Critical exponents describe the behavior of physical quantities near continuousphase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems at thermal equilibrium, the critical exponents depend only on:

• the dimension of the system
• the range of the interaction
• thespin dimension

These properties of critical exponents are supported by experimental data. Analytical results can be theoretically achieved inmean field theory in high dimensions or when exact solutions are known such as the two-dimensionalIsing model. The theoretical treatment in generic dimensions requires therenormalization group approach or, for systems at thermal equilibrium, theconformal bootstrap techniques.Phase transitions and critical exponents appear in many physical systems such as water at thecritical point, in magnetic systems, in superconductivity, in percolation and in turbulent fluids.The critical dimension above which mean field exponents are valid varies with the systems and can even be infinite.

The control parameter that drivesphase transitions is often temperature but can also be other macroscopic variables like pressure or an external magnetic field. For simplicity, the following discussion works in terms of temperature; the translation to another control parameter is straightforward. The temperature at which the transition occurs is called thecritical temperatureT_c. We want to describe the behavior of a physical quantityf in terms of apower law around the critical temperature; we introduce thereduced temperature

${\displaystyle \tau :=\frac{T-T_{\mathrm{c}}}{T_{\mathrm{c}}}}$

which is zero at thephase transition, and define the critical exponent${\displaystyle k}$ as:

${\displaystyle k\stackrel{\text{def}}{=}\underset{\tau \to 0}{lim}\frac{\log |f(\tau )|}{\log |\tau |}}$

This results in the power law we were looking for:

${\displaystyle f(\tau )\propto {\tau }^k,\tau \to 0}$

It is important to remember that this represents the asymptotic behavior of the functionf(τ) asτ → 0.

More generally one might expect

${\displaystyle f(\tau )=A{\tau }^k\left(1+b{\tau }^{k_1}+\cdots \right)}$

Let us assume that the system at thermal equilibrium has two different phases characterized by anorder parameterΨ, which vanishes at and aboveT_c.

Consider thedisordered phase (τ > 0),ordered phase (τ < 0) andcritical temperature (τ = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It is also another standard convention to use superscript/subscript + (−) for the disordered (ordered) state. In generalspontaneous symmetry breaking occurs in the ordered phase.

The following entries are evaluated atJ = 0 (except for theδ entry)

The critical exponents can be derived from the specific free energyf(J,T) as a function of the source and temperature. The correlation length can be derived from thefunctionalF[J;T]. In many cases, the critical exponents defined in the ordered and disordered phases are identical.

When the upper critical dimension is four, these relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors. These do not appear in dimensions arbitrarily close to but not exactly four, which can be used asa way around this problem.^[1]

The classicalLandau theory (also known asmean field theory) values of the critical exponents for a scalar field (of which theIsing model is the prototypical example) are given by

${\displaystyle \alpha ={\alpha }^{\mathrm{\prime }}=0,\beta =\frac{1}{2},\gamma ={\gamma }^{\mathrm{\prime }}=1,\delta =3}$

If we add derivative terms turning it into a mean fieldGinzburg–Landau theory, we get

${\displaystyle \eta =0,\nu =\frac{1}{2}}$

One of the major discoveries in the study of critical phenomena is that mean field theory of critical points is only correct when the space dimension of the system is higher than a certain dimension called theupper critical dimension which excludes the physical dimensions 1, 2 or 3 in most cases. The problem with mean field theory is that the critical exponents do not depend on the space dimension. This leads to a quantitative discrepancy below the critical dimensions, where the true critical exponents differ from the mean field values. It can even lead to a qualitative discrepancy at low space dimension, where a critical point in fact can no longer exist, even though mean field theory still predicts there is one. This is the case for the Ising model in dimension 1 where there is no phase transition. The space dimension where mean field theory becomes qualitatively incorrect is called the lower critical dimension.

The most accurately measured value ofα is −0.0127(3) for the phase transition ofsuperfluidhelium (the so-calledlambda transition). The value was measured on a space shuttle to minimize pressure differences in the sample.^[2] This value is in a significant disagreement with the most precise theoretical determinations^[3][4][5] coming from high temperature expansion techniques,Monte Carlo methods and theconformal bootstrap.^[6]

Critical exponents can be evaluated viaMonte Carlo methods of lattice models. The accuracy of this first principle method depends on the available computational resources, which determine the ability to go to the infinite volume limit and to reduce statistical errors. Other techniques rely on theoretical understanding of critical fluctuations. The most widely applicable technique is therenormalization group. Theconformal bootstrap is a more recently developed technique, which has achieved unsurpassed accuracy for theIsing critical exponents.

