= Solution
The <scaling hypothesis for critical phenomena> says that the singular part of the <free-energy density> near a continuous <thermodynamic critical point> becomes a generalized homogeneous function of the relevant thermal and magnetic variables. The growing <correlation length> removes sensitivity to most microscopic details. Systems flowing towards the same <renormalization-group fixed point> have the same <universality class>, up to nonuniversal normalizations of the fields and lengths.
A change of length scale by $b>1$ gives, when ordinary <hyperscaling> holds,
$$
f_s(t,h)=b^{-d}f_s(b^{y_t}t,b^{y_h}h).
$$
Choosing $b=|t|^{-1/y_t}$ gives $f_s=|t|^{d/y_t}g_\pm(h/|t|^{y_h/y_t})$. More generally the specified homogeneous form uses exponents $2-\alpha$ and $\Delta$ without presupposing <hyperscaling>. Its thermodynamic derivatives determine the <critical exponents>. Analytic backgrounds are separated from this singular part.
Throughout the calculation, fractional powers of the thermal variable mean $|t|$, with different scaling functions above and below the transition. The spontaneous <magnetization> exponent uses the ordered branch $t<0$ with $h\to0^+$ after the <thermodynamic limit>. Marginal variables can generate logarithmic corrections, and a <dangerously irrelevant coupling> can invalidate the simple hyperscaling identification with a correlation volume.
Back to article page
