Critical three-state Potts model

The quantum three-state Potts model is a lattice Hamiltonian that has a discrete $S_{3}$ permutation symmetry. It is the version of the quantum clock model with three degrees of freedom on each lattice site.

Hamiltonian

The Hamiltonian of this model is given by

$H=-J(\sum _{\langle i,j\rangle }(Z_{i}^{\dagger }Z_{j}+Z_{i}Z_{j}^{\dagger })+g\sum _{j}(X_{j}+X_{j}^{\dagger }))$

The first term couples degrees of freedom on nearest neighbour sites in the lattice. $X$ and $Z$ are $3\times 3$ clock matrices satisfying $X^{3}=Z^{3}=1$ and same-site commutation relation $ZX=\omega XZ$ where $\omega =-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}$. Here $J$ and $g$ are positive parameters.

This Hamiltonian is symmetric under any permutation of the three $Z$ eigenstates on each site, as long as the same permutation is done on every site. Thus it is said to have a global $S_{3}$ symmetry. A $\mathbb {Z} _{3}$ subgroup of this symmetry is generated by the unitary operator $\prod _{j}X_{j}$.

One Dimensional Phase Transition

In one dimension, the model has two gapped phases, the ordered phase and the disordered phase. The ordered phase occurs at $0<g<1$ and is characterised by a nonzero ground state expectation value of the order parameter $Z_{j}$ at any site $j$. The ground state in this phase explicitly breaks the global $S_{3}$ symmetry and is thus three-fold degenerate. The disordered phase occurs at $g>1$ and is characterised by a single ground state.

In between these two phases is a phase transition at $g=1$. At this particular value of $g$, the Hamiltonian is gapless. 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. This means that the low energy degrees of freedom are symmetric under the Virasoro algebra. This particular conformal field theory is called the critical three-state Potts model. It has a central charge of $c=4/5$, 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.

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. Each field in the theory is either a W-algebra primary field, or a descendant of a W primary field generated by acting on a primary field with W-algebra generators. The holomorphic W primaries are given by $1,\epsilon ,\sigma _{1},\sigma _{2},\psi _{1},\psi _{2}$ with scaling dimensions (Virasoro $L_{0}$ eigenvalue) $0,2/5,1/15,1/15,2/3,2/3$ respectively. The antiholomorphic W primaries similarly are given by $1,{\bar {\epsilon }},{\bar {\sigma }}_{1},{\bar {\sigma }}_{2},{\bar {\psi }}_{1},{\bar {\psi }}_{2}$ with the same scaling dimensions ( eigenvalues of ${\bar {L}}_{0}$). A field in the critical three-state Potts model is a combination of a holomorphic and anti-holomorphic field. This theory is considered to be the simplest minimal model with a non-diagonal partition function when expressed in terms of Virasoro characters. The partition function is, however, diagonal when expressed in terms of W-algebra characters.

The operators $\sigma _{1},\sigma _{2},\psi _{1},\psi _{2}$ are charged under the global $\mathbb {Z} _{3}$ symmetry. That is, under a global global $\mathbb {Z} _{3}$ transformation, they pick up phases $\sigma _{a}\to e^{2\pi ia/3}\sigma _{a}$ and $\psi _{a}\to e^{2\pi ia/3}\psi _{a}$ for $a=1,2$.

The critical three-state Potts model is one of the two conformal field theories that exist with central charge $c=4/5$ and is modularly invariant. 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 $\mathbb {Z} _{2}$ orbifold transformation.