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 (ie 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 ${\displaystyle r^{-d+2-\eta }}$.

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, ${\displaystyle Sym_{n}}$ 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.

dimension	Symmetries	${\displaystyle \alpha }$	${\displaystyle \beta }$	${\displaystyle \gamma }$	${\displaystyle \delta }$	${\displaystyle \nu }$	${\displaystyle \eta }$	class
2	${\displaystyle Sym_{3}}$	1/3	1/9	13/9		5/6		3-state Potts
2	${\displaystyle Sym_{4}}$	2/3	1/12	7/6		2/3		Ashkin-Teller (4-state Potts)
1	1		0	1		1		Ordinary percolation
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)		0.689(10)	−0.0944(28)
5	1		0.830(10)	1.185(5)		0.569(5)
6+	1	−1	1	1	2	1/2	0
1	1	0.159464(6)	0.276486(8)	2.277730(5)	0.159464(6)	1.096854(4)	0.313686(8)	Directed percolation
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
3	${\displaystyle {\mathcal {O}}(3)}$	−0.12(1)	0.366(2)	1.395(5)		0.707(3)	0.035(2)	Heisenberg
2	${\displaystyle Sym_{2}}$	0	1/8	7/4	15	1	1/4	2D Ising
3	${\displaystyle Sym_{2}}$	0.11007(7)	0.32653(10)	1.2373(2)	4.7893(8)	0.63012(16)	0.03639(15)	3D Ising
								Local linear interface
all	any	0	1/2	1	3	1/2	0	Mean field
								Molecular beam epitaxy
								Gaussian free field
3	${\displaystyle {\mathcal {O}}(2)}$	−0.0146(8)	0.3485(2)	1.3177(5)	4.780(2)	0.67155(27)	0.0380(4)	XY

• 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.