Editing Universality class

{{Short description|Collection of models with the same renormalization group flow limit}}
{{More references|date=December 2017}}
In [[statistical mechanics]], a '''universality class''' is a collection of [[mathematical model]]s which share a single [[scale invariance|scale-invariant]] limit under the process of [[renormalization group]] flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, [[asymptotic]] phenomena such as [[critical exponent]]s will be the same for all models in the class.

Some well-studied universality classes are the ones containing the [[Ising model]] or the [[percolation theory]] at their respective [[phase transition]] points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes will have a lower and upper [[critical dimension]]: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog of [[mean-field theory]] (this dimension is 4d for Ising or for directed percolation, and 6d for undirected percolation).

==List of critical exponents==
Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its [[reduced temperature]] <math>\tau</math>, its [[order parameter]] measuring how much of the system is in the "ordered" phase, the [[specific heat]], and so on.

*The exponent <math>\alpha</math> is the exponent relating the specific heat C to the reduced temperature: we have <math>C = \tau^{-\alpha}</math>. The specific heat will usually be singular at the critical point, but the minus sign in the definition of <math>\alpha</math> allows it to remain positive.
*The exponent <math>\beta</math> relates the order parameter <math>\Psi</math> to the temperature. Unlike most critical exponents it is assumed positive, since the order parameter will usually be zero at the critical point. So we have <math>\Psi = |\tau|^{\beta}</math>.
*The exponent <math>\gamma</math> relates the temperature with the system's response to an external driving force, or source field. We have <math>d\Psi/dJ = \tau^{-\gamma}</math>, with J the driving force.
*The exponent <math>\delta</math> relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have <math>J = \Psi^\delta</math> (hence <math>\Psi = J^{1/\delta}</math>), with the same meanings as before.
*The exponent <math>\nu</math> relates the size of correlations (i.e. patches of the ordered phase) to the temperature; away from the critical point these are characterized by a [[correlation length]] <math>\xi</math>. We have <math>\xi = \tau^{-\nu}</math>.
*The exponent <math>\eta</math> measures the size of correlations at the critical temperature. It is defined so that the [[correlation function]] scales as <math>r^{-d+2-\eta}</math>.
*The exponent <math>\sigma</math>, used in [[percolation theory]], measures the size of the largest clusters (roughly, the largest ordered blocks) at 'temperatures' (connection probabilities) below the critical point. So <math>s_{\max} \sim (p_c - p)^{-1/\sigma}</math>.
*The exponent <math>\tau</math>, also from [[percolation theory]], measures the number of size ''s'' clusters far from <math>s_{\max}</math> (or the number of clusters at criticality): <math>n_s \sim s^{-\tau} f(s/s_{\max})</math>, with the <math>f</math> factor removed at critical probability.

For symmetries, the group listed gives the symmetry of the order parameter. The group <math>\mathrm{Dih}_n</math> is the [[dihedral group]], the symmetry group of the ''n''-gon, <math>S_n</math> is the ''n''-element [[symmetric group]], <math>\mathrm{Oct}</math> is the [[octahedral group]], and <math>O(n)</math> is the [[orthogonal group]] in ''n'' dimensions. '''1''' is the [[trivial group]].

