exact density-to-potential maps becomes apparent also in observables such as the photon number or the polarizability of the electronic subsystem. We finally show that a change in functional variables can make these observables behave more smoothly and exemplify that the density-to-potential maps can give us physical insights into the behavior of coupled electron-photon systems by identifying a very large polarizability due to ultra-strong electron-photon coupling.