Universality class explained

In statistical mechanics, a universality class is a collection of mathematical models which share a single 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 exponents 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

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

• The exponent

\alpha is the exponent relating the specific heat C to the reduced temperature: we have C = \tau ^{-\alpha} . The specific heat will usually be singular at the critical point, but the minus sign in the definition of \alpha allows it to remain positive.

• The exponent

\beta relates the order parameter \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 \Psi = |\tau| ^{\beta} .

• The exponent

\gamma relates the temperature with the system's response to an external driving force, or source field. We have d\Psi/dJ = \tau ^{-\gamma} , with J the driving force.

• The exponent

\delta relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have J = \Psi ^{\delta} (hence \Psi = J ^{1/\delta} ), with the same meanings as before.

• The exponent

\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 \xi . We have \xi = \tau ^{-\nu} .

• The exponent

η measures the size of correlations at the critical temperature. It is defined so that the correlation function scales as r ^{-d+2-η} .

• The exponent

\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 s _{max} \sim (p _{c} - p) ^{-1/\sigma} .

• The exponent

\tau , also from percolation theory, measures the number of size s clusters far from s _{max} (or the number of clusters at criticality): n _{s} \sim s ^{-\tau} f(s/s _{max} ) , with the f factor removed at critical probability.

For symmetries, the group listed gives the symmetry of the order parameter. The group

Dih _{n} is the dihedral group, the symmetry group of the n-gon, S _{n} is the n-element symmetric group, Oct is the octahedral group, and O(n) is the orthogonal group in n dimensions. 1 is the trivial group.

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

External links

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

Notes and References

1. Book: Fajardo . Juan A. B. . Universality in Self-Organized Criticality . 2008 . Granada .
2. 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. 10.1088/2399-6528/abd8e9. 2008.08258 . 2021JPhCo...5a5008F . 2399-6528. free.
3. Luis . Edwin . de Assis . Thiago . Ferreira . Silvio . Andrade . Roberto . Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one-dimension . Physical Review E . 2019 . 99 . 2 . 022801 . 10.1103/PhysRevE.99.022801 . 30934348 . 1812.03114 . 2019PhRvE..99b2801L . 91187266 .