Universality classes: Difference between revisions
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==Directed percolation== | ==Directed percolation== | ||
==Ising== | ==Ising== | ||
The Hamiltonian of the Ising model is | |||
\begin{equation} | |||
{\cal H}=\sum{<i,j>}S_iS_j | |||
\end{equation} | |||
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} | |||
==Local linear interface== | ==Local linear interface== | ||
==Mean-field== | ==Mean-field== | ||
Revision as of 13:01, 20 July 2011
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| name | |||
| 3-state Potts | |||
| Ashkin-Teller | |||
| Chiral | |||
| Directed percolation | |||
| Ising | |||
| Local linear interface | |||
| Mean-field | |||
| Molecular beam epitaxy | |||
| Random-field |
3-state Potts
Ashkin-Teller
Chiral
Directed percolation
Ising
The Hamiltonian of the Ising model is \begin{equation} {\cal H}=\sum{<i,j>}S_iS_j \end{equation} 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}