Linear Response Formula Sheet

Linear response predicts the first change in a detector observable caused by a weak source. The formulas are compact only after the source sign, coupled operator, detector, reference state, Fourier transform, contact terms, and limiting protocol have been fixed.

This sheet uses one convention throughout:

$$\begin{aligned} H_{\mathrm{pert}}(t) &= -f(t)B, \\ \chi_{AB}^{\mathrm R}(t) &= \frac{i}{\hbar} \theta(t) \\ &\quad\times \langle[A(t),B(0)]\rangle_0. \end{aligned}$$

With the transform $e^{+i\omega t}$, a passive Hermitian autochannel has

$$\operatorname{Im} \chi_{AA}^{\mathrm R}(\omega) \geq 0 \qquad (\omega>0).$$

Changing the source sign or retarded-kernel sign changes intermediate formulas. It must not change measurable absorbed power after all definitions are transformed consistently.

Canonical Scope

This page is a lookup and consistency sheet. It collects:

• the source-to-detector response kernel;
• space-time and Fourier-domain forms;
• Lehmann and response-spectral representations;
• absorption, causality, and Kramers–Kronig checks;
• equilibrium fluctuation–dissipation conversions;
• static Kubo–Mori response and noncommuting limits;
• contact terms, conductivity, diffusion, and common source dictionaries;
• compact oscillator and two-level benchmarks.

Kubo Formula owns the density-operator derivation and physical applications. Retarded and Advanced Response owns the full analytic structure. Susceptibilities owns units, protocols, tensor channels, and named susceptibilities. Fluctuation–Dissipation Theorem owns the KMS derivation. Transport Coefficients Preview owns hydrodynamic and transport interpretation.

Minimum Response Ledger

Before quoting a susceptibility, state:

1. the unperturbed generator and reference state;
2. the perturbation $H_{\mathrm{pert}}$, including its sign;
3. the source $f_b$ and its units;
4. the coupled operator $B_b$;
5. the detector $A_a$ and whether it depends explicitly on the source;
6. total, density, per-site, per-volume, and internal-index normalizations;
7. real-space, momentum-space, real-frequency, energy, or Matsubara variables;
8. Fourier signs and factors of $2\pi$ and $\hbar$;
9. the order of thermodynamic, zero-frequency, zero-wavevector, and regulator limits;
10. whether the desired object is isolated dynamical response or an equilibrated thermodynamic derivative.

The ordered label $\chi_{AB}$ means “response of $A$ to a source coupled through $B$.” It does not generally equal $\chi_{BA}$.

Master Kubo Formula

Let a multicomponent source couple as

$$H_{\mathrm{pert}}(t) = -\sum_b \int d^d r\, f_b(\mathbf r,t) B_b(\mathbf r).$$

For a detector $A_a(\mathbf r,t)$ with no explicit source dependence,

$$\begin{aligned} \delta\langle A_a(\mathbf r,t) \rangle &={} \\[-0.2em] &\quad \sum_b \int_{-\infty}^{\infty} dt' \int d^d r' \\ &\quad\times \chi_{ab}^{\mathrm R} (\mathbf r,t;\mathbf r',t') \\ &\quad\times f_b(\mathbf r',t') + O(f^2), \end{aligned}$$

where

$$\begin{aligned} \chi_{ab}^{\mathrm R} (\mathbf r,t;\mathbf r',t') &={} \\[-0.2em] &\quad \frac{i}{\hbar} \theta(t-t') \\ &\quad\times \bigl\langle \bigl[ A_a(\mathbf r,t), \\ &\qquad B_b(\mathbf r',t') \bigr] \bigr\rangle_0. \end{aligned}$$

The operators evolve with the unperturbed generator. The expectation value is evaluated in the unperturbed state.

The operational definition is

$$\chi_{ab}^{\mathrm R} = \left. \frac{ \delta\langle A_a\rangle_f }{ \delta f_b } \right|_{f=0}$$

when $A_a$ has no explicit dependence on $f_b$. Consequently,

$$[\chi_{ab}] = \frac{[A_a]}{[f_b]}.$$

One-line sign check

In the interaction picture,

$$\delta\rho_I(t) = \frac{i}{\hbar} \int_{t_0}^{t} dt'\, f(t') [B_I(t'),\rho_0].$$

Cyclicity of the trace gives

$$\begin{aligned} \delta\langle A(t)\rangle &= \frac{i}{\hbar} \int_{t_0}^{t} dt'\, f(t') \\ &\quad\times \langle[A_I(t),B_I(t')]\rangle_0. \end{aligned}$$

This is the quickest way to recover the response sign from $H_{\mathrm{pert}}=-fB$ instead of relying on memory.

