Critical phenomenon

A critical phenomenon is scale-invariant behavior near a continuous phase transition, described by universal critical exponents.

An order parameter vanishes in a symmetric phase and becomes nonzero when that symmetry is spontaneously broken.

Nonconserved relaxational order-parameter dynamics has ${\partial }_t\psi =-\Gamma \delta F/\delta \psi$ plus thermal noise. Its zero-wavenumber mode may decay.

Conserved relaxational order-parameter dynamics has ${\partial }_t\varphi =M{\nabla }^2(\delta F/\delta \varphi )$ plus conserved thermal noise. Its relaxation rate vanishes as $q^2$ at small wavenumber.

A hydrodynamic mode is a long-wavelength collective mode whose decay rate tends to zero because of a conservation law. Coupling to fast nonconserved variables renormalizes its diffusivity and eigenvector.

Landau theory expands an effective free energy analytically in powers of an order parameter, constrained by symmetry.

Landau-Ginzburg theory supplements the local Landau potential with gradient terms and treats the order parameter as a spatial field.

For a quadratic kernel $K(t,k)\sim t+k^2$, the singular heat-capacity correction is proportional to

For an ordinary quartic Landau transition, the heat-capacity and order-parameter exponents are $\alpha =0$ and $\beta =1/2$. At a tricritical sextic point they are $\alpha =1/2$ and $\beta =1/4$.

A tricritical point is where a line of continuous transitions meets a line of first-order transitions; in a ${\mathbb{Z}}_2$ Landau expansion both quadratic and quartic coefficients vanish and a positive sextic term stabilizes the free energy.

The upper critical dimension is the dimension above which the Gaussian or mean-field fixed point controls critical exponents. It is four for the ordinary Ising critical point and three for its tricritical point.

A Wilsonian renormalization-group step integrates out short-distance modes, rescales coordinates to restore the cutoff, and rescales fields to normalize a chosen kinetic term.

At the Gaussian fixed point all interaction couplings vanish and scaling dimensions follow from the quadratic theory.

Below four dimensions, the scalar quartic theory has an interacting Wilson-Fisher fixed point that controls the Ising universality class.

A linearized renormalization-group eigenvector with positive eigenvalue grows under coarse-graining and is relevant; a negative eigenvalue is irrelevant and a zero eigenvalue is marginal.

Systems belong to one universality class when their long-distance renormalization-group flows approach the same fixed point.

An irrelevant coupling is dangerously irrelevant when setting it to zero makes a scaling function singular or removes the term needed to stabilize the ordered phase.

Homogeneity at a fixed point relates thermodynamic critical exponents to the correlation-length exponent $\nu$ and anomalous dimension $\eta$.

When the singular free energy in one correlation volume is of order one, $\alpha =2-d\nu$. Hyperscaling can fail above the upper critical dimension because of a dangerously irrelevant coupling.

Each broken continuous internal-symmetry generator normally produces a massless Goldstone mode; for a symmetry breaking pattern $G\to H$, their number is $\dim G-\dim H$ in the standard relativistic setting.