# nLab universality class

Contents

## Surveys, textbooks and lecture notes

This entry is about the phenomenon of universality in physics. See also universal property for a (different) concept of a similar name in mathematics.

# Contents

## Idea

In physics, universality is a term which means the following:

Under the proper conditions, different systems can exhibit the same behaviour, as measured by quantitative indices, if they meet the same qualitative criteria. Sets of systems which are equivalent in this manner are known as universality classes.

If we have some complicated phenomenon we can’t understand directly, and we figure out (or make a good stab at guessing) the universality class to which it belongs, we can make testable predictions about the complicated thing by working with a simpler member of that universality class. Membership in a universality class depends on properties like how many spatial dimensions a system lives in, symmetries and the like. People have identified universality classes, with varying degrees of rigour. Lots of them have names, sometimes even with dashes inside; the ones we understand less well and aren’t so familiar with get abstruse symbols for labels instead. A non-exhaustive tabulation of these labels might look something like this:

Stable DistributionsEquilibriumRandom MatricesNon-EquilibriumExtreme-Value DistributionsDynamical Maps
Gaussian2D IsingUnitaryDirected percolationGumbel1D Feigenbaum
Cauchy2D PottsOrthogonalDynamic percolationFréchet2D Volume-preserving
Lévy2D Tricritical IsingSymplecticCDPWeibull$\vdots$
2D Tricritical Potts$U(N+M)/U(N)\times U(M)$TDP
2D Other RSOS$Sp(N+M)/Sp(N)\times Sp(M)$Manna
3D Ising$Sp(2N)$Edwards–Wilkinson
3D Potts$SO(2N)$KPZ
$\vdots$$O(N+M) / O(N) \times O(M)$$\vdots$
$SO(2N) / U(N)$
$Sp(2N) / U(N)$

The general concept of renormalization is really important here. The universality classes we understand best correspond to fixed points of renormalization-group transforms.

The first column, “stable distributions”, basically comes from the central limit theorem and the ways in which the conditions necessary for it to apply can fail to obtain. The middle column with all the funny symbols comes from Élie Cartan‘s classification of symmetric spaces. The 2D part of the equilibrium statistical systems can be given a taxonomy based on the ADE classification of Dynkin diagrams and conformal field theories. As Cardy (2010) writes,

[T]he study of the representation theory of the Virasoro algebra gives a way of classifying all possible CFTs and thereby universality classes in 2d. The breakthrough in this direction came following the seminal 1984 paper of Belavin, Polyakov and Zamolodchikov (BPZ) in which they showed that, for certain special rational values of $c \lt 1$, the CFT closes with only a finite number of representations of the Virasoro algebra, and, for, these cases, all the critical exponent?s and multi-point correlation functions are calculable. Shortly thereafter Friedan, Qiu and Shenker showed that unitary CFTs (corresponding to local, positive definite Boltzmann weights) are a subset of this list, with $c = 1 - 6/m(m + 1)$ and $m$ an integer $\geq 3$. This gives rise to what might be termed the ‘conformal periodic table’. The first few examples may be identified with well-known universality classes. The ‘hydrogen atom’ of CFT is the scaling limit of the critical Ising model, ‘helium’ is the tricritical Ising model, and so on. Note, however, that at the next value of $c = 4/5$ two possible ‘isotopes’ arise. In the second, corresponding to the critical 3-state Potts model, not all the scaling dimensions allowed by BPZ in fact occur, but some of those that do actually appear twice. In fact the constraint of unitarity is not sufficient to determine exactly which representations actually occur in a given CFT. The answer to this is provided by demanding consistency of the theory on a torus, by interchanging the interpretations of space and imaginary time []. For the torus, this is a modular transformation, and the requirement of modular invariance has become another powerful tool in classifying CFTs, completely solved in the case $c \lt 1$ by Cappelli, Itzykson and Zuber.

Some relationships between columns can be identified. For example, dynamical surface growth and random matrix theory are, unexpectedly, linked: the eigenvalue distributions of the Gaussian Unitary? and Gaussian Orthogonal Ensembles? show up as surface heights in Kardar-Parisi-Zhang phenomena. See (Takeuchi et al.).

## Timeline

The modern terminology of this subject dates to the late 1960s and early ’70s. By 1968, the quantitative concurrence of critical exponent?s experimentally observed in magnetic materials and fluid phase transitions was sufficiently well established that it was reasonable to ask, “To what extent are the observed values of the critical-point exponents universal?” (Fisher 1968; for a more recent presentation of relevant empirical figures, see Sethna 2006, p. 267). At the 1970 Enrico Fermi summer school in Varenna, Italy, Leo P. Kadanoff summarized recent results with a “hypothesis of universality” which the conference proceedings record as follows:

All phase transition problems can be divided into a small number of different classes depending upon the dimensionality of the system and the symmetries of the order state. Within each class, all phase transitions have identical behaviour in the critical region, only the names of the variables are changed. (Kadanoff 1971)

The following is a capsule history tabulating some important events in the historical development of the modern understanding of universality.

