  We reveal a divergent issue associated with the mean-field theory for Bose
gases in optical lattices constructed by the widely used straightforward
mean-field decoupling of the hopping term, where the corresponding mean-field
Hamiltonian generally assumes no lower energy bound once the spatial dependence
of the mean-field superfluid order parameter is taken into account. Via a
systematic functional integral approach, we solve this issue by establishing a
general finite temperature mean-field theory that can treat any possible
spatial dependence of the order parameter without causing the divergent issue.
Interestingly, we find the theory generally assumes an intrinsic non-hermitian
structure that originates from the indefiniteness of the hopping matrix of the
system. Within this theory, we develop an efficient approach for investigating
the physics of the system at finite temperature, where properties of the system
can be calculated via straightforward investigation on the saddle points of an
effective potential function for the order parameter. We illustrate our
approach by investigating the finite temperature superfluid transition of Bose
gases in optical lattices. Since the underlying finite temperature mean-field
theory is quite general, this approach can be straightforwardly applied to
investigate the finite temperature properties of related systems with phases
possessing complex spatial structures.
