Absence of dimensional crossover at the zero-temperature transition in a Bose system

The quantum lattice-dynamic model for structural phase transitions and the Bose-gas system are compared in the large-$n$ limit in order to clarify the notion of dimensional crossover at the $T=0\mathrm{}$ transition. While in the first case the zero-point fluctuations induce a $d$-to-($d+1\mathrm{}$) dimensional crossover, it is found that in the Bose gas there are no quantum fluctuations of the order parameter in the ground state and there is no dimensional crossover.