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| '''Universality classes''' are groups of [[Idealised models | models]] that have the same set of [[critical exponents]]
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| | | {| border="1" |
| :{| border="1"
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| | dimension ||<math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || <math>\delta</math> ||<math>\nu</math> || <math>\eta</math> || class
| | | <math>d</math> || <math>n</math> || <math>\sigma</math> || name |
| |- | | |- |
| | || || || || || || || 3-state Potts | | | || || || 3-state Potts |
| |- | | |- |
| | || || || || || || ||Ashkin-Teller | | | || || ||Ashkin-Teller |
| |- | | |- |
| | || || || || || || ||Chiral | | | || || ||Chiral |
| |- | | |- |
| | || || || || || || ||Directed percolation | | | || || ||Directed percolation |
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| | 2 || 0 || 1/8 || 7/4 || || 1 || 1/4 || 2D Ising | | | || || ||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 |
| |- | | |- |
| | || || || || || || ||Local linear interface | | | || || ||Mean-field |
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| | any || 0 || 1/2 || 1 || 3 || 1/2 || 0 || Mean-field
| | | || || ||Molecular beam epitaxy |
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| | || || || || || || ||Molecular beam epitaxy
| | | || || ||Random-field |
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| | || || || || || || ||Random-field | |
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| | 3 || −0.0146(8) || 0.3485(2) || 1.3177(5) || 4.780(2) ||0.67155(27) || 0.0380(4) || XY
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| |} | | |} |
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| where
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| *<math>\alpha</math> is known as the [[Critical exponents#Heat capacity exponent| heat capacity exponent]]
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| *<math>\beta</math> is known as the [[Critical exponents#Magnetic order parameter exponent | magnetic order parameter exponent]]
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| *<math>\gamma</math> is known as the [[Critical exponents#Susceptibility exponent |susceptibility exponent ]]
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| *<math>\delta</math> is known as the [[Critical exponents#Equation of state exponent |equation of state exponent ]]
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| *<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length exponent]]
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| *<math>\eta</math> is known as the anomalous dimension in the critical correlation function.
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| =Derivations=
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| ==3-state Potts== | | ==3-state Potts== |
| [[Potts model]]
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| ==Ashkin-Teller== | | ==Ashkin-Teller== |
| [[Ashkin-Teller model]]
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| ==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> |
| :<math>
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| \alpha=0 | | \alpha=0 |
| </math> | | </math> |
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| (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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| \delta=15 | | \delta=15 |
| </math> | | </math> |
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| along with <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>:
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| :<math>
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| \nu=1
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| </math>
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|
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| :<math>
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| \eta = 1/4
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| </math>
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| In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and [[Renormalisation group]] analysis provide accurate estimates <ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 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)]</ref>:
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|
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| :<math>
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| \nu=0.63012(16)
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| </math>
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|
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| :<math>
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| \alpha=0.1096(5)
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| </math>
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|
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| :<math>
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| \beta= 0.32653(10)
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| </math>
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|
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| :<math>
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| \gamma=1.2373(2)
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| </math>
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|
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| :<math>
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| \delta=4.7893(8)
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| </math>
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|
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| :<math>
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| \eta =0.03639(15)
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| </math>
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| with a critical temperature of <math>k_BT_c = 4.51152786~S </math><ref>[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>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.
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| ==Local linear interface== | | ==Local linear interface== |
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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>) |
| ====Equation of state exponent: <math>\delta</math>====
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| (final result: <math>\delta=3</math>)
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| ====Correlation length exponent: <math>\nu</math>====
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| (final result: <math>\nu=1/2</math>)
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| ====Correlation function exponent: <math>\eta</math>====
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| (final result: <math>\eta=0</math>)
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| ==Molecular beam epitaxy== | | ==Molecular beam epitaxy== |
| | ==See also== |
| | *[[Critical exponents]] |
| ==Random-field== | | ==Random-field== |
| ==XY==
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| For the three dimensional [[XY model]] one has the following [[critical exponents]]<ref name="Campostrini2001" >[http://dx.doi.org/10.1103/PhysRevB.63.214503 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)]</ref>:
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|
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| :<math>
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| \nu=0.67155(27)
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| </math>
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|
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| :<math>\alpha = -0.0146(8)</math>
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|
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| :<math>
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| \beta= 0.3485(2)
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| </math>
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|
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| :<math>
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| \gamma=1.3177(5)
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| </math>
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|
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| :<math>
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| \delta=4.780(2)
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| </math>
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| :<math>
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| \eta =0.0380(4)
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| </math>
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| =References=
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| <references/>
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| [[category: Renormalisation group]] | | [[category: Renormalisation group]] |