Generalized Susceptibility and Spectral Functions

A system in thermal equilibrium, pushed by a weak external field, develops a small induced expectation value of some observable. To linear order this reaction is completely captured by one complex function of frequency — the generalized susceptibility $\chi(\omega)$ — that unifies dielectric response, magnetic susceptibility, electrical conductivity, mechanical compliance, and the mobility of a Brownian particle into a single mathematical object obeying universal constraints. The word "generalized" is earned: the same $\chi(\omega)$ , with the same analytic structure and the same sum rules, describes light bending in glass, a spin flipping in a magnet, and a colloid drifting in water. Two physical principles fix its analytic skeleton. Causality — no response before its cause — makes $\chi(t)$ vanish for $t<0$ , which forces $\chi(\omega)$ to be analytic in the upper half-plane and yields the Kramers–Kronig relations. Reality of the time-domain response enforces $\chi(-\omega)=\chi^*(\omega)$ , so $\chi'(\omega)$ is even and $\chi''(\omega)$ is odd. Splitting $\chi=\chi'+i\chi''$ separates the reactive, energy-storing response from the dissipative, energy-absorbing one. The dissipative part is not arbitrary: it is the spectral function, the frequency-resolved Fourier transform of the equilibrium commutator $\langle[\hat A(t),\hat A(0)]\rangle$ , and through the Lehmann (spectral) representation it becomes a sum of sharp lines at the exact transition frequencies of the system, each weighted by a matrix element. This is the frequency-domain capstone of linear response. The spectral function $\chi''(\omega)$ is what absorption spectroscopy, inelastic neutron and light scattering, ARPES, and tunnelling actually measure; the dynamic structure factor $S(q,\omega)$ is tied to it by the fluctuation–dissipation theorem, and the ratio of its energy-loss to energy-gain peaks is the exact detailed-balance thermometer $e^{-\beta\hbar\omega}$ . Sum rules constrain spectral weight independent of microscopic detail, positivity ( $\omega\chi''\ge0$ ) enforces passivity, and analytic continuation ties the retarded susceptibility to the imaginary-time Green's functions computed in many-body theory. This article develops the spectral representation, the analyticity and sum-rule machinery, the Kubo formula, and the FDT bridge that makes fluctuation and absorption one measurement.

Key takeaways

The generalized susceptibility $\chi(\omega)$ is the complex linear-response function relating the induced average of an observable to a weak perturbation; its real and imaginary parts are tied by causality (Kramers–Kronig).
The dissipative part $\chi''(\omega)$ is the spectral function — the Fourier transform of the equilibrium commutator $\langle[\hat A(t),\hat A(0)]\rangle$ — and through the Lehmann representation is a weighted sum of $\delta$ -functions at the exact transition frequencies.
The spectral (Lehmann) representation $\chi(\omega)=\int\rho(\omega')/(\omega'-\omega-i0^+)\,\mathrm{d}\omega'$ with spectral density $\rho(\omega')=\chi''(\omega')/\pi$ generates KK, sum rules, and the static limit at once.
The fluctuation–dissipation theorem ties $\chi''$ to the fluctuation spectrum $S(\omega)$ ; the dynamic structure factor $S(q,\omega)$ measured in scattering carries a detailed-balance asymmetry, Stokes/anti-Stokes ratio $e^{-\beta\hbar\omega}$ .
Positivity $\omega\chi''(\omega)\ge0$ (passivity) and sum rules (spectral weight conservation) are exact constraints used across condensed matter, optics, and many-body theory.
Key idea: A weak perturbation $H'=-f(t)\hat A$ gives $\langle\delta A(t)\rangle=\int_{-\infty}^t\chi(t-t')f(t')\,\mathrm{d}t'$ ; the causal kernel's transform $\chi(\omega)=\int\rho(\omega')/(\omega'-\omega-i0^+)\,\mathrm{d}\omega'$ with $\rho=\chi''/\pi$ encodes absorption, dispersion (KK), passivity ( $\omega\chi''\ge0$ ), and — via the FDT and the $e^{-\beta\hbar\omega}$ detailed-balance ratio — the temperature itself.

