Universality classes
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
- \(\alpha\) is known as the heat capacity exponent
- \(\beta\) is known as the magnetic order parameter exponent
- \(\gamma\) is known as the susceptibility exponent
- \(\nu\) is known as the correlation length
- \(\eta\) is known as...
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[edit] Derivations
[edit] 3-state Potts
[edit] Ashkin-Teller
[edit] Chiral
[edit] Directed percolation
[edit] Ising
The Hamiltonian of the Ising model is
\( H=\sum_{}S_i S_j \)
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
\[ \nu=1 \]
\[ \eta = 1/4 \]
In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates
\[ \nu=0.63012(16) \]
\[ \alpha=0.1096(5) \]
\[ \beta= 0.32653(10) \]
\[ \gamma=1.2373(2) \]
\[ \delta=4.7893(8) \]
\[ \eta =0.03639(15) \]
with a critical temperature of \(k_BT_c = 4.51152786~S \)
[edit] Local linear interface
[edit] Mean-field
The critical exponents of are derived as follows
[edit] Heat capacity exponent: \(\alpha\)
(final result: \(\alpha=0\))
[edit] Magnetic order parameter exponent: \(\beta\)
(final result: \(\beta=1/2\))
[edit] Susceptibility exponent: \(\gamma\)
(final result: \(\gamma=1\))
[edit] Molecular beam epitaxy
[edit] Random-field
[edit] XY
For the three dimensional XY model one has the following critical exponents
\[ \nu=0.67155(27) \]
\[\alpha = -0.0146(8)\]
\[ \beta= 0.3485(2) \]
\[ \gamma=1.3177(5) \]
\[ \delta=4.780(2) \]
\[ \eta =0.0380(4) \]
[edit] References
- ↑ Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review 180 pp. 594-600 (1969)
- ↑ Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E 65 066127 (2002)
- ↑ A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General 29 pp. 5727-5733 (1996)
- ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4
- ↑ Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B 63 214503 (2001)
