Specific heat of Bosons in a lattice

We present a theoretical study of specific heat of bosons ($C_v$) in a simple cubic lattice. We have studied the non-interacting bosons and the Tonks gas. For both cases, the $C_v$ above the bose condensation temperature shows considerable temperature dependence compared to that of free bosons. For Tonks gas, we find that the low-temperature specific heat increases as the system gets closer to the Mott transition.