Excitonic supersolid in quantum Hall graphene bilayers

We study the ground state of two graphene sheets separated by a distance $d$ in the quantum Hall regime where the top layer has electrons and the bottom layer has holes as carriers. We obtain a rich mean-field phase diagram as a function of distance $d$ and the partial filling factor $\nu_e=\nu_h=\nu$ for different Landau levels. We find that the ground state in high Landau levels at large $d$ is a generalized Wigner crystal that includes anisotropic stripe and bubble states, and at small $d$ the ground state is a uniform excitonic condensate. We show that for a wide range of partial filling factors $0\le \nu \leq 1/2$, at intermediate values of $d$, the ground state has interlayer phase coherence as well as a lattice structure, i.e. it is an excitonic supersolid. We discuss the predictions for signatures of such a state in transport and optical experiments.