  Lattice gases in the strongly correlated regime have been proven to simulate
quantum magnetic models under certain conditions: the mapping of the
double-well system onto the Lipkin-Meshkov-Glick spin model is a paradigmatic
case. A suitable definition of the length in the Hilbert space of the system
leads to the concept of a correlation length, whose divergence is a
characteristic property of continuous quantum phase transitions. We calculate
the finite-size scaling of some observables like e.g. the magnetization or the
population imbalance, as well as of the Schmidt gap, obtaining in this way the
critical exponents associated to such transitions. The systematic definition of
the Schmidt gap in extended Hamiltonians provides a good tool to analyze the
set of critical exponents associated to transitions in systems formed by a
larger number of traps. This demonstrates, thus, the potential use of
mesoscopic Bose-Einstein condensates as quantum simulators of condensed matter
systems.
