  Atomic many-body phase transitions and quantum criticality have recently
attracted much attention in non-standard optical lattices. Here we perform an
experimental study of finite-temperature superfluid transition of bosonic atoms
confined in a three dimensional triangular lattice, whose structure can be
continuously deformed to dimensional crossover regions including quasi-one and
two dimensions. This non-standard lattice system provides a versatile platform
to investigate many-body correlated phases. For the three dimensional case, we
find that the finite temperature superfluid transition agrees quantitatively
with the Gutzwiller mean field theory prediction, whereas tuning towards
reduced dimensional cases, both quantum and thermal fluctuation effects are
more dramatic, and the experimental measurement for the critical point becomes
strongly deviated from the mean field theory. We characterize the fluctuation
effects in the whole dimension crossover process. Our experimental results
imply strong many-body correlations in the system beyond mean field
description, paving a way to study quantum criticality near Mott-superfluid
transition in finite temperature dimension-crossover lattices.