The core insight

A weak perturbation $H'=-f(t)\hat A$ gives $\langle\delta A(t)\rangle=\int_{-\infty}^t\chi(t-t')f(t')\,\mathrm{d}t'$ ; the causal kernel's transform admits the spectral representation $\chi(\omega)=\int\rho(\omega')/(\omega'-\omega-i0^+)\,\mathrm{d}\omega'$ with density $\rho(\omega')=\chi''(\omega')/\pi$ , which encodes absorption, dispersion (Kramers–Kronig), passivity $\omega\chi''\ge0$ , and — through the fluctuation–dissipation theorem and the detailed-balance ratio $e^{-\beta\hbar\omega}$ — the temperature.

Where this lives today

Spectral functions are the common language of modern spectroscopy. Angle-resolved photoemission (ARPES) measures the single-particle spectral function $A(\mathbf{k},\omega)=-2\,\mathrm{Im}\,G^R$ directly, mapping quasiparticle dispersions, lifetimes (peak widths), and the "kinks" where electrons couple to phonons or spin fluctuations in cuprate superconductors; the sum rule $\int A\,\mathrm{d}\omega=1$ guarantees consistency of the extracted band structure. Inelastic neutron scattering reads the dynamic structure factor $S(q,\omega)$ to chart magnon and phonon dispersions and their damping, and its Stokes/anti-Stokes asymmetry is a routine local thermometer. Optical conductivity, dielectric loss, and Raman spectra are all $\chi''$ in disguise, tied to the reactive response by Kramers–Kronig. The same object anchors computation. Quantum Monte Carlo and dynamical mean-field theory produce imaginary-time (Matsubara) correlators; the physically measurable spectral function is their analytic continuation, and the notoriously ill-posed inversion is regularized precisely by spectral positivity ( $\rho\ge0$ ) and the sum rules derived here. In quantum-information and circuit-QED settings, the spectral function of a qubit's environment — its noise spectrum weighted by detailed balance — sets decoherence rates, and engineered spectral densities are how dissipation is designed rather than merely tolerated. Wherever "density of excitations," "spectral weight," or "noise spectrum" appears with quantitative intent, the generalized susceptibility's spectral function is the object being invoked.

The physical picture

Push a damped pendulum with an oscillating force. Part of your push swings it back and forth in phase, storing and returning energy each cycle — the reactive response, $\chi'(\omega)$ . Part fights friction and is lost as heat, out of phase — the dissipative response, $\chi''(\omega)$ . Resonances show up as peaks in $\chi''$ and sharp swings in $\chi'$ . The generalized susceptibility is just this in-phase/out-of-phase decomposition, promoted to an exact statement about any equilibrium observable driven by any conjugate field. Causality is the hidden scaffolding. Because the system can only respond after being pushed, the time-domain kernel is one-sided, and a one-sided function has a transform that is the boundary value of a function analytic in the upper half-plane. That single fact chains absorption to dispersion everywhere — measure the absorption and you can compute the refraction. The dissipative part is more than a companion, though: it is literally the spectrum of the system's spontaneous fluctuations. A system at temperature $T$ jiggles on its own, and the frequencies at which it jiggles hardest are exactly the frequencies at which it most readily drinks energy from a drive. Fluctuation and dissipation are one dynamical fingerprint seen from two sides. There is a thermometer buried in the asymmetry. Scatter a probe off the system — a neutron, a photon — and it can either lose energy to the system (Stokes, creating an excitation) or gain energy from a thermally pre-existing excitation (anti-Stokes). At finite temperature the anti-Stokes line is always weaker, because it requires the excitation to be there already, and the exact ratio of anti-Stokes to Stokes intensity is $e^{-\beta\hbar\omega}$ — pure detailed balance. Read that ratio and you have measured the temperature of the sample from the shape of its scattering spectrum alone.

Figure

The dynamic structure factor S(q,ω): Stokes and anti-Stokes peaks whose intensity ratio e^(−βℏω) is a thermometer

Figure

The spectral function χ″(ω) as a comb of transition lines (Lehmann), broadening to a resonance with damping

The core equations

The framework begins, as always, with weak-field linear response and a causal kernel.

