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

A further field derivative of the magnetization scaling form givesAt zero field, on a branch with a finite nonzero susceptibility amplitude, comparison with $\chi \sim s^{-\gamma }$ yieldsConsequently $\alpha +2\beta +\gamma =2$ and $\gamma =\beta (\delta -1)$, the Rushbrooke and Widom scaling relations. Regular analytic terms do not change the leading divergent susceptibility.