Scaling relations among critical exponents

Algebraic relations (Rushbrooke, Widom, Fisher, hyperscaling) among critical exponents under scaling hypotheses near criticality.

Scaling relations among critical exponents

Scaling relations are constraints among critical exponents that follow from the assumption that near criticality there is a single diverging length scale and that the singular part of thermodynamics is approximately scale invariant.

Scaling hypothesis (one common form)

Assume the singular free energy density satisfies a homogeneity relation

f_s(t,h)=b^{-d}\, f_s(b^{y_t}t,\, b^{y_h}h)

for any scale factor $b>0$ , with relevant exponents $y_t,y_h$ . This perspective is naturally produced by a renormalization-group transformation .

Equivalently, one often writes a reduced scaling form

f_s(t,h)=|t|^{2-\alpha}\,\Phi_\pm\!\left(\frac{h}{|t|^\Delta}\right)

for some scaling function $\Phi_\pm$ and gap exponent $\Delta$ .

Core scaling relations (common set)

These relations are expected to hold in many systems at a continuous transition:

Widom relation

\gamma=\beta(\delta-1).

Rushbrooke relation

\alpha+2\beta+\gamma=2.

Fisher relation (links correlations to susceptibility)

\gamma=\nu(2-\eta).

This uses the definition of $\eta$ from the critical decay of the two-point correlation function and the divergence of the correlation length .

Josephson / hyperscaling relation (dimension-dependent)

2-\alpha=d\,\nu.

This relation can fail when hyperscaling is violated (for instance, above an upper critical dimension, or when dangerous irrelevant variables are present).

Useful derived consequences

Combining Fisher + hyperscaling with standard definitions yields

\beta=\frac{\nu}{2}\,(d-2+\eta),

under the same set of assumptions.