\langle\delta A(t)\rangle=\int_{-\infty}^{\infty}\chi(t-t')\,f(t')\,\mathrm{d}t',\qquad \chi(\tau)=0\ \text{for}\ \tau<0 \qquad (1)

where: $\langle\delta A(t)\rangle$ — induced deviation of observable $\hat A$ ; $f(t)$ — generalized force conjugate to $\hat A$ via $H'=-f(t)\hat A$ ; $\chi(\tau)$ — causal (retarded) response function. Setting $\hat A$ to a density, spin, or current makes $\chi$ the dielectric, magnetic, or conductivity response, respectively. The Fourier transform is the generalized susceptibility, analytic in the upper half-plane and split into reactive and dissipative parts.

\chi(\omega)=\int_0^{\infty}\chi(\tau)\,e^{i\omega\tau}\,\mathrm{d}\tau=\chi'(\omega)+i\chi''(\omega) \qquad (2)

where: $\chi'(\omega)$ — reactive part, even in $\omega$ ; $\chi''(\omega)$ — dissipative part, odd in $\omega$ ; analyticity for $\mathrm{Im}\,\omega>0$ follows from causality. The reactive part stores and returns energy; the dissipative part is where energy is absorbed. The dissipative part is the spectral function — the Fourier transform of the equilibrium commutator — and, resolved in energy eigenstates, becomes the Lehmann representation: a comb of lines at the exact transition frequencies.

\chi''(\omega)=\frac{\pi}{\hbar Z}\sum_{m,n}e^{-\beta E_n}\,|\langle m|\hat A|n\rangle|^2\,\big[\delta(\omega-\omega_{mn})-\delta(\omega+\omega_{mn})\big] \qquad (3)

where: $\omega_{mn}=(E_m-E_n)/\hbar$ — exact transition frequency; $|\langle m|\hat A|n\rangle|^2$ — oscillator strength (transition matrix element); $Z=\sum_n e^{-\beta E_n}$ — partition function. Each pair of levels contributes a sharp line; a finite lifetime broadens each $\delta$ into a Lorentzian of width $\gamma$ , which is what ARPES and neutron scattering record. The spectral (Kramers–Kronig / Lehmann) representation writes the full complex susceptibility as a Cauchy integral of the spectral density, generating dispersion, absorption, and the static limit from one object.

\chi(\omega)=\int_{-\infty}^{\infty}\frac{\rho(\omega')}{\omega'-\omega-i0^+}\,\mathrm{d}\omega',\qquad \rho(\omega')=\frac{\chi''(\omega')}{\pi} \qquad (4)

where: $\rho(\omega')$ — spectral density; the $-i0^+$ prescription places the poles just below the real axis (retarded), enforcing causality. Using $1/(x-i0^+)=\mathcal{P}(1/x)+i\pi\delta(x)$ splits this into the real part (a principal-value dispersion integral — the Kramers–Kronig relation) and the imaginary part $\chi''(\omega)=\pi\rho(\omega)$ , recovering the spectral function. The fluctuation–dissipation theorem ties the spectral function to the correlation spectrum through the exact detailed-balance factor; the classical limit is the familiar Nyquist form.

S(\omega)\equiv\int_{-\infty}^{\infty}\!\langle\hat A(t)\hat A(0)\rangle\,e^{i\omega t}\,\mathrm{d}t=2\hbar\,[1+n_B(\omega)]\,\chi''(\omega)\ \xrightarrow{\hbar\omega\ll k_BT}\ \frac{2k_BT}{\omega}\chi''(\omega) \qquad (5)

where: $S(\omega)$ — power spectrum of equilibrium fluctuations; $n_B(\omega)=1/(e^{\beta\hbar\omega}-1)$ — Bose factor; $\beta=1/k_BT$ . Detailed balance $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ is baked in — the origin of the Stokes/anti-Stokes asymmetry. Positivity and sum rules are the exact constraints that survive any approximation. The first-moment ( $f$ -sum) rule fixes total spectral weight through an equal-time commutator.

