Critical exponent

Critical exponents describe the behaviour 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. The experimental 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. Phase transitions and critical exponents appear also in percolation systems.

Definition

Phase transitions occur at a certain temperature, called the critical temperature \( T_c \) . We want to describe the behaviour of a physical quantity f in terms of a power law around the critical temperature. So we introduce the reduced temperature \( \tau := (T-T_c)/T_c \) , which is zero at the phase transition, and define the critical exponent k.

\( k \, \stackrel{\text{def}}{=} \, \lim_{\tau \to 0}{\log |f(\tau)| \over \log |\tau|} \text{.} \)

This results in the power law we were looking for.

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

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

More generally one might expect

\( f(\tau)=A \tau^k (1+b\tau ^{k_1} + \cdots) \text{.} \)

The most important critical exponents

Above and below \(T_c \) the system has two different phases characterized by an order parameter \( \Psi \), which vanishes at and above \(T_c \).

Let us consider the disordered phase (\( \tau > 0 \)), ordered phase (\( \tau < 0 \) ) and critical temperature (\( \tau = 0 \)) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/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.

Keys
\( \Psi\)	order parameter \( ( \frac{\rho-\rho_c}{\rho_c} \) for the liquid-gas critical point, magnetization for the Curie point,etc.)
\( \tau\)	\( \frac{T-T_c}{T_c} \)
\( f \)	specific free energy
\( C \)	specific heat; \( -T\frac{\partial^2 f}{\partial T^2} \)
\( J \)	ource field (e.g. \( \frac{P-P_c}{P_c} \) where P is the pressure and Pc the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field H for the Curie point )
\( \chi \)	the susceptibility/compressibility/etc.; \( \frac{\partial \Psi}{\partial J} \)
\( \xi\)	correlation length
\( d \)	the number of spatial dimensions
\( \left\langle \psi(\vec{x}) \psi(\vec{y}) \right\rangle \)	the correlation function
r	spatial distance

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

Critical exponents for \( \tau > 0\) > 0 (disordered phase)
Greek letter	relation
\( \alpha \)	\( C \propto \tau^{-\alpha}\)
\( \gamma \)	\( \chi \propto \tau^{-\gamma} \)
\(\nu \)	\( \xi \propto \tau^{-\nu}\)

Critical exponents for \( \tau < 0\) < 0 (ordered phase)
Greek letter	relation
\( \alpha^\prime \)	\( C \propto (-\tau)^{-\alpha^\prime} \)
\( \beta \)	\( \Psi \propto (-\tau)^{\beta}\)
\( \gamma^\prime \)	\( \chi \propto (-\tau)^{-\gamma^\prime}\)
\( \nu^\prime\)	\( \xi \propto (-\tau)^{-\nu^\prime}\)

Critical exponents for \( \tau = 0\) = 0
\( \delta \)	\( J \propto \Psi^\delta\)
\( \eta \)	\( \left\langle \psi(0) \psi(r) \right\rangle \propto r^{-d+2-\eta} \)

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 theory

The classical Landau theory (aka mean field theory) values for a scalar field are

\( \alpha = \alpha^\prime = 0\, \)
\( \beta = \frac{1}{2}\, \)
\( \gamma = \gamma^\prime = 1\, \)
\( \delta = 3\, \)

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

\( \eta = 0\, \)

\( \nu = \frac{1}{2}\, \)

One of the major discoveries in the study of critical points is that mean field theory is completely wrong in the vicinity of critical points in two and three dimensions. In four dimensions, we have logarithmic corrections.
Experimental values

The most accurately measured value of \( \alpha is −0.0127 \) for the phase transition of superfluid helium (the so-called lambda transition). The value was measured in a satellite to minimize pressure differences in the sample (see here). This result agrees with theoretical prediction obtained by variational perturbation theory (see here or here).
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 \( a^\Delta \) for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.
Scaling relations

\( \alpha \equiv \alpha' \)
\( \gamma \equiv \gamma' \)
\( \nu \equiv \nu' \)

T\(hus, the exponents above and below the critical temperature, respectively, have identical values. This is understandable, since the respective scaling functions, \( f_\pm(k\xi ,\dots) \) , originally defined for \( k\xi \ll 1 \) , should become identical if extrapolated to \( k\xi \gg 1\, \) .

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

\( \nu d = 2 - \alpha = 2\beta + \gamma = \beta(\delta + 1) = \gamma \frac{\delta + 1}{\delta - 1}\, \)

\( 2 - \eta = \frac{\gamma}{\nu} = d \frac{\delta - 1}{\delta + 1} \)

These equations imply that there are only two independent exponents, e.g., \( \,\nu \) and \( \eta\, \) . All this follows from the theory of the renormalization group.
Anisotropy

There are some anisotropic systems where the correlation length is direction dependent.
Multicritical points

More complex behaviour may occur at multicritical points, at the border or on intersections of critical manifolds.
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, \( \tau_{\mathrm{char}} \) , of a system diverges as \( \tau_{\mathrm{char}}\propto \xi^z \) , 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.
Transport properties

Critical exponents also exist for transport quantities like viscosity and heat conductivity.
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 also

Rushbrooke inequality
Widom scaling

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 (Read online at [1])
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

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