Correlation Function

A correlation function turns the vague idea of "fluctuations" into a measurable curve. Take a fluctuating field — the local magnetization, density, or any order parameter $\phi(\mathbf{r})$ — and ask: if $\phi$ is unusually large at the origin, how likely is it to be large a distance $r$ away? The two-point connected correlation function $G(r)=\langle\delta\phi(\mathbf{0})\,\delta\phi(\mathbf{r})\rangle$ answers exactly that. In a homogeneous, isotropic system it depends only on the separation $r$ , and its shape is a fingerprint of the phase the system is in. Deep in a disordered phase, correlations are short-ranged: $G$ dies exponentially over a distance $\xi$ , the correlation length, which is just a few lattice spacings. As a continuous transition is approached, $\xi$ swells without bound and the decay slows until, exactly at $T_c$ , the exponential is replaced by a pure power law $G(r)\sim r^{-(d-2+\eta)}$ . Fluctuations of every size coexist — this scale-free behaviour is the microscopic engine of critical phenomena, scaling, and universality. Two identities make $G(r)$ indispensable. First, its spatial integral is a thermodynamic susceptibility: $\chi=\beta\int\mathrm{d}^d r\,G(r)$ . A soft, easily polarized system is precisely one with long, slowly decaying correlations — response and correlation are two faces of the same coin. Second, the Fourier transform of $G$ is the static structure factor $S(\mathbf{k})$ , the quantity a scattering experiment reads out directly. Critical opalescence — the milky glow of a fluid near its critical point — is the divergence of $S(\mathbf{k}\to0)$ made visible to the naked eye. The critical power law $G(r)\sim r^{-(d-2+\eta)}$ defines the anomalous dimension $\eta$ . Ornstein–Zernike (mean-field) theory predicts $\eta=0$ ; fluctuations beyond mean field generate a small nonzero $\eta$ that, with $\nu$ , fixes every other critical exponent through the scaling relations. The correlation function is the bridge from the microscopic Hamiltonian to the thermodynamics to the exponents measured in the laboratory.

Key takeaways

The connected correlation function $G(r)=\langle\delta\phi(\mathbf{0})\,\delta\phi(\mathbf{r})\rangle$ measures how a fluctuation at one point statistically influences one a distance $r$ away.
Away from criticality it decays exponentially, $G(r)\sim e^{-r/\xi}$ , defining the correlation length $\xi$ ; at $T_c$ , $\xi\to\infty$ and $G(r)\sim r^{-(d-2+\eta)}$ becomes scale-free.
Its spatial integral is the susceptibility, $\chi=\beta\int\mathrm{d}^d r\,G(r)=\beta S(0)$ (the fluctuation–response sum rule); its Fourier transform is the structure factor $S(\mathbf{k})$ measured by scattering.
The anomalous dimension $\eta$ corrects the Ornstein–Zernike form and is an independent critical exponent set by the universality class; Fisher's relation reads $\gamma=\nu(2-\eta)$ .
Broken continuous symmetry gives massless Goldstone modes and power-law transverse correlations, which destroy long-range order in $d\le2$ (Mermin–Wagner).
Key idea: $G(r)$ is the master object of criticality — its decay length is $\xi$ , its $T_c$ power law defines $\eta$ , its integral is $\chi$ , and its Fourier transform is the measurable $S(\mathbf{k})$ .

The core insight

The two-point connected correlation function $G(r)=\langle\delta\phi(\mathbf{0})\,\delta\phi(\mathbf{r})\rangle$ is the master object of critical phenomena: its decay length is the correlation length $\xi$ , its $r\to\infty$ power law at $T_c$ defines the anomalous dimension $\eta$ , its spatial integral is the susceptibility, and its Fourier transform is the measurable structure factor $S(\mathbf{k})$ .