\omega\,\chi''(\omega)\ge0\ \ (\text{passivity}),\qquad \int_{-\infty}^{\infty}\!\omega\,\chi''(\omega)\,\frac{\mathrm{d}\omega}{\pi}=\frac{1}{\hbar}\big\langle[\dot{\hat A},\hat A]\big\rangle \qquad (6)

where: $\dot{\hat A}=\tfrac{i}{\hbar}[H,\hat A]$ — Heisenberg time derivative. For a density response this becomes the Thomas–Reiche–Kuhn $f$ -sum rule $\int\omega S(q,\omega)\,\mathrm{d}\omega=\hbar q^2 N/2m$ : spectral weight can shift in frequency but its first moment is pinned by a commutator, an exact benchmark for any spectral-function calculation.

Derivation AThe spectral representation and Kramers-Kronig from analyticity of χ(ω)

The goal is to build the entire analytic apparatus — the spectral (Lehmann) representation, the Kramers–Kronig relations, and the identification $\chi''=\pi\rho$ — from the single fact that a causal, retarded susceptibility is analytic in the upper half-plane with poles pushed infinitesimally below the real axis. The strategy is to write $\chi(\omega)$ as a Cauchy integral of its own spectral density with the retarded $-i0^+$ prescription, then apply the Sokhotski–Plemelj identity to separate the principal-value (dispersive) and delta-function (absorptive) pieces. The reward is that dispersion, absorption, and the static limit all fall out of one integral, which is why the spectral representation is the master formula of the whole subject.

Step 1. Posit the retarded spectral representation $\chi(\omega)=\int_{-\infty}^\infty\rho(\omega')/(\omega'-\omega-i0^+)\,\mathrm{d}\omega'$ , with the poles just below the real axis so $\chi$ is analytic for $\mathrm{Im}\,\omega>0$ . Why: causality means $\chi(t)=0$ for $t<0$ , and the $-i0^+$ placement of the poles is exactly the analytic-in-the-upper-half-plane condition written in the frequency integral.

Step 2. Apply the Sokhotski–Plemelj identity $1/(x-i0^+)=\mathcal{P}(1/x)+i\pi\delta(x)$ with $x=\omega'-\omega$ : $\chi(\omega)=\mathcal{P}\!\int\dfrac{\rho(\omega')}{\omega'-\omega}\,\mathrm{d}\omega'+i\pi\,\rho(\omega)$ . Why: the distributional identity cleanly separates the real (principal-value) part from the imaginary (on-shell delta) part of the propagator.

Step 3. Read off the imaginary part: $\chi''(\omega)=\pi\,\rho(\omega)$ , i.e. the spectral density is the dissipative response up to $\pi$ . Why: this identifies $\rho(\omega)=\chi''(\omega)/\pi$ — the spectral function is precisely the absorptive part, the object measured in every scattering and absorption experiment.

Step 4. Substitute $\rho=\chi''/\pi$ back into the real part to get the Kramers–Kronig relation $\chi'(\omega)=\tfrac1\pi\mathcal{P}\!\int\chi''(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . Why: dispersion is now the Hilbert transform of absorption — the same relation the contour derivation gives, here as a corollary of the spectral representation.

Step 5. Set $\omega=0$ to obtain the static susceptibility as a spectral integral, $\chi(0)=\tfrac2\pi\int_0^\infty\chi''(\omega')/\omega'\,\mathrm{d}\omega'$ , using that $\chi''$ is odd. Why: the thermodynamic (zero-frequency) response is the inverse-frequency-weighted integral of all dissipative spectral weight — tying equilibrium thermodynamics to the full dynamical spectrum in one line. *Verified: verify_aq-sm-generalized-susceptibility-spectral-functions.py confirms $\chi''=\pi\rho$ , the Sokhotski–Plemelj split, the KK relation, and the static sum rule on the damped-oscillator spectral density.*

Go deeper · Derivation B · 1 minDetailed balance and the Stokes/anti-Stokes ratio from the Lehmann representation

Derivation BDetailed balance and the Stokes/anti-Stokes ratio from the Lehmann representation

The goal is to show that the correlation spectrum obeys the exact quantum detailed-balance relation $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ , and that this fixes both the fluctuation–dissipation theorem $S=2\hbar[1+n_B]\chi''$ and the Stokes/anti-Stokes intensity ratio measured in scattering. The strategy is to write the power spectrum in the energy eigenbasis, integrate the time exponential to get a comb of $\delta$ -functions, and then relabel the summation indices $m\leftrightarrow n$ to expose how the Boltzmann weight transforms under $\omega\to-\omega$ . The pivot is that swapping which level is initial and which is final flips the sign of the transferred energy while re-weighting by the Boltzmann factor.

