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

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

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 generated 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 quantum three-state Potts model.

One-dimensional model

There are various analytical methods that can be used to study the quantum clock model specifically in one dimension.

Kramers–Wannier duality

A nonlocal mapping of clock matrices known as the Kramers–Wannier duality transformation can be done as follows: $${\begin{aligned}{\tilde {X_{j}}}&=Z_{j}^{\dagger }Z_{j+1}\\{\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}&=X_{j+1}\end{aligned}}$$ Then, in terms of the newly defined clock matrices with tildes, which obey the same algebraic relations as the original clock matrices, the Hamiltonian is simply $H=-Jg\sum _{j}({\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}+g^{-1}{\tilde {X}}_{j}^{\dagger }+{\textrm {h.c.}})$. This indicates that the model with coupling parameter $g$ is dual to the model with coupling parameter $g^{-1}$, and establishes a duality between the ordered phase and the disordered phase.

Note that there are some subtle considerations at the boundaries of the one dimensional chain; as a result of these, the degeneracy and $\mathbb {Z} _{N}$ symmetry properties of phases are changed under the Kramers–Wannier duality. A more careful analysis involves coupling the theory to a $\mathbb {Z} _{N}$ gauge field; fixing the gauge reproduces the results of the Kramers Wannier transformation.

Phase transition

When $g=1$, the quantum clock model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. This self-dual theory generally represents a phase transition between the ordered phase at $0<g<1$ and the disordered phase at $1<g$.

Jordan–Wigner transformation

Another nonlocal mapping known as the Jordan Wigner transformation can be used to express the theory in terms of parafermions.