A well-known 1985 paper by Fateev and Zamolodchikov "Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems" discusses the phase diagram of the $Z_N$-invariant clock model in 2d, which is a generalization of the 2d Ising model. The case $N=2$ corresponds to the Ising model, the case $N=3$ is a special case of the Potts model. Already for N=4 the phase diagram is more complicated that for the Ising model: there are 3 phases, instead of 2. The phase diagram for N=5 is also shown. But it is not clear from the paper which transition lines are 1st order and which ones are 2nd order (both for $N=4$ and $N=5$). Has there been any further studies of the phase diagram of the clock model, and what does the phase diagram look like for $N>3$?
I guess I should clarify that by a clock model I do not mean the most obvious generalization of the Ising model with a single parameter, but the most general nearest-neighbor model which for $N>3$ has more than one parameter. Say, for $N=4$ and $N=5$ it has a 2-dimensional parameter space, and the phase diagram involves a KT phase and two phases with a nonzero correlation length. There are transitions between all three of them.