Where this lives today

Every modern determination of a critical exponent runs through the correlation function. Small-angle neutron scattering on magnets like RbMnF $_3$ , X-ray scattering on binary alloys, and dynamic light scattering on near-critical fluids all fit the Ornstein–Zernike lineshape to extract $\xi(T)$ and, from the $T_c$ lineshape, the anomalous dimension $\eta$ . On the theory side, the conformal-bootstrap program computes $\eta$ for the 3D Ising model to six-figure precision by exploiting the scale invariance of $G(r)$ at $T_c$ — the same power law that first appeared in Ornstein and Zernike's 1914 optics paper now anchors one of the most precise calculations in statistical physics.

The physical picture

Picture spins on a lattice. Flip one and ask how far the disturbance travels before thermal noise washes it out. Deep in the disordered phase the answer is a few lattice spacings: $\xi$ is small and $G(r)$ collapses exponentially. As the temperature is tuned toward $T_c$ , cooperative regions grow; fluctuating domains of every size appear, and $\xi$ swells until at $T_c$ correlated patches span the entire sample. The correlation function is the spatial memory of a fluctuation — how far its influence reaches. The area under $G(r)$ — its zero-wavevector Fourier component $S(0)$ — is the total mean-square fluctuation of the order parameter, and by the sum rule this equals the susceptibility $\chi$ . So the statement "the system is easy to polarize" and the statement "correlations are long-ranged" are literally the same statement. This is why every critical divergence travels in a pack: as $\xi\to\infty$ , $\chi$ , the specific heat, and the scattered light intensity all diverge together. The cleanest way to see $G(r)$ is to scatter radiation off the sample. The intensity at wavevector transfer $\mathbf{k}$ is $S(\mathbf{k})$ , the Fourier transform of $G(r)$ . Long-wavelength density fluctuations near a critical point scatter strongly at small angles — so strongly that a transparent fluid turns milky. The eye is, quite literally, measuring the divergence of $G$ .

Figure

Correlation function G(r) — exponential decay for T greater than T_c versus scale-free power law at T_c

Figure

Scattering as a Fourier camera — S(k) is the transform of G(r), forward scattering diverges near T_c

The core equations

Every statement below starts from one object. The connected two-point function subtracts the disconnected piece $\langle\phi\rangle\langle\phi\rangle$ so that it measures genuine statistical dependence and vanishes at large separation even in an ordered phase where $\langle\phi\rangle\ne0$ .

G(\mathbf{r},\mathbf{r}')=\langle\phi(\mathbf{r})\phi(\mathbf{r}')\rangle-\langle\phi(\mathbf{r})\rangle\langle\phi(\mathbf{r}')\rangle \qquad (1)

where: $\phi(\mathbf{r})$ — local order-parameter field; $\langle\cdot\rangle$ — thermal average; $G$ — connected (truncated) correlator, equal to the covariance of the fluctuations. In a translationally invariant system it depends only on $\mathbf{r}-\mathbf{r}'$ , and if isotropic only on $r=|\mathbf{r}-\mathbf{r}'|$ . Writing $\delta\phi=\phi-\langle\phi\rangle$ makes the fluctuation content explicit: $G$ is the covariance of the field with itself, displaced by $\mathbf{r}$ .

G(r)=\langle\delta\phi(\mathbf{0})\,\delta\phi(\mathbf{r})\rangle,\qquad \delta\phi=\phi-\langle\phi\rangle \qquad (2)

where: $\delta\phi$ — fluctuation about the mean; $r$ — separation. This is the form scattering measures, because radiation couples to $\delta\phi$ , not to the uniform background. The experimentally accessible quantity is the spatial Fourier transform, the static structure factor. It is what a diffractometer records: intensity as a function of wavevector transfer.

