Universality classes

$d$	$n$	$\sigma$	name
			3-state Potts
			Ashkin-Teller
			Chiral
			Directed percolation
			Ising
			Local linear interface
			Mean-field
			Molecular beam epitaxy
			Random-field

Ising

The Hamiltonian of the Ising model is

${\cal {H}}=\sum {<i,j>}S_{i}S_{j}$

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

The order parameter is \begin{equation} m=\sum_i S_i \end{equation}

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are \begin{equation} \alpha=0 \end{equation} (In fact, the specific hear diverges logarithmically with the critical temperature)

\begin{equation} \beta=\frac{1}{8} \end{equation} \begin{equation} \gamma=\frac{7}{4} \end{equation} \begin{equation} \delta=15 \end{equation}

See also

• Critical exponents