Critical three-state Potts model

The critical three-state Potts model 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.

The modular partition function of the critical three-state Potts model is given by

::$Z\; =\; |\backslash chi\_\{1,1\}\; +\; \backslash chi\_\{4,1\}|^2\; +\; |\backslash chi\_\{2,1\}\; +\; \backslash chi\_\{3,1\}|^2\; +\; 2|\backslash chi\_\{4,3\}|^2+2|\backslash chi\_\{3,3\}|^2$

Here $\backslash chi\_\{r,s\}\; (q)\; \backslash equiv\; \backslash textrm\{Tr\}\_\{(r,s)\}\; (q^\{L\_0-c/24\})$ refers to the Virasoro character, found by taking the trace over the Verma module generated from the Virasoro primary operator labeled by integers $r,\; s$. The labeling $(r,\; s)$ is a standard convention for primary operators of the 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 $1,\; \backslash epsilon,\; \backslash sigma\_1,\; \backslash sigma\_2,\; \backslash psi\_1,\; \backslash psi\_2$. The local antiholomorphic W primaries similarly are given by $1,\; \backslash bar\{\backslash epsilon\},\; \backslash bar\{\backslash sigma\}\_1,\; \backslash bar\{\backslash sigma\}\_2,\; \backslash bar\{\backslash psi\}\_1,\; \backslash bar\{\backslash psi\}\_2$ 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 $(3,1)$ primary, are not primaries of the W algebra.

class="wikitable" |+ Chiral W primaries in the critical 3-state Potts model
Primary	Dimension	$\backslash mathbb\{Z\}\_3$ charge !Kac Label
$1$	0	0 |(1,1)+(4,1)
$\backslash epsilon$	2/5	0 |(2,1)+(3,1)
$\backslash psi\_1$	2/3	1 |(1,3)
$\backslash psi\_2$	2/3	-1 |(1,3)
$\backslash sigma\_1$	1/15	1 |(3,3)
$\backslash sigma\_2$	1/15	-1 |(3,3)

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 $\backslash chi\_1\; =\; \backslash chi\_\{1,1\}\; +\; \backslash chi\_\{4,1\}$ and $\backslash chi\_\{\backslash epsilon\}\; =\; \backslash chi\_\{2,1\}\; +\; \backslash chi\_\{3,1\}$, we can write

::$Z\; =\; |\backslash chi\_\{1\}|^2\; +\; |\backslash chi\_\{\backslash epsilon\}|^2\; +\; |\backslash chi\_\{\backslash psi\_1\}|^2+|\backslash chi\_\{\backslash psi\_2\}|^2+|\backslash chi\_\{\backslash sigma\_1\}|^2+|\backslash chi\_\{\backslash sigma\_2\}|^2$

The operators $\backslash sigma\_1,\; \backslash sigma\_2,\; \backslash psi\_1,\; \backslash psi\_2$ are charged under the action of a global $\backslash mathbb\{Z\}\_3$ symmetry. That is, under a global global $\backslash mathbb\{Z\}\_3$ transformation, they pick up phases $\backslash sigma\_a\; \backslash to\; e^\{2\backslash pi\; i\; a/3\}\; \backslash sigma\_a$ and $\backslash psi\_a\; \backslash to\; e^\{2\backslash pi\; i\; a/3\}\; \backslash psi\_a$ for $a\; =\; 1,2$. The fusion rules governing the operator product expansions involving these fields respect the action of this $\backslash mathbb\{Z\}\_3$ transformation. There is also a charge conjugation symmetry that interchanges $\backslash sigma\_1\; \backslash leftrightarrow\; \backslash sigma\_2,\; \backslash psi\_1\; \backslash leftrightarrow\; \backslash psi\_2$. Sometimes the notation $\backslash sigma,\; \backslash sigma^\backslash dagger,\; \backslash psi,\; \backslash psi^\backslash dagger$ is used in the literature instead of $\backslash sigma\_1,\; \backslash sigma\_2,\; \backslash psi\_1,\; \backslash psi\_2$.

The critical three-state Potts model is one of the two modularly invariant conformal field theories that exist with central charge $c=\; 4/5$. 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 $\backslash mathbb\{Z\}\_2$ orbifold transformation to the latter.

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

:$H\; =\; -J(\backslash sum\_\{\; \backslash langle\; i,\; j\; \backslash rangle\}\; (Z^\backslash dagger\_i\; Z\_\{j\}+\; Z\_i\; Z^\{\backslash dagger\}\_\{j\})\; +\; g\; \backslash sum\_j\; (X\_j\; +\; X^\backslash dagger\_j)\; )$

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

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 $\backslash mathbb\{Z\}\_3$ subgroup of this symmetry is generated by the unitary operator $\backslash prod\_j\; X\_j$.

In one dimension, the model has two gapped phases, the ordered phase and the disordered phase. The ordered phase occurs at

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 $j$ map to the corresponding field positions $z\; ,\; \backslash bar\{z\}$ in space-time, and non-universal real number prefactors are ignored.

• $Z\_j\; \backslash sim\; \backslash Phi\_\{\backslash sigma\_1,\; \backslash bar\{\backslash sigma\}\_1\}\; (z,\backslash bar\; z)$, the $\backslash frac\{2\}\{15\}$-dimensional field composed of holomorphic and anti-holomorphic parts $\backslash sigma\_1(z)$ and $\backslash bar\; \backslash sigma\_1(\backslash bar\; z)$
• $Z\_j^\backslash dagger\; \backslash sim\; \backslash Phi\_\{\backslash sigma\_2,\; \backslash bar\{\backslash sigma\}\_2\}\; (z,\backslash bar\; z)$
• $Z\_j\; Z\_\{j+1\}^\backslash dagger\; -\; \backslash frac\{1\}\{2\}(X\_j\; +\; X\_\{j+1\})\; +\; \backslash textrm\{h.c.\}\; \backslash sim\; \backslash Phi\_\{\backslash epsilon,\; \backslash bar\{\backslash epsilon\}\}(z,\backslash bar\; z)$ . As can be seen in the lattice language, adding this operator to every site of the Hamiltonian has the effect of tuning $g$ away from 1. This operator is called the thermal operator, because in the classical statistical mechanics analog of the quantum lattice model, tuning $g$ would be equivalent to changing temperature away from the critical temperature.
• $-Z\_j\; Z\_\{j+1\}^\backslash dagger\; -\; \backslash frac\{1\}\{2\}(X\_j\; +\; X\_\{j+1\})\; +\; \backslash textrm\{h.c.\}\; +\; \backslash frac\{4\}\{3\}+\; \backslash frac\{2\backslash sqrt\{3\}\}\{\backslash pi\}\; \backslash sim\; T(z)\; +\; \backslash bar\; T(\backslash bar\; z)$, the dimension-2 stress-energy tensor field.
• $Z\_j(2-3\backslash omega^2\; X\_j\; -\; 3\backslash omega\; X\_j^2)\; -2Z\_j^\backslash dagger\; (Z\_\{j-1\}^\backslash dagger\; +Z\_\{j+1\}^\backslash dagger)\; \backslash sim\; \backslash psi\_1(z)\; \backslash bar\; \backslash psi\_1(\backslash bar\; z)$

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Category:Conformal field theory