\hat G(\mathbf{k})=\int \mathrm{d}^d r\; G(r)\,e^{-i\mathbf{k}\cdot\mathbf{r}}\equiv S(\mathbf{k}) \qquad (3)

where: $\mathbf{k}$ — wavevector probed by scattering; $S(\mathbf{k})$ — static structure factor. Small $k$ probes long-wavelength fluctuations; large $k$ probes short-range structure. Truncating the Landau free energy at quadratic order in the fluctuations gives the Ornstein–Zernike propagator — a Lorentzian in $k$ whose width is the inverse correlation length.

\hat G_{\mathrm{OZ}}(\mathbf{k})=\frac{k_B T}{c\,(k^2+\xi^{-2})},\qquad \xi^2=\frac{c}{a} \qquad (4)

where: $c$ — stiffness (the coefficient of $\tfrac12 c(\nabla\phi)^2$ in the Landau functional); $a\propto(T-T_c)$ — mass term; $\xi$ — correlation length. Because $a\propto t$ , the OZ theory predicts $\xi\propto|t|^{-1/2}$ , i.e. mean-field $\nu=\tfrac12$ . Exactly at $T_c$ the OZ Lorentzian's inverse-transform loses its scale and $G$ becomes a pure power law; the deviation of that power from the naive $d-2$ defines the anomalous dimension.

G(r)\;\sim\;\frac{1}{r^{d-2+\eta}}\qquad (r\ll\xi,\;T\to T_c) \qquad (5)

where: $\eta$ — anomalous (Fisher) exponent, $=0$ in mean field and $\approx0.036$ for the 3D Ising class; $d$ — spatial dimension. Integrating $G$ over all space collapses the double sum of fluctuations into the susceptibility — the fluctuation–response sum rule.

\chi=\beta\int \mathrm{d}^d r\; G(r)=\beta\,S(\mathbf{k}=0) \qquad (6)

where: $\chi$ — response of $\langle\phi\rangle$ to its conjugate field $h$ ; $\beta=1/k_BT$ . Because $S(0)$ is the integrated fluctuation, a divergence of $\chi$ is a divergence of the total fluctuation. Combining the sum rule with the critical scaling form of $G$ yields Fisher's relation, tying the thermodynamic exponent $\gamma$ to the correlation exponents $\nu,\eta$ .

\gamma=\nu(2-\eta) \qquad (7)

where: $\gamma$ — susceptibility exponent, $\chi\sim|t|^{-\gamma}$ ; $\nu$ — correlation-length exponent, $\xi\sim|t|^{-\nu}$ . Mean field ( $\eta=0,\nu=\tfrac12$ ) gives $\gamma=1$ .

Derivation AOrnstein–Zernike correlation function in real space

The goal is to invert the Gaussian propagator $\hat G(\mathbf{k})=k_BT/[c(k^2+\xi^{-2})]$ back to position space and read off two limits: the exponentially screened decay that defines $\xi$ away from criticality, and the scale-free power law $r^{-(d-2)}$ at $T_c$ that fixes the mean-field value $\eta=0$ . The calculation is the Euclidean massive (Yukawa) propagator with mass $\xi^{-1}$ , and the $d=3$ case reduces to the familiar screened-Coulomb form.

Step 1. Write the inverse transform $G(r)=\dfrac{k_BT}{c}\displaystyle\int\frac{\mathrm{d}^d k}{(2\pi)^d}\,\frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{k^2+\xi^{-2}}$ . Why this integral: it is the Green's function of the operator $(-\nabla^2+\xi^{-2})$ , the massive Helmholtz/Yukawa kernel.

Step 2. Take the massless (critical) limit $\xi\to\infty$ : the integrand becomes $1/k^2$ , whose transform is the Coulomb Green's function $G(r)=\dfrac{k_BT}{c}\dfrac{A_d}{r^{d-2}}$ with $A_d=\Gamma(d/2-1)/(4\pi^{d/2})$ . Why this matters: this is the mean-field critical correlation, so the power is exactly $d-2$ , i.e. $\eta=0$ .