Explicit Source Dependence and Contact Terms

If the measured operator depends on the source, the commutator is not the full functional derivative. Write

$$A_a[f] = A_a^{(0)} + \sum_b \mathcal C_{ab}f_b + O(f^2).$$

For a local instantaneous dependence,

$$\begin{aligned} K_{ab}^{\mathrm R} (\mathbf r,t;\mathbf r',t') &={} \\[-0.2em] &\quad \chi_{A_a^{(0)}B_b}^{\mathrm R} (\mathbf r,t;\mathbf r',t') \\ &\quad+ \langle\mathcal C_{ab}\rangle_0 \delta^{(d)}(\mathbf r-\mathbf r') \\ &\qquad{}\times \delta(t-t'). \end{aligned}$$

The full response is

$$\delta\langle A_a\rangle = \sum_b K_{ab}^{\mathrm R} *f_b.$$

Depending on context, $\mathcal C$ is called a contact, diamagnetic, stress, or seagull term. Derive it from the source-dependent Hamiltonian or observable. Omitting it can violate gauge invariance, continuity constraints, and sum rules.

Stationarity and Fourier Dictionary

If

$$[H_0,\rho_0] = 0,$$

then

$$\chi^{\mathrm R}(t,t') = \chi^{\mathrm R}(t-t').$$

For a translation-invariant state and geometry,

$$\chi^{\mathrm R} (\mathbf r,t;\mathbf r',t') = \chi^{\mathrm R} (\mathbf r-\mathbf r',t-t').$$

This sheet uses

$$\begin{aligned} X(\mathbf q,\omega) &={} \\[-0.2em] &\quad \int_{-\infty}^{\infty} dt \int d^d r\, e^{i\omega t-i\mathbf q\cdot\mathbf r} \\ &\quad\times X(\mathbf r,t), \\ X(\mathbf r,t) &={} \\[-0.2em] &\quad \int\frac{d\omega}{2\pi} \int\frac{d^d q}{(2\pi)^d} \\ &\quad\times e^{-i\omega t+i\mathbf q\cdot\mathbf r} X(\mathbf q,\omega). \end{aligned}$$

The convolution becomes

$$\delta\langle A_a(\mathbf q,\omega) \rangle = \sum_b K_{ab}^{\mathrm R} (\mathbf q,\omega) f_b(\mathbf q,\omega).$$

For

$$f(t) = \operatorname{Re} \left[ f_0e^{-i\omega t} \right],$$

the steady first-order response is

$$\delta\langle A(t)\rangle = \operatorname{Re} \left[ \chi_{AB}^{\mathrm R}(\omega) f_0e^{-i\omega t} \right].$$

Under this transform:

• retarded kernels are analytic for $\operatorname{Im}z>0$;
• stable retarded poles lie below the real axis;
• the retarded Lehmann denominator carries $+i0$;
• $E=\hbar\omega$ converts angular frequency to energy.

Lehmann Lookup

For

$$H_0\lvert n\rangle = E_n\lvert n\rangle, \qquad \rho_0 = \sum_n p_n\lvert n\rangle\langle n\rvert,$$

define

$$A_{nm} = \langle n\lvert A\rvert m\rangle, \qquad B_{mn} = \langle m\lvert B\rvert n\rangle.$$

Then

$$\chi_{AB}^{\mathrm R}(\omega) = \sum_{n,m} \frac{ (p_m-p_n) A_{nm}B_{mn} }{ \hbar\omega + E_n-E_m + i0 }.$$

The four ingredients are transition energies, detector matrix elements, source matrix elements, and population differences. A state can exist in the spectrum yet remain absent from a chosen response because either operator matrix element vanishes.

For numerical work, replace $i0$ by a finite width only after declaring whether it represents instrumental resolution, physical damping, a finite observation window, or visualization. A broadening parameter is not generated by the exact finite-system formula.

