  We investigate the ground state of a balanced electron-hole system in the
quantum Hall regime using mean-field theory and obtain a rich phase diagram as
a function of interlayer distance d and the filling factor within a layer. We
identify an excitonic condensate phase, an excitonic supersolid phase, as well
as uncorrelated Wigner crystal states. We find that balanced electron-hole
system exhibits a supersolid phase a wide range of filling factors, with
different crystal structure ground states. We obtain the ground state stiffness
in the excitonic phases and show that the phase transitions from a uniform
condensate to a supersolid is accompanied by a marked change in the stiffness.
Our results provide the first semi-quantitative determination excitonic
supersolid phase diagram and properties.
