Correlation Functions and Linear Response

Correlation functions describe how two observables fluctuate together in space or time. Linear response describes how an observable changes when a weak external source is applied. The two are related in equilibrium, but they are different quantities. Their operator order, normalization, and physical interpretation must remain explicit.

This chapter helps you choose the function needed for a scattering spectrum, a susceptibility, a transport coefficient, or a test of spatial order. Begin with Correlation Functions Overview for the main distinctions.

Required background. Enter with a declared state space and operator algebra from Many-Body Hilbert Spaces and Operators, pure- and mixed-state expectation values from Density Operators, and time evolution from the Heisenberg Picture.

Helpful background. Quantum Statistical Mechanics and Thermal Density Operators supply equilibrium conventions when needed. The Interacting Systems and Approximation Methods gateway helps when a correlator must be approximated. Thermodynamic Limit supplies finite-size and order-of-limits discipline. Field operators, density and current operators, translation symmetry, and Fourier transforms are branch-specific preparation.

Fluctuations and driven response

In a stationary state, an ordinary correlation is $C_{AB}(t)=\langle A(t)B(0)\rangle$. Its connected form subtracts $\langle A\rangle\langle B\rangle$. Reversing the operator order generally gives a different function; time ordering, symmetrization, and a retarded commutator are further distinct choices.

For a perturbation $H'(t)=-f(t)B$, define the causal susceptibility by

$$\begin{aligned} \delta\langle A(t)\rangle &=\int_{-\infty}^{\infty}dt'\,\chi^R_{AB}(t-t')f(t'),\\ \chi^R_{AB}(t) &=\frac{i}{\hbar}\theta(t)\langle[A(t),B(0)]\rangle. \end{aligned}$$

Here the measured observable is taken to have no explicit source dependence; when it does, include the corresponding contact term. The positive prefactor matches the minus sign in the source coupling. A retarded Green function defined with $-i/\hbar$ has the opposite convention and cannot be inserted unchanged as this susceptibility.

The step function makes the response vanish before the perturbation. It does not make the ordinary correlation vanish there. For spectra, also specify the Fourier sign, whether the variable is energy or angular frequency, and which elastic or disconnected pieces are retained. For transport and static response, the order of the large-system, zero-momentum, zero-frequency, and vanishing-broadening limits can change the result.

Choose the quantity you need

Choose the branch that matches the observable and probe.

1. Learn the common language. Start with Correlation Functions Overview. It distinguishes the correlation hierarchy, operator order, connected subtraction, spatial and temporal decay, and response functions without replacing the focused treatments.
2. Build the static branch. Read Equal-Time Correlations for spin, density, one-body, and pair diagnostics. Continue to Connected Correlation Functions for cumulants, clustering, phase mixtures, and correlation lengths.
3. Transform static or dynamic correlations into probe variables. Structure Factors develops momentum-resolved static and dynamic weight, scattering conventions, elastic pieces, and moments. It follows both equal-time and time-dependent preparation.
4. Build the dynamical branch. Time-Dependent Correlations develops ordinary two-time functions, Lehmann weights, stationarity, detailed balance, dephasing, and recurrence.
5. Choose the single-particle branch only for number-changing propagation. After field operators and grand-canonical notation, Green Functions in Many-Body QM treats particle addition and removal, poles, continua, and the bridge to Matsubara functions. A Green function is not a synonym for every two-point correlator.
6. Choose the causal-response branch for a weak external source. Retarded and Advanced Response develops commutator support, analyticity, and dispersion relations. Kubo Formula then connects a specified source to a specified detector, including contact terms and noncommuting static limits.
7. Name the response channel. Susceptibilities specializes the general kernel to magnetic, density, compressibility, pairing, and matrix-valued responses with units and protocols explicit.
8. Interpret frequency-resolved weight. Spectral Functions compares exact lines, continua, quasiparticle peaks, linewidths, and measured intensity across several channels. Read the relevant Green-function, structure-factor, or retarded-response branch first.
9. Use equilibrium identities only under their hypotheses. Fluctuation–Dissipation Theorem relates ordered or symmetrized equilibrium fluctuations to absorptive response after the convention and KMS assumptions are fixed. Sum Rules supplies exact spectral moments and commutator checks.
10. Enter transport last. Transport Coefficients Preview requires conserved densities or currents, Kubo response, thermodynamic-limit discipline, and a declared ballistic, diffusive, anomalous, or localized regime. It is a bridge, not a full transport theory.

