# 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 $\displaystyle{ N }$ states on each site. The Hamiltonian of this model is

$\displaystyle{ H = -J \left( \sum_{ \langle i, j \rangle} (Z^\dagger_i Z_j + Z_i Z^\dagger_j ) + g \sum_j (X_j + X^\dagger_j) \right) }$

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 $\displaystyle{ N \times N }$ 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.

The model obeys a global $\displaystyle{ \mathbb{Z}_N }$ symmetry, which is generated by the unitary operator $\displaystyle{ U_X = \prod_j X_j }$ where the product is over every site of the lattice. In other words, $\displaystyle{ U_X }$ commutes with the Hamiltonian.

When $\displaystyle{ N=2 }$ the quantum clock model is identical to the transverse-field Ising model. When $\displaystyle{ N=3 }$ 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: \displaystyle{ \begin{align}\tilde{X_j} &= Z^\dagger_j Z_{j+1} \\ \tilde{Z}^\dagger_j \tilde{Z}_{j+1} &= X_{j+1} \end{align} } 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 $\displaystyle{ H = -Jg \sum_j ( \tilde{Z}^\dagger_j \tilde{Z}_{j+1} + g^{-1}\tilde{X}^\dagger_{j} + \textrm{h.c.} ) }$. This indicates that the model with coupling parameter $\displaystyle{ g }$ is dual to the model with coupling parameter $\displaystyle{ g^{-1} }$, 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 $\displaystyle{ \mathbb{Z}_N }$ symmetry properties of phases are changed under the Kramers–Wannier duality. A more careful analysis involves coupling the theory to a $\displaystyle{ \mathbb{Z}_N }$ gauge field; fixing the gauge reproduces the results of the Kramers Wannier transformation.

### Phase transition

When $\displaystyle{ g =1 }$, 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 $\displaystyle{ 0 \lt g \lt 1 }$ and the disordered phase at $\displaystyle{ 1 \lt g }$.

### Jordan–Wigner transformation

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