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 neighbor 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. When formula_21, the model is again equivalent to the Ising model. When formula_22, 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: formula_23 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_24. This indicates that the model with  coupling parameter formula_15 is dual to the model with coupling parameter formula_26, 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_27 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. For formula_29, there is a unique phase transition from the ordered phase to the disordered  phase at formula_30. The model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. For formula_22, there are two phase transition points at formula_32 and formula_33. 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 formula_34, while perturbative study found that the essential singularity behaves as formula_35 where formula_36 goes from formula_37 to formula_38 as formula_1 increases from formula_40 to formula_41. 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.