Critical exponent

Critical exponents are observed in second-order phase transitions. They characterize the power law behavior of many physical quantities as a function of

\tau \equiv (T-T_{c})/T_c

where $T$ is the temperature and $T_{c}$ its critical value, at which a second-order phase transition is observed. 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 (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-ρ_c)/ρ_c for the liquid-gas critical point, magnetization for the Curie point,etc.)
τ	(T-T_c)/T_c
C	specific heat; $-T\frac{\partial^2 F}{\partial T^2}$
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, 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 ~ ψ^δ
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 diemnsions, however, the power laws are modified by logarithmic factors. This problem does not appear in 3.99 dimensions, though.

α=α'

γ=γ'

ν=ν'

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 $\alpha$ 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). 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 $\beta=\gamma/(\delta-1)$ , $\nu=\gamma/(2-\eta)$ .

The critical exponents can be computed from conformal field theory

External link

Hagen Kleinert, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-76 (Read online at ^*)

Kritischer Exponent

Phase changes | Conformal field theory | Renormalization group

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