 Solution

ID: 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 gives

χ = \partial_{h} m = s^{2 - α - 2Δ} g_{m, \pm}^{'} (x) .

At zero field, on a branch with a finite nonzero susceptibility amplitude, comparison with

χ \sim s^{- γ}

yields

γ = 2Δ - 2 + α = Δ - β .

Consequently

α + 2 β + γ = 2

and

γ = β (δ - 1)

, the Rushbrooke and Widom scaling relations. Regular analytic terms do not change the leading divergent susceptibility.

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