Quantum clock model

The quantum clock model is a quantum lattice model. It is a generalisation of the transverse-field Ising model . It is defined on a lattice with $N$ states on each site. The Hamiltonian of this model is

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

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

$Z_{j}X_{k}=e^{{\frac {2\pi i}{N}}\delta _{j,k}}X_{k}Z_{j}$ and $X_{j}^{N}=Z_{j}^{N}=1$

where $\delta _{j,k}$ 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.

The model obeys a global $\mathbb {Z} _{N}$ symmetry, which is implemented by the unitary operator $U_{X}=\prod _{j}X_{j}$ where the product is over every site of the lattice. In other words, $U_{X}$ commutes with the Hamiltonian.

When $N=2$ the quantum clock model is identical to the transverse field Ising model. When $N=3$ the quantum clock model is equivalent to the three-state Potts model.