Step 1. Write $S(\omega)=\tfrac{2\pi}{Z}\sum_{m,n}e^{-\beta E_n}|\langle m|\hat A|n\rangle|^2\,\delta(\omega-\omega_{mn})$ by inserting complete sets and integrating the time exponential. Why: the equilibrium thermal average over energy eigenstates turns the two-time correlation into a manifestly positive comb of transition lines weighted by initial-state Boltzmann factors.

Step 2. Evaluate $S(-\omega)$ and relabel $m\leftrightarrow n$ : the delta becomes $\delta(\omega-\omega_{mn})$ again, the matrix element is unchanged (Hermiticity), but the Boltzmann weight is now $e^{-\beta E_m}=e^{-\beta E_n}e^{-\beta\hbar\omega_{mn}}$ . Why: relabelling exchanges initial and final states, and the extra Boltzmann factor $e^{-\beta\hbar\omega}$ is exactly the population difference between them — the origin of detailed balance.

Step 3. Conclude $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ (the KMS condition). Why: the energy-gain (anti-Stokes) process needs a thermally excited initial state, so it is suppressed relative to the energy-loss (Stokes) process by precisely the Boltzmann population ratio.

Step 4. Form the commutator spectrum $2\hbar\chi''(\omega)=S(\omega)-S(-\omega)=(1-e^{-\beta\hbar\omega})S(\omega)$ and invert: $S(\omega)=2\hbar\,\chi''(\omega)/(1-e^{-\beta\hbar\omega})=2\hbar[1+n_B(\omega)]\chi''(\omega)$ . Why: the commutator is the difference of forward and backward spectra, so dividing it out yields the fluctuation–dissipation theorem; expanding for $\hbar\omega\ll k_BT$ recovers the classical $2k_BT\chi''/\omega$ .

Step 5. Identify the scattering observable: the ratio of anti-Stokes to Stokes intensity is $S(-\omega)/S(\omega)=e^{-\beta\hbar\omega}$ , a thermometer read directly from the spectrum. Why: measuring the peak-height ratio at energy transfer $\hbar\omega$ returns $\beta=1/k_BT$ with no calibration — Landau–Placzek and Raman thermometry rest on exactly this identity. *Verified: verify_aq-sm-generalized-susceptibility-spectral-functions.py confirms $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ , the FDT assembly $S=2\hbar[1+n_B]\chi''$ , its classical limit, and the ratio $e^{-2}\approx0.135$ at $\hbar\omega=2k_BT$ .*

Limiting cases

The formalism specializes to familiar cases in known limits:

Case	Result	Meaning
Static response	$\chi(0)=\dfrac2\pi\int_0^\infty\dfrac{\chi''(\omega)}{\omega}\,\mathrm{d}\omega$	Zero-frequency KK: thermodynamic susceptibility is the low-energy spectral weight.
Classical (high- $T$ ) FDT	$S(\omega)=\dfrac{2k_BT}{\omega}\,\chi''(\omega)$	$2\hbar[1+n_B]\to2k_BT/\omega$ ; recovers Nyquist noise and the Einstein relation.
Quantum / zero temperature	$S(\omega)=2\hbar\,\Theta(\omega)\,\chi''(\omega)$	$n_B\to0$ : only spontaneous emission survives, the spectrum is one-sided.
Single relaxation (Debye/OU)	$\chi''(\omega)=\chi_0\dfrac{\omega\tau}{1+\omega^2\tau^2}$	Lorentzian absorption peaked at $\omega=1/\tau$ ; the Ornstein–Uhlenbeck case.
Damped oscillator	$\chi''(\omega)=\dfrac{1}{m}\dfrac{\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}$	Resonant absorption at $\omega\approx\omega_0$ of width $\gamma$ ; optical and mechanical response.

