Valley-selective topologically ordered states in irradiated bilayer graphene

Gapless bilayer graphene is susceptible to a variety of spontaneously gapped states. As predicted by theory and observed by experiment, the ground state is, however, topologically trivial, because a valley-independent gap is energetically favorable. Here, we show that under the application of interlayer electric field and circularly polarized light, one valley can be selected to exhibit the original interaction instability while the other is frozen out. Tuning this Floquet system stabilizes multiple competing topologically ordered states, distinguishable by edge transport and circular dichroism. Notably, quantized charge, spin, and valley Hall conductivities coexist in one stabilized state.