Cavity-mediated electron-electron interactions: Renormalizing Dirac states in graphene

Embedding materials in optical cavities has emerged as a strategy for tuning material properties. Here, we develop a nonperturbative quantum electrodynamical approach based on a photon-free self-consistent Hartree-Fock framework to model the coupling between cavity photons and electrons and crystals. We apply this approach to graphene coupled to cavity photons of different polarizations. Photons introduce nonlocal electron-electron interactions, solely due to the quantum nature of light, that lead to substantial renormalization of Dirac bands. The nonlocal interactions induced by anisotropic linearly polarized photons give rise to wedge-shaped bands and Dirac gap. When isotropic cavity photon modes are introduced, the Dirac cones remain gapless, but a Fermi velocity renormalization yet indicates the role of nonlocal interactions. This theoretical framework paves the way for revealing nonperturbative quantum effects in strongly coupled light-matter systems and allows for a more comprehensive discovery of cavity-induced phenomena.