Universality classes

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

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)

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

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

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Mean-field

The critical exponents of are derived as follows :

Heat capacity exponent: $\alpha$

(final result: $\alpha =0$)

Magnetic order parameter exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}

(final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=1/2} )

Susceptibility exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}

(final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=1} )

See also

• Critical exponents