We obtain the phase diagram of a Bose-Fermi mixture of hardcore spinless Bosons and spin-polarized Fermions with nearest neighbor intra-species interaction and on-site inter-species repulsion in an optical lattice at half-filling using a slave-boson
mean-field theory. We show that such a system can have four possible phases which are a) supersolid Bosons coexisting with Fermions in the Mott state, b) Mott state of Bosons coexisting with Fermions in a metallic or charge-density wave state, c) a metallic Fermionic state coexisting with superfluid phase of Bosons, and d) Mott insulating state of Fermions and Bosons. We chart out the phase diagram of the system and provide analytical expressions for the phase boundaries within mean-field theory. We demonstrate that the transition between these phases are generically first order with the exception of that between the supersolid and the Mott states which is a continuous quantum phase transition. We also obtain the low-energy collective excitations of the system in these phases. Finally, we study the particle-hole excitations in the Mott insulating phase and use it to determine the dynamical critical exponent $z$ for the supersolid-Mott insulator transition. We discuss experiments which can test our theory.