{| class="wikitable"
|-
!class
! dimension !! Symmetry !!<math>\alpha</math> !! <math>\beta</math> !! <math>\gamma</math> !! <math>\delta</math> !!<math>\nu</math> !! <math>\eta</math> 
|-   align="center"
|3-state [[Potts model|Potts]]
| 2 ||<math>S_3</math>||{{sfrac|1|3}} || {{sfrac|1|9}} || {{sfrac|13|9}}|| 14 || {{sfrac|5|6}} ||  {{sfrac|4|15}}
|-   align="center"
|Ashkin–Teller (4-state Potts)
| 2 ||<math>S
_4</math>||{{sfrac|2|3}} || {{sfrac|1|12}} || {{sfrac|7|6}} || 15 || {{sfrac|2|3}} || {{sfrac|1|4}} 
|-   align="center"
| rowspan="6" |[[Percolation critical exponents|Ordinary percolation]]
| 1 || 1 || 1 || 0 || 1 || <math>\infty</math> || 1 || 1 
|-   align="center"
| 2 || 1 || &minus;{{sfrac|2|3}} || {{sfrac|5|36}} || {{sfrac|43|18}} || {{sfrac|91|5}} || {{sfrac|4|3}} || {{sfrac|5|24}} <!-- || [[Percolation critical exponents|Ordinary percolation]] -->
|-   align="center"
| 3 || 1 || &minus;0.625(3) || 0.4181(8) || 1.793(3) || 5.29(6) || 0.87619(12) || 0.46(8) or 0.59(9) <!-- || [[Percolation critical exponents|Ordinary percolation]] -->
|-   align="center"
| 4 || 1 || &minus;0.756(40) || 0.657(9) || 1.422(16) || 3.9 or 3.198(6) || 0.689(10) || &minus;0.0944(28) <!-- || [[Percolation critical exponents|Ordinary percolation]] -->
|-   align="center"
| 5 || 1 || ≈ &minus;0.85 || 0.830(10) || 1.185(5) || 3.0 || 0.569(5) || &minus;0.075(20) or &minus;0.0565 <!-- || [[Percolation critical exponents|Ordinary percolation]] -->
|-   align="center"
| 6{{sup|+}} || 1 || &minus;1 || 1 || 1 || 2 || {{sfrac|1|2}} || 0 <!-- || [[Percolation critical exponents|Ordinary percolation]] -->
|-   align="center"
| rowspan="4" |[[Directed percolation]]
| 1 || 1 || 0.159464(6) || 0.276486(8) || 2.277730(5) || 0.159464(6) || 1.096854(4) || 0.313686(8)
|-   align="center"
| 2 || 1 || 0.451 || 0.536(3) || 1.60 || 0.451 || 0.733(8) || 0.230 <!-- || [[Directed percolation]] -->
|-   align="center"
| 3 || 1 || 0.73 || 0.813(9) || 1.25 || 0.73 || 0.584(5) || 0.12 <!-- || [[Directed percolation]] -->
|-   align="center"
| 4{{sup|+}} || 1 || &minus;1 || 1 || 1 || 2 || {{sfrac|1|2}} || 0 <!-- || [[Directed percolation]] -->
|- align="center"
| rowspan="4" | [[Conserved directed percolation]] (Manna, or "local linear interface")
| 1 || 1 ||  || 0.28(1) ||  || 0.14(1) || 1.11(2)<ref name="Fajardo">{{cite book |last1=Fajardo |first1=Juan A. B. |title=Universality in Self-Organized Criticality |date=2008 |location=Granada |url=http://hera.ugr.es/tesisugr/17706312.pdf}}</ref> || 0.34(2)<ref name="Fajardo"/> <!-- || Conserved directed percolation -->
|-   align="center"
| 2 || 1 ||  || 0.64(1) || 1.59(3) || 0.50(5) || 1.29(8) || 0.29(5) <!-- || Conserved directed percolation -->
|-   align="center"
| 3 || 1 ||  || 0.84(2) || 1.23(4) || 0.90(3) || 1.12(8) || 0.16(5) <!-- || Conserved directed percolation -->
|-   align="center"
| 4{{sup|+}} || 1 ||  || 1 || 1 || 1 || 1 || 0 <!-- || Conserved directed percolation -->
|- align="center"
| rowspan="2" |[[Protected percolation]]
| 2 || 1 ||  || 5/41<ref name=":0">{{Cite journal|last1=Fayfar|first1=Sean|last2=Bretaña|first2=Alex|last3=Montfrooij|first3=Wouter|date=2021-01-15|title=Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems|journal=Journal of Physics Communications|volume=5|issue=1|pages=015008|doi=10.1088/2399-6528/abd8e9|arxiv=2008.08258 |bibcode=2021JPhCo...5a5008F |issn=2399-6528|doi-access=free}}</ref>|| 86/41<ref name=":0" />|| ||  || 
|- align="center"
| 3 || 1 ||  || 0.28871(15)<ref name=":0" />|| 1.3066(19)<ref name=":0" />|| ||  || 
|- align="center"
| rowspan="2" |[[Ising critical exponents|Ising]]
| 2 ||<math>\mathbb{Z}_2</math>|| 0 ||{{sfrac|1|8}}||{{sfrac|7|4}}|| 15 || 1 ||{{sfrac|1|4}}
|- align="center"
| 3 ||<math>\mathbb{Z}_2</math>|| 0.11008(1) || 0.326419(3) || 1.237075(10) || 4.78984(1) || 0.629971(4) || 0.036298(2)
|- align="center"
|[[XY model|XY]]
| 3 ||<math>O(2)</math>||-0.01526(30) || 0.34869(7) || 1.3179(2) || 4.77937(25) ||0.67175(10) || 0.038176(44)
|- align="center"
|[[Heisenberg model (classical)|Heisenberg]]
| 3 ||<math>O(3)</math>||&minus;0.12(1) || 0.366(2) || 1.395(5) ||  || 0.707(3) || 0.035(2)
|-   align="center"
|[[Mean-field theory|Mean field]]
| all || any || 0 || {{sfrac|1|2}}   || 1  || 3 || {{sfrac|1|2}} || 0 
|- align="center"
|[[Molecular beam epitaxy]]<ref>{{cite journal |last1=Luis |first1=Edwin |last2=de Assis |first2=Thiago |last3=Ferreira |first3=Silvio |last4=Andrade |first4=Roberto |title=Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one-dimension |journal=Physical Review E |year=2019 |volume=99 |issue=2 |page=022801 |doi=10.1103/PhysRevE.99.022801 |pmid=30934348 |arxiv=1812.03114 |bibcode=2019PhRvE..99b2801L |s2cid=91187266 }}</ref>
|  ||  ||  ||  ||  ||  ||  || 
|- align="center"
|[[Gaussian free field]]
|  ||  ||  ||  ||  ||  ||  || 
|}

==References==
{{Reflist}}

==External links==
* [http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes  Universality classes] from Sklogwiki
* Zinn-Justin, Jean (2002). ''Quantum field theory and critical phenomena'', Oxford, Clarendon Press (2002), {{ISBN|0-19-850923-5}}
* {{Cite journal|arxiv=cond-mat/0205644|doi=10.1103/RevModPhys.76.663|title=Universality classes in nonequilibrium lattice systems|year=2004|last1=Ódor|first1=Géza|journal=Reviews of Modern Physics|volume=76|issue=3|pages=663–724|bibcode=2004RvMP...76..663O|s2cid=96472311}}
* {{cite journal|arxiv=cond-mat/9701018|doi=10.1088/0305-4470/30/24/036|title=Critical Exponents of the Four-State Potts Model|year=1997|last1=Creswick|first1=Richard J.|last2=Kim|first2=Seung-Yeon|journal=Journal of Physics A: Mathematical and General|volume=30|issue=24|pages=8785–8786|s2cid=16687747}}


[[Category:Critical phenomena]]
[[Category:Renormalization group]]
[[Category:Scale-invariant systems]]