Correlations, Susceptibilities, and Correlation Lengths

Static response and spatial correlation are two views of equilibrium fluctuations. In a classical Gibbs ensemble a susceptibility is the spatial integral of a connected equal-time correlation, multiplied by inverse temperature. Quantum static response generally requires an additional imaginary-time integral; that distinction is made explicitly below. A correlation length can be read from asymptotic decay, a screening pole, or a second moment, but these estimators agree only when one isolated long-distance scale dominates. None is automatically a real-time quasiparticle mass.

The scaling relations among correlations, susceptibilities, and correlation lengths are reviewed with their critical assumptions in Fisher 1967, pp. 615–730.

Required background. Partition Functions and Thermodynamic Response supplies the source derivatives. Connected Correlators and Cumulants fixes connected normalization. Helpful background. Critical Exponents, Scaling Relations, and Hyperscaling Caveats develops the scaling limit used near criticality.

Connected correlations and response

Begin with a classical Gibbs ensemble at fixed temperature and regulator, and a source-independent real scalar observable $\mathcal O(\mathbf x)$ . In a translation-invariant state, define

C(\mathbf x)= \langle\mathcal O(\mathbf x)\mathcal O(\mathbf0)\rangle -\langle\mathcal O\rangle^2.

Couple a uniform source through $H_h=H-hQ$ , where $Q=\int_V\mathrm d^{d-1}x\,\mathcal O(\mathbf x)$ . Differentiating the normalized weight $e^{-\beta H_h}/Z_h$ gives $\partial_h\langle Q\rangle=\beta\langle(\delta Q)^2\rangle$ . In a periodic volume, translation invariance reduces the double spatial integral of the covariance to $V$ times a single integral. Consequently

\chi =\frac{\partial\langle\mathcal O\rangle}{\partial h} =\beta\int_V\mathrm d^{d-1}x\,C(\mathbf x),

where $\langle\mathcal O\rangle$ denotes the spatially averaged density. If the source also changes the operator or action explicitly, a contact term must be added. On a finite periodic lattice the same identity holds as a sum, with the zero mode and volume normalization stated explicitly.

The structure factor is

S(\mathbf k)=\int\mathrm d^{d-1}x\, e^{-i\mathbf k\cdot\mathbf x}C(\mathbf x).

This spatial Fourier definition is even in $\mathbf k$ for a parity-invariant real channel. For the classical ensemble, $\chi=\beta S(\mathbf0)$ after any required contact subtraction.

Quantum isothermal response

For a quantum Gibbs state $\rho_h=Z_h^{-1}e^{-\beta H_h}$ with the same linear source, define $\mathcal O(\tau,\mathbf x)=e^{\tau H_h}\mathcal O(\mathbf x)e^{-\tau H_h}$ . The operator-exponential derivative developed on Partition Functions and Thermodynamic Response instead gives

\chi =\int_0^\beta\!\mathrm d\tau\int_V\!\mathrm d^{d-1}x\, \langle\delta\mathcal O(\tau,\mathbf x) \delta\mathcal O(0,\mathbf0)\rangle.

The state re-equilibrates at the same temperature as $h$ changes. This is an isothermal susceptibility, not automatically the zero-frequency retarded response of an isolated system. The quantum integral and its classical reduction are derived in Kubo 1957, § 3, pp. 576–577, Eqs. (3.16) and (3.21). If chemical potentials are present, the imaginary-time generator is the full $H_h-\sum_a\mu_aQ_a$ .

Define the static spatial correlator and its Fourier transform by

\begin{aligned} C_{\mathrm{stat}}(\mathbf x) &=\int_0^\beta\!\mathrm d\tau\, \langle\delta\mathcal O(\tau,\mathbf x) \delta\mathcal O(0,\mathbf0)\rangle,\\ G_E(0,\mathbf k) &=\int\!\mathrm d^{d-1}x\, e^{-i\mathbf k\cdot\mathbf x}C_{\mathrm{stat}}(\mathbf x). \end{aligned}

Thus $\chi=G_E(0,\mathbf0)$ , the zero Matsubara-frequency, zero spatial-momentum correlator. The equal-time structure factor $S(\mathbf k)$ has no imaginary-time integral. Its zero-momentum value obeys $\chi=\beta S(0)$ in a quantum state if the integrated source observable $Q$ commutes with the full Gibbs generator; otherwise that substitution is generally false.

For example, a free quantum scalar mode with $\omega_{\mathbf k}^2=\mathbf k^2+m^2$ has

S(\mathbf k)=\frac{1}{2\omega_{\mathbf k}} \coth\frac{\beta\omega_{\mathbf k}}2, \qquad G_E(0,\mathbf k)=\frac{1}{\omega_{\mathbf k}^2}.

The first expression follows from $\langle q^2\rangle=(n_B+1/2)/\omega$ for a canonically normalized oscillator; the second is the inverse static quadratic kernel. Multiplying the first by $\beta$ recovers the second only in the classical high-temperature limit for that mode. Below, $C$ and $S$ denote the classical spatial correlator and structure factor. To study quantum static screening, apply the spatial definitions to $C_{\mathrm{stat}}$ and $G_E(0,\mathbf k)$ instead, with the operator channel and contact terms fixed.

