  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.
