Orderings at a finite temperature in bosonic Hubbard model with confinement potentials

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We investigated finite temperature properties of quasi-2D dilute-bosonic-gas system with confinement potential by numerically solving the projected Gross-Pitaevskii equation. The exponent of phase correlation functions at the boundary between coherent and in-coherent region takes the same values of the KT phase transition. We also studied the orderings of the extended hard-core bosonic-Hubbard model on the square lattice by performing the quantum Monte Carlo simulations. In this system, the chemical potential specially increases along the x-axis. The results indicated that the checker board solid state and superfluid state are well specially separated and the particle density jumps at hc, where hc corresponds to the critical values of the first order transition in the systems with uniform potential. This behavior was not changed when we changed the potential shapes. This suggests that the static quantities such as particle density, local compressibility, and local stiffness, can be explained by the local density approximation.