Exponential, screening, and second-moment lengths

For an isotropic continuum channel with one isolated simple screening pole, large radial separation gives

C(r)\sim A\,r^{-(d-2)/2}e^{-r/\xi_{\mathrm{exp}}}.

Equivalently $S(\mathbf k)$ has a nearest singularity at $\mathbf k^2=-\xi_{\mathrm{exp}}^{-2}$ . At small real momentum an Ornstein–Zernike form reads

S(\mathbf k)\simeq \frac{S(0)}{1+\xi_2^2\mathbf k^2+O(k^4)},

with second-moment length

\xi_2^2 =\frac{1}{2(d-1)} \frac{\int\mathrm d^{d-1}x\,\mathbf x^2 C(\mathbf x)} {\int\mathrm d^{d-1}x\,C(\mathbf x)}.

On a periodic lattice with $L$ sites in each direction, use lattice spacing one. A common estimator avoids coordinate wrapping:

\xi_{2,L} =\frac{1}{2\sin(\pi/L)} \left[ \frac{S(\mathbf0)}{S(\mathbf k_{\min})}-1 \right]^{1/2}, \qquad \mathbf k_{\min}=\frac{2\pi}{L}(1,0,\ldots).

The displayed length is in lattice units; multiply it by the lattice spacing for a physical length. This estimator assumes an approximately isotropic single-pole small-momentum form. Direction-dependent estimators are required on anisotropic lattices; a multi-exponential channel need not yield $\xi_2=\xi_{\mathrm{exp}}$ .

Gaussian check and critical scaling

For the static Gaussian functional

\mathcal F[\phi] =\frac12\int\mathrm d^{d-1}x \left[(\nabla\phi)^2+m_s^2\phi^2\right],

with probability proportional to $e^{-\beta\mathcal F}$ and $m_s^2>0$ , the covariance is

S(\mathbf k)=\frac{T}{\mathbf k^2+m_s^2}.

Therefore

\xi=\frac1{m_s}, \qquad \chi=\beta S(0)=\frac1{m_s^2}

for the displayed source normalization. This is a screening mass: it controls static spatial decay. A pole mass comes from a real-time frequency singularity, and a curvature mass from a local effective-potential Hessian. Interactions can separate all three.

Near a continuous classical transition at $T_c>0$ , let $t=(T-T_c)/T_c$ . For separations large compared with the regulator and a scaling regime with $2-\eta>0$ , the singular long-distance part has the form

C(r;t)\sim \frac{1}{r^{d-3+\eta}}\, \mathcal C(r/\xi), \qquad \xi\sim\lvert t\rvert^{-\nu}, \qquad \chi\sim\xi^{2-\eta}.

The susceptibility power follows by integrating the spatial scaling form out to $r\sim\xi$ ; short-distance contributions give a nonsingular background. This is a classical finite-temperature scaling statement, not a prescription for omitting imaginary time at a quantum critical point. Finite volume cuts off the divergence when $\xi$ approaches $L$ . Extracting a critical exponent therefore requires a joint finite-size and scaling-window analysis; fitting a large but finite $\xi$ at one volume does not establish a critical point.

Validation checklist

Confirm that connected rather than full correlations are integrated.
Distinguish classical equal-time covariance from quantum imaginary-time integrated response.
State the source sign and any contact or zero-mode subtraction.
Compare exponential and second-moment estimators over increasing volumes.
Vary the fit window and include more than one screening state when required.
Test anisotropy by rotating $\mathbf k_{\min}$ .
Keep screening, curvature, pole, and damping scales distinct.
Near criticality, demonstrate $L/\xi$ control rather than extrapolating a single volume.

The finite-volume and limit distinctions are summarized in the shared equilibrium table. Gaussian determinants and the failure of quadratic control continue on Gaussian Fluctuations and Effective Free Energy.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: How do Euclidean configurations, correlations, Gaussian fluctuations, phases, and thermodynamic limits relate?

A Euclidean weight defines correlations; a positive Hessian licenses Gaussian fluctuations locally, while phase coexistence and spontaneous breaking require a separately ordered thermodynamic and source limit. Solid connections show the primary relation; dashed outlines or arrows mark qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Exercise

For $S(k)=A/(k^2+m_1^2)+B/(k^2+m_2^2)$ with $0<m_1<m_2$ and $A,B>0$ , compare the exponential and second-moment lengths.

Solution

The large-distance decay is controlled by the nearest pole, so $\xi_{\mathrm{exp}}=1/m_1$ . Expanding at small $k$ gives $\xi_2^2=(A/m_1^4+B/m_2^4)/(A/m_1^2+B/m_2^2)$ . Unless the light pole dominates the zero-momentum weight, $\xi_2<1/m_1$ . The difference is a diagnostic of multiple scales, not a contradiction.

References

Fisher, Michael E. “The Theory of Equilibrium Critical Phenomena.” Reports on Progress in Physics 30, no. 2 (1967): 615–730. doi:10.1088/0034-4885/30/2/306.
Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems.” Journal of the Physical Society of Japan 12, no. 6 (1957): 570–586. doi:10.1143/JPSJ.12.570; Open PDF.