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Potts Universality Classes

Overview

The $q$-state Potts model generalises the Ising model ($q = 2$) to a spin variable $\sigma_i \in \{1, 2, \ldots, q\}$ with Hamiltonian

\[H = -J \sum_{\langle i,j \rangle} \delta_{\sigma_i, \sigma_j}\]

where $\delta$ is the Kronecker delta. The global symmetry group is $S_q$ (permutation of the $q$ colours).

In $d = 2$ the phase transition is second-order for $q \leq 4$ and first-order for $q > 4$. For $q = 2, 3, 4$ the critical exponents are known exactly via the Coulomb gas mapping.

$q = 2$ –- Ising

The $q = 2$ Potts model is equivalent to the Ising model with a rescaled coupling. See Ising universality class.

$q = 3$ –- Three-State Potts ($d = 2$)

Symmetry: $S_3$ (permutation of 3 colours).

ExponentValuePhysical meaningReference
$\alpha$$1/3$Specific heat: $C \sim \lvert T - T_c\rvert^{-1/3}$Baxter (1973); den Nijs (1979)
$\beta$$1/9$Order parameterAlexander (1975); Nienhuis (1984)
$\gamma$$13/9$SusceptibilityScaling relation
$\nu$$5/6$Correlation lengthCoulomb gas
$\eta$$4/15$Anomalous dimensionCoulomb gas
$\delta$$14$Critical isotherm$\delta = 1 + \gamma/\beta$
$c$$4/5$Central charge (CFT)Minimal model $\mathcal{M}(5,6)$

Scaling Relations

\[\alpha + 2\beta + \gamma = \tfrac{1}{3} + \tfrac{2}{9} + \tfrac{13}{9} = 2 \quad\checkmark\]

\[\gamma = \nu(2 - \eta) = \tfrac{5}{6}\bigl(2 - \tfrac{4}{15}\bigr) = \tfrac{13}{9} \quad\checkmark\]

\[2 - \alpha = d\nu \implies \tfrac{5}{3} = 2 \cdot \tfrac{5}{6} \quad\checkmark\]

$q = 4$ –- Four-State Potts ($d = 2$)

Symmetry: $S_4$.

The $q = 4$ case is marginal: the transition is second-order but sits at the boundary of the first-order regime. Logarithmic corrections to scaling appear, making numerical extraction of exponents notoriously difficult.

ExponentValueNoteReference
$\alpha$$2/3$Strong specific-heat divergenceBaxter (1973)
$\beta$$1/12$Very small — order parameter onset is slowden Nijs (1979)
$\gamma$$7/6$SusceptibilityScaling relation
$\nu$$2/3$Correlation lengthCoulomb gas
$\eta$$1/4$Same anomalous dimension as Ising $d = 2$—
$\delta$$15$Same as Ising $d = 2$$\delta = 1 + \gamma/\beta$
$c$$1$Central charge (CFT)Orbifold $c = 1$ theory

Logarithmic Corrections

At $q = 4$ the RG $\beta$-function has a double zero at the fixed point, leading to multiplicative logarithmic corrections of the form

\[M \sim (T_c - T)^{1/12} \lvert\ln(T_c - T)\rvert^{1/8}\]

These corrections make finite-size scaling analysis substantially harder than for $q = 2$ or $q = 3$.

Scaling Relations

\[\alpha + 2\beta + \gamma = \tfrac{2}{3} + \tfrac{1}{6} + \tfrac{7}{6} = 2 \quad\checkmark\]

\[2 - \alpha = d\nu \implies \tfrac{4}{3} = 2 \cdot \tfrac{2}{3} \quad\checkmark\]

QAtlas API

using QAtlas

# q = 3, d = 2: exact (Rational{Int})
e3 = QAtlas.fetch(Universality(:Potts3), CriticalExponents(); d=2)
# (β = 1//9, ν = 5//6, γ = 13//9, η = 4//15, ...)

# q = 4, d = 2: exact (Rational{Int})
e4 = QAtlas.fetch(Universality(:Potts4), CriticalExponents(); d=2)
# (β = 1//12, ν = 2//3, γ = 7//6, η = 1//4, ...)

References

  • R. J. Baxter, "Potts model at the critical temperature", J. Phys. C 6, L445 (1973) –- exact critical temperature.
  • R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982), Ch. 12.
  • B. Nienhuis, "Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas", J. Stat. Phys. 34, 731 (1984) –- Coulomb gas derivation of exact exponents.
  • M. P. M. den Nijs, J. Phys. A 12, 1857 (1979) –- exponent relations via Coulomb gas.
  • J. Salas, A. D. Sokal, "Logarithmic corrections and finite-size scaling in the two-dimensional 4-state Potts model", J. Stat. Phys. 88, 567 (1997) –- log corrections.

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