Scaling relation for critical exponents

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

If magnetic data collapse with independently specified critical exponents $\beta ,\gamma$, the ordinary scaling identities infer $\Delta =\beta +\gamma$, $\delta =1+\gamma /\beta$, and the displayed exponents. The Fisher scaling relation then gives $\eta =2-\gamma /\nu$. These are conditional inferences, not five additional independent measurements. Hyperscaling is needed for the spatial exponents, and subtraction can make small inferred $\alpha$ or $\eta$ unusually sensitive to uncertainty. A slope on linear axes is not a fitted logarithmic exponent.

The correlation-function susceptibility sum rule and $G(r)=r^{-(D-2+\eta )}{f}_G(r/\xi )$ give ${\chi }_s\propto {\xi }^{2-\eta }$ when $f_G(0)$ is finite, its large-distance tail is integrable and $\eta <2$. With correlation-length critical exponent $\nu$, this gives the displayed relation for the magnetic-susceptibility critical exponent. Microscopic distances contribute a regular background rather than the divergent critical power.

A connected correlation function with critical anomalous dimension $\eta$ scales as $G(r)={\xi }^{-(D-2+\eta )}\mathcal{G}(r/\xi )$. If its large-distance shape is $s^{-(D-1)/2}{e}^{-s}$, substitution of $s=r/\xi$ produces the displayed amplitude. The extra ${\xi }^{-\eta }$ relative to the massive Gaussian field correlation tail is necessary for matching at $r$ of order $\xi$. Integrating over space gives magnetic susceptibility proportional to ${\xi }^{2-\eta }$; omitting that factor would instead make the long-distance contribution scale as ${\xi }^2$.

The scaling hypothesis treats the singular equilibrium free-energy density as a generalized homogeneous function of thermal and field-like scaling variables. Choosing the blocking factor to make the thermal variable order one gives $F_s=|t{|}^{D/y_t}{f}_{\pm }(h/|t{|}^{y_h/y_t})$. It concerns the singular contribution after subtracting analytic backgrounds. Dangerously irrelevant couplings or marginal logarithms can qualify this simple two-variable form. Differentiation yields scaling relations for critical exponents.

Differentiate $f_s=|t{|}^{2-\alpha }{g}_{f,\pm }(h/|t{|}^{\Delta })$ with respect to its conjugate field. A finite nonzero one-sided ordered-branch amplitude gives $\beta =2-\alpha -\Delta$. To approach a finite critical isotherm as $t\to 0$ at fixed small $h$, the large-argument function must behave as $g_m(x)\sim C\mathrm{sgn}(x)|x{|}^{1/\delta }$. Cancelling the remaining thermal power yields $\beta =\Delta /\delta$. An analytic background, branch selection and thermodynamic limit must be distinguished from the singular free energy being differentiated.

For a scalar quartic Landau free energy $r_{t}t{m}^2/2+u{m}^4/4-hm$ with $r_t,u>0$, rescaling $m=(r_t/u{)}^{1/2}|t{|}^{1/2}\psi$ gives this minimized scaling form, with $a=r_t^2/u$ and $b=\sqrt{u}/r_t^{3/2}$. The two functions minimize $\pm {\psi }^2/2+{\psi }^4/4-H\psi$. In particular $f_{+}(0)=0$ and $f_{-}(0)=-1/4$. Below the transition the equilibrium field dependence has a cusp at zero; pure-phase derivatives are one-sided. The form gives order-parameter critical exponent $\beta =1/2$ and magnetic-susceptibility critical exponent $\gamma =1$.

The Widom scaling relation isIt follows by differentiating the homogeneous critical equation of state with respect to the external field.

The Rushbrooke scaling relation isIt follows from the homogeneous scaling form of the singular free energy.

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.