Critical Exponent Glossary

Canonical treatment: Critical Exponents and Scaling owns the derivations, scaling hypotheses, finite-size analysis, quantum extensions, exercises, and references. This page is a notation and definition lookup.

A critical exponent specifies an asymptotic law for a named observable, a named scaling field, and a stated path toward a continuous critical point. Amplitudes and regular backgrounds are not encoded by the exponent.

Primary Exponents

Let $t=(T-T_c)/T_c$ , let $h$ be the ordering field, and let $d$ be the spatial dimension.

Symbol	Defining asymptotic law	Path and qualification
$\alpha$	$C_{\mathrm s}\sim\lvert t\rvert^{-\alpha}$	$h=0$ ; $\alpha=0$ may mean a jump or logarithm
$\beta_{\mathrm{op}}$	$M\sim(-t)^{\beta_{\mathrm{op}}}$	ordered side, $t\to0^-$ , $h=0$
$\gamma$	$\chi\sim\lvert t\rvert^{-\gamma}$	$h=0$
$\delta$	$M\sim\operatorname{sgn}(h)\lvert h\rvert^{1/\delta}$	critical isotherm, $t=0$
$\nu$	$\xi\sim\lvert t\rvert^{-\nu}$	$h=0$
$\eta$	$G(r)\sim r^{-(d-2+\eta)}$	critical point and scaling regime
$z$	$\tau\sim\xi^z$ , or $\Delta\sim\xi^{-z}$	dynamic universality class must be specified

Here $\beta_{\mathrm{op}}$ distinguishes the order-parameter exponent from inverse temperature $\beta=1/(k_{\mathrm B}T)$ .

RG and Auxiliary Symbols

Symbol	Meaning
$y_t$	thermal scaling eigenvalue; $y_t=1/\nu$
$y_h$	ordering-field eigenvalue; $y_h=(d+2-\eta)/2$ under ordinary isotropic scaling
$\omega$	leading positive correction-to-scaling exponent, often from an irrelevant eigenvalue $y_i=-\omega$
$\phi$	crossover exponent; its definition depends on the pair of competing scaling fields
$\psi$	activated-dynamics exponent when $\ln\tau\sim\xi^\psi$ replaces finite- $z$ scaling

Common Scaling Relations

For an ordinary isotropic critical point below its upper critical dimension, without dangerous irrelevant variables,

\alpha+2\beta_{\mathrm{op}}+\gamma=2, \qquad \gamma=\beta_{\mathrm{op}}(\delta-1),

\gamma=\nu(2-\eta), \qquad 2-\alpha=d\nu.

These are conditional consistency relations, not definitions. Mean-field behavior above an upper critical dimension, long-range interactions, anisotropy, disorder, multiple length scales, or dangerous irrelevant variables can modify the assumptions.

Finite-Size and Quantum Lookup

At criticality, representative finite-size forms are

M(L)\sim L^{-\beta_{\mathrm{op}}/\nu}, \qquad \chi(L)\sim L^{\gamma/\nu}, \qquad \Delta(L)\sim L^{-z}.

A dimensionless quantity such as a Binder ratio commonly approaches a scale-invariant value, with crossing drift controlled by irrelevant fields and analytic corrections. At a quantum critical point the thermal length scales as $\xi_\tau\sim\xi^z$ ; the effective classical dimensionality $d+z$ is useful only when the quantum-to-classical mapping and its assumptions apply.

Notation Collisions

Symbol	Possible meanings
$\beta$	inverse temperature or order-parameter exponent
$\eta$	anomalous dimension, viscosity, or infinitesimal regulator
$\delta$	critical-isotherm exponent, Dirac delta, or small variation
$\nu$	correlation-length exponent, frequency, or filling factor
$z$	dynamic exponent, complex coordinate, or fugacity

Always define the symbol locally and state the limiting path.