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 formula_1 states on each site. The Hamiltonian of this model is
Here, the subscripts refer to lattice sites, and the sum formula_3 is done over pairs of nearest neighbour sites formula_4 and formula_5. The clock matrices formula_6 and formula_7 are formula_8 generalisations of the Pauli matrices satisfying
where formula_11 is 1 if formula_12 and formula_13 are the same site and zero otherwise. formula_14 is a prefactor with dimensions of energy, and formula_15 is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.
The model obeys a global formula_16 symmetry, which is generated by the unitary operator formula_17 where the product is over every site of the lattice. In other words, formula_18 commutes with the Hamiltonian.
When formula_19 the quantum clock model is identical to the transverse-field Ising model. When formula_20 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:
formula_21
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 formula_22. This indicates that the model with coupling parameter formula_15 is dual to the model with coupling parameter formula_24, 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 formula_25 symmetry properties of phases are changed under the Kramers–Wannier duality. A more careful analysis involves coupling the theory to a formula_16 gauge field; fixing the gauge reproduces the results of the Kramers Wannier transformation.
Phase transition.
When formula_27, 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 formula_28 and the disordered phase at formula_29.
Jordan–Wigner transformation.
Another nonlocal mapping known as the Jordan Wigner transformation can be used to express the theory in terms of parafermions.