The realization of systems of spatially separated electrons and holes, in the search of the equilibrium excitonic condensate foreseen by , has gone on for more than two decades, but so far the condensate has eluded experimental detection . It has been recently suggested that a substantial enhancement of the electron-hole attraction, to favour the condensate, could be obtained in coupled graphene bilayers ; a couple of experimental realizations has promptly followed . Such systems have a quadratic energy dispersion on a wide density range and a twofold valley degeneracy in each bilayer.
We have embarked in extensive QMC simulations of an electron-hole bilayer with mass symmetry and twofold valley degeneracy (in each layer), to partially mimic the situation encountered in the coupled graphene bilayers proposed in . Our goal is to provide benchmark results for the electron-hole bilayer investigating the qualitative and quantitative effects of the valley degeneracy, inferred by comparison with the conventional electron-hole bilayer possessing only spin degeneracy .
We have studied systems of 168 particles, for in-layer densities corresponding to 1≤rs ≤8 and inter-layer distances 1≤d/aB ≤4. We find an excitonic condensate for rs≳1 at intermediate distances, whereas a quadriexciton fluid  and a plasma fluid are stable respectively at smaller and larger distances. The region of stability of the excitonic condensate is significantly shrunk with respect to the system without valley degeneracy . Moreover we observe for the first time the formation of quadriexcitons, first predicted by Wang and Kittel  45 years ago.
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