  Many zero temperature transitions, involving the deviation in the value of a
$U(1)$ conserved charge from a quantized value, are described by the dilute
Bose gas quantum critical point. On such transitions, we study the consequences
of perturbations which break the symmetry down to $Z_N$ in $d$ spatial
dimensions. For the case $d=1$, $N=2$, we obtain exact, finite temperature,
multicritical crossover functions by a mapping to an integrable lattice model.
