Quantum clock model

The quantum clock model is a quantum lattice model.¹ 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

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

Here, the subscripts refer to lattice sites, and the sum ${\displaystyle \sum _{\langle i,j\rangle }}$ 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 neighbor 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. When ${\displaystyle N=4}$, the model is again equivalent to the Ising model. When ${\displaystyle N>4}$, strong evidences have been found that the phase transitions exhibited in these models should be certain generalizations ² of Kosterlitz–Thouless transition, whose physical nature is still largely unknown.

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

A nonlocal mapping of clock matrices known as the Kramers–Wannier duality transformation can be done as follows:³ $${\displaystyle {\begin{aligned}{\tilde {X_{j}}}&=Z_{j}^{\dagger }Z_{j+1}\\{\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}&=X_{j+1}\end{aligned}}}$$ 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 ${\displaystyle H=-Jg\sum _{j}({\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}+g^{-1}{\tilde {X}}_{j}^{\dagger }+{\textrm {h.c.}})}$. This indicates that the model with coupling parameter ${\displaystyle g}$ 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.

For ${\displaystyle N=2,3,4}$, there is a unique phase transition from the ordered phase to the disordered phase at ${\displaystyle g=1}$. The model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. For ${\displaystyle N>4}$, there are two phase transition points at ${\displaystyle g_{1}<1}$ and ${\displaystyle g_{2}=1/g_{1}>1}$. Strong evidences have been found that these phase transitions should be a class of generalizations² of Kosterlitz–Thouless transition. The KT transition predicts that the free energy has an essential singularity that goes like ${\displaystyle e^{-{\tfrac {c}{\sqrt {|g-g_{c}|}}}}}$, while perturbative study found that the essential singularity behaves as ${\displaystyle e^{-{\tfrac {c}{|g-g_{c}|^{\sigma }}}}}$ where ${\displaystyle \sigma }$ goes from ${\displaystyle 0.2}$ to ${\displaystyle 0.5}$ as ${\displaystyle N}$ increases from ${\displaystyle 5}$ to ${\displaystyle 9}$. The physical pictures⁴ of these phase transitions are still not clear.

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