Critical exponent

Critical exponents describe the behavior of physical quantities near continuous phase 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

the dimension of the system
the range of the interaction
the spin dimension

These properties of critical exponents are supported by experimental data. Analytical results can be theoretically achieved in mean field theory for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the renormalization group approach. Phase transitions and critical exponents appear also in percolation systems. However, here the critical dimension above which mean field exponents are valid is 6 and higher dimensions.^[1] Mean field critical exponents are also valid for random graphs, such as Erdős–Rényi graphs, which can be regarded as infinite dimensional systems.^[2]

Definition

Phase transitions occur at a certain temperature, 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. So we introduce the reduced temperature

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

which is zero at the phase transition, and define the critical exponent $k$ :

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

This results in the power law we were looking for:

f(\tau )\propto \tau ^{k}\,,\quad \tau \approx 0

It is important to remember that this represents the asymptotic behavior of the function $f (τ)$ as $τ \to 0$ .

More generally one might expect

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

The most important critical exponents

Below $T c$ the system has two different phases characterized by an order parameter $Ψ$ , which vanishes at and above $T c$ .

Let us 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. We have spontaneous symmetry breaking in the ordered phase. So, we will arbitrarily take any solution in the phase.

Definitions
$Ψ$	order parameter (e.g. $ρ - ρ c / ρ c$ for the liquid–gas critical point, magnetization for the Curie point, etc.)
$τ$	$T - T c / T c$
$f$	specific free energy
$C$	specific heat; $- T \partial 2 f / \partial T 2$
$J$	source field (e.g. $P - P c / P c$ where $P$ is the pressure and $P c$ the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field $H$ for the Curie point)
$χ$	the susceptibility, compressibility, etc.; $\partial ψ / \partial J$
$ξ$	correlation length
$d$	the number of spatial dimensions
$⟨ ψ (x \to) ψ (y \to)⟩$	the correlation function
$r$	spatial distance

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

Critical exponents for $τ > 0$ (disordered phase)
Greek letter	relation
$α$	$C \propto τ - α$
$γ$	$χ \propto τ - γ$
$ν$	$ξ \propto τ - ν$

Critical exponents for $τ < 0$ (ordered phase)
Greek letter	relation
$α'$	$C \propto (- τ) - α'$
$β$	$Ψ \propto (- τ) β$
$γ'$	$χ \propto (- τ) - γ'$
$ν'$	$ξ \propto (- τ) - ν'$

Critical exponents for $τ = 0$
Greek letter	relation
$δ$	$J \propto Ψ δ$
$η$	$⟨ ψ (0) ψ (r)⟩ \propto r - d + 2 - η$

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. This problem does not appear in 3.99 dimensions, though.

Mean field critical exponents of Ising-like systems

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

\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

\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 four or higher (which unfortunately excludes many of the experimentally relevant cases). This dimension is called the upper critical dimension. The problem with mean field theory is that the critical exponents do not depend on the space dimension. This leads to a quantitative discrepancy in space dimensions 2 and 3, where the true critical exponents differ from the mean field values. It leads to a qualitative discrepancy in space dimension 1, where a critical point in fact no longer exists, even though mean field theory still predicts there is one. The space dimension where mean field theory becomes qualitatively incorrect is called the lower critical dimension.

Experimental values

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.^[3] This value is in a significant disagreement with the most precise theoretical determination by a combination of Monte Carlo and high temperature expansion techniques. Other techniques give results in agreement in the experiment but are less precise.^[4]

Scaling functions

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.

Scaling relations

It was believed for a long time that the critical exponents were the same above and below the critical temperature, e.g. $α \equiv α'$ or $γ \equiv γ'$ . 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.^[5]

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

{\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.

Anisotropy

There are some anisotropic systems where the correlation length is direction dependent. For percolation see Dayan et al.^[6]

Directed percolation can be also regarded as anisotropic percolation. In this case the critical exponents are different and the upper critical dimension is 5.^[7]

Multicritical points

More complex behavior may occur at multicritical points, at the border or on intersections of critical manifolds. For a simple model with multicritical points see reference.^[8]

Static versus dynamic properties

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 $τ char \propto ξ z$ , 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. For critical exponents for dynamics in percolation systems see reference.^[1]

The critical exponents can be computed from conformal field theory.

Transport properties

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 city traffic.^[9]

Self-organized criticality

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

Percolation Theory

Phase transitions and critical exponents appear also in percolation processes where the concentration of occupied sites or links play the role of temperature. See percolation critical exponents. For percolation the critical exponents are different from Ising. For example, in the mean field $\delta =2$ for percolation^[1] compared to $\delta =3$ for Ising.

