Bose-Hubbard models capture physics of cold atoms in optical lattices.
Despite their simplicity, they are not exactly analytically solvable in any dimension.

In the first part of the talk, I will discuss basic properties of the 1D
Bose-Hubbard model focusing on physically meaningful observables such as
the ground state energy, the variance of on-site atom number occupation,
and the nearest-neighbor correlation function. I will present analytical
results coming from very high-order perturbative expansions and
carefully compare them to numerical simulations.

In the second part of the talk, I will show how the position of the
critical point of the 2D Bose-Hubbard model can be efficiently obtained
from the variance of on-site atom number occupation and the
nearest-neighbor correlation function. The analogies between the
experimental studies of the lambda transition in liquid helium-4 and our
theoretical studies of the superfluid-Mott insulator transition of the
2D Bose-Hubbard model will be discussed.