Z N model

The formula_1 model (also known as the clock model) is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrable only on one and two-dimensional lattices, in several special cases.
Definition.
The formula_1 model is defined by assigning a spin value at each node formula_3 on a graph, with the spins taking values formula_4, where formula_5. The spins therefore take values in the form of complex roots of unity. Roughly speaking, we can think of the spins assigned to each node of the formula_1 model as pointing in any one of formula_7 equidistant directions. The Boltzmann weights for a general edge formula_8 are:
where formula_10 denotes complex conjugation and the formula_11 are related to the interaction strength along the edge formula_8. Note that formula_13 and formula_14 is often set to 1. The (real valued) Boltzmann weights are invariant under the transformations formula_15 and formula_16, analogous to universal rotation and reflection respectively.
Self-dual critical solution.
There is a class of solutions to the formula_1 model defined on an in general anisotropic square lattice. If the model is self-dual in the Kramers–Wannier sense and thus critical, and the lattice is such that there are two possible 'weights' formula_18 and formula_19 for the two possible edge orientations, we can introduce the following parametrization in formula_20:
Requiring the duality relation and the star–triangle relation, which ensures integrability, to hold, it is possible to find the solution:
with formula_24. This particular case of the formula_1 model is often called the FZ model in its own right, after V.A. Fateev and A.B. Zamolodchikov who first calculated this solution. The FZ model approaches the XY model in the limit as formula_26. It is also a special case of the chiral Potts model and the Kashiwara–Miwa model.
Solvable special cases.
As is the case for most lattice models in statistical mechanics, there are no known exact solutions to the formula_1 model in three dimensions. In two dimensions, however, it is exactly solvable on a square lattice for certain values of formula_7 and/or the 'weights' formula_29. Perhaps the most well-known example is the Ising model, which admits spins in two opposite directions (i.e. formula_30). This is precisely the formula_1 model for formula_32, and therefore the formula_1 model can be thought of as a generalization of the Ising model. Other exactly solvable models corresponding to particular cases of the formula_1 model include the three-state Potts model, with formula_35 and formula_36, where formula_37 is a certain critical value (FZ), and the critical Askin–Teller model where formula_38.
Quantum version.
A quantum version of the formula_39 clock model can be constructed in a manner analogous to the transverse-field Ising model. The Hamiltonian of this model is the following:
Here, the subscripts refer to lattice sites, and the sum formula_41 is done over pairs of nearest neighbour sites formula_42 and formula_43. The clock matrices formula_44 and formula_45 are generalisations of the Pauli matrices satisfying
and 
where formula_48 is 1 if formula_49 and formula_50 are the same site and zero otherwise. formula_51 is a prefactor with dimensions of energy, and formula_52 is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.