A system is far from any phase transition. You measure the connected correlation function G(r) at increasing separations r. What behavior should you expect?
AG(r) grows with r because fluctuations accumulate over distance
BG(r) remains roughly constant because the system is in equilibrium
CG(r) decays exponentially, falling off as exp(−r/ξ) with a finite correlation length
DG(r) decays as a power law because equilibrium systems always have scale-free correlations
Far from a critical point, fluctuations are short-ranged: the connected correlation function decays exponentially with a finite correlation length ξ. This means distant parts of the system are essentially statistically independent. Power-law decay (option D) only occurs near a critical point, where ξ diverges and the system becomes scale-free — that is precisely what makes critical phenomena special and hard to treat perturbatively.
Question 2 Multiple Choice
In a neutron scattering experiment, the measured structure factor S(q) shows a sharp, narrow peak at a particular wavevector q₀. What does this indicate about the system?
AThe correlation length ξ is very short, so fluctuations are localized near q₀
BLong-range spatial order exists with periodicity 2π/q₀, because S(q) is the Fourier transform of the density-density correlation function
CThe system is near a critical point, because sharp features in S(q) signal diverging correlations
DThe two-point function G(r) is identically zero except at the distance corresponding to q₀
The structure factor S(q) = ∫G(r)exp(iq·r)dr is the Fourier transform of the density-density correlation function. A sharp Bragg peak at q₀ signals long-range periodic order (like a crystal) with spatial period 2π/q₀ — the correlations persist to large r without decaying. A broad, diffuse peak would indicate short-range order with correlation length ξ ~ 1/Δq. Option C is wrong: a critical point produces a broad divergence near q = 0 (long-wavelength fluctuations), not a sharp peak.
Question 3 True / False
The full correlator ⟨A(r)B(r')⟩ being large means A(r) and B(r') are strongly correlated.
TTrue
FFalse
Answer: False
The full correlator ⟨A(r)B(r')⟩ being large only means both observables have large average values — it may simply equal ⟨A(r)⟩⟨B(r')⟩. The **connected** correlator G(r,r') = ⟨A(r)B(r')⟩ − ⟨A(r)⟩⟨B(r')⟩ measures the actual statistical correlation. If G = 0, the two observables fluctuate independently regardless of how large their means are. The connected correlator subtracts out the trivial contribution from the mean values, leaving only the covariance — the true measure of whether fluctuations at r and r' are linked.
Question 4 True / False
Near a critical point, the correlation length ξ diverges and the connected correlation function G(r) decays as a power law rather than exponentially.
TTrue
FFalse
Answer: True
This is the defining signature of criticality. Far from a phase transition, G(r) ~ exp(−r/ξ) with finite ξ, meaning fluctuations at large separations are uncorrelated. At the critical point, ξ diverges and exponential decay is replaced by power-law decay G(r) ~ r^(−(d−2+η)), where η is a critical exponent. The power-law form is scale-free — there is no characteristic length — which is why critical systems look the same at all scales (self-similarity) and why they require renormalization group methods rather than perturbative treatments.
Question 5 Short Answer
Why is the connected correlation function G(r,r') = ⟨A(r)B(r')⟩ − ⟨A⟩⟨B⟩ the natural measure of spatial correlations, rather than the full correlator ⟨A(r)B(r')⟩?
Think about your answer, then reveal below.
Model answer: The connected correlator isolates the genuine statistical dependence between fluctuations. The full correlator ⟨A(r)B(r')⟩ includes a contribution ⟨A⟩⟨B⟩ that would be present even if the two locations fluctuated completely independently. By subtracting this 'trivial' product of means, the connected correlator is zero exactly when the fluctuations at r and r' are independent, and nonzero only when knowing A(r) actually gives information about B(r'). It is the covariance of the fields, directly analogous to the covariance of random variables in probability theory.
The connection to partition function derivatives makes this precise: G(r,r') = δ²ln(Z)/δh(r)δh(r') — the second cumulant, not the second moment. In statistics, cumulants (connected correlators) encode genuine dependencies, while raw moments mix dependencies with mean effects. The same principle applies in field theory: only connected correlators give you the correlation length and the physics of fluctuations.