Critical exponent

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Critical exponents describe the behaviour of physical quantities near continuous phase transitions. Remarkable about them is that they are universal, i.e. do not depend on 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 were found in experiments. 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.

Contents

[edit] Definition

Phase transitions occour at a certain temperature, called the critical temperature Tc. 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 τ: = (T − Tc) / Tc, which is zero at the phase transition, and define the critical exponent k.

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

This results in the power law we were looking for.

f(\tau) \propto \tau^k,\ \ \tau\approx 0

[edit] The most important critical exponents

Above and below Tc the system has two different phases characterized by an order parameter Ψ, which vanishes at and above Tc.

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

Keys
Ψ order parameter (ρ − ρc)/ρc for the liquid-gas critical point, magnetization for the Curie point,etc.)
τ (T − Tc)/Tc
C specific heat; -T\frac{\partial^2 F}{\partial T^2}
J source field (e.g. (P − Pc)/Pc where P is the pressure and Pc the critical pressure for the liquid-gas critical point, the magnetic field H for the Curie point )
χ the susceptibility/compressibility/etc.; \frac{\partial \Psi}{\partial J}
ξ correlation length
d the number of spatial dimensions
\left\langle \psi(\vec{x}) \psi(\vec{y}) \right\rangle the correlation function

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 ~ τ−α
γ χ ~ τ−γ
ν ξ ~ τ−ν
Critical exponents for τ < 0 (ordered phase)
Greek letter relation
α' C ~ (−τ)−α'
β Ψ ~ (−τ)β
γ' χ ~ (−τ)−γ'
ν' ξ ~ (−τ)−ν'
Critical exponents for τ = 0
δ J ~ ψδ
η \left\langle \psi(0) \psi(r) \right\rangle \sim r^{-d+2-\eta}

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.

\alpha \equiv \alpha'
\gamma \equiv \gamma'
\nu \equiv \nu'

Thus, the exponents above and below the critical temperature, repectively, 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\,.

The classical (Landau theory aka mean field theory) values are

α = α' = 0
β = 1/2
γ = γ' = 1
δ = 3

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

η = 0
ν = 1 / 2


The most accurately measured value of α 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).

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

\beta\equiv\gamma/(\delta-1),\,
\nu\equiv\gamma/(2-\eta)\,,

and a lot of similar relations, which implies that there are only two independent exponents, e.g., \,\nu and \eta\,. All this follows from the theory of the renormalization group.

[edit] 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_{\, char.}, of a system diverges as \tau_{\,char.}=\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.

[edit] See also

[edit] External links