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<h2 id="list-of-critical-exponents">List of critical exponents</h2> <p>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 <a href="/facts/Reduced_temperature/0saS6U1I">reduced temperature</a> τ {\displaystyle \tau } , its <a href="/facts/Order_parameter/eLP2BY8R">order parameter</a> measuring how much of the system is in the "ordered" phase, the <a href="/facts/Specific_heat/9P0U6YgI">specific heat</a>, and so on. </p> <ul><li>The exponent α {\displaystyle \alpha } is the exponent relating the specific heat C to the reduced temperature: we have C = τ − α {\displaystyle C=\tau ^{-\alpha }} . The specific heat will usually be singular at the critical point, but the minus sign in the definition of α {\displaystyle \alpha } allows it to remain positive.</li> <li>The exponent β {\displaystyle \beta } relates the order parameter Ψ {\displaystyle \Psi } 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 Ψ = | τ | β {\displaystyle \Psi =|\tau |^{\beta }} .</li> <li>The exponent γ {\displaystyle \gamma } relates the temperature with the system's response to an external driving force, or source field. We have d Ψ / d J = τ − γ {\displaystyle d\Psi /dJ=\tau ^{-\gamma }} , with J the driving force.</li> <li>The exponent δ {\displaystyle \delta } relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have J = Ψ δ {\displaystyle J=\Psi ^{\delta }} (hence Ψ = J 1 / δ {\displaystyle \Psi =J^{1/\delta }} ), with the same meanings as before.</li> <li>The exponent ν {\displaystyle \nu } 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 <a href="/facts/Correlation_length/8mmhdp8J">correlation length</a> ξ {\displaystyle \xi } . We have ξ = τ − ν {\displaystyle \xi =\tau ^{-\nu }} .</li> <li>The exponent η {\displaystyle \eta } measures the size of correlations at the critical temperature. It is defined so that the <a href="/facts/Correlation_function/8mmhdp8J">correlation function</a> scales as r − d + 2 − η {\displaystyle r^{-d+2-\eta }} .</li> <li>The exponent σ {\displaystyle \sigma } , used in <a href="/facts/Percolation_theory/Bop4vG7u">percolation theory</a>, measures the size of the largest clusters (roughly, the largest ordered blocks) at 'temperatures' (connection probabilities) below the critical point. So s max ∼ ( p c − p ) − 1 / σ {\displaystyle s_{\max }\sim (p_{c}-p)^{-1/\sigma }} .</li> <li>The exponent τ {\displaystyle \tau } , also from <a href="/facts/Percolation_theory/Bop4vG7u">percolation theory</a>, measures the number of size <i>s</i> clusters far from s max {\displaystyle s_{\max }} (or the number of clusters at criticality): n s ∼ s − τ f ( s / s max ) {\displaystyle n_{s}\sim s^{-\tau }f(s/s_{\max })} , with the f {\displaystyle f} factor removed at critical probability.</li></ul> <p>For symmetries, the group listed gives the symmetry of the order parameter. The group D i h n {\displaystyle \mathrm {Dih} _{n}} is the <a href="/facts/Dihedral_group/1PUlRh4M">dihedral group</a>, the symmetry group of the <i>n</i>-gon, S n {\displaystyle S_{n}} is the <i>n</i>-element <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a>, O c t {\displaystyle \mathrm {Oct} } is the <a href="/facts/Octahedral_group/qLqhYxgm">octahedral group</a>, and O ( n ) {\displaystyle O(n)} is the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> in <i>n</i> dimensions. 