# Phase diagram of the $Z_N$-invariant clock model

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.

• 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. Nov 7 '16 at 22:32
• I did numerical simulations (DMRG) for this model. You might be interested: arXiv:1602.07814. Mar 9 '18 at 17:26

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.