Critical exponent

The control parameter that drives phase 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 the critical temperature T_c. We want to describe the behavior of a physical quantity f in terms of a power law around the critical temperature; we introduce the reduced temperature

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

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

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

This results in the power law we were looking for:

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

It is important to remember that this represents the asymptotic behavior of the function f(τ) as τ → 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 has two different phases characterized by an order parameter Ψ, which vanishes at and above T_c.

Consider the disordered phase (τ > 0), ordered phase (τ < 0) and critical 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 general spontaneous symmetry breaking occurs in the ordered phase.

The following entries are evaluated at J = 0 (except for the δ entry)

The critical exponents can be derived from the specific free energy f(J,T) as a function of the source and temperature. The correlation length can be derived from the functional F[J;T].

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 as a way around this problem.[3]

The classical Landau theory (also known as mean field theory) values of the critical exponents for a scalar field (of which the Ising model is the prototypical example) are given by

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

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

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

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 the upper 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 of α is −0.0127(3) for the phase transition of superfluid helium (the so-called lambda transition). The value was measured on a space shuttle to minimize pressure differences in the sample.[4] This value is in a significant disagreement with the most precise theoretical determinations[5][6][7] coming from high temperature expansion techniques, Monte Carlo methods and the conformal bootstrap.[8]

Unsolved problem in physics: Explain the discrepancy between the experimental and theoretical determinations of the heat capacity critical exponent α for the superfluid transition in Helium-4.[8] (more unsolved problems in physics)

Critical exponents can be evaluated via Monte Carlo simulations 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 the renormalization group. The conformal bootstrap is a more recently developed technique, which has achieved unsurpassed accuracy for the Ising 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 an infrared 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 of a will be equivalent to rescaling operators and source fields by a factor of a^Δ for some Δ. 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′ or γ ≡ 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 exponents γ and γ′ are not identical.[9]

Critical exponents are denoted by Greek letters. They fall into universality classes and obey the scaling relations

${\displaystyle {\begin{aligned}\nu d&=2-\alpha =2\beta +\gamma =\beta (\delta +1)=\gamma {\frac {\delta +1}{\delta -1}}\\2-\eta &={\frac {\gamma }{\nu }}=d{\frac {\delta -1}{\delta +1}}\end{aligned}}}$

These equations imply that there are only two independent exponents, e.g., ν and η. All this follows from the theory of the renormalization group.

There are some anisotropic systems where the correlation length is direction dependent. For percolation see Dayan et al.[10]

Directed percolation can be also regarded as anisotropic percolation. In this case the critical exponents are different and the upper critical dimension is 5.[11]

More complex behavior may occur at multicritical 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 as τ_char ∝ ξ ^z, with a dynamical exponent z. Moreover, the large static universality classes of equivalent models with identical static critical exponents decompose into smaller dynamical universality classes, if one demands that also the dynamical exponents are identical.

The critical exponents can be computed from conformal field theory.

See also anomalous scaling dimension.

Critical exponents also exist for transport quantities like viscosity and heat conductivity. A recent study suggests that critical exponents of percolation play an important role in urban traffic.[12]

Critical exponents also exist for self organized criticality for dissipative systems.

Phase transitions and critical exponents appear also in percolation processes where the concentration of occupied sites or links play the role of temperature. The simplest example is perhaps percolation in a two dimensional square lattice. Sites are randomly occupied withprobability p. For small values of p the occupied sites form only small clusters. At a certain threshold pc a giant cluster is formed, and we have a second-order phase transition.[1][13] See percolation critical exponents. For percolation the critical exponents are different from Ising. For example, in the mean field ${\displaystyle \delta =2}$ for percolation[1] compared to ${\displaystyle \delta =3}$ for Ising. In network theory, the strength of interactions between communities has been found to behave analogous to an external field in magnets near the phase transition or as ghost field in percolation.[14]

• 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); Paperback ISBN 981-02-4658-7
• Toda, M., Kubo, R., N. Saito, Statistical Physics I, Springer-Verlag (Berlin, 1983); Hardcover ISBN 3-540-11460-2
• J.M.Yeomans, Statistical Mechanics of Phase Transitions, Oxford Clarendon Press
• H. E. Stanley Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971
• A. Bunde and S. Havlin (editors), Fractals in Science, Springer, 1995
• A. Bunde and S. Havlin (editors), Fractals and Disordered Systems, Springer, 1996
• 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. Delamotte Critical exponents can be different on the two sides of a transition: A generic mechanism https://arxiv.org/abs/1508.07852