1 is the <a href="/facts/Trivial_group/XPjQkQFm">trivial group</a>. </p> <table><tbody><tr><th>class</th><th>dimension</th><th>Symmetry</th><th> α {\displaystyle \alpha } </th><th> β {\displaystyle \beta } </th><th> γ {\displaystyle \gamma } </th><th> δ {\displaystyle \delta } </th><th> ν {\displaystyle \nu } </th><th> η {\displaystyle \eta } </th></tr><tr><td>3-state <a href="/facts/Potts_model/JlSQMYH1">Potts</a></td><td>2</td><td> S 3 {\displaystyle S_{3}} </td><td>1/3</td><td>1/9</td><td>13/9</td><td>14</td><td>5/6</td><td>4/15</td></tr><tr><td>Ashkin–Teller (4-state Potts)</td><td>2</td><td> S 4 {\displaystyle S_{4}} </td><td>2/3</td><td>1/12</td><td>7/6</td><td>15</td><td>2/3</td><td>1/4</td></tr><tr><td rowspan="6"><a href="/facts/Percolation_critical_exponents/IzTN4b0N">Ordinary percolation</a></td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td><td> ∞ {\displaystyle \infty } </td><td>1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>−2/3</td><td>5/36</td><td>43/18</td><td>91/5</td><td>4/3</td><td>5/24</td></tr><tr><td>3</td><td>1</td><td>−0.625(3)</td><td>0.4181(8)</td><td>1.793(3)</td><td>5.29(6)</td><td>0.87619(12)</td><td>0.46(8) or 0.59(9)</td></tr><tr><td>4</td><td>1</td><td>−0.756(40)</td><td>0.657(9)</td><td>1.422(16)</td><td>3.9 or 3.198(6)</td><td>0.689(10)</td><td>−0.0944(28)</td></tr><tr><td>5</td><td>1</td><td>≈ −0.85</td><td>0.830(10)</td><td>1.185(5)</td><td>3.0</td><td>0.569(5)</td><td>−0.075(20) or −0.0565</td></tr><tr><td>6+</td><td>1</td><td>−1</td><td>1</td><td>1</td><td>2</td><td>1/2</td><td>0</td></tr><tr><td rowspan="4"><a href="/facts/Directed_percolation/xvRnByH7">Directed percolation</a></td><td>1</td><td>1</td><td>0.159464(6)</td><td>0.276486(8)</td><td>2.277730(5)</td><td>0.159464(6)</td><td>1.096854(4)</td><td>0.313686(8)</td></tr><tr><td>2</td><td>1</td><td>0.451</td><td>0.536(3)</td><td>1.60</td><td>0.451</td><td>0.733(8)</td><td>0.230</td></tr><tr><td>3</td><td>1</td><td>0.73</td><td>0.813(9)</td><td>1.25</td><td>0.73</td><td>0.584(5)</td><td>0.12</td></tr><tr><td>4+</td><td>1</td><td>−1</td><td>1</td><td>1</td><td>2</td><td>1/2</td><td>0</td></tr><tr><td rowspan="4">Conserved directed percolation (Manna, or "local linear interface")</td><td>1</td><td>1</td><td></td><td>0.28(1)</td><td></td><td>0.14(1)</td><td>1.11(2)<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></td><td>0.34(2)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></td></tr><tr><td>2</td><td>1</td><td></td><td>0.64(1)</td><td>1.59(3)</td><td>0.50(5)</td><td>1.29(8)</td><td>0.29(5)</td></tr><tr><td>3</td><td>1</td><td></td><td>0.84(2)</td><td>1.23(4)</td><td>0.90(3)</td><td>1.12(8)</td><td>0.16(5)</td></tr><tr><td>4+</td><td>1</td><td></td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td></tr><tr><td rowspan="2">Protected percolation</td><td>2</td><td>1</td><td></td><td>5/41<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></td><td>86/41<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></td><td></td><td></td><td></td></tr><tr><td>3</td><td>1</td><td></td><td>0.28871(15)<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></td><td>1.3066(19)<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></td><td></td><td></td><td></td></tr><tr><td rowspan="2"><a href="/facts/Ising_critical_exponents/yuafGFnU">Ising</a></td><td>2</td><td> Z 2 {\displaystyle \mathbb {Z} _{2}} </td><td>0</td><td>1/8</td><td>7/4</td><td>15</td><td>1</td><td>1/4</td></tr><tr><td>3</td><td> Z 2 {\displaystyle \mathbb {Z} _{2}} </td><td>0.11008(1)</td><td>0.326419(3)</td><td>1.237075(10)</td><td>4.78984(1)</td><td>0.629971(4)</td><td>0.036298(2)</td></tr><tr><td><a href="/facts/XY_model/KT2GB1qW">XY</a></td><td>3</td><td> O ( 2 ) {\displaystyle O(2)} </td><td>-0.01526(30)</td><td>0.34869(7)</td><td>1.3179(2)</td><td>4.77937(25)</td><td>0.67175(10)</td><td>0.038176(44)</td></tr><tr><td><a href="/facts/Heisenberg_model_(classical)/RpyLfhNX">Heisenberg</a></td><td>3</td><td> O ( 3 ) {\displaystyle O(3)} </td><td>−0.12(1)</td><td>0.366(2)</td><td>1.395(5)</td><td></td><td>0.707(3)</td><td>0.035(2)</td></tr><tr><td><a href="/facts/Mean-field_theory/tJrwSURE">Mean field</a></td><td>all</td><td>any</td><td>0</td><td>1/2</td><td>1</td><td>3</td><td>1/2</td><td>0</td></tr><tr><td><a href="/facts/Molecular_beam_epitaxy/zIhe0lMv">Molecular beam epitaxy</a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Gaussian_free_field/QHSvpJjn">Gaussian free field</a></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes">Universality classes</a> from Sklogwiki</li> <li>Zinn-Justin, Jean (2002). <i>Quantum field theory and critical phenomena</i>, Oxford, Clarendon Press (2002), <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-850923-5</li> <li>Ódor, Géza (2004). "Universality classes in nonequilibrium lattice systems". <i>Reviews of Modern Physics</i>. 