G_{c} (x) = ⟨ m (x) m (0)⟩ - ⟨ m ⟩^{2} .

(1)

Above the transition at zero field the subtraction is zero; in an ordered phase it is essential. For a dimensionless source coupled to the total magnetization

M = \int d^{d} x m (x)

, differentiating

Z

twice gives

χ = V^{- 1} (⟨ M^{2} ⟩ - ⟨ M ⟩^{2})

. Translation invariance then proves the correlation-function susceptibility sum rule

χ = \int d^{d} x G_{c} (x) .

(2)

Insert the scaling form, use spherical coordinates and set

s = r / ξ

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

(3)

For

η < 2

, finite nonzero

g (0)

and an integrable large-

s

tail, the last integral tends to a finite nonzero constant. Microscopic separations add an analytic response background. Thus the singular magnetic susceptibility obeys

χ_{s} ≍ ξ^{2 - η}

, and at zero field

ξ ≍ ∣ t ∣^{- ν}

gives the Fisher scaling relation

γ = (2 - η) ν .

(4)

A dimensional magnetic field would add a nonsingular inverse-temperature prefactor. Integrating the full unsubtracted ordered-phase correlation function instead would yield a volume-divergent term