  We study the phase diagram, both at zero and finite temperature, in a class
of $\mathbb{Z}_q$ models with infinite range interactions. We are able to
identify the transitions between a symmetry-breaking and a trivial phase by
using a mean-field approach and a perturbative expansion. We perform our
analysis on a Hamiltonian with $2p$-body interactions and we find first-order
transitions for any $p>1$; in the case $p=1$, the transitions are first-order
for $q=3$ and second-order otherwise. In the infinite-range case there is no
trace of gapless incommensurate phase but, when the transverse field is
maximally chiral, the model is in a symmetry-breaking phase for arbitrarily
large fields. We analytically study the transtion in the limit of infinite $q$,
where the model possesses a continuous $U(1)$ symmetry.
