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.
model is defined by assigning a spin value at each node
on a graph, with the spins taking values
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
The Boltzmann weights for a general edge
denotes complex conjugation and the
are related to the interaction strength along the edge
The (real valued) Boltzmann weights are invariant under the transformations
, analogous to universal rotation and reflection respectively.
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'
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
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.
As is the case for most lattice models in statistical mechanics, there are no known exact solutions to the
In two dimensions, however, it is exactly solvable on a square lattice for certain values of
Perhaps the most well-known example is the Ising model, which admits spins in two opposite directions (i.e.
Other exactly solvable models corresponding to particular cases of the
model include the three-state Potts model, with
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
are generalisations of the Pauli matrices satisfying and where
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.