Universality classes

Universality classes are groups of models that have the same set of critical exponents

 dimension $\alpha$ $\beta$ $\gamma$ $\delta$ $\nu$ $\eta$ class 3-state Potts Ashkin-Teller Chiral Directed percolation 2 0 1/8 7/4 1 1/4 2D Ising 3 0.1096(5) 0.32653(10) 1.2373(2) 4.7893(8) 0.63012(16) 0.03639(15) 3D Ising Local linear interface any 0 1/2 1 3 1/2 0 Mean-field Molecular beam epitaxy Random-field 3 −0.0146(8) 0.3485(2) 1.3177(5) 4.780(2) 0.67155(27) 0.0380(4) XY

where

Derivations

Ising

The Hamiltonian of the Ising model is

$H=\sum_{}S_i S_j$

where $S_i=\pm 1$ and the summation runs over the lattice sites.

The order parameter is $m=\sum_i S_i$

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are

$\alpha=0$

(In fact, the specific heat diverges logarithmically with the critical temperature)

$\beta=\frac{1}{8}$

$\gamma=\frac{7}{4}$

$\delta=15$

along with [1]:

$\nu=1$
$\eta = 1/4$

In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates [2]:

$\nu=0.63012(16)$
$\alpha=0.1096(5)$
$\beta= 0.32653(10)$
$\gamma=1.2373(2)$
$\delta=4.7893(8)$
$\eta =0.03639(15)$

with a critical temperature of $k_BT_c = 4.51152786~S$[3]. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

Mean-field

The critical exponents of are derived as follows [4]:

Heat capacity exponent: $\alpha$

(final result: $\alpha=0$)

Magnetic order parameter exponent: $\beta$

(final result: $\beta=1/2$)

Susceptibility exponent: $\gamma$

(final result: $\gamma=1$)

Equation of state exponent: $\delta$

(final result: $\delta=3$)

Correlation length exponent: $\nu$

(final result: $\nu=1/2$)

Correlation function exponent: $\eta$

(final result: $\eta=0$)

XY

For the three dimensional XY model one has the following critical exponents[5]:

$\nu=0.67155(27)$
$\alpha = -0.0146(8)$
$\beta= 0.3485(2)$
$\gamma=1.3177(5)$
$\delta=4.780(2)$
$\eta =0.0380(4)$