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Ornstein–Zernike form

Phenomenological real- and Fourier-space asymptotics of connected correlations in a phase with finite correlation length, including the Lorentzian small- structure factor.
Ornstein–Zernike form

The Ornstein–Zernike (OZ) form is a standard asymptotic description of how connected correlations decay in a translation-invariant phase with a finite correlation length ξ\xi (see ).

Setup: connected correlations and structure factor

Let G(r)G(r) denote the connected two-point correlation function of an observable/order-parameter field:

G(r)  =  ⟨ϕ(0)ϕ(r)⟩−⟨ϕ⟩2, G(r) \;=\; \langle \phi(0)\phi(r)\rangle - \langle \phi\rangle^2,

as in .

Its Fourier transform (often called the structure factor) is

S(k)  =  ∑reik⋅r G(r)(lattice),S(k)  =  ∫Rdeik⋅r G(r) ddr(continuum), S(k) \;=\; \sum_{r} e^{i k\cdot r}\, G(r) \quad \text{(lattice)}, \qquad S(k) \;=\; \int_{\mathbb{R}^d} e^{i k\cdot r}\, G(r)\, d^dr \quad \text{(continuum)},

matching the notion in . The zero-mode S(0)S(0) is proportional to the susceptibility.

OZ small-kk ansatz (Fourier space)

The core OZ statement is a quadratic expansion of the inverse structure factor near k=0k=0:

S(k)−1  ≈  S(0)−1 (1+(kξ)2)(∣k∣ξ≪1), S(k)^{-1} \;\approx\; S(0)^{-1}\,\bigl(1 + (k\xi)^2\bigr) \qquad (|k|\xi \ll 1),

equivalently

S(k)  ≈  S(0)1+(kξ)2. S(k) \;\approx\; \frac{S(0)}{1 + (k\xi)^2}.

This is the “Lorentzian” line shape: a single correlation length ξ\xi controls the width of S(k)S(k) around k=0k=0.

A common operational definition extracted from this expansion is the second-moment correlation length:

ξ2  =  −12d ∇k2S(k)∣k=0S(0), \xi^2 \;=\; -\frac{1}{2d}\,\frac{\nabla_k^2 S(k)\big|_{k=0}}{S(0)},

when S(k)S(k) is smooth near 00.

Real-space asymptotic implied by OZ

Inverting the Lorentzian form yields exponential decay with an algebraic prefactor. In dd dimensions,

G(r)  ∼  Ar(d−1)/2 e−r/ξ,r→∞, G(r) \;\sim\; \frac{A}{r^{(d-1)/2}}\,e^{-r/\xi}, \qquad r\to\infty,

for some amplitude AA (model- and observable-dependent).

Interpretation:

  • ξ\xi sets the exponential decay scale.
  • The power-law prefactor is the generic large-rr behavior of the inverse Fourier transform of a simple pole/“massive” propagator.

Relation to critical scaling

Near a critical point (see ), the OZ form is often refined to incorporate the anomalous dimension η\eta (see ):

S(k)  =  k−2+η F(kξ), S(k) \;=\; k^{-2+\eta}\,F(k\xi),

for a scaling function FF. At criticality ξ=∞\xi=\infty, this implies

S(k)∼k−2+η,G(r)∼r−(d−2+η). S(k) \sim k^{-2+\eta}, \qquad G(r) \sim r^{-(d-2+\eta)}.

Scaling relations then connect exponents, e.g. γ=(2−η)ν\gamma=(2-\eta)\nu (see ).

Prerequisites