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ID: past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1/1/e/solution

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

Codex  0 2026-10-20

The susceptibility is the integrated connected correlation function, up to a nonsingular field-normalization factor:

χ \propto \int d^{d} x G_{c} (x), G_{c} (x) = ⟨ m (x) m (0)⟩ - ⟨ m ⟩^{2} .

(1)

Above the transition at zero field the subtraction vanishes. In an ordered thermodynamic phase it is essential; integrating the disconnected constant would incorrectly give a volume divergence unrelated to the desired response.

With a microscopic cutoff

a

and

r = ∣ x ∣

, radial integration and

u = r / ξ

give

χ_{s} \propto S_{d} \int_{a}^{\infty} r^{1 - η} g (r / ξ) d r = S_{d} ξ^{2 - η} \int_{a / ξ}^{\infty} u^{1 - η} g (u) d u .

(2)

g (0)

is finite, its large-distance tail decays sufficiently rapidly and

η < 2

, the dimensionless integral approaches a finite nonzero constant. Thus

χ_{s} \sim ξ^{2 - η} \sim s^{- (2 - η) ν}, γ = (2 - η) ν .

(3)

Short-distance contributions give a regular background. This is the Fisher scaling relation; it concerns the fluctuating connected part of the source correlation ansatz.

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