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 neighbor 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. When $N=4$, the model is again equivalent to the Ising model. When $N>4$, strong evidences have been found that the phase transitions exhibited in these models should be certain generalizations of Kosterlitz–Thouless transition, whose physical nature is still largely unknown.

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

For $N=2,3,4$, there is a unique phase transition from the ordered phase to the disordered phase at $g=1$. The model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. For $N>4$, there are two phase transition points at $g_{1}<1$ and $g_{2}=1/g_{1}>1$. Strong evidences have been found that these phase transitions should be a class of generalizations of Kosterlitz–Thouless transition. The KT transition predicts that the free energy has an essential singularity that goes like $e^{-{\tfrac {c}{\sqrt {|g-g_{c}|}}}}$, while perturbative study found that the essential singularity behaves as $e^{-{\tfrac {c}{|g-g_{c}|^{\sigma }}}}$ where $\sigma$ goes from $0.2$ to $0.5$ as $N$ increases from $5$ to $9$. The physical pictures of these phase transitions are still not clear.

Jordan–Wigner transformation

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