Example: three probes of one electron model

Consider three experiments on the same interacting electron model. Elastic or energy-resolved scattering from density couples to a density structure factor after form factors, connected or elastic subtraction, and resolution are declared. Photoemission instead probes a removal spectral function built from a single-particle Green function, with matrix elements and occupations separating intrinsic weight from measured intensity. Optical conductivity requires a current source and detector, the retarded current response, contact terms, a Kubo limit, and a distinction among regular weight, a Drude contribution, and finite-size broadened lines. The Hamiltonian is the same; the operator pair, ordering, trace sector, normalization, and permissible inference are not.

Continue from a correlation to its interpretation

Use Finite-Temperature Methods for imaginary-time and Matsubara descriptions, Quasiparticles and Collective Modes for excitation interpretation, and Phases, Order, and Criticality for bulk-order claims. Dynamical Correlation Functions Numerically treats finite-time and numerical resolution.

The Correlation Function Definitions and Linear Response Formula Sheet provide compact reminders. For a named material, continue to Transport, Response, and Optics.

Common pitfalls

“Equal-time means static response.” An equal-time covariance is not generally the zero-frequency retarded susceptibility; equilibrium identities and limiting procedures are required.

“Time ordered means causal.” A time-ordered Green function and a retarded commutator have different definitions, analytic roles, and measurement dictionaries.

“Stationary means thermal.” Time-translation invariance does not imply a Gibbs state, detailed balance, or the KMS condition.

“Connected means interacting or entangled.” Connected subtraction removes products of lower moments. Noninteracting states can have connected correlations, and connected correlations alone do not certify entanglement.

“A spectral peak is automatically a quasiparticle.” A defensible interpretation needs the operator channel, system-size behavior, intrinsic width, nearby continuum, sum rules, and instrumental resolution.

“Artificial broadening is a lifetime.” Broadening finite-system delta lines improves visualization or continuation stability; it is not an intrinsic decay rate unless independently justified.

“The static limit is unique.” The limits $q\to0$ and $\omega\to0$ may not commute, and contact terms, conservation laws, boundaries, and the thermodynamic limit can change the result.

“Fluctuation–dissipation applies to every driven state.” Its standard form requires equilibrium or the stated KMS conditions. A nonequilibrium steady state needs a separate analysis.

“A sum rule validates the entire spectrum.” A correct integrated moment is a strong constraint, not proof of pointwise accuracy, correct linewidths, or a unique reconstruction.

“Linear response means weak interactions.” It means expansion in a weak external source around a declared reference state; the unperturbed system may be strongly interacting.

Exercises

Exercise 1: Route three observables

Choose the minimum route for (a) antiferromagnetic spatial order in a finite spin lattice, (b) an electron-removal spectrum, and (c) dc conductivity. For each, name one convention or limit that must be declared.

Solution

For (a), use Correlation Functions Overview → Equal-Time Correlations → Connected Correlation Functions → Structure Factors; declare the ordering wavevector, normalization, boundary conditions, elastic or disconnected subtraction, and finite-size scaling before a bulk-order claim. For (b), prepare field operators and the ensemble, then use Time-Dependent Correlations → Green Functions → Spectral Functions; declare the removal convention, Fourier normalization, chemical-potential zero, matrix elements, and resolution. For (c), prepare current operators, then use Retarded Response → Kubo Formula → Susceptibilities → Sum Rules → Transport Preview; declare contact terms and the order of thermodynamic, $q\to0$, $\omega\to0$, and vanishing-broadening limits.

Exercise 2: Repair a response claim

Repair the statement: “The equal-time variance is large, so the system has a large causal susceptibility at zero frequency.”

Solution

The variance specifies an equal-time fluctuation in a declared state. A causal susceptibility is a retarded response to a named source, with operator order, contact terms, held-fixed variables, and a limiting protocol. An equilibrium fluctuation relation may connect particular static quantities under its hypotheses, while the dynamic fluctuation–dissipation theorem relates frequency-resolved fluctuations to absorptive response. Without those assumptions and limits, the inference does not follow.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, CRC Press (1990).
• R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, 2nd ed., Springer (1991).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998).