We investigate zero temperature and finite temperature properties of the Bose Hubbard Model in the hard core limit using Random Phase Approximation (RPA) and Cluster Mean Field Theory (CMFT). We show that our RPA calculations are able to capture quantum and thermal fluctuations significantly better than CMFT.

The cluster mean-field with density matrix renormalization (CMFT + DMRG) method which combines the simplicity of the mean-field theory and the numerical power of the density-matrix renormalization group method is applied to understand the quantum phases of the one-dimensional Bose–Hubbard models. We show that the CMFT + DMRG method is an effective numerical technique with moderate computational resources to determine relevant order parameters and correlation functions of large one-dimensional systems. We apply the CMFT + DMRG for the Bose Hubbard and extended Bose Hubbard models to account for the superfluid, Mott insulator, and density wave phases in these models. Our results are in good agreement with the known phase diagram of these models, demonstrating the efficacy of this method.