  We investigate the temperature-dependent behavior emerging in the vicinity of
the superfluid (SF) to Mott-insulator (MI) transition of interacting bosons in
a two-dimensional optical lattice, described by the Bose-Hubbard model. The
equilibrium phase diagram at finite temperature is computed using the cluster
mean-field (CMF) theory including a finite cluster-size scaling. The SF, MI,
and normal fluid (NF) phases are characterized as well as the transition or
crossover temperatures between them are estimated by computing physical
quantities such as the superfluid fraction, compressibility and sound velocity
using the CMF method. We find that the non-local correlations included in a
finite cluster, when extrapolated to infinite size, leads to quantitative
agreement of the phase boundaries with quantum Monte Carlo (QMC) results as
well as with experiments. Moreover, we show that the von Neumann entanglement
entropy within a cluster corresponds to the system's entropy density and that
it is enhanced near the SF-MI quantum critical point (QCP) and at the SF- NF
boundary. The behavior of the transition lines near this QCP, at and away from
the particle-hole (p-h) symmetric point located at the Mott-tip, is also
discussed. Our results obtained by using the CMF theory can be tested
experimentally using the quantum gas microscopy method.