Response Spectral Density

Define

$$\begin{aligned} \rho_{AB}(\omega) \equiv{}& \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \\ &\times \langle[A(t),B(0)]\rangle_0. \end{aligned}$$

For $\operatorname{Im}z\ne0$ and adequate high-frequency behavior,

$$\chi_{AB}(z) = \int_{-\infty}^{\infty} d\omega'\, \frac{ \rho_{AB}(\omega') }{ \omega'-z }.$$

The retarded boundary value is

$$\begin{aligned} \chi_{AB}^{\mathrm R}(\omega) ={}& \operatorname{PV} \int_{-\infty}^{\infty} d\omega'\, \frac{ \rho_{AB}(\omega') }{ \omega'-\omega } \\ &+ i\pi\rho_{AB}(\omega). \end{aligned}$$

Therefore

$$\operatorname{Im} \chi_{AB}^{\mathrm R}(\omega) = \pi\rho_{AB}(\omega).$$

For a Hermitian autochannel in a stationary passive equilibrium state,

$$\begin{aligned} \rho_{AA}(-\omega) &= -\rho_{AA}(\omega), \\ \operatorname{Re} \chi_{AA}^{\mathrm R}(-\omega) &= \operatorname{Re} \chi_{AA}^{\mathrm R}(\omega), \\ \operatorname{Im} \chi_{AA}^{\mathrm R}(-\omega) &= -\operatorname{Im} \chi_{AA}^{\mathrm R}(\omega), \end{aligned}$$

and

$$\operatorname{Im} \chi_{AA}^{\mathrm R}(\omega) \geq 0 \qquad (\omega>0).$$

These parity and positivity statements do not apply component by component to a generic cross-susceptibility.

Absorption and Passivity

For a Hermitian source channel driven harmonically,

$$\overline P = \frac{\omega}{2} \lvert f_0\rvert^2 \operatorname{Im} \chi_{BB}^{\mathrm R}(\omega).$$

Thus the imaginary part is absorptive under the conventions of this sheet. The real part is reactive.

For several source components, define the dissipative matrix

$$\boldsymbol{\chi}'' \equiv \frac{ \boldsymbol{\chi}^{\mathrm R} - \boldsymbol{\chi}^{\mathrm R\dagger} }{ 2i }.$$

Then

$$\overline P = \frac{\omega}{2} \mathbf f_0^\dagger \boldsymbol{\chi}'' \mathbf f_0.$$

Passivity requires $\boldsymbol{\chi}''$ to be positive semidefinite for $\omega>0$. An individual off-diagonal entry can have either sign.

Causality and Dispersion

Causal support,

$$\chi_{AB}^{\mathrm R}(t) = 0 \qquad (t<0),$$

implies upper-half-plane analyticity. When no subtraction is needed,

$$\begin{aligned} \operatorname{Re} \chi_{AB}^{\mathrm R}(\omega) &= \frac1\pi \operatorname{PV} \\ &\quad\times \int_{-\infty}^{\infty} d\omega'\, \frac{ \operatorname{Im} \chi_{AB}^{\mathrm R}(\omega') }{ \omega'-\omega }, \end{aligned}$$

and

$$\begin{aligned} \operatorname{Im} \chi_{AB}^{\mathrm R}(\omega) &= -\frac1\pi \operatorname{PV} \\ &\quad\times \int_{-\infty}^{\infty} d\omega'\, \frac{ \operatorname{Re} \chi_{AB}^{\mathrm R}(\omega') }{ \omega'-\omega }. \end{aligned}$$

Slow high-frequency decay requires subtracted dispersion relations. A finite experimental frequency window does not generally support an accurate direct Hilbert transform without tail modeling.

The advanced kernel is

$$\chi_{AB}^{\mathrm A}(t) = -\frac{i}{\hbar} \theta(-t) \langle[A(t),B(0)]\rangle_0.$$

For a channel matrix on the real axis,

$$\boldsymbol{\chi}^{\mathrm A}(\omega) = \boldsymbol{\chi}^{\mathrm R\dagger}(\omega).$$

Equilibrium Fluctuation–Dissipation Dictionary

Let

$$\delta A = A-\langle A\rangle_\beta, \qquad \delta B = B-\langle B\rangle_\beta.$$

Using angular-frequency spectra without a $2\pi$ prefactor, define

$$\begin{aligned} S_{AB}^{>}(\omega) &\equiv \int_{-\infty}^{\infty} dt\, e^{i\omega t} \langle\delta A(t)\delta B(0)\rangle_\beta, \\ S_{AB}^{<}(\omega) &\equiv \int_{-\infty}^{\infty} dt\, e^{i\omega t} \langle\delta B(0)\delta A(t)\rangle_\beta. \end{aligned}$$

For one equilibrium thermal generator and a compatible operator pair,

$$S_{AB}^{<}(\omega) = e^{-\beta\hbar\omega} S_{AB}^{>}(\omega).$$

The absorptive response is

$$\operatorname{Im} \chi_{AB}^{\mathrm R}(\omega) = \frac{ S_{AB}^{>}(\omega) - S_{AB}^{<}(\omega) }{ 2\hbar }.$$

