parafermion CFT, is a conformal field theory in two dimensions.
It is a minimal model with central charge
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.
[1][2][3] The critical three-state Potts model has a central charge of
, 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
refers to the Virasoro character, found by taking the trace over the Verma module generated from the Virasoro primary operator labeled by integers
is a standard convention for primary operators of the
The local antiholomorphic W primaries similarly are given by
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).
The fusion rules governing the operator product expansions involving these fields respect the action of this
There is also a charge conjugation symmetry that interchanges
The critical three-state Potts model is one of the two modularly invariant conformal field theories that exist with central charge
The other such theory is the tetracritical Ising model, which has a diagonal partition function in terms of Virasoro characters.
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
The first term couples degrees of freedom on nearest neighbour sites in the lattice.
subgroup of this symmetry is generated by the unitary operator
and is characterised by a nonzero ground state expectation value of the order parameter
The ground state in this phase explicitly breaks the global
and is characterised by a single ground state.
, the Hamiltonian is gapless with a ground state energy of
In other words, in the limit of an infinitely long chain, the lowest energy eigenvalues of the Hamiltonian are spaced infinitesimally 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.
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.
map to the corresponding field positions
in space-time, and non-universal real number prefactors are ignored.