 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2013/paper-1/6/c/solution

Differentiate the equation of state at fixed temperature:

χ = \frac{1}{a + 3 b m ^{2}} .

Above the transition, zero-field equilibrium has

m = 0

. Below it the stable pure phases have

m^{2} = - a / b

. Thus

χ (t) = \frac{1}{a _{0} t} (t > 0), χ (t) = \frac{1}{2 a _{0} ∣ t ∣} (t < 0), t \to 0^{+} lim \frac{χ ( t )}{χ ( - t )} = 2.

This is the mean-field magnetic-susceptibility amplitude ratio. Below the transition, take a one-sided response in a selected ordered phase: the globally minimizing magnetization switches discontinuously between opposite phases when

h

crosses zero. Its two-sided derivative at precisely zero field is therefore not the finite pure-phase magnetic susceptibility used in the ratio.

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