Critical temperatures of hard-core boson model on square lattice within Bethe approximation

We consider the inclusion of short-range correlations for a two-dimensional model of local bosons on a square lattice in the framework of the Bethe approximation for clusters of 2 and 4 sites. Explicit equations are obtained for determining the critical temperatures of charge and superfluid ordering and their solutions are considered for various ratios of the charge-charge correlation parameter and the transfer integral. It is shown that taking into account short-range correlations for temperatures of charge ordering leads to the appearance of a critical concentration of bosons, limiting the region of existence of solutions like charge ordering. For superfluid ordering, when short-range correlations are taken into account, the critical temperature is reduced down to zero values at half-filling. The phase diagram of the model of local bosons is constructed with allowance for phase separation within the framework of Maxwell's construction, and it is shown that taking into account short-range correlations in the Bethe approximation quantitatively approximates the form of the phase diagram to the results of the quantum Monte Carlo method.