  We study a fermionic two-band model with the interband transition resonantly
coupled to a cavity. This model was recently proposed to explain
cavity-enhanced charge transport, but a thorough characterization of the closed
system, in particular localization of various excitations, is lacking. In this
work, using exact diagonalization, we characterize the system by its spectrum
under various filling factors and variable disorder. As in the Dicke model, the
effective light-matter coupling scales with the square root of the system size.
However, there is an additional factor that decreases with increasing doping
density. The transition from the weak-coupling regime to the strong-coupling
regime occurs when the effective light-matter coupling is larger than the
electronic bandwidth. Here, the formation of exciton-polaritons is accompanied
by the formation of bound excitons. Photon spectral functions exhibit
significant weights on the in-gap states between the polaritons, even without
disorder. Finally, while the localization of electron-hole excitations in a
disordered system is lifted by strong coupling, the same is not true for free
charges, which remain localized at strong and even ultrastrong coupling. Based
on this finding, we discuss scenarios for charge transport.
