The ${\displaystyle Z_{N}}$ 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 ${\displaystyle Z_{N}}$ model is defined by assigning a spin value at each node ${\displaystyle r}$ on a graph, with the spins taking values ${\displaystyle s_{r}=\exp {\frac {2\pi iq}{N}}}$, where ${\displaystyle q\in \{0,1,\ldots ,N-1\}}$. 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 ${\displaystyle Z_{N}}$ model as pointing in any one of ${\displaystyle N}$ equidistant directions. The Boltzmann weights for a general edge ${\displaystyle rr'}$ are:

${\displaystyle w\left(r,r'\right)=\sum _{k=0}^{N-1}x_{k}^{\left(rr'\right)}\left(s_{r}s_{r'}^{*}\right)^{k}}$

where ${\displaystyle *}$ denotes complex conjugation and the ${\displaystyle x_{k}^{\left(rr'\right)}}$ are related to the interaction strength along the edge ${\displaystyle rr'}$. Note that ${\displaystyle x_{k}^{\left(rr'\right)}=x_{N-k}^{\left(rr'\right)}}$ and ${\displaystyle x_{0}}$ are often set to 1. The (real valued) Boltzmann weights are invariant under the transformations ${\displaystyle s_{r}\rightarrow \omega ^{k}s_{r}}$ and ${\displaystyle s_{r}\rightarrow s_{r}^{*}}$, analogous to universal rotation and reflection respectively.

Self-dual critical solution

There is a class of solutions to the ${\displaystyle Z_{N}}$ 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' ${\displaystyle x_{k}^{1}}$ and ${\displaystyle x_{k}^{2}}$ for the two possible edge orientations, we can introduce the following parametrization in ${\displaystyle \alpha }$:

${\displaystyle x_{n}^{1}=x_{n}\left(\alpha \right)}$

${\displaystyle x_{n}^{2}=x_{n}\left(\pi -\alpha \right)}$–

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

${\displaystyle x_{n}\left(\alpha \right)=\prod _{k=0}^{n-1}{\frac {\sin \left(\pi k/N+\alpha /2N\right)}{\sin \left[\pi \left(k+1\right)/N-\alpha /2N\right]}}}$

with ${\displaystyle x_{0}=1}$. This particular case of the ${\displaystyle Z_{N}}$ 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 ${\displaystyle N\rightarrow \infty }$. 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 ${\displaystyle Z_{N}}$ model in three dimensions. In two dimensions, however, it is exactly solvable on a square lattice for certain values of ${\displaystyle N}$ and/or the 'weights' ${\displaystyle x_{k}}$. Perhaps the most well-known example is the Ising model, which admits spins in two opposite directions (i.e. ${\displaystyle s_{r}=\pm 1}$). This is precisely the ${\displaystyle Z_{N}}$ model for ${\displaystyle N=2}$, and therefore the ${\displaystyle Z_{N}}$ model can be thought of as a generalization of the Ising model. Other exactly solvable models corresponding to particular cases of the ${\displaystyle Z_{N}}$ model include the three-state Potts model, with ${\displaystyle N=3}$ and ${\displaystyle x_{1}=x_{2}=x_{c}}$, where ${\displaystyle x_{c}}$ is a certain critical value (FZ), and the critical Askin–Teller model where ${\displaystyle N=4}$.

Quantum version

A quantum version of the ${\displaystyle Z_{N}}$ clock model can be constructed in a manner analogous to the transverse-field Ising model. The Hamiltonian of this model is the following:

${\displaystyle H=-J(\sum _{\langle i,j\rangle }(Z_{i}^{\dagger }Z_{j}+Z_{i}Z_{j}^{\dagger })+g\sum _{j}(X_{j}+X_{j}^{\dagger }))}$

Here, the subscripts refer to lattice sites, and the sum ${\displaystyle \sum _{\langle i,j\rangle }}$ is done over pairs of nearest neighbour sites ${\displaystyle i}$ and ${\displaystyle j}$. The clock matrices ${\displaystyle X_{j}}$ and ${\displaystyle Z_{j}}$ are generalisations of the Pauli matrices satisfying

${\displaystyle Z_{j}X_{k}=e^{{\frac {2\pi i}{N}}\delta _{j,k}}X_{k}Z_{j}}$

and

${\displaystyle X_{j}^{N}=Z_{j}^{N}=1}$

where ${\displaystyle \delta _{j,k}}$ is 1 if ${\displaystyle j}$ and ${\displaystyle k}$ are the same site and zero otherwise. ${\displaystyle J}$ is a prefactor with dimensions of energy, and ${\displaystyle g}$ 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 ${\displaystyle Z_{N}}$-models", Physics Letters A, 92, pp. 37–39
• M.A. Rajabpour and J. Cardy (2007); "Discretely holomorphic parafermions in lattice ${\displaystyle Z_{N}}$ models" J. Phys. A 22 40, 14703–14714