Universality class

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 ${\displaystyle \tau }$, its order parameter measuring how much of the system is in the "ordered" phase, the specific heat, and so on.

• The exponent ${\displaystyle \alpha }$ is the exponent relating the specific heat C to the reduced temperature: we have ${\displaystyle C=\tau ^{-\alpha }}$. The specific heat will usually be singular at the critical point, but the minus sign in the definition of ${\displaystyle \alpha }$ allows it to remain positive.
• The exponent ${\displaystyle \beta }$ relates the order parameter ${\displaystyle \Psi }$ 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 ${\displaystyle \Psi =|\tau |^{\beta }}$.
• The exponent ${\displaystyle \gamma }$ relates the temperature with the system's response to an external driving force, or source field. We have ${\displaystyle d\Psi /dJ=\tau ^{-\gamma }}$, with J the driving force.
• The exponent ${\displaystyle \delta }$ relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have ${\displaystyle J=\Psi ^{\delta }}$ (hence ${\displaystyle \Psi =J^{1/\delta }}$), with the same meanings as before.
• The exponent ${\displaystyle \nu }$ 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 correlation length ${\displaystyle \xi }$. We have ${\displaystyle \xi =\tau ^{-\nu }}$.
• The exponent ${\displaystyle \eta }$ measures the size of correlations at the critical temperature. It is defined so that the correlation function scales as ${\displaystyle r^{-d+2-\eta }}$.
• The exponent ${\displaystyle \sigma }$, 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 ${\displaystyle s_{max}\sim (p_{c}-p)^{-1/\sigma }}$.
• The exponent ${\displaystyle \tau }$, also from percolation theory, measures the number of size s clusters far from ${\displaystyle s_{max}}$ (or the number of clusters at criticality): ${\displaystyle n_{s}\sim s^{-\tau }f(s/s_{max})}$, with the ${\displaystyle f}$ factor removed at critical probability.

For symmetries, the group listed gives the symmetry of the order parameter. The group ${\displaystyle Dih_{n}}$ is the dihedral group, the symmetry group of the n-gon, ${\displaystyle S_{n}}$ is the n-element symmetric group, ${\displaystyle Oct}$ is the octahedral group, and ${\displaystyle O(n)}$ is the orthogonal group in n dimensions. 1 is the trivial group.

class	dimension	Symmetry	${\displaystyle \alpha }$	${\displaystyle \beta }$	${\displaystyle \gamma }$	${\displaystyle \delta }$	${\displaystyle \nu }$	${\displaystyle \eta }$
3-state Potts	2	${\displaystyle S_{3}}$	1/3	1/9	13/9	14	5/6	4/15
Ashkin–Teller (4-state Potts)	2	${\displaystyle S_{4}}$	2/3	1/12	7/6	15	2/3	1/4
Ordinary percolation	1	1	1	0	1	${\displaystyle \infty }$	1	1
	2	1	−2/3	5/36	43/18	91/5	4/3	5/24
	3	1	−0.625(3)	0.4181(8)	1.793(3)	5.29(6)	0.87619(12)	0.46(8) or 0.59(9)
	4	1	−0.756(40)	0.657(9)	1.422(16)	3.9 or 3.198(6)	0.689(10)	−0.0944(28)
	5	1	≈ −0.85	0.830(10)	1.185(5)	3.0	0.569(5)	−0.075(20) or −0.0565
	6+	1	−1	1	1	2	1/2	0
Directed percolation	1	1	0.159464(6)	0.276486(8)	2.277730(5)	0.159464(6)	1.096854(4)	0.313686(8)
	2	1	0.451	0.536(3)	1.60	0.451	0.733(8)	0.230
	3	1	0.73	0.813(9)	1.25	0.73	0.584(5)	0.12
	4+	1	−1	1	1	2	1/2	0
Conserved directed percolation (Manna, or "local linear interface")	1	1		0.28(1)		0.14(1)	1.11(2)[1]	0.34(2)[1]
	2	1		0.64(1)	1.59(3)	0.50(5)	1.29(8)	0.29(5)
	3	1		0.84(2)	1.23(4)	0.90(3)	1.12(8)	0.16(5)
	4+	1		1	1	1	1	0
Protected percolation	2	1		5/41[2]	86/41[2]
	3	1		0.28871(15)[2]	1.3066(19)[2]
Ising	2	${\displaystyle \mathbb {Z} _{2}}$	0	1/8	7/4	15	1	1/4
	3	${\displaystyle \mathbb {Z} _{2}}$	0.11008(1)	0.326419(3)	1.237075(10)	4.78984(1)	0.629971(4)	0.036298(2)
XY	3	${\displaystyle O(2)}$	-0.01526(30)	0.34869(7)	1.3179(2)	4.77937(25)	0.67175(10)	0.038176(44)
Heisenberg	3	${\displaystyle O(3)}$	−0.12(1)	0.366(2)	1.395(5)		0.707(3)	0.035(2)
Mean field	all	any	0	1/2	1	3	1/2	0
Molecular beam epitaxy[3]
Gaussian free field

• Universality classes from Sklogwiki
• Zinn-Justin, Jean (2002). Quantum field theory and critical phenomena, Oxford, Clarendon Press (2002), ISBN 0-19-850923-5
• Ódor, Géza (2004). "Universality classes in nonequilibrium lattice systems". Reviews of Modern Physics. 76 (3): 663–724. arXiv:cond-mat/0205644. Bibcode:2004RvMP...76..663O. doi:10.1103/RevModPhys.76.663. S2CID 96472311.
• Creswick, Richard J.; Kim, Seung-Yeon (1997). "Critical Exponents of the Four-State Potts Model". Journal of Physics A: Mathematical and General. 30 (24): 8785–8786. arXiv:cond-mat/9701018. doi:10.1088/0305-4470/30/24/036. S2CID 16687747.