Editing Universality classes
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{{Stub-general}} | |||
{| border="1" | |||
|- | |- | ||
| <math>d</math> || <math>n</math> || <math>\sigma</math> || <math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || class | |||
|- | |- | ||
| | | || || || || || || 3-state Potts | ||
|- | |- | ||
| || || | | || || || || || ||Ashkin-Teller | ||
|- | |- | ||
| || | | || || || || || ||Chiral | ||
|- | |- | ||
| || || | | || || || || || ||Directed percolation | ||
|- | |- | ||
| | | || || || 0 || <math>1/8</math> || <math>7/4</math> || 2D Ising | ||
|- | |- | ||
| | | || || || 0 || <math>1/8</math> || <math>7/4</math> || 3D Ising | ||
|- | |- | ||
| || | | || || || || || ||Local linear interface | ||
|- | |- | ||
| || || ||0 || <math>1/2</math> || 1 ||Mean-field | |||
|- | |- | ||
| || | | || || || || || ||Molecular beam epitaxy | ||
|- | |- | ||
| || || | | || || || || || ||Random-field | ||
|} | |} | ||
==3-state Potts== | ==3-state Potts== | ||
==Ashkin-Teller== | ==Ashkin-Teller== | ||
==Chiral== | ==Chiral== | ||
==Directed percolation== | ==Directed percolation== | ||
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In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are | In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are | ||
<math> | |||
\alpha=0 | \alpha=0 | ||
</math> | </math> | ||
(In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]]) | (In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]]) | ||
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</math> | </math> | ||
In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and [[Renormalisation group]] analysis provide accurate estimates: | |||
:<math> | :<math> | ||
\nu= | \nu=0.6298 (5) | ||
</math> | </math><ref name="Hasenbusch">[http://dx.doi.org/10.1103/PhysRevB.59.11471 M. Hasenbusch, K. Pinn and S. Vinti "Critical exponents of the three-dimensional Ising universality class from finite-size scaling with standard and improved actions", Physical Review B '''59''' pp. 11471-11483 (1999)]</ref> | ||
:<math> | :<math> | ||
\ | \alpha=0.108(5) | ||
</math> | </math> <ref name="Kolesik"> [http://dx.doi.org/10.1016/0378-4371(94)00302-A Miroslav Kolesik and Masuo Suzuki "Accurate estimates of 3D Ising critical exponents using the coherent-anomaly method", Physica A: Statistical and Theoretical Physics '''215''' pp. 138-151 (1995)]</ref> | ||
:<math> | :<math> | ||
\ | \beta= 0.3269(6) | ||
</math> | </math> <ref name="Talapov">[http://dx.doi.org/10.1088/0305-4470/29/17/042 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)]</ref> | ||
:<math> | :<math> | ||
\ | \gamma=1.237(4) | ||
</math> | </math><ref name="Kolesik"> </ref> | ||
:<math> | :<math> | ||
\ | \delta=4.77(5) | ||
</math> | </math><ref name="Kolesik"> </ref> | ||
:<math> | :<math> | ||
\eta =0.0366(8) | |||
</math><ref name="Hasenbusch"> </ref> | |||
\eta =0. | |||
</math> | |||
with a critical temperature of <math>K_c = J/k_BT_c = 0.2216544</math><ref name="Talapov"> </ref>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections. | |||
==Local linear interface== | ==Local linear interface== | ||
==Mean-field== | ==Mean-field== | ||
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====Susceptibility exponent: <math>\gamma</math>==== | ====Susceptibility exponent: <math>\gamma</math>==== | ||
(final result: <math>\gamma=1</math>) | (final result: <math>\gamma=1</math>) | ||
==Molecular beam epitaxy== | ==Molecular beam epitaxy== | ||
==See also== | |||
*[[Critical exponents]] | |||
==Random-field== | ==Random-field== | ||
== | ==References== | ||
<references/> | <references/> | ||
[[category: Renormalisation group]] | [[category: Renormalisation group]] |