Equivalently,

$$S_{AB}^{>}(\omega) = \frac{ 2\hbar }{ 1-e^{-\beta\hbar\omega} } \operatorname{Im} \chi_{AB}^{\mathrm R}(\omega).$$

For a Hermitian autochannel,

$$\begin{aligned} S_{AA}^{\mathrm{sym}}(\omega) &\equiv \frac12 \int_{-\infty}^{\infty} dt\, e^{i\omega t} \\ &\quad\times \bigl\langle \bigl\{ \delta A(t), \delta A(0) \bigr\} \bigr\rangle_\beta. \end{aligned}$$

The fluctuation–dissipation theorem is

$$S_{AA}^{\mathrm{sym}}(\omega) = \hbar \coth\left( \frac{\beta\hbar\omega}{2} \right) \operatorname{Im} \chi_{AA}^{\mathrm R}(\omega).$$

In the classical low-frequency regime,

$$\beta\hbar\lvert\omega\rvert \ll 1,$$

this becomes

$$S_{AA}^{\mathrm{sym}}(\omega) \simeq \frac{ 2k_{\mathrm B}T }{ \omega } \operatorname{Im} \chi_{AA}^{\mathrm R}(\omega).$$

The full theorem requires thermal equilibrium, not merely a stationary state. For a grand-canonical ensemble, number-changing operators must use a consistent generator so that the detailed-balance exponent includes chemical work correctly.

Static Thermodynamic Response

For

$$H_f = H_0-fB,$$

the isothermal self-susceptibility is

$$\chi_T^{BB} \equiv \left. \frac{ \partial\langle B\rangle_f }{ \partial f } \right|_{f=0}.$$

In the canonical ensemble,

$$\chi_T^{BB} = \int_0^\beta d\lambda\, \left\langle \delta B(-i\hbar\lambda) \delta B(0) \right\rangle_\beta,$$

where

$$B(-i\hbar\lambda) = e^{\lambda H_0} B e^{-\lambda H_0}.$$

Equivalently,

$$\chi_T^{BB} = \frac1\hbar \int_0^{\beta\hbar} d\tau\, \langle\delta B(\tau)\delta B(0)\rangle_\beta.$$

If

$$[B,H_0] = 0,$$

then

$$\chi_T^{BB} = \beta \left( \langle B^2\rangle_\beta - \langle B\rangle_\beta^2 \right).$$

For a nondegenerate zero-temperature ground state,

$$\chi_{T=0}^{BB} = 2 \sum_{n>0} \frac{ \lvert\langle n\lvert B\rvert0\rangle\rvert^2 }{ E_n-E_0 }.$$

The thermodynamic derivative need not equal the zero-frequency isolated retarded response. In particular, if $B$ is conserved,

$$\chi_{BB}^{\mathrm R}(t) = 0,$$

while $\chi_T^{BB}$ can remain nonzero because equilibration changes statistical weights among sectors.

Order of Limits

For a spatial response, distinguish

$$\chi_{\mathrm{static}} = \lim_{\mathbf q\to0} \lim_{\omega\to0} \chi(\mathbf q,\omega)$$

from

$$\chi_{\mathrm{uniform}} = \lim_{\omega\to0} \lim_{\mathbf q\to0} \chi(\mathbf q,\omega).$$

Hydrodynamic poles and conservation laws can make these limits unequal. Transport may additionally require an explicit order for

$$\begin{aligned} L&\to\infty, & \eta&\to0^+, \\ \mathbf q&\to0, & \omega&\to0. \end{aligned}$$

A finite isolated spectrum usually gives delta functions, recurrences, or a vanishing uniform commutator response rather than a bulk dc coefficient.

Common Source and Detector Channels

Physical source	Coupled operator	Typical detector	Essential convention
force $F$	position $x$	position or velocity	$H_{\mathrm{pert}}=-Fx$
spin field $h^\beta$ in energy units	spin $S^\beta$	spin $S^\alpha$	add $g\mu_{\mathrm B}$ when converting to magnetic-field units
local chemical potential $\delta\mu$	number density $n$	density $n$	distinguish $\partial n/\partial\mu$ from $n^{-2}\partial n/\partial\mu$
vector potential $\mathbf A$	paramagnetic current $\mathbf J^p$	physical current $\mathbf j[\mathbf A]$	include the diamagnetic or stress term
pairing source $\eta$	pair operator and its adjoint	pair amplitude	declare charge, phase, and Nambu normalization
strain or metric source	stress tensor	stress tensor	include instantaneous elastic terms

The source should be defined by its Hamiltonian coupling, not only by a verbal label.

