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Universality classes

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Universality classes are groups of models that have the same set of critical exponents

dimension \alpha \beta \gamma \delta \nu \eta class
3-state Potts
Ashkin-Teller
Chiral
Directed percolation
2 0 1/8 7/4 1 1/4 2D Ising
3 0.1096(5) 0.32653(10) 1.2373(2) 4.7893(8) 0.63012(16) 0.03639(15) 3D Ising
Local linear interface
0 1/2 1 Mean-field
Molecular beam epitaxy
Random-field
3 −0.0146(8) 0.3485(2) 1.3177(5) 4.780(2) 0.67155(27) 0.0380(4) XY

where

Derivations[edit]

3-state Potts[edit]

Potts model

Ashkin-Teller[edit]

Ashkin-Teller model

Chiral[edit]

Directed percolation[edit]

Ising[edit]

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

along with [1]:


\nu=1

\eta = 1/4


In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates [2]:


\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_BT_c = 4.51152786~S [3]. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

Local linear interface[edit]

Mean-field[edit]

The critical exponents of are derived as follows [4]:

Heat capacity exponent: \alpha[edit]

(final result: \alpha=0)

Magnetic order parameter exponent: \beta[edit]

(final result: \beta=1/2)

Susceptibility exponent: \gamma[edit]

(final result: \gamma=1)

Molecular beam epitaxy[edit]

Random-field[edit]

XY[edit]

For the three dimensional XY model one has the following critical exponents[5]:


\nu=0.67155(27)
\alpha = -0.0146(8)

\beta= 0.3485(2)

\gamma=1.3177(5)

\delta=4.780(2)

\eta =0.0380(4)

References[edit]