The  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  model is defined by assigning a spin value at each node  on a graph, with the spins taking values , where . 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  model as pointing in any one of  equidistant directions. The Boltzmann weights for a general edge  are:

where  denotes complex conjugation and the  are related to the interaction strength along the edge . Note that  and  are often set to 1. The (real valued) Boltzmann weights are invariant under the transformations  and , analogous to universal rotation and reflection respectively.

Self-dual critical solution

There is a class of solutions to the  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'  and  for the two possible edge orientations, we can introduce the following parametrization in :

–
Requiring the duality relation and the star–triangle relation, which ensures integrability, to hold, it is possible to find the solution:

with . This particular case of the  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 . 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  model in three dimensions. In two dimensions, however, it is exactly solvable on a square lattice for certain values of  and/or the 'weights' . Perhaps the most well-known example is the Ising model, which admits spins in two opposite directions (i.e. ). This is precisely the  model for , and therefore the  model can be thought of as a generalization of the Ising model. Other exactly solvable models corresponding to particular cases of the  model include the three-state Potts model, with  and , where  is a certain critical value (FZ), and the critical Askin–Teller model where .

Quantum version

A quantum version of the  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  is done over pairs of nearest neighbour sites  and . The clock matrices  and  are generalisations of the Pauli matrices satisfying

and 

where  is 1 if  and  are the same site and zero otherwise.  is a prefactor with dimensions of energy, and  is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.

References 

 V. A. Fateev and A. B. Zamolodchikov (1982); "Self-dual solutions of the star-triangle relations in -models", Physics Letters A, 92, pp. 37–39
 M.A. Rajabpour and J. Cardy (2007); "Discretely holomorphic parafermions in lattice  models" J. Phys. A 22 40, 14703–14714

Spin models
Exactly solvable models
Lattice models
Statistical mechanics