Past exam of the phase transitions and collective phenomena course of the University of Cambridge / 2012 / Paper 1 / 1 / d

Hyperscaling adds the physical assumption that the singular free energy associated with one correlation volume is of order $k_B{T}_c$:At fixed scaled field $x=h/s^{\Delta }$, this predicts $f_s\sim s^{d\nu }$, whereas the given thermodynamic form has power $2-\alpha$. HenceThe product $g_f(x)[g_{\xi }(x){]}^d$ can be a finite dimensionless function of fixed $x$; it need not be constant. The homogeneity of $\xi$ alone is insufficient: the correlation-volume free-energy assumption supplies the connection. Above an upper critical dimension, a dangerously irrelevant coupling can invalidate this argument; at a marginal dimension logarithms may qualify pure powers.