Universality classes: Difference between revisions

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The Hamiltonian of the Ising model is  
The Hamiltonian of the Ising model is  


\begin{equation}
 
{\cal H}=\sum{<i,j>}S_iS_j
{\cal H}=\sum{<i,j>}S_iS_j
\end{equation}
 


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

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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


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

Local linear interface

Mean-field

Molecular beam epitaxy

See also

Random-field