 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2022/3/c/solution

The self-consistency equation requires that the mean spin computed using the single-site Boltzmann distribution equals the spin

S

used to form the effective field. With

h = 4 J S + B

⟨ s ⟩_{h} = \frac{\sum _{s = - 1, 0, 1} s e ^{β h s}}{\sum _{s = - 1, 0, 1} e ^{β h s}} = \frac{e ^{β h} - e ^{- β h}}{1 + e ^{β h} + e ^{- β h}} .

Thus

S = \frac{2 sinh [ β ( 4 J S + B )]}{1 + 2 cosh [ β ( 4 J S + B )]} .

Equivalently, differentiating part (b) gives

N^{- 1} \partial F_{MF} / \partial S = 4 J [S - ⟨ s ⟩_{h}]

, so stationarity of the Helmholtz free energy in the mean-field approximation is exactly this condition. Among multiple stationary points one must select the stable minimum.

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