Universality classes

2020-10-21T11:28:09Z

159.149.173.87: I inserted missing mean-field critical exponents (delta, nu, eta) both in the table and in the description below

'''Universality classes''' are groups of [[Idealised models | models]] that have the same set of [[critical exponents]]

:{| border="1"
|-
| dimension ||<math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || <math>\delta</math> ||<math>\nu</math> || <math>\eta</math> || class
|-
| || || || || || || || 3-state Potts
|-
| || || || || || || ||Ashkin-Teller
|-
| || || || || || || ||Chiral
|-
| || || || || || || ||Directed percolation
|-
| 2 || 0 || <math>1/8</math> || <math>7/4</math> || || 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
|-
| any || 0 || <math>1/2</math> || 1 || 3 || <math>1/2</math> || 0 || 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
*<math>\alpha</math> is known as the [[Critical exponents#Heat capacity exponent| heat capacity exponent]]
*<math>\beta</math> is known as the [[Critical exponents#Magnetic order parameter exponent | magnetic order parameter exponent]]
*<math>\gamma</math> is known as the [[Critical exponents#Susceptibility exponent |susceptibility exponent ]]
*<math>\delta</math> is known as the [[Critical exponents#Equation of state exponent |equation of state exponent ]]
*<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length exponent]]
*<math>\eta</math> is known as...
=Derivations=
==3-state Potts==
[[Potts model]]
==Ashkin-Teller==
[[Ashkin-Teller model]]
==Chiral==
==Directed percolation==
==Ising==
The [[Hamiltonian]] of the [[Ising model]] is

<math>
H=\sum_{<i,j>}S_i S_j
</math>

where <math>S_i=\pm 1</math> and the summation runs over the lattice sites.

The [[Order parameters | order parameter]] is
<math>
m=\sum_i S_i
</math>

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are

:<math>
\alpha=0
</math>

(In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]])

<math>
\beta=\frac{1}{8}
</math>

<math>
\gamma=\frac{7}{4}
</math>

<math>
\delta=15
</math>

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>:

:<math>
\nu=1
</math>

:<math>
\eta = 1/4
</math>

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>:

:<math>
\nu=0.63012(16)
</math>

:<math>
\alpha=0.1096(5)
</math>

:<math>
\beta= 0.32653(10)
</math>

:<math>
\gamma=1.2373(2)
</math>

:<math>
\delta=4.7893(8)
</math>

:<math>
\eta =0.03639(15)
</math>

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.

==Local linear interface==
==Mean-field==
The [[critical exponents]] of are derived as follows <ref>Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4 </ref>:
====Heat capacity exponent: <math>\alpha</math>====
(final result: <math>\alpha=0</math>)
====Magnetic order parameter exponent: <math>\beta</math>====
(final result: <math>\beta=1/2</math>)
====Susceptibility exponent: <math>\gamma</math>====
(final result: <math>\gamma=1</math>)
====Equation of state exponent: <math>\delta</math>====
(final result: <math>\delta=3</math>)
====Correlation length exponent: <math>\nu</math>====
(final result: <math>\nu=1/2</math>)
====Correlation function exponent: <math>\eta</math>====
(final result: <math>\eta=0</math>)
==Molecular beam epitaxy==
==Random-field==
==XY==
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>:

:<math>
\nu=0.67155(27)
</math>

:<math>\alpha = -0.0146(8)</math>

:<math>
\beta= 0.3485(2)
</math>

:<math>
\gamma=1.3177(5)
</math>

:<math>
\delta=4.780(2)
</math>

:<math>
\eta =0.0380(4)
</math>
=References=
<references/>
[[category: Renormalisation group]]

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Universality classes