Common trapχ″ is the spectral object, and passivity constrains ωχ″ — not χ″ itself

Conventions multiply and mislead. Some texts define $\chi$ via $H'=-f\hat A$ , others via $+f\hat A$ , flipping the sign of $\chi''$ . The response, relaxation, and correlation functions differ by time derivatives and factors of $\beta$ — they are not interchangeable. Passivity constrains $\omega\chi''(\omega)\ge0$ , not $\chi''$ alone: since $\chi''$ is odd, it is negative for $\omega<0$ , and only the product $\omega\chi''$ must be non-negative. Finally, the classical FDT $S=2k_BT\chi''/\omega$ is only the high-temperature limit of $S=2\hbar[1+n_B]\chi''$ ; using it where $\hbar\omega\gtrsim k_BT$ discards zero-point noise and the Stokes/anti-Stokes asymmetry.

Worked examples

Example 1 — The damped-oscillator spectral function: peak, width, and passivity. This computes a real spectral function and verifies its exact constraints. Take a single damped mode $\chi(\omega)=(1/m)/(\omega_0^2-\omega^2-i\gamma\omega)$ with $\omega_0=1$ , $\gamma=0.1$ , $m=1$ (units of $\omega_0$ ).

Step 1. Extract the spectral function: $\chi''(\omega)=\dfrac{1}{m}\dfrac{\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}$ . At resonance $\omega=\omega_0=1$ : $\chi''(\omega_0)=\dfrac{\gamma\omega_0}{\gamma^2\omega_0^2}=\dfrac{1}{\gamma\omega_0}=\dfrac{1}{0.1}=\mathbf{10.0}$ .

Step 2. Find the width: the loss peak has full width at half maximum $\approx\gamma=0.1$ for weak damping, so weaker damping gives a taller, narrower line — the peak height scales as $1/(m\gamma\omega_0)$ .

Step 3. Check passivity by sampling: at $\omega=+0.5$ , $\chi''=0.1\cdot0.5/[(0.75)^2+(0.05)^2]=0.05/0.565=0.0885>0$ , and $\omega\chi''=0.044>0$ ; at $\omega=-0.5$ , $\chi''=-0.0885<0$ but $\omega\chi''=+0.044>0$ . Passivity $\omega\chi''\ge0$ holds on both sides — as required for an equilibrium system that can only absorb.

Step 4. Verify the static sum rule numerically: $\chi(0)=\tfrac2\pi\int_0^\infty\chi''(\omega)/\omega\,\mathrm{d}\omega$ . The integrand is $\tfrac1m\,\gamma/[(\omega_0^2-\omega^2)^2+\gamma^2\omega^2]$ ; the integral returns $1/(m\omega_0^2)=\mathbf{1.0}$ , matching $\chi(0)=1/(m\omega_0^2)$ read directly from $\chi(\omega{=}0)$ . The spectral weight reconstructs the static response.

Example 2 — Reading temperature off a scattering spectrum. This turns the detailed-balance ratio into a working thermometer. A neutron-scattering experiment on a magnet at temperature $T$ finds an excitation at energy transfer $\hbar\omega=2k_BT$ ; measure the anti-Stokes/Stokes peak-height ratio.

Step 1. Detailed balance gives the ratio directly: $\dfrac{S(-\omega)}{S(\omega)}=e^{-\beta\hbar\omega}=e^{-\hbar\omega/k_BT}$ .

Step 2. Insert $\hbar\omega=2k_BT$ : $e^{-2}=\mathbf{0.135}$ . The energy-gain (anti-Stokes) peak is only 13.5% as intense as the energy-loss (Stokes) peak.

Step 3. Invert to thermometry: if instead the measured ratio were $R=0.5$ , then $\hbar\omega/k_BT=-\ln R=\ln2=0.693$ , so $T=\hbar\omega/(0.693\,k_B)$ . For an excitation at $\hbar\omega=1$ meV this gives $k_BT=1.44\,\mathrm{meV}$ , i.e. $T=1.44\times10^{-3}\,\mathrm{eV}/(8.617\times10^{-5}\,\mathrm{eV/K})\approx\mathbf{16.7\ K}$ .

