Finite temperature correlations in the Bose-Hubbard model: application of the Gauge $P$ representation

We study ultracold Bose gases in periodic potentials as described by the Bose-Hubbard model. In 1D and at finite temperature, we simulate ultracold Bose gases in imaginary time with the gauge $P$ representation. We study various quantities including the Luttinger parameter $K$, which is important for locating the boundaries of the Mott insulator lobes, and find a simple relation for the kinetic energy part of the Bose-Hubbard Hamiltonian. We show that for J=0, the stepwise pattern of the average number of particles per lattice site versus the chemical potential vanishes at temperatures above $T\approx 0.1U$. Also, at chemical potential $\mu =0.5U$ and temperature $T=0.5U$ by increasing $J$, the relative value of the number fluctuation decreases and approaches that of a coherent state.