Magnetic and Density Lookups

For a spin source in energy units,

$$H_{\mathrm{pert}}(t) = -\sum_{j,\beta} h_j^\beta(t) S_j^\beta,$$

the response is

$$\delta\langle S_i^\alpha(\omega) \rangle = \sum_{j,\beta} \chi_{S_i^\alpha S_j^\beta}^{\mathrm R}(\omega) h_j^\beta(\omega).$$

Magnetic susceptibility quoted per tesla acquires the appropriate powers of $g\mu_{\mathrm B}$ and any volume or molar normalization.

For a local chemical-potential source,

$$H_{\mathrm{pert}}(t) = -\int d^d r\, \delta\mu(\mathbf r,t) n(\mathbf r),$$

one has

$$\delta n(\mathbf q,\omega) = \chi_{nn}^{\mathrm R}(\mathbf q,\omega) \delta\mu(\mathbf q,\omega).$$

Two common compressibilities are

$$\kappa_n = \left( \frac{\partial n}{\partial\mu} \right)_T, \qquad \kappa_T = \frac1{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T.$$

At exactly $\mathbf q=0$, a conserved total number has no finite-frequency isolated self-response. The thermodynamic compressibility is obtained from the static equilibrated protocol.

Conductivity Lookup

For continuum particles of signed charge $Q$ in a uniform vector potential,

$$H[\mathbf A] = \sum_i \frac{ (\mathbf p_i-Q\mathbf A)^2 }{ 2m } + V.$$

Expanding gives

$$H[\mathbf A] = H_0 - \mathbf A\cdot\mathbf J^p + \frac{NQ^2}{2m} \mathbf A^2.$$

The physical current density is

$$\mathbf j[\mathbf A] = -\frac1V \frac{\partial H}{\partial\mathbf A} = \mathbf j^p - \frac{nQ^2}{m} \mathbf A.$$

Define

$$\begin{aligned} \Pi_{\alpha\beta}^{\mathrm R}(\omega) &\equiv \frac1V \chi_{J_\alpha^pJ_\beta^p}^{\mathrm R}(\omega), \\ D_{\alpha\beta} &\equiv \frac{nQ^2}{m} \delta_{\alpha\beta}. \end{aligned}$$

Since

$$\mathbf E(\omega) = i\omega\mathbf A(\omega),$$

the optical conductivity is

$$\sigma_{\alpha\beta}(\omega) = \frac{i}{\omega+i0} \left[ D_{\alpha\beta} - \Pi_{\alpha\beta}^{\mathrm R}(\omega) \right].$$

On a lattice, $D_{\alpha\beta}$ is the appropriate kinetic or stress expectation obtained by differentiating the gauged Hamiltonian. It is not generally $nQ^2/m$.

A standard decomposition is

$$\operatorname{Re} \sigma(\omega) = \pi D_{\mathrm{Dr}} \delta(\omega) + \sigma_{\mathrm{reg}}(\omega).$$

The Drude weight $D_{\mathrm{Dr}}$ is the surviving zero-frequency weight after paramagnetic and diamagnetic contributions are combined. It is not the bare tensor $D_{\alpha\beta}$.

Diffusion and Einstein Lookup

For one conserved density in a diffusive regime,

$$\begin{aligned} \chi_{nn}^{\mathrm R}(\mathbf q,\omega) &= \chi \frac{ Dq^2 }{ Dq^2-i\omega }, \\ \chi &= \left( \frac{\partial n}{\partial\mu} \right)_T. \end{aligned}$$

The pole is

$$\omega = -iDq^2.$$

It reproduces the noncommuting limits

$$\begin{aligned} \lim_{\mathbf q\to0} \lim_{\omega\to0} \chi_{nn}^{\mathrm R} &= \chi, \\ \lim_{\omega\to0} \lim_{\mathbf q\to0} \chi_{nn}^{\mathrm R} &= 0. \end{aligned}$$

For carriers of signed charge $Q$, consistent number-current normalization gives the Einstein relation

$$\sigma = Q^2\chi D.$$

These formulas require a genuine diffusive window, local equilibrium, and the same conserved density in $\chi$, $D$, and $\sigma$.

Reciprocity Lookup

Adjoint relations follow from definitions. Onsager–Casimir reciprocity requires equilibrium microscopic reversibility and correctly transformed time-reversal-odd controls.

