Critical Exponent Glossary

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. See Critical Exponents and Scaling for the derivations and scaling hypotheses.

Exponents at a thermal critical point

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, $t\to0^-$; choose a broken-symmetry branch after the thermodynamic limit
$\gamma$	$\chi\sim\lvert t\rvert^{-\gamma}$	$h=0$
$\delta$	$M\sim\operatorname{sgn}(h)\lvert h\rvert^{1/\delta}$	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
$z$	$\tau\sim\xi^z$; quantum gap $\Delta\sim\hbar/\tau$	thermal relaxation and quantum excitation exponents need not describe the same dynamics

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

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

In a finite symmetry-preserving state, the signed mean order parameter can vanish identically. The $M(L)$ law then refers to an absolute or root-mean-square estimator with the stated scaling behavior, not that zero mean.

A dimensionless quantity such as a Binder ratio can approach a scale-invariant value, with crossing drift from corrections to scaling. At a quantum critical point, the correlation time scales as $\xi_\tau\sim\xi^z$, while the thermal time extent is $\beta\hbar$. Fixed-aspect scaling keeps their finite-size ratio fixed; it does not automatically eliminate excited-state thermal populations. Ground-state calculations require convergence as $\beta\Delta_L$ becomes sufficiently large.

Under a single-scale quantum hyperscaling hypothesis, replace $d$ by $d+z$ in the free-energy hyperscaling relation. The equal-time order-parameter correlator is then $G(r)\sim r^{-(d+z-2+\eta)}$, not the thermal $d-2+\eta$ form above. The exponent called $\alpha$ in this relation describes a derivative with respect to the relevant tuning field; it need not be the physical temperature-specific-heat exponent at the quantum critical point.

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.

Example: testing hyperscaling

The two-dimensional thermal Ising exponents $\nu=1$, $\eta=1/4$, $\beta_{\mathrm{op}}=1/8$, and $\gamma=7/4$ satisfy $\gamma=\nu(2-\eta)$ and $2\beta_{\mathrm{op}}+\gamma=2$. Its $\alpha=0$ accompanies a logarithmic specific-heat singularity. By contrast, inserting mean-field $\nu=1/2$, $\alpha=0$ into $2-\alpha=d\nu$ works only at $d=4$; ordinary hyperscaling fails above the upper critical dimension.

References

• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996), doi:10.1017/CBO9781316036440.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).