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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.

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  • $\begingroup$ 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. $\endgroup$ Nov 7 '16 at 22:32
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    $\begingroup$ I did numerical simulations (DMRG) for this model. You might be interested: arXiv:1602.07814. $\endgroup$
    – Christophe
    Mar 9 '18 at 17:26
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The phase diagram for $q > 3$ clock model consists of a disordered phase, an $XY$-like phase and a fully ordered phase. A 2018 paper by Chatterjee, Puri and Rual cover this [1]. Essentially, the upper phase transition temperature converges to the $XY$ transition temperature fairly quickly, while a lower transition decays towards zero as $1/q^2$. In the $XY$-like phase, the order parameter correlations decay algebraically.

Ostlund (1981) [2] is another enlightening paper. He looks at the asymmetric clock model (there's a bias in the $x$-direction) and has some interesting phase diagrams included. By biasing one direction, Ostlund shows that the lower transition temperature decreases while the upper transition temperature is relatively constant.

For $q > 4$, you have the lower transition being first order. The upper transitions are $XY$-like.

[1] https://link.aps.org/pdf/10.1103/PhysRevE.98.032109

[2] https://link.aps.org/pdf/10.1103/PhysRevB.24.398

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