If $A$ and $B$ have time-reversal parities $\epsilon_A$ and $\epsilon_B$, a common convention gives

$$\chi_{AB}^{\mathrm R} (\omega;\mathcal B) = \epsilon_A\epsilon_B \chi_{BA}^{\mathrm R} (\omega;-\mathcal B).$$

Momentum, current, pseudovector, and complex-basis indices can add component transformations. Reversing operator labels while leaving a magnetic field or rotation unchanged is not a reciprocity test.

Matsubara Route

For an equilibrium bosonic observable channel, a Matsubara calculation samples an analytic response function at

$$\nu_n = \frac{2\pi n}{\beta}, \qquad n\in\mathbb Z.$$

The retarded response is the upper real-axis boundary of the same analytic function after all overall signs, factors of $\hbar$, static zero-mode terms, and contact terms have been matched.

Symbolic replacement

$$i\nu_n \longrightarrow \hbar\omega+i0$$

is meaningful only after that analytic function has been identified. Recovering a real-frequency spectrum from finitely many noisy Matsubara values is an ill-posed inverse problem, not a direct substitution.

Thermal Green Functions defines imaginary-time ordering. Spectral Representation owns the exact continuation bridge and static bosonic terms. Analytic Continuation owns numerical inference.

Exact Benchmark: Harmonic Oscillator

Let

$$\begin{aligned} H_0 &= \frac{p^2}{2m} + \frac12m\Omega^2x^2, \\ H_{\mathrm{pert}}(t) &= -F(t)x. \end{aligned}$$

Since

$$[x(t),x(0)] = -\frac{i\hbar}{m\Omega} \sin(\Omega t),$$

the retarded response is

$$\chi_{xx}^{\mathrm R}(t) = \theta(t) \frac{ \sin(\Omega t) }{ m\Omega }.$$

Its frequency-domain form is

$$\chi_{xx}^{\mathrm R}(\omega) = \frac1m \frac1{ \Omega^2-(\omega+i0)^2 }.$$

The static response is

$$\chi_{xx}^{\mathrm R}(0) = \frac1{m\Omega^2},$$

and the positive-frequency absorptive line is

$$\operatorname{Im} \chi_{xx}^{\mathrm R}(\omega) = \frac{\pi}{2m\Omega} \delta(\omega-\Omega).$$

This benchmark checks the source sign, pole side, static spring compliance, spectral weight, and positive absorption simultaneously.

Exact Benchmark: Two-Level System

Take

$$H_0 = \frac{\hbar\Omega}{2} \sigma_z, \qquad H_{\mathrm{pert}}(t) = -f(t)\sigma_x.$$

At inverse temperature $\beta$, define the population difference

$$\Delta p = \tanh\left( \frac{\beta\hbar\Omega}{2} \right).$$

The transverse response is

$$\begin{aligned} \chi_{xx}^{\mathrm R}(\omega) ={}& \frac{\Delta p}{\hbar} \biggl[ \frac1{\omega+\Omega+i0} \\ &\quad - \frac1{\omega-\Omega+i0} \biggr] \\ ={}& \frac{2\Omega\Delta p}{\hbar} \frac1{ \Omega^2-(\omega+i0)^2 }. \end{aligned}$$

Thus

$$\chi_{xx}^{\mathrm R}(0) = \frac{ 2\Delta p }{ \hbar\Omega },$$

and

$$\operatorname{Im} \chi_{xx}^{\mathrm R}(\omega) = \frac{\pi\Delta p}{\hbar} \delta(\omega-\Omega).$$

Positive temperature gives $\Delta p>0$ and absorption. Population inversion gives $\Delta p<0$ and gain.

Calculation Routes

Route	Direct output	Main caution
exact diagonalization	discrete Lehmann weights	finite spectra require declared broadening and limit order
real-time evolution	causal kernel or ordinary correlator	finite time causes windowing and frequency convolution
equilibrium fluctuations	ordered or symmetrized spectra	fluctuation–dissipation conversion requires thermal equilibrium
Matsubara calculation	imaginary-axis samples	continuation is ill posed with finite noisy data
perturbation theory	approximate analytic kernel	preserve causality, symmetries, and contact terms consistently
experiment	convolved detector response	de-embed source calibration, resolution, backgrounds, and units

Reliability Checklist

A response result should pass as many of these checks as apply:

1. $H_{\mathrm{pert}}$ is written explicitly.
2. Source and detector labels are not interchanged.
3. The dimensions equal detector divided by source.
4. The time-domain kernel vanishes before the source acts.
5. Retarded poles lie in the lower half-plane for a stable passive system.
6. Hermitian autochannels obey the expected real and imaginary parity.
7. Positive-frequency absorption is nonnegative in a passive equilibrium state.
8. Kramers–Kronig relations hold with any required subtractions.
9. Equal-time commutators and known sum rules are satisfied.
10. Contact terms are derived from the same source-coupled Hamiltonian.
11. Conserved uniform modes and order-of-limits issues are stated.
12. Static thermodynamic and isolated dynamical protocols are not conflated.
13. Any broadening, window, or continuation prior is reported.
14. The linear regime is checked by varying source amplitude.

