Engineering topological phase transitions in multi-layer graphene coupled to a chiral cavity

Strong coupling between materials and cavity fields can significantly influence the electronic properties of the materials, altering the phases of matter. Recently chiral cavities have been theoretically shown to induce a topological band gap in graphene giving rise to cavity Chern insulator phase. Here we study Bernal stacked graphene subject to enhanced chiral vacuum fluctuations in a tHz cavity. We demonstrate both analytically and numerically that bilayer graphene in a chiral cavity undergoes a gate-tunable topological phase transition between valley-chern and full-chern phases of the vacuum band, resulting in a cavity realization of the honeycomb Haldane model. We also extend the recent concept of topological light-matter hybridization gaps, which relate the Berry phase to the properties of exchanged photons and matter, to multilayer graphene structures. This extension reveals an anomalous cavity chern insulator phase where the chern number of the band is zero, yet chiral edge modes still exist, resulting in a cavity analogue of the anomalous Floquet chern insulator.