Correlation length and scaling hypotheses

Define reduced temperature $t=(T-T_c)/T_c$ and set the conjugate ordering field to zero for thermal exponents. The spontaneous order parameter below the transition behaves as $|t|^\beta$, the susceptibility as $|t|^{-\gamma}$, and correlation length as $|t|^{-\nu}$. On the critical isotherm instead, order scales as the field magnitude to power $1/\delta$. These are different paths in parameter space; replacing one by another changes the question being asked. Below the transition, response refers to a selected phase, with its field removed after the thermodynamic limit.

The heat-capacity exponent alpha characterizes a singular contribution superposed on a generally nonzero analytic background. Alpha zero may mean a finite jump or a logarithmic singularity, and negative alpha allows a finite heat capacity with a singular cusp. Amplitudes can differ on opposite sides of the transition even when exponents agree. An arbitrary smooth function can imitate a power over a narrow interval, so asymptotic ranges and independently located critical parameters are essential to a meaningful exponent estimate.

For a short-range scalar system with one dominant diverging length, the connected correlation has a scaling form: a critical power times a function of separation divided by correlation length. The exponent eta measures the departure from Gaussian spatial decay. Correlations are approximately critical for separations between microscopic spacing and the correlation length, then decay more rapidly outside that range in the disordered phase.

Integrating the connected correlation over d-dimensional space supplies the zero-wavevector susceptibility, apart from the declared thermal-energy and spin normalization. The radial measure contributes $r^{d-1}$, and the correlation contributes $r^{-(d-2+\eta)}$, leaving an integral proportional to $\xi^{2-\eta}$ when that infrared contribution dominates. Combining this with the thermal length divergence gives Fisher's exponent relation. The microscopic contribution remains as a nonsingular background rather than a new divergent power.

If one correlated volume contributes an order-one amount to the singular dimensionless free energy, then its density scales as $\xi^{-d}$. This gives $2-\alpha=d\nu$, called hyperscaling. It is not merely another algebraic rewriting of the thermal homogeneity assumption. It uses the additional claim that the correlation length controls the singular free-energy volume scale without an extra dangerous coupling or length.

Above the quartic model's upper critical dimension, ordinary hyperscaling fails even when several other exponent identities remain valid. At the upper critical dimension logarithmic factors modify naive pure powers. A Berezinskii–Kosterlitz–Thouless transition instead has an essential length singularity on its disordered side, so the usual finite nu power-law description is inappropriate. Universality concerns asymptotic classes specified by symmetry, dimension, range, and relevant perturbations, not a universal power law shared by every transition.