Step 3. Restore finite $\xi$ and evaluate the large- $r$ asymptotics by saddle point: $G(r)\propto e^{-r/\xi}/r^{(d-1)/2}$ . Why the extra power: the Gaussian curvature of the exponent at the saddle contributes the algebraic prefactor on top of the exponential envelope.

Step 4. Specialize to $d=3$ , where the radial integral is elementary: $\displaystyle\int_0^\infty\!\sin(kr)e^{-r/\xi}\,\mathrm{d}r$ gives $G(r)=\dfrac{k_BT}{4\pi c}\dfrac{e^{-r/\xi}}{r}$ , reducing to $k_BT/(4\pi c\,r)$ at criticality. Why $d=3$ is clean: the 3D Yukawa transform is exactly $4\pi/(k^2+\xi^{-2})$ , with no special functions. *Verified: the $d=3$ form $e^{-r/\xi}/(4\pi c\,r)$ and the closed 3D Yukawa transform are checked symbolically in verify_aq-sm-correlation-function.py; they reproduce Landau–Lifshitz §146 and Kardar Ch. 3.*

Go deeper · Derivation B · 1 minSusceptibility divergence and Fisher's relation

Derivation BSusceptibility divergence and Fisher's relation

The goal is to derive Fisher's scaling relation $\gamma=\nu(2-\eta)$ purely from the sum rule $\chi=\beta\int\mathrm{d}^d r\,G(r)$ and the critical scaling form $G(r)=r^{-(d-2+\eta)}\,\mathcal{G}(r/\xi)$ . The key move is to feed the scaling form into the integral, rescale distances by $\xi$ , and let the correlation length carry all the temperature dependence. Nothing in this argument uses a specific model — it holds for every universality class.

Step 1. Insert the scaling form into the sum rule; the angular integral is a constant and the radial part is $\chi\propto\displaystyle\int_0^\infty\!\mathrm{d}r\,r^{d-1}\,r^{-(d-2+\eta)}\,\mathcal{G}(r/\xi)=\int_0^\infty\!\mathrm{d}r\,r^{1-\eta}\,\mathcal{G}(r/\xi)$ . Why only the radial part: isotropy makes the angular integral an overall constant that cannot affect an exponent.

Step 2. Rescale $u=r/\xi$ , so each factor of $r$ becomes $\xi u$ : $\chi\propto\xi^{2-\eta}\displaystyle\int_0^\infty\!\mathrm{d}u\,u^{1-\eta}\,\mathcal{G}(u)$ . Why the integral is finite: $\mathcal{G}(u)$ cuts off exponentially at large $u$ , so the $u$ -integral is a pure universal number with no hidden temperature dependence.

Step 3. Use $\xi\sim|t|^{-\nu}$ : $\chi\sim\xi^{2-\eta}\sim|t|^{-\nu(2-\eta)}$ . Matching to the definition $\chi\sim|t|^{-\gamma}$ gives $\gamma=\nu(2-\eta)$ . Why this is nontrivial: it connects a purely thermodynamic exponent $\gamma$ to two exponents ( $\nu,\eta$ ) defined entirely through the correlation function.

Step 4. Check the mean-field case: $\eta=0,\nu=\tfrac12$ give $\gamma=1$ , the Landau value; the 3D Ising values $\nu\approx0.630,\eta\approx0.036$ give $\gamma\approx1.237$ , matching the accepted $1.24$ . Why the check matters: it verifies the relation across both the mean-field fixed point and the fluctuation-dominated 3D class. *Verified: the sum-rule scaling, Fisher relation, and both the MF and 3D-Ising numerical checks are in verify_aq-sm-correlation-function.py.*

Go deeper · Derivation C · 1 minGoldstone power-law correlations in the ordered phase

Derivation CGoldstone power-law correlations in the ordered phase

The goal is to show that breaking a continuous symmetry produces a massless transverse mode whose correlations decay only algebraically, and to extract the lower critical dimension $d_\ell=2$ below which those fluctuations destroy long-range order (Mermin–Wagner). Start from an $O(n)$ order parameter that has picked a direction, $\langle\boldsymbol\phi\rangle=\phi_0\hat e_1$ , and split fluctuations into a longitudinal (massive) part and $n-1$ transverse (Goldstone) parts.

