Critical three-state Potts model
The three-state Potts CFT, also known as the formula_1 parafermion CFT, is a conformal field theory in two dimensions. It is a minimal model with central charge formula_2. It is considered to be the simplest minimal model with a non-diagonal partition function in Virasoro characters, as well as the simplest non-trivial CFT with the W-algebra as a symmetry. Properties. The critical three-state Potts model has a central charge of formula_3, and thus belongs to the discrete family of unitary minimal models with central charge less than one. These conformal field  theories are fully classified and for the most part well-understood. The modular partition function of the critical three-state Potts model is given by Here formula_5 refers to the Virasoro character, found by taking the trace over the Verma module generated from the Virasoro primary operator labeled by integers formula_6. The labeling formula_7 is a standard convention for primary operators of the formula_8 minimal models. Furthermore, the critical three-state Potts model is symmetric not only under the Virasoro algebra, but also under an enlarged algebra called the W-algebra that includes the Virasoro algebra as well as some  spin-3 currents. The local holomorphic W primaries are given by formula_9. The local antiholomorphic W primaries similarly are given by formula_10 with the same scaling dimensions. Each field in the theory is either a combination of a holomorphic and antiholomorphic W-algebra primary field, or a descendant of such a field generated by acting with W-algebra generators. Some primaries of the Virasoro algebra, such as the formula_11 primary, are not primaries of the W algebra. The partition function is diagonal when expressed in terms of W-algebra characters (where traces are taken over irreducible representations of the W algebra , instead of over irreducible representations of the Virasoro algebra). Since formula_12 and formula_13, we can write The operators formula_15 are charged under the action of a global formula_1 symmetry. That is, under a global global formula_1 transformation, they pick up phases formula_18 and formula_19 for formula_20. The fusion rules governing the operator product expansions involving these fields respect the action of this formula_1 transformation. There is also a charge conjugation symmetry that interchanges formula_22. Sometimes the notation formula_23 is used in the literature instead of formula_15. The critical three-state Potts model is one of the two modularly invariant conformal  field theories that exist with central charge formula_25. The other such theory is the tetracritical Ising model, which has a diagonal partition function in terms of Virasoro characters. It is possible to obtain the critical three-state Potts model from the tetracritical Ising model by applying a formula_26 orbifold transformation to the latter. Lattice Hamiltonians. The critical three-state Potts conformal field theory can be realised as the low energy effective theory at the phase transition of the one-dimensional quantum three-state Potts model. The Hamiltonian of the quantum three-state Potts model is given by Here formula_28 and  formula_29 are positive parameters. The first term couples degrees of freedom on nearest neighbour sites in the lattice. formula_30 and formula_31 are formula_32 clock matrices satisfying formula_33 and same-site commutation relation formula_34 where formula_35. This Hamiltonian is symmetric under any permutation of the three formula_36 eigenstates on each site, as long as the same permutation is done on every site. Thus it is said to have a global formula_37 symmetry. A formula_1 subgroup of this symmetry is generated by the unitary operator formula_39. In one dimension, the model has two gapped phases, the ordered phase and the disordered phase . The ordered phase occurs at formula_40 and is characterised by a nonzero ground state expectation value of the order parameter formula_41 at any site formula_42. The ground state in this phase explicitly breaks the global formula_37 symmetry and is thus three-fold degenerate. The disordered phase occurs at formula_44 and is characterised by a single ground state. In between these two phases is a phase transition at formula_45. At this particular value of formula_29, the Hamiltonian is gapless with a ground state energy of formula_47, where formula_48 is the length of the chain. In other words, in the limit of  an infinitely long chain, the lowest energy eigenvalues of the Hamiltonian are spaced infinitessimally close to each other. As is the case for most one dimensional gapless theories, it is possible to describe the low energy physics of the 3-state Potts model using a 1+1 dimensional conformal field theory; in this particular lattice model that conformal field theory is none other than the critical three-state Potts model. Lattice operator correspondence. Under the flow of renormalisation group, lattice operators in the quantum three-state Potts model flow to fields in the conformal field theory. In general, understanding  which operators flow to what fields is difficult and not obvious. Analytical and numerical arguments suggest a correspondence between a few lattice operators and CFT fields as follows. Lattice indices formula_42 map to the corresponding field positions formula_50 in space-time, and non-universal real number prefactors are ignored.