In light of the critical scalings, we can reexpress all thermodynamic quantities in terms of dimensionless quantities. Close enough to the critical point, everything can be reexpressed in terms of certain ratios of the powers of the reduced quantities. These are the scaling functions.

The origin of scaling functions can be seen from the renormalization group. The critical point is aninfrared fixed point. In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group. This basically means that rescaling the system by a factor ofa will be equivalent to rescaling operators and source fields by a factor ofa^Δ for someΔ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.

It was believed for a long time that the critical exponents were the same above and below the critical temperature, e.g.α ≡α′ orγ ≡γ′. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then the exponentsγ andγ′ are not identical.^[7]

Critical exponents are denoted by Greek letters. They fall intouniversality classes and obey thescaling andhyperscaling relations

These equations imply that there are only two independent exponents, e.g.,ν andη. All this follows from the theory of therenormalization group.^{[clarification needed]}

Phase transitions and critical exponents also appear inpercolation processes where the concentration of "occupied" sites or links of a lattice are the control parameter of the phase transition (compared to temperature in classical phase transitions in physics). One of the simplest examples is Bernoulli percolation in a two dimensional square lattice. Sites are randomly occupied with probability${\displaystyle p}$. A cluster is defined as a collection of nearest neighbouring occupied sites. For small values of${\displaystyle p}$ the occupied sites form only small local clusters. At thepercolation threshold${\displaystyle p_c\approx 0.5927}$ (also called critical probability) a spanning cluster that extends across opposite sites of the system is formed, and we have a second-order phase transition that is characterized by universal critical exponents.^[8][9] For percolation theuniversality class is different from the Ising universality class. For example, the correlation length critical exponent is${\displaystyle \nu =4/3}$ for 2D Bernoulli percolation compared to${\displaystyle \nu =1}$ for the 2D Ising model. For a more detailed overview, seePercolation critical exponents.

There are someanisotropic systems where the correlation length is direction dependent.

Directed percolation can be also regarded as anisotropic percolation. In this case the critical exponents are different and the upper critical dimension is 5.^[10]

More complex behavior may occur atmulticritical points, at the border or on intersections of critical manifolds. They can be reached by tuning the value of two or more parameters, such as temperature and pressure.

The above examples exclusively refer to the static properties of a critical system. However dynamic properties of the system may become critical, too. Especially, the characteristic time,τ_char, of a system diverges asτ_char ∝ξ^z, with adynamical exponentz. Moreover, the largestatic universality classes of equivalent models with identical static critical exponents decompose into smallerdynamical universality classes, if one demands that also the dynamical exponents are identical.

The equilibrium critical exponents can be computed fromconformal field theory.

See alsoanomalous scaling dimension.

Critical exponents also exist for self organized criticality fordissipative systems.

• Universality class for the numerical values of critical exponents
• Complex networks
• Random graphs
• Rushbrooke inequality
• Widom scaling
• Conformal bootstrap
• Ising critical exponents
• Percolation critical exponents
• Network science
• Percolation theory
• Graph theory

• Hagen Kleinert and Verena Schulte-Frohlinde,Critical Properties of φ⁴-Theories,World Scientific (Singapore, 2001); PaperbackISBN 981-02-4658-7
• Toda, M., Kubo, R., N. Saito,Statistical Physics I, Springer-Verlag (Berlin, 1983); HardcoverISBN 3-540-11460-2
• J.M.Yeomans,Statistical Mechanics of Phase Transitions, Oxford Clarendon Press
• H. E. StanleyIntroduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971
• Universality classes from Sklogwiki
• Zinn-Justin, Jean (2002).Quantum field theory and critical phenomena, Oxford, Clarendon Press (2002),ISBN 0-19-850923-5
• Zinn-Justin, J. (2010)."Critical phenomena: field theoretical approach" Scholarpedia article Scholarpedia, 5(5):8346.
• D. Poland, S. Rychkov, A. Vichi,"The Conformal Bootstrap: Theory, Numerical Techniques, and Applications", Rev.Mod.Phys. 91 (2019) 015002,http://arxiv.org/abs/1805.04405
• F. Leonard and B. DelamotteCritical exponents can be different on the two sides of a transition: A generic mechanism, Phys. Rev. Lett. 115, 200601 (2015),https://arxiv.org/abs/1508.07852,

• Phase transitions
• Critical phenomena
• Renormalization group

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