Step 4. Draw the lesson: the shape of the scattering spectrum encodes the temperature through the exact detailed-balance factor $e^{-\beta\hbar\omega}$ , with no calibration standard. A scattering spectrum is a thermometer, which is why Stokes/anti-Stokes ratios are used to measure local temperatures in Raman microscopy and neutron spectroscopy.

Origins · 1 minspectral function and how this was discovered

spectral function

concept — the dissipative response resolved in frequency · The spectral function is the frequency-resolved dissipative part of a generalized susceptibility, χ″(ω), equivalently the Fourier transform of an equilibrium commutator or correlation. Its Lehmann (spectral) representation writes it as a weighted comb of δ-functions at the exact transition frequencies of the many-body system, each weighted by a matrix element; finite lifetimes broaden the lines into Lorentzians. The concept was formalized through Kubo's 1957 linear-response theory and the Callen–Welton 1951 fluctuation–dissipation theorem, and it is the object that absorption spectroscopy, inelastic neutron and light scattering, and angle-resolved photoemission (ARPES) directly measure. Its sum rules conserve spectral weight, its positivity ωχ″≥0 enforces passivity, and its analytic continuation connects the measured retarded response to the imaginary-time Matsubara correlators computed in many-body theory — making it the bridge between what is calculated and what is seen.

History

1928

— Nyquist relates resistor voltage noise to dissipation, an early spectral-function result

1951

— Callen and Welton prove the fluctuation–dissipation theorem, linking χ″ to the fluctuation spectrum

1954

— Van Hove introduces the dynamic structure factor S(q,ω) for neutron scattering

1957

— Kubo formulates linear response and the spectral (Lehmann-type) representation of susceptibilities

1959

— Martin and Schwinger cast equilibrium correlation functions in the KMS / spectral framework

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 (fluctuation–dissipation, response functions) · Kardar, Statistical Physics of Particles, ch. on response and correlations · Landau & Lifshitz, Statistical Physics Part 1, §123–126 (generalized susceptibility, dissipation) · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 15 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 10 · Kubo, Toda & Hashitsume, Statistical Physics II, Ch. 4 · D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions.

Provenance. Equations and limits follow Landau–Lifshitz §123–126, Pathria & Beale Ch. 15, Kubo–Toda–Hashitsume Ch. 4, and Forster. A verifier recomputes: (1) the spectral representation and Sokhotski–Plemelj split giving $\chi''=\pi\rho$ and the KK relation; (2) the Lehmann/detailed-balance relation $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ and its assembly into $S=2\hbar[1+n_B]\chi''$ ; (3) the classical limit $S\to2k_BT\chi''/\omega$ ; (4) the damped-oscillator $\chi''(\omega_0)=1/(\gamma\omega_0)=10$ , the passivity check $\omega\chi''\ge0$ , and the static sum rule $\chi(0)=1/(m\omega_0^2)=1$ by numerical integration; (5) the thermometry numbers $e^{-2}=0.135$ at $\hbar\omega=2k_BT$ and $T\approx16.7$ K for ratio $0.5$ at $\hbar\omega=1$ meV. Standard textbook values (TRK $f$ -sum rule $\hbar q^2 N/2m$ , Nyquist $4k_BTR$ , ARPES sum rule $\int A\,\mathrm{d}\omega=1$ ) are quoted.

Builds on: The Fluctuation-Dissipation Theorem · The Langevin Equation · The Ornstein-Uhlenbeck Process · Linear algebra of correlation functions. Leads to: Kubo formula and transport coefficients · The Einstein Relation · Brownian Motion · Many-body Green's functions.

Why are the Stokes and anti-Stokes peaks unequal, and how does temperature drop out of a "formal" spectral representation?

The spectral representation itself is formal, but the weighting of its lines is thermal. In the Lehmann sum each transition $|n\rangle\to|m\rangle$ carries the Boltzmann factor $e^{-\beta E_n}$ of its initial state. Energy loss (Stokes) starts from a lower state; energy gain (anti-Stokes) starts from a thermally excited higher state, which is rarer by $e^{-\beta\hbar\omega}$ . Relabelling the sum shows $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ exactly. So temperature enters not through the analytic structure (which is pure causality) but through the equilibrium populations weighting the spectral lines — the "formal" representation is thermal because the ensemble average is.

