When

b = κ_{1} = 0

, stationarity in

ϕ

gives

\frac{δ F}{δ ϕ} = a ϕ + \frac{c}{2} p^{2} = 0, ϕ = - \frac{c}{2 a} p^{2} .

Substituting this value, or completing the square in the Gaussian integral over

ϕ

, gives

F [p] = \int [\frac{A}{2} p^{2} + \frac{B}{4} p^{4} + \frac{κ}{2} ∣\nabla p ∣^{2}] d r, B = B - \frac{c ^{2}}{2 a} .

When

A > 0

and the quartic term is negligible, each Fourier mode is Gaussian with the Ornstein--Zernike correlation function

S_{p} (q) = \frac{k _{B} T}{A + κ q ^{2}}

up to the chosen Fourier normalization. Its correlation length is

ξ_{p} = κ / A

ii

Solution

The eliminated field obeys

ϕ = - c p^{2} / (2 a)

, so its connected correlation function is

C_{ϕ} (r) = \frac{c ^{2}}{4 a ^{2}} [⟨ p^{2} (0) p^{2} (r)⟩ - ⟨ p^{2} ⟩^{2}] .

For the zero-mean Gaussian field of part (i), Wick theorem gives

C_{ϕ} (r) = \frac{c ^{2}}{2 a ^{2}} C_{p} (r)^{2} .

It is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if

C_{p} (r)

has exponential factor

e^{- r / ξ_{p}}

, then

C_{ϕ} (r)

has

e^{- 2 r / ξ_{p}}

ξ_{p} /2

, with the algebraic prefactor also squared.

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