YearContributorsEvent
1733de MoivreSuggestion of what we now know as the central limit theorem
1822Cagniard de la TourObservation of critical point at end of liquid-gas coexistence curve
1873–75van der Waals, MaxwellEquation of state for non-ideal gases, and its application to phase transitions
1893van der WaalsUse of critical exponents to characterize a phase transition (Levelt Sengers 1976)
1895P. CurieExperimental observation of demagnetization by heat; pointing out similarity between liquid-gas and ferromagnetic transitions
1896–1900VerschaffeltDiscovery that the critical exponents predicted from van der Waals theory do not match experiment (Levelt Sengers 1976)
1901LyapunovDemonstration of why the central limit theorem works
1908SmoluchowskiDiscovery that critical opalescence is due to density fluctuations across many scales
1937LandauGeneral mean-field theory of phase transitions
1944OnsagerExact solution of 2D Ising model, illuminating the breakdown of analyticity and providing a valuable comparison point for later approximation techniques
1947–49Feynman, Schwinger, TomonagaWorkable renormalization techniques for “sweeping infinities under the rug” in quantum electrodynamics
1954–56Stueckelberg, Gell-Mann, Bogoliubov et al.Discovery that the “renormalization group” (RG) captured an important formal property of quantum field theory
1955WignerEigenvalue statistics of random matrices
1970–72Kadanoff, Wilson et al.Development of RG concept, discovery that it explains universal properties of phase transitions like critical exponents
1976DoiSecond-quantization formalism for classical stochastic many-particle dynamics
1976–79FeigenbaumObservation and use of RG to explain universal scaling exponents in 1D dynamical maps
1977Hohenberg, HalperinClassification of critical phenomena in dynamic continuum theories
1978–80Grassberger, Sundermeyer, de la Torre, Cardy, SugarConnection of Reggeon field theory? with directed percolation
1980Libchaber et al.Observation of Feigenbaum’s constants in experimental systems, starting with convection of mercury
1981–82Janssen, GrassbergerFormulation of DP conjecture, setting out the likely criteria for phase transitions to belong to the directed-percolation class
1984Belavin, Polyakov, Zamolodchikovconformal field theory in 2D
1986Zamolodchikovc-theorem for 2D theories, identifying RG fixed points with CFTs of specific central charges
1986Kardar, Parisi, ZhangKPZ equation for dynamical surface growth by deposition, showing that RG is applicable to its far-from-equilibrium dynamics
1996–97Zirnbauer, AltlandSymmetry classes (not exactly the same thing as universality classes, but clearly part of the same picture) for random matrices, following Cartan’s classification of symmetric spaces
1998–99Freedman, et al.First indications that the AdS/CFT correspondence and related implementations of gauge/gravity duality could lead to a “holographic $c$-theorem” generalising Zamolodchikov’s work to higher dimensions

General sources:

• T. Tao (2010), “A second draft of a non-technical article on universality” (web)

Statistical distributions:

Equilibrium critical phenomena:

• B. Berche et al. (2009), “Critical phenomena: 150 years since Cagniard de la Tour” J. Phys. Studies 13: 3201, arXiv:0905.1886.
• J. P. Sethna (2006), Statistical Physics: Entropy, Order Parameters, and Complexity. Oxford University Press. Available on the author’s website. See in particular chapter 12.
• Kardar’s Statistical Physics of Particles and Statistical Physics of Fields
• Henkel’s Conformal Invariance and Critical Phenomena (1999)
• P. A. Pearce (1994), “Recent progress in solving A–D–E lattice models” Physica A 205: 15–30. (pdf) This one brings in Temperley-Lieb algebra?s, two-colour braid monoid algebras and Reidemeister moves.
• J. Cardy (2010), The Ubiquitous ‘c’: From the Stefan–Boltzmann Law to Quantum Information, J.Stat.Mech. 2010: P10004, (arXiv:1008.2331)
• A. Cappelli and J.-B. Zuber (2010), A-D-E Classification of Conformal Field Theories Scholarpedia, 5(4): 10314 (web). Also available as arXiv:0911.3242.
• Michael E. Fisher (1968), Renormalization of Critical Exponents by Hidden Variables Physical Review 176: 257–72. (web)
• L. P. Kadanoff (1971), Critical Behavior. Universality and Scaling in Proceedings of the International School of Physics Enrico Fermi, Course LI (27 July – 8 August 1970). Edited by M. S. Green.
• J. M. H. Levelt Sengers (1976), Critical Exponents at the Turn of the Century Physica A 82: 319–51. (web)

Random matrix theory:

• Yan Fyodorov (2011) Random matrix theory. Scholarpedia, 6(3):9886 (web)
• this conversation at the n-Category Café on the Tenfold Way.

Nonequilibrium critical phenomena:

• P. C. Hohenberg and B. I. Halperin (1977). “Theory of dynamic critical phenomena” Rev. Mod. Phys. 49: 435. (web)
• Nonequilibrium Phase Transitions (2008) by Henkel et al.
• Vollmayr-Lee’s talks on the “field theory approach to diffusion-limited reactions”, Boulder School for Condensed Matter and Materials Physics (2009), in particular lecture 4
• G. Odor (2004), “Universality classes in nonequilibrium lattice systems” Rev. Mod. Phys. 76: 663. arXiv:cond-mat/0205644v7
• I. Corwin (2011), “The Kardar-Parisi-Zhang equation and universality class” arXiv:1106.1596.
• Takeuchi et al. Growing interfaces uncover universal fluctuations behind scale invariance Sci Rep. (Nature) 1 34, (arXiv:1108.2118)

Dynamical maps and chaos theory:

• P. Cvitanovic, ed. Universality in Chaos (1983).