External links and literature

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.
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

References

1 2 3 Bunde, Armin; Havlin, Shlomo (1996). "Percolation I". Fractals and Disordered Systems. Springer, Berlin, Heidelberg. pp. 59–114. doi:10.1007/978-3-642-84868-1_2. ISBN 9783642848704.
↑ Cohen, Reuven; Havlin, Shlomo (2010). "Introduction". Complex Networks: Structure, Robustness and Function. Cambridge University Press. pp. 1–6. doi:10.1017/cbo9780511780356.001. ISBN 9780521841566.
↑ Lipa, J. A.; Nissen, J.; Stricker, D.; Swanson, D.; Chui, T. (2003). "Specific heat of liquid helium in zero gravity very near the lambda point". Physical Review B. 68 (17): 174518. arXiv:cond-mat/0310163. Bibcode:2003PhRvB..68q4518L. doi:10.1103/PhysRevB.68.174518.
↑ Vicari, Ettore (2007). Critical phenomena and renormalization-group ﬂow of multi-parameter Φ⁴ ﬁeld theories (PDF). The XXV International Symposium on Lattice Field Theory, July 30 - August 4, 2007, Regensburg, Germany. p. 7 (Table 2). arXiv:0709.1014v2.
↑ Leonard, F.; Delamotte, B. (2015). "Critical exponents can be different on the two sides of a transition". Phys. Rev. Lett. 115 (20): 200601. arXiv:1508.07852. Bibcode:2015PhRvL.115t0601L. doi:10.1103/PhysRevLett.115.200601.
↑ Dayan, I.; Gouyet, J.F.; Havlin, S. (1991). "Percolation in multi-layered structures". J. Phys. A. 24 (6): L287. Bibcode:1991JPhA...24L.287D. doi:10.1088/0305-4470/24/6/007.
↑ Kinzel, W. (1982). Deutscher, G., ed. "Directed Percolation". Percolation and Processes. Bristol: Adam Hilger Pub. Co.
↑ Majdandzic, A.; Podobnik, B.; Buldyrev, S.V.; Kenett, D.Y.; Havlin, S.; Stanley, H.E. (2014). "Spontaneous recovery in dynamical networks". Nature Physics. 10: 34. Bibcode:2014NatPh..10...34M. doi:10.1038/nphys2819.
↑ Zeng, Guanwen; Li, Daqing; Gao, Liang; Gao, Ziyou; Havlin, Shlomo (2017-09-10). "Switch of critical percolation modes in dynamical city traffic". arXiv:1709.03134. Bibcode:2017arXiv170903134Z.

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[:0-1] 1 2 3 Bunde, Armin; Havlin, Shlomo (1996). "Percolation I". Fractals and Disordered Systems. Springer, Berlin, Heidelberg. pp. 59–114. doi:10.1007/978-3-642-84868-1_2. ISBN 9783642848704.

[2] Cohen, Reuven; Havlin, Shlomo (2010). "Introduction". Complex Networks: Structure, Robustness and Function. Cambridge University Press. pp. 1–6. doi:10.1017/cbo9780511780356.001. ISBN 9780521841566.

[3] Lipa, J. A.; Nissen, J.; Stricker, D.; Swanson, D.; Chui, T. (2003). "Specific heat of liquid helium in zero gravity very near the lambda point". Physical Review B. 68 (17): 174518. arXiv:cond-mat/0310163. Bibcode:2003PhRvB..68q4518L. doi:10.1103/PhysRevB.68.174518.

[4] Vicari, Ettore (2007). Critical phenomena and renormalization-group ﬂow of multi-parameter Φ⁴ ﬁeld theories (PDF). The XXV International Symposium on Lattice Field Theory, July 30 - August 4, 2007, Regensburg, Germany. p. 7 (Table 2). arXiv:0709.1014v2.

[5] Leonard, F.; Delamotte, B. (2015). "Critical exponents can be different on the two sides of a transition". Phys. Rev. Lett. 115 (20): 200601. arXiv:1508.07852. Bibcode:2015PhRvL.115t0601L. doi:10.1103/PhysRevLett.115.200601.

[6] Dayan, I.; Gouyet, J.F.; Havlin, S. (1991). "Percolation in multi-layered structures". J. Phys. A. 24 (6): L287. Bibcode:1991JPhA...24L.287D. doi:10.1088/0305-4470/24/6/007.

[7] Kinzel, W. (1982). Deutscher, G., ed. "Directed Percolation". Percolation and Processes. Bristol: Adam Hilger Pub. Co.

[8] Majdandzic, A.; Podobnik, B.; Buldyrev, S.V.; Kenett, D.Y.; Havlin, S.; Stanley, H.E. (2014). "Spontaneous recovery in dynamical networks". Nature Physics. 10: 34. Bibcode:2014NatPh..10...34M. doi:10.1038/nphys2819.

[9] Zeng, Guanwen; Li, Daqing; Gao, Liang; Gao, Ziyou; Havlin, Shlomo (2017-09-10). "Switch of critical percolation modes in dynamical city traffic". arXiv:1709.03134. Bibcode:2017arXiv170903134Z.