Common Mistakes

Memorizing a sign without the perturbation

The sign of $\chi^{\mathrm R}$ is tied to $H_{\mathrm{pert}}$. Re-derive it from the interaction-picture density operator when conventions differ.

Swapping source and detector

$\chi_{AB}$ is the response of $A$ to the source coupled through $B$. Cross-response is not generally symmetric.

Using an anticommutator for observable response

Fermionic single-particle Green functions often use anticommutators. A physical linear response of observables uses the ordinary commutator, even when the microscopic constituents are fermions.

Mixing energy and angular frequency

Replacing $\omega$ by $E$ requires $E=\hbar\omega$, including Jacobians in delta functions and spectral normalizations.

Dropping the contact term

Current, stress, and gauge responses often depend explicitly on the source. The state-mediated commutator alone is then incomplete.

Calling every zero-frequency quantity static

Thermodynamic derivatives, $\omega\to0$, $\mathbf q\to0$, finite-size adiabatic response, and dc transport are different protocols.

Applying fluctuation–dissipation outside equilibrium

Stationarity alone does not imply KMS detailed balance. Driven, aging, generalized, and multi-reservoir states require separate analysis.

Assigning positivity to a cross-component

Passivity constrains the dissipative quadratic form. It does not force every off-diagonal imaginary part to be positive.

Treating artificial broadening as a lifetime

A Lorentzian plotting width is not evidence for intrinsic decay unless its physical origin and convergence are established.

Extracting dc transport from a finite isolated spectrum

The thermodynamic and long-time limits are part of the definition. A finite spectrum can carry delta peaks and recurrences instead of a smooth dc coefficient.

Exercises

Exercise 1: Recover the Kubo sign

Starting from

$$H_{\mathrm{pert}}(t) = -f(t)B,$$

show that the first-order response of $A$ contains

$$+\frac{i}{\hbar} \theta(t-t') \langle[A(t),B(t')]\rangle_0.$$

Solution

The interaction-picture density operator obeys

$$i\hbar \frac{d\rho_I}{dt} = [H_{\mathrm{pert},I}(t),\rho_I(t)].$$

Replacing $\rho_I$ on the right by $\rho_0$ gives

$$\begin{aligned} \delta\rho_I(t) &= -\frac{i}{\hbar} \int_{t_0}^{t} dt'\, [H_{\mathrm{pert},I}(t'),\rho_0] \\ &= \frac{i}{\hbar} \int_{t_0}^{t} dt'\, f(t') [B_I(t'),\rho_0]. \end{aligned}$$

Therefore

$$\begin{aligned} \delta\langle A(t)\rangle &= \operatorname{Tr} \left[ A_I(t)\delta\rho_I(t) \right] \\ &= \frac{i}{\hbar} \int_{t_0}^{t} dt'\, f(t') \\ &\quad\times \langle[A_I(t),B_I(t')]\rangle_0. \end{aligned}$$

Extending the integral over all $t'$ inserts $\theta(t-t')$ and produces the stated retarded kernel.

Exercise 2: Oscillator static and absorptive checks

For the oscillator benchmark, verify the static limit and the sign of the positive-frequency spectral line.

Solution

Setting $\omega=0$ in

$$\chi_{xx}^{\mathrm R}(\omega) = \frac1m \frac1{ \Omega^2-(\omega+i0)^2 }$$

gives

$$\chi_{xx}^{\mathrm R}(0) = \frac1{m\Omega^2}.$$

This is the classical displacement per applied static force. Using

$$\frac1{x+i0} = \operatorname{PV}\frac1x - i\pi\delta(x),$$

the pole at $\omega=\Omega$ contributes

$$\operatorname{Im} \chi_{xx}^{\mathrm R}(\omega) = \frac{\pi}{2m\Omega} \delta(\omega-\Omega)$$

for positive frequency. Its coefficient is positive, as passivity requires.

Exercise 3: Classical fluctuation–dissipation limit

Derive the classical low-frequency limit of the symmetrized fluctuation–dissipation theorem.