Step 1. Decompose $\boldsymbol\phi=(\phi_0+\sigma)\hat e_1+\boldsymbol\pi$ with $\boldsymbol\pi\perp\hat e_1$ . Why the split: rotating the order parameter (a $\boldsymbol\pi$ deformation) must cost zero energy at zero wavevector, because it is a symmetry operation on the ground state.

Step 2. The transverse part of the quadratic action carries only the gradient term, $\beta\mathcal{H}_\perp=\tfrac12\int\frac{\mathrm{d}^d k}{(2\pi)^d}\,c\,k^2\,|\boldsymbol\pi_{\mathbf{k}}|^2$ . Why no mass term: a mass would penalize a uniform rotation, contradicting the Goldstone theorem.

Step 3. Equipartition gives the massless propagator $\langle|\pi_{\mathbf{k}}|^2\rangle=k_BT/(c\,k^2)$ , which diverges as $k\to0$ ; transforming back, $G_\perp(r)\sim r^{-(d-2)}$ . Why it is long-ranged: the $1/k^2$ pole at the origin is the hallmark of algebraic, not exponential, decay.

Step 4. Estimate the total transverse fluctuation $\langle\pi^2\rangle\sim\int\mathrm{d}^d k/k^2$ ; the infrared part $\int_0 k^{d-3}\mathrm{d}k$ converges only for $d>2$ , and diverges for $d\le2$ . Why this kills order: an infrared-divergent $\langle\pi^2\rangle$ means the order parameter cannot stay pinned to a direction — this is the Mermin–Wagner result $d_\ell=2$ . *Verified: the Goldstone propagator, the $d=3$ convergence versus $d=2$ divergence, and $d_\ell=2$ are checked numerically in verify_aq-sm-correlation-function.py, consistent with Kardar's $O(n)$ treatment.*

Limiting cases

The shape of $G(r)$ depends sharply on where the system sits relative to $T_c$ and on the symmetry of the ordered state.

Case	Result	Meaning
Disordered, $T>T_c$	$G(r)\sim e^{-r/\xi}/r^{(d-1)/2}$ , $\xi<\infty$	Ornstein–Zernike exponential decay with finite range $\xi$ .
Critical, $T=T_c$	$G(r)\sim r^{-(d-2+\eta)}$	Scale-free power law; $\xi=\infty$ ; mean field gives $\eta=0$ , i.e. $r^{-(d-2)}$ .
Ordered, broken continuous symmetry	$G_\perp(r)\sim r^{-(d-2)}$ (Goldstone)	Transverse correlations are power-law for all $T<T_c$ ; destroy order in $d\le2$ .
Ordered, discrete symmetry	$G_c(r)\sim e^{-r/\xi}$	Connected part decays exponentially; the full correlator tends to $\langle\phi\rangle^2\ne0$ .

Worked examples

Example 1 — Ratio of correlations at two separations (3D OZ). In three dimensions the OZ form is $G(r)=\dfrac{k_BT}{4\pi c}\dfrac{e^{-r/\xi}}{r}$ . Compare $G$ at $r=2\xi$ and at $r=\xi/2$ .

Step 1. Form the ratio; the prefactor $k_BT/4\pi c$ cancels: $\dfrac{G(2\xi)}{G(\xi/2)}=\dfrac{(\xi/2)}{(2\xi)}\,e^{-(2\xi-\xi/2)/\xi}=\dfrac{1}{4}\,e^{-3/2}$ .

Step 2. Evaluate: $\tfrac14 e^{-1.5}=0.25\times0.2231=\mathbf{0.0558}$ . Moving from half a correlation length to two correlation lengths cuts the correlation by a factor $\approx18$ — most of it from the exponential, a smaller part from the $1/r$ prefactor. The correlation at $2\xi$ is about 5.6 % of that at $\xi/2$ .

