Questions: Correlation Functions and Spatial Correlations

Question 1

In the disordered phase (T > T_c) of an Ising-like system, what best describes the two-point correlation function G(r) = ⟨σ(0)σ(r)⟩ for large r?

• A. G(r) → m² (the square of the mean magnetization), reflecting long-range order even above T_c
• B. G(r) − m² decays exponentially as e^{−r/ξ}, where ξ is a finite correlation length — spins behave nearly independently beyond a few lattice spacings
• C. G(r) decays as a power law r^{−(d−2+η)}, indicating scale-free correlations throughout the entire disordered phase
• D. G(r) = 0 everywhere in the disordered phase because the mean magnetization is zero

Answer: (B)

Above T_c, the system is disordered (m = 0) and correlations decay exponentially: G(r) ~ exp(−r/ξ), where ξ is the correlation length. Beyond a distance of order ξ, knowing the value of one spin gives essentially no information about a distant spin. The correlation length ξ is small at high temperature and grows as T approaches T_c from above. Power-law decay (option C) is the behavior exactly at T_c — not in the disordered phase. Option D confuses the mean magnetization being zero with the correlation function being identically zero.

Question 2

At exactly the critical temperature T_c, what happens to the correlation length ξ and the functional form of G(r)?

• A. ξ reaches its maximum finite value and G(r) transitions from exponential to linear decay
• B. ξ diverges to infinity and G(r) decays as a power law r^{−(d−2+η)}, with no characteristic length scale — correlations extend over all scales simultaneously
• C. ξ diverges but G(r) still decays exponentially, just arbitrarily slowly
• D. ξ = 0 at T_c, meaning only nearest-neighbor correlations survive at the critical point

Answer: (B)

At T_c, the correlation length diverges (ξ → ∞) and the exponential decay gives way to power-law decay: G(r) ~ r^{−(d−2+η)}, where η is a critical exponent. A power law has no characteristic length — there is no scale beyond which correlations vanish. This means the system looks statistically the same at any scale of magnification: it is scale-invariant. This scale invariance is why critical phenomena exhibit universality independent of microscopic details, and it is why the system appears turbid (critical opalescence): density fluctuations are correlated at all wavelengths simultaneously.

Question 3

Near the critical temperature, the divergence of the correlation length ξ directly explains why macroscopic response functions such as magnetic susceptibility and specific heat also diverge at T_c.

• T. True
• F. False

Answer: True

The susceptibility χ = (1/kT) ∫ [G(r) − m²] d^dr is the spatial integral of the connected correlation function. When ξ diverges, this integral extends to arbitrarily large distances and the susceptibility diverges. The specific heat is similarly related to energy-energy correlations, which are also governed by ξ. The diverging correlation length is the unifying explanation for all the divergences at T_c: when a system has correlations extending over all scales, it responds dramatically to small perturbations because an enormous number of degrees of freedom are effectively coupled together.

Question 4

In the ordered phase (T ≪ T_c), the two-point correlation function G(r) decays exponentially to zero for large r, just as it does in the disordered phase above T_c.

• T. True
• F. False

Answer: False

This is a crucial distinction. Above T_c (disordered), m = 0 and G(r) decays exponentially to zero — no long-range order. Below T_c (ordered), m ≠ 0 and G(r) → m² at large r, reflecting long-range order: even widely separated spins remain correlated on average. The function does not go to zero in the ordered phase; it approaches a positive constant. The connected correlation function G(r) − m² may still decay exponentially below T_c far from the transition, but G(r) itself saturates rather than vanishing.

Question 5

What physical phenomenon, observable in the laboratory, directly reveals the divergence of the correlation length near a critical point? How does the Fourier transform relationship between G(r) and scattering data explain it?

Model answer: Critical opalescence — the milky, cloudy appearance of fluids near their liquid-gas critical point — directly reveals the diverging correlation length. Scattering experiments measure the structure factor S(q) = ∫ G(r) e^{iq·r} d^dr, the Fourier transform of G(r). A large correlation length means G(r) is significant over a wide spatial range; its Fourier transform therefore has a sharp peak near q = 0 (long-wavelength scattering). This corresponds to density fluctuations correlated over long distances, which scatter light of all wavelengths — producing the milky opalescent appearance.

The Fourier relationship bridges theory and experiment: you cannot directly observe G(r) in a lab, but you can measure S(q) by recording the angular distribution of scattered X-rays, neutrons, or light. The critical divergence of S(q → 0) appears as anomalously strong forward scattering — exactly what is observed as opalescence. This connection was observed experimentally long before the theoretical framework for critical phenomena existed, and explaining it was one of the major successes of renormalization group theory. The structure factor also reveals the correlation length directly: ξ can be extracted from the width of the scattering peak via a Lorentzian fit.