  We derive a general criterion for determining the onset of superradiant phase
transition in electronic bands coupled to a cavity field, with possibly
electron-electron interactions. For longitudinal superradiance in 2D or genuine
1D systems, we prove that it is always prevented, thereby extending existing
no-go theorems. Instead, a superradiant phase transition can occur to a
nonuniform transverse cavity field and we give specific examples in
non-interacting models, either through Fermi surface nesting or parabolic band
touching. Investigating the resulting time-reversal symmetry breaking
superradiant states, we find in the former case Fermi surface lifting down to
four Dirac points on a square lattice model, with topologically protected
zero-modes, and in the latter case topological bands with non-zero Chern number
on an hexagonal lattice.