Example 2 — How much the susceptibility grows on approaching $T_c$ . For the 3D Ising class, $\chi\sim|t|^{-\gamma}$ with $\gamma=\nu(2-\eta)$ and $\nu=0.630,\eta=0.036$ , giving $\gamma=1.237$ . Compare $t_1=10^{-2}$ with $t_2=10^{-3}$ .

Step 1. Compute $\gamma$ from Fisher's relation: $\gamma=0.630\times(2-0.036)=0.630\times1.964=\mathbf{1.237}$ .

Step 2. Ratio of susceptibilities: $\dfrac{\chi(t_2)}{\chi(t_1)}=\left(\dfrac{t_1}{t_2}\right)^{\gamma}=10^{1.237}=\mathbf{17.3}$ . A tenfold approach toward $T_c$ multiplies the susceptibility — and hence the integrated fluctuation and the forward-scattered intensity — by about 17. In mean field ( $\gamma=1$ ) the same step would give exactly 10; the extra factor is the fluctuation enhancement that opalescence makes visible.

Origins · 1 mincorrelation function and how this was discovered

correlation function

the statistical fingerprint of a phase · The two-point correlation function grew from Josiah Willard Gibbs's ensemble averages into the central diagnostic of condensed matter. Leonard Ornstein and Frits Zernike gave it its critical form in 1914 while explaining critical opalescence; Michael Fisher, Leo Kadanoff, and Kenneth Wilson later turned the anomalous decay exponent η into a cornerstone of the renormalization group. Today the connected correlator — measured by neutron, X-ray, and light scattering — is how experimentalists read a system's correlation length, anomalous dimension, and universality class directly off a diffraction pattern.

History

1914

— Ornstein and Zernike give the correlation function its screened form, explaining critical opalescence

1937

— Landau formulates the order-parameter description of phase transitions

1944

— Onsager's exact 2D Ising solution yields η = 1/4, a nonzero anomalous dimension

1964

— Fisher relates the susceptibility exponent to the correlation function, γ = ν(2 − η)

1966

— Mermin, Wagner and Hohenberg prove continuous symmetry cannot break in d ≤ 2

1971

— Wilson's renormalization group derives η from the fixed point

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 13–14 (correlations and scaling) · Kardar, Statistical Physics of Fields, Ch. 2–3 (Gaussian theory, correlation functions) · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 7 (fluctuations) · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 10 (correlations, response, dissipation) · Landau & Lifshitz, Statistical Physics Part 1, §116, §146 (correlation of fluctuations, Ornstein–Zernike).

Provenance. A verifier should: reproduce the Gaussian/OZ propagator $\hat G(k)=k_BT/[c(k^2+\xi^{-2})]$ and its $d=3$ real-space form $e^{-r/\xi}/(4\pi c\,r)$ (Kardar Ch. 3, Landau–Lifshitz §146); check the susceptibility sum rule and Fisher relation $\gamma=\nu(2-\eta)$ (Pathria & Beale Ch. 14); confirm mean-field $\eta=0,\nu=1/2,\gamma=1$ and 3D Ising $\eta\approx0.036,\nu\approx0.63,\gamma\approx1.24$ (Pelissetto–Vicari review); verify the Goldstone propagator $k_BT/(ck^2)$ and Mermin–Wagner $d_\ell=2$ . All numbers are standard textbook values.

Builds on: Order Parameter · Landau Theory of Phase Transitions · Continuous Phase Transitions. Leads to: Correlation Length and Its Divergence · Critical Exponents · Scaling Hypothesis and Scaling Relations · Goldstone Modes.

What is the difference between the connected, truncated, and full correlators?

