Critical exponents

Definitions of the standard critical exponents and what physical singularities they quantify near a continuous phase transition.

Critical exponents

Critical exponents quantify how thermodynamic observables and correlations become singular as the system approaches a critical point. They are central to universality : many microscopic models share the same exponents.

Let $t=(T-T_c)/T_c$ and let $h$ be the field conjugate to the order parameter (e.g., magnetic field for magnetization).

Standard thermodynamic exponents

Heat capacity exponent $\alpha$

For the constant-volume heat capacity (or its analog),

C \sim |t|^{-\alpha}.

(Connects to heat capacity at constant volume .)

Order parameter exponent $\beta$

For $h=0$ and $t\to 0^-$ (below criticality),

m(t,0)\sim (-t)^\beta.

Susceptibility exponent $\gamma$

For $h=0$ and $t\to 0$ ,

\chi(t,0)=\frac{\partial m}{\partial h}(t,0)\sim |t|^{-\gamma}.

Critical isotherm exponent $\delta$

At $t=0$ (critical temperature),

m(0,h)\sim |h|^{1/\delta}\,\mathrm{sign}(h)\quad (h\to 0).

Correlation exponents

Correlation length exponent $\nu$

The correlation length diverges as

\xi(t,0)\sim |t|^{-\nu}.

Anomalous dimension exponent $\eta$

At criticality, the two-point correlation function typically has power-law decay:

\langle \phi(0)\phi(r)\rangle_c \sim \frac{1}{|r|^{d-2+\eta}} \quad (|r|\to\infty,\ t=0),

for an appropriate local field $\phi$ (e.g., spin component).

Where these come from (organizing principle)

Many exponents can be read from the singular part of the free energy density (see statistical free energy ) and from how coarse-graining changes effective couplings under a renormalization-group transformation .

Standard thermodynamic exponents

Heat capacity exponent α\alphaα

Order parameter exponent β\betaβ

Susceptibility exponent γ\gammaγ

Critical isotherm exponent δ\deltaδ