Universality classes

$d$	$n$	$\sigma$	$\alpha$	$\beta$	$\gamma$	class
						3-state Potts
						Ashkin-Teller
						Chiral
						Directed percolation
			0	$1/8$	$7/4$	2D Ising
			0	$1/8$	$7/4$	3D Ising
						Local linear interface
			0	$1/2$	1	Mean-field
						Molecular beam epitaxy
						Random-field
						XY

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)

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

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

$\delta =15$

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 $k_{B}T_{c}=4.51152786~S$. 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: $\gamma =1$)