What is the difference between the spectral function χ″ and the power spectrum S(ω)?

They carry the same information but differ by the detailed-balance factor. $\chi''(\omega)$ is the dissipative response — the antisymmetric combination $S(\omega)-S(-\omega)=2\hbar\chi''$ built from the commutator, and it is odd in $\omega$ . $S(\omega)$ is the fluctuation spectrum — the correlation-function transform, manifestly positive, and asymmetric because of detailed balance. The FDT relates them: $S(\omega)=2\hbar[1+n_B(\omega)]\chi''(\omega)$ . Classically ( $\hbar\omega\ll k_BT$ ) they become proportional, $S\approx2k_BT\chi''/\omega$ , and the distinction blurs — but quantum-mechanically $S$ knows about zero-point motion and $\chi''$ does not, so they genuinely differ.

Why does the static susceptibility come from an inverse-frequency integral of χ″?

Setting $\omega=0$ in the Kramers–Kronig relation and using that $\chi''$ is odd gives $\chi(0)=\tfrac2\pi\int_0^\infty\chi''(\omega')/\omega'\,\mathrm{d}\omega'$ . Physically, the static (thermodynamic) response is how the system settles under a constant field, and that settling is the accumulated effect of all its dynamical modes weighted by how slowly they respond — hence the $1/\omega'$ weighting, which emphasizes low-frequency spectral weight. Because passivity gives $\chi''/\omega'\ge0$ , the integral is manifestly positive: the static susceptibility of a stable system cannot be negative, a thermodynamic stability statement recovered from the spectrum.

What does it mean that the spectral function is a sum of delta functions?

For an isolated finite quantum system the exact spectral function is a comb: $\chi''(\omega)=\pi\sum$ (weights) $\times[\delta(\omega-\omega_{mn})-\delta(\omega+\omega_{mn})]$ , one sharp line at each transition frequency $\omega_{mn}=(E_m-E_n)/\hbar$ , with weight equal to the squared matrix element (oscillator strength). Real systems couple to a continuum or have finite lifetimes, which broadens each $\delta$ into a Lorentzian of width $\gamma\sim1/\tau$ . So a measured absorption or ARPES peak of finite width is a broadened spectral line, and its area is the oscillator strength while its width is the inverse lifetime — both physical observables read straight off the spectral function.

How does the generalized susceptibility connect to many-body Green's functions?

The retarded susceptibility $\chi(\omega)$ is the analytic continuation to real frequency of the imaginary-time (Matsubara) correlation function computed diagrammatically. Many-body theory calculates $\chi(i\omega_n)$ on the discrete Matsubara frequencies $\omega_n=2\pi n/\beta\hbar$ ; continuing $i\omega_n\to\omega+i0^+$ gives the measurable retarded response, with spectral density $\rho=\chi''/\pi$ playing the role of the analytic-continuation kernel. The single-particle spectral function $A(\mathbf{k},\omega)=-2\,\mathrm{Im}\,G^R(\mathbf{k},\omega)$ measured by ARPES is exactly this object for the electron Green's function, and its sum rule $\int A\,\mathrm{d}\omega=1$ is the anticommutation-relation constraint. The generalized susceptibility is thus the bridge between what theory computes (Matsubara) and what experiment sees (retarded spectra).

What do sum rules actually constrain, and why are they useful?

Sum rules fix moments of the spectral function to exact equal-time (thermodynamic or commutator) quantities that are far easier to compute than the full spectrum. The zeroth moment (static sum rule) gives $\chi(0)$ ; the first moment ( $f$ -sum rule) gives $\int\omega\chi''\,\mathrm{d}\omega/\pi=\langle[\dot{\hat A},\hat A]\rangle/\hbar$ , which for density counts particles (Thomas–Reiche–Kuhn). Their power is twofold: they let you calibrate absolute spectral intensities from a measured lineshape, and they test any approximate calculation — a diagrammatic or numerical spectral function that violates a sum rule is provably wrong. Because spectral weight can move in frequency but not appear or vanish, sum rules also track physics like the superconducting condensate's "missing weight."