ZN model

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

::$w\backslash left(r,r\text{'}\backslash right)=\backslash sum\_\{k=0\}^\{N-1\}x\_\{k\}^\{\backslash left(rr\text{'}\backslash right)\}\backslash left(s\_\{r\}s\_\{r\text{'}\}^*\backslash right)^k$

where $*$ denotes complex conjugation and the $x\_\{k\}^\{\backslash left(rr\text{'}\backslash right)\}$ are related to the interaction strength along the edge $rr\text{'}$. Note that $x\_\{k\}^\{\backslash left(rr\text{'}\backslash right)\}=x\_\{N-k\}^\{\backslash left(rr\text{'}\backslash right)\}$ and $x\_0$ are often set to 1. The (real valued) Boltzmann weights are invariant under the transformations $s\_r\; \backslash rightarrow\; \backslash omega^k\; s\_r$ and $s\_r\; \backslash rightarrow\; s^\{*\}\_\{r\}$, analogous to universal rotation and reflection respectively.

There is a class of solutions to the $Z\_N$ 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' $x\_k^1$ and $x\_k^2$ for the two possible edge orientations, we can introduce the following parametrization in $\backslash alpha$:

::$x\_n^1=x\_\{n\}\backslash left(\backslash alpha\backslash right)$

::$x\_n^2=x\_\{n\}\backslash left(\backslash pi-\backslash alpha\backslash right)$–

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

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

with $x\_0=1$. This particular case of the $Z\_N$ 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 $N\backslash rightarrow\backslash infty$. 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 $Z\_N$ model in three dimensions. In two dimensions, however, it is exactly solvable on a square lattice for certain values of $N$ and/or the 'weights' $x\_\{k\}$. Perhaps the most well-known example is the Ising model, which admits spins in two opposite directions (i.e. $s\_r=\backslash pm\; 1$). This is precisely the $Z\_N$ model for $N=2$, and therefore the $Z\_N$ model can be thought of as a generalization of the Ising model. Other exactly solvable models corresponding to particular cases of the $Z\_N$ model include the three-state Potts model, with $N=3$ and $x\_1=x\_2=x\_c$, where $x\_c$ is a certain critical value (FZ), and the critical Askin–Teller model where $N=4$.

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

:$H\; =\; -J(\backslash sum\_\{\; \backslash langle\; i,\; j\; \backslash rangle\}\; (Z^\backslash dagger\_i\; Z\_\{j\}+\; Z\_i\; Z^\{\backslash dagger\}\_\{j\})\; +\; g\; \backslash sum\_j\; (X\_j\; +\; X^\backslash dagger\_j)\; )$

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

:$Z\_j\; X\_k\; =\; e^\{\backslash frac\{2\backslash pi\; i\; \}\{N\}\backslash delta\_\{j,k\}\}\; X\_k\; Z\_j$

and

:$X\_j^N\; =\; Z\_j^N\; =\; 1$

where $\backslash delta\_\{j,k\}$ is 1 if $j$ and $k$ are the same site and zero otherwise. $J$ is a prefactor with dimensions of energy, and $g$ is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.

{{Reflist}}

• V. A. Fateev and A. B. Zamolodchikov (1982); "Self-dual solutions of the star-triangle relations in $Z\_N$-models", Physics Letters A, 92, pp. 37–39
• [https://arxiv.org/PS_cache/arxiv/pdf/0708/0708.3772v3.pdf M.A. Rajabpour and J. Cardy (2007); "Discretely holomorphic parafermions in lattice $Z\_N$ models"] J. Phys. A 22 40, 14703–14714

Category:Spin models

Category:Exactly solvable models

Category:Lattice models

Category:Statistical mechanics