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

• $\alpha$ is known as the heat capacity exponent
• $\beta$ is known as the magnetic order parameter exponent
• $\gamma$ is known as the susceptibility exponent
• $\delta$ is known as the equation of state exponent
• $\nu$ is known as the correlation length exponent
• $\eta$ is known as the anomalous dimension in the critical correlation function.

Derivations

3-state Potts

Potts model

Ashkin-Teller

Ashkin-Teller model

Ising

The Hamiltonian of the Ising model is

$H=\sum_{<i,j>}S_i S_j$

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 :

$\nu=1$

$\eta = 1/4$

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

$\nu=0.63012(16)$

$\alpha=0.1096(5)$

$\beta= 0.32653(10)$

$\gamma=1.2373(2)$

$\delta=4.7893(8)$

$\eta =0.03639(15)$

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

Mean-field

The critical exponents of are derived as follows :

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:

$\nu=0.67155(27)$

$\alpha = -0.0146(8)$

$\beta= 0.3485(2)$

$\gamma=1.3177(5)$

$\delta=4.780(2)$

$\eta =0.0380(4)$