The full correlator is $\langle\phi(\mathbf 0)\phi(\mathbf r)\rangle$ ; the connected (or truncated, or cumulant) correlator subtracts the product of means, $\langle\phi(\mathbf 0)\rangle\langle\phi(\mathbf r)\rangle$ . Away from an ordered phase both agree because $\langle\phi\rangle=0$ . In the ordered phase they differ by the constant $\langle\phi\rangle^2$ : the full correlator plateaus there, while the connected one decays to zero and correctly reports the range of genuine fluctuations. The connected correlator is the one that integrates to the susceptibility and Fourier-transforms to the scattering structure factor (which excludes the forward Bragg peak).

How is the structure factor S(k) actually measured?

In a scattering experiment (light, X-ray, or neutron) a beam of wavevector $\mathbf k_{\rm in}$ hits the sample and is deflected to $\mathbf k_{\rm out}$ ; the wavevector transfer is $\mathbf k=\mathbf k_{\rm out}-\mathbf k_{\rm in}$ , with magnitude $k=(4\pi n/\lambda)\sin(\theta/2)$ for scattering angle $\theta$ . The measured intensity at that angle is proportional to $S(\mathbf k)$ , the Fourier transform of $G(r)$ . Scanning the angle scans $k$ : small angles probe long-wavelength fluctuations (small $k$ , near the diverging forward peak), large angles probe short-range structure. Fitting the Ornstein–Zernike Lorentzian $S(k)\propto1/(k^2+\xi^{-2})$ to the data yields the correlation length directly.

Why does the correlation function decay exponentially away from criticality but as a power law at T_c?

$G(r)\sim e^{-r/\xi}$ contains an explicit length, $\xi$ . At $T_c$ the correlation length diverges, removing the only scale in the problem. A scale-free function of a single variable must be a pure power law, so $G(r)\to r^{-(d-2+\eta)}$ . Physically, at $T_c$ fluctuating domains exist on every length from the lattice spacing up to the system size; there is no characteristic size, hence no characteristic decay length. This scale invariance is exactly what the renormalization group formalizes as a fixed point.

What sets the value of the anomalous dimension η?

$\eta$ measures how much the critical decay departs from the naive Gaussian power $d-2$ . In mean-field (Ornstein–Zernike) theory $\eta=0$ exactly, because the Gaussian propagator inverts to $r^{-(d-2)}$ . Below the upper critical dimension $d_c=4$ , interactions between fluctuations renormalize the field and generate a small positive $\eta$ : about $0.036$ for the 3D Ising class, $1/4$ for 2D Ising. Its value depends only on dimensionality and order-parameter symmetry — the universality class — not on microscopic details, which is why measuring $\eta$ classifies a transition.

How can response and correlation be "the same thing"?

The fluctuation–response sum rule $\chi=\beta\int\mathrm{d}^d r\,G(r)$ says the susceptibility (how much the order parameter responds to a small conjugate field) equals the total integrated fluctuation. The reason is a single differentiation of the partition function: $\partial\langle\phi\rangle/\partial h$ produces the second cumulant of the spatially integrated field, which is exactly the integral of $G$ . So a system that responds strongly to a field is one whose spontaneous fluctuations are large and long-ranged. This is the static (equal-time) version of the fluctuation–dissipation theorem, and it is why $\chi$ and $\xi$ diverge together at $T_c$ .

Do Goldstone modes appear at every phase transition?

No — only when a continuous symmetry is spontaneously broken. Breaking a discrete symmetry (like the up/down $\mathbb Z_2$ of the Ising magnet) produces a massive longitudinal mode and exponentially decaying connected correlations, with no gapless excitation. Breaking a continuous symmetry (like the rotational $O(n)$ symmetry of a Heisenberg magnet or a superfluid's phase) leaves directions in order-parameter space that cost no energy to rotate; those are the Goldstone modes, with a massless $1/k^2$ propagator and power-law transverse correlations. Their infrared divergence in $d\le2$ is what forbids continuous-symmetry order in low dimensions.