76 (3): 663–724. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/cond-mat/0205644">cond-mat/0205644</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004RvMP...76..663O">2004RvMP...76..663O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FRevModPhys.76.663">10.1103/RevModPhys.76.663</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:96472311">96472311</a>.</li> <li>Creswick, Richard J.; Kim, Seung-Yeon (1997). "Critical Exponents of the Four-State Potts Model". <i>Journal of Physics A: Mathematical and General</i>. 30 (24): 8785–8786. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/cond-mat/9701018">cond-mat/9701018</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0305-4470%2F30%2F24%2F036">10.1088/0305-4470/30/24/036</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16687747">16687747</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fajardo, Juan A. B. (2008). Universality in Self-Organized Criticality (PDF). Granada.{{cite book}}: CS1 maint: location missing publisher (link) <a href="http://hera.ugr.es/tesisugr/17706312.pdf" target="_blank">http://hera.ugr.es/tesisugr/17706312.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Fajardo, Juan A. B. (2008). Universality in Self-Organized Criticality (PDF). Granada.{{cite book}}: CS1 maint: location missing publisher (link) <a href="http://hera.ugr.es/tesisugr/17706312.pdf" target="_blank">http://hera.ugr.es/tesisugr/17706312.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Fayfar, Sean; Bretaña, Alex; Montfrooij, Wouter (2021-01-15). "Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems". Journal of Physics Communications. 5 (1): 015008. arXiv:2008.08258. Bibcode:2021JPhCo...5a5008F. doi:10.1088/2399-6528/abd8e9. ISSN 2399-6528. <a href="https://doi.org/10.1088%2F2399-6528%2Fabd8e9" target="_blank">https://doi.org/10.1088%2F2399-6528%2Fabd8e9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Fayfar, Sean; Bretaña, Alex; Montfrooij, Wouter (2021-01-15). "Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems". Journal of Physics Communications. 5 (1): 015008. arXiv:2008.08258. Bibcode:2021JPhCo...5a5008F. doi:10.1088/2399-6528/abd8e9. ISSN 2399-6528. <a href="https://doi.org/10.1088%2F2399-6528%2Fabd8e9" target="_blank">https://doi.org/10.1088%2F2399-6528%2Fabd8e9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Fayfar, Sean; Bretaña, Alex; Montfrooij, Wouter (2021-01-15). "Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems". Journal of Physics Communications. 5 (1): 015008. arXiv:2008.08258. Bibcode:2021JPhCo...5a5008F. doi:10.1088/2399-6528/abd8e9. ISSN 2399-6528. <a href="https://doi.org/10.1088%2F2399-6528%2Fabd8e9" target="_blank">https://doi.org/10.1088%2F2399-6528%2Fabd8e9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Fayfar, Sean; Bretaña, Alex; Montfrooij, Wouter (2021-01-15). "Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems". Journal of Physics Communications. 5 (1): 015008. arXiv:2008.08258. Bibcode:2021JPhCo...5a5008F. doi:10.1088/2399-6528/abd8e9. ISSN 2399-6528. <a href="https://doi.org/10.1088%2F2399-6528%2Fabd8e9" target="_blank">https://doi.org/10.1088%2F2399-6528%2Fabd8e9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Luis, Edwin; de Assis, Thiago; Ferreira, Silvio; Andrade, Roberto (2019). "Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one-dimension". Physical Review E. 99 (2): 022801. arXiv:1812.03114. Bibcode:2019PhRvE..99b2801L. doi:10.1103/PhysRevE.99.022801. PMID 30934348. S2CID 91187266. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>