  Mesoscopic interacting Bose-Einstein condensates confined in a few traps
display phase transitions that cannot be explained with a mean field theory. By
describing each trap as an effective site of a Bose-Hubbard model and using the
Schwinger representation of spin operators, these systems can be mapped to spin
models. We show that it is possible to define correlations between bosons in
such a way that critical behavior is associated to the divergence of a
correlation length accompanied by a gapless spectrum in the thermodynamic
limit. The latter is now defined as the limit in which the mean field analysis
becomes valid. Such description provides critical exponents to the associated
phase transitions and encompasses the notion of universality demonstrating thus
the potential use of mesoscopic Bose-Einstein condensates as quantum simulators
of condensed matter systems.
