Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 72 / 2 / 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,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 $h\to 0^{+}$ 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.