Quantum clock model

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 generalizations^[2] 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 pictures^[4] 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.