Solution

For small $x$,

$$\coth x = \frac1x + O(x).$$

Taking

$$x = \frac{\beta\hbar\omega}{2}$$

gives

$$\hbar \coth\left( \frac{\beta\hbar\omega}{2} \right) = \frac{2}{\beta\omega} + O(\hbar^2\omega).$$

Since $\beta^{-1}=k_{\mathrm B}T$,

$$S_{AA}^{\mathrm{sym}}(\omega) \simeq \frac{ 2k_{\mathrm B}T }{ \omega } \operatorname{Im} \chi_{AA}^{\mathrm R}(\omega).$$

The approximation is local to $\beta\hbar|\omega|\ll1$; it is not valid across an arbitrary quantum spectrum.

Exercise 4: Conserved quantity paradox

Suppose $[B,H_0]=0$. Show why the isolated retarded self-response vanishes while the equilibrium isothermal susceptibility can be nonzero.

Solution

Conservation gives

$$B(t) = B,$$

so

$$\chi_{BB}^{\mathrm R}(t) = \frac{i}{\hbar} \theta(t) \langle[B,B]\rangle_0 = 0.$$

In a canonical equilibrium family with $H_f=H_0-fB$,

$$\chi_T^{BB} = \beta \left( \langle B^2\rangle - \langle B\rangle^2 \right),$$

which is nonzero whenever different conserved sectors have fluctuating thermal weights. The retarded protocol evolves one isolated state; the thermodynamic protocol compares re-equilibrated states.

Exercise 5: Diffusive order of limits

Evaluate both iterated limits of

$$\chi_{nn}^{\mathrm R}(\mathbf q,\omega) = \chi \frac{ Dq^2 }{ Dq^2-i\omega }.$$

Solution

At fixed nonzero $\mathbf q$,

$$\lim_{\omega\to0} \chi_{nn}^{\mathrm R}(\mathbf q,\omega) = \chi.$$

Taking $\mathbf q\to0$ afterward leaves $\chi$. At fixed nonzero $\omega$,

$$\lim_{\mathbf q\to0} \chi_{nn}^{\mathrm R}(\mathbf q,\omega) = 0.$$

Taking $\omega\to0$ afterward leaves zero. The pole at $\omega=-iDq^2$ makes the origin path dependent and encodes conservation of the uniform mode.

Exercise 6: Diamagnetic contact term

For

$$H[\mathbf A] = \sum_i \frac{ (\mathbf p_i-Q\mathbf A)^2 }{ 2m } + V,$$

derive the source-dependent physical current and identify the contact term.

Solution

Expanding the kinetic energy gives

$$H[\mathbf A] = H_0 - \mathbf A\cdot\mathbf J^p + \frac{NQ^2}{2m} \mathbf A^2.$$

The physical current density follows from differentiating the Hamiltonian:

$$\begin{aligned} \mathbf j[\mathbf A] &= -\frac1V \frac{\partial H}{\partial\mathbf A} \\ &= \mathbf j^p - \frac{nQ^2}{m} \mathbf A. \end{aligned}$$

The second term is instantaneous and therefore contributes

$$-\frac{nQ^2}{m} \delta_{\alpha\beta} \delta(t-t')$$

to the current response to $\mathbf A$. Combining it with the paramagnetic current commutator yields the gauge-consistent conductivity kernel.

Cross-Links

• Symbols and Conventions
• Correlation Function Definitions
• Matsubara Frequency Table
• Operator Identities
• Kubo Formula
• Retarded and Advanced Response
• Susceptibilities
• Fluctuation–Dissipation Theorem
• Sum Rules
• Transport Coefficients Preview
• Fluctuations and Susceptibilities
• Thermal Green Functions
• Spectral Representation
• Bosonic and Fermionic Matsubara Frequencies
• Analytic Continuation
• Driven Many-Body Systems
• Linear Response Preview
• Fourier-Transform Conventions

References

1. R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I”, Journal of the Physical Society of Japan 12, 570–586 (1957).
2. H. B. Callen and T. A. Welton, “Irreversibility and Generalized Noise”, Physical Review 83, 34–40 (1951).
3. R. Kubo, “The Fluctuation-Dissipation Theorem”, Reports on Progress in Physics 29, 255–284 (1966).
4. L. Onsager, “Reciprocal Relations in Irreversible Processes. I”, Physical Review 37, 405–426 (1931).
5. L. P. Kadanoff and P. C. Martin, “Hydrodynamic Equations and Correlation Functions”, Annals of Physics 24, 419–469 (1963).
6. J. M. Luttinger, “Theory of Thermal Transport Coefficients”, Physical Review 135, A1505–A1514 (1964).
7. G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
8. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, CRC Press (1990).
9. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
10. A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).