16: Quantum Linear Response Theory

A.3.1 Interaction picture

We work in the interaction picture, in which operators evolve with \(H_{0}\) and the density matrix evolves with the perturbation. The interaction-picture counterpart of any Schrödinger-picture operator \(O\) is

\[ O_{I}(t)=e^{i H_{0} t / \hbar} O e^{-i H_{0} t / \hbar} . \tag{A.5} \label{A.5} \]

The perturbation Hamiltonian in the interaction picture is therefore

\[ H_{I}(t)=-f(t) B_{I}(t), \quad B_{I}(t)=e^{i H_{0} t / \hbar} B e^{-i H_{0} t / \hbar} . \tag{A.6} \label{A.6} \]

The interaction-picture density matrix obeys the von Neumann equation:

\[ \frac{d \rho_{I}}{d t}=-\frac{i}{\hbar}\left[H_{I}(t), \rho_{I}(t)\right] . \tag{A.7} \label{A.7} \]

A.3.2 First-order perturbation theory

Integrating Eq. (A.7) from \(t=-\infty\) with initial condition \(\rho_{I}(-\infty)=\rho_{0}\), and retaining only the first-order term in \(f\) :

\[ \rho_{I}(t) \approx \rho_{0}+\delta \rho_{I}(t), \quad \delta \rho_{I}(t)=-\frac{i}{\hbar} \int_{-\infty}^{t} d t^{\prime}\left[H_{I}\left(t^{\prime}\right), \rho_{0}\right] . \tag{A.8} \label{A.8} \]

The change in the expectation value of \(A\) is

\[ \begin{align*} \delta\langle A(t)\rangle & =\operatorname{Tr}\left[\delta \rho_{I}(t) A_{I}(t)\right] \\[4pt] & =-\frac{i}{\hbar} \int_{-\infty}^{t} d t^{\prime} f\left(t^{\prime}\right) \operatorname{Tr}\left[\rho_{0}\left[A_{I}(t), B_{I}\left(t^{\prime}\right)\right]\right] \tag{A.9} \label{A.9} \end{align*} \]

where we used the cyclic property of the trace: \(\operatorname{Tr}\left(\left[H_{I}\left(t^{\prime}\right), \rho_{0}\right] A_{I}(t)\right)=\operatorname{Tr}\left(\rho_{0}\left[A_{I}(t), B_{I}\left(t^{\prime}\right)\right]\right)\).

The cyclic property \(\operatorname{Tr}(X Y Z)=\operatorname{Tr}(Z X Y)\) gives

\[ \operatorname{Tr}\left(\left[H_{I}\left(t^{\prime}\right), \rho_{0}\right] A_{I}(t)\right)=\operatorname{Tr}\left(H_{I}\left(t^{\prime}\right) \rho_{0} A_{I}(t)\right)-\operatorname{Tr}\left(\rho_{0} H_{I}\left(t^{\prime}\right) A_{I}(t)\right)=f\left(t^{\prime}\right) \operatorname{Tr}\left(\rho_{0}\left[A_{I}(t), B_{I}\left(t^{\prime}\right)\right]\right) . \nonumber \]

Since \(H_{0}\) is time-independent, the commutator depends only on \(t-t^{\prime}\), and we may write \(\left[A_{I}(t), B_{I}\left(t^{\prime}\right)\right]=\left[A\left(t-t^{\prime}\right), B(0)\right]\) (using translational invariance of the equilibrium average). Extending the upper limit to \(\infty\) with a Heaviside function, we define the retarded susceptibility:

\[ \chi_{A B}\left(t-t^{\prime}\right)=\frac{i}{\hbar} \theta\left(t-t^{\prime}\right)\left\langle\left[A\left(t-t^{\prime}\right), B(0)\right]\right\rangle_{0} \tag{A.10} \label{A.10} \]

so that \(\delta\langle A(t)\rangle=\int_{-\infty}^{\infty} d t^{\prime} \chi_{A B}\left(t-t^{\prime}\right) f\left(t^{\prime}\right)\).

The \(\theta\left(t-t^{\prime}\right)\) factor enforces causality: the response at time \(t\) can only depend on the perturbation at earlier times \(t^{\prime}<t\).

A.4.1 Derivation of mean power dissipation

Consider a real, monochromatic driving force (written as a sum to keep \(f(t)\) real):

\[ f(t)=f_{0} \cos (\omega t)=\frac{f_{0}}{2}\left(e^{-i \omega t}+e^{i \omega t}\right), \quad f_{0} \in \mathbb{R} . \tag{A.13} \label{A.13} \]

The instantaneous power delivered to the system by the perturbation \(H^{\prime}=-f(t) B\) is

\[ P(t)=\frac{d\left\langle H^{\prime}\right\rangle}{d t}=-\dot{f}(t)\langle B(t)\rangle, \tag{A.14} \label{A.14} \]

where we used \(\frac{d}{d t}\left\langle H^{\prime}\right\rangle=\left\langle\partial H^{\prime} / \partial t\right\rangle=-\dot{f}(t)\langle B(t)\rangle\) (the Hellmann-Feynman theorem applied to the time derivative of the perturbation).

Because \(H^{\prime}=-f(t) B\), we have \(\partial H^{\prime} / \partial t=-\dot{f}(t) B\), so \(P=\langle\partial H / \partial t\rangle=-\dot{f}(t)\langle B\rangle\). The minus sign is consistent with the convention that work done on the system is positive when \(\dot{f}\) and \(\langle B\rangle\) have opposite signs.

The linear response gives \(\langle B(t)\rangle=\chi(\omega) \frac{f_{0}}{2} e^{-i \omega t}+\) c.c., and \(\dot{f}(t)=-i \omega \frac{f_{0}}{2} e^{-i \omega t}+\) c.c. Multiplying and time-averaging over one period (the cross terms at \(\pm 2 \omega\) vanish):

\[ \begin{align*} \langle P\rangle & =-\frac{f_{0}^{2}}{4}\left[(-i \omega) \chi(\omega)+(i \omega) \chi^{*}(\omega)\right] \\[4pt] & =\frac{f_{0}^{2}}{4} \cdot i \omega\left[\chi(\omega)-\chi^{*}(\omega)\right]=\frac{f_{0}^{2}}{4} \cdot i \omega \cdot 2 i \chi^{\prime \prime}(\omega) . \tag{A.15} \label{A.15} \end{align*} \]

\[ \langle P\rangle=\frac{\omega f_{0}^{2}}{2} \chi^{\prime \prime}(\omega) . \tag{A.16} \label{A.16} \]

The passivity condition (energy is absorbed from the drive, not emitted) requires \(\langle P\rangle \geq 0\) for \(\omega>0\), which implies \(\chi^{\prime \prime}(\omega) \geq 0\) for \(\omega>0\).

Comparison with the complex-amplitude convention: Some texts write \(f(t)=f_{0} e^{-i \omega t}+\) c.c. (with \(\left|f_{0}\right|^{2}\) replacing \(f_{0}^{2} / 4\) ). In that convention, the mean power is \(\langle P\rangle=2 \omega\left|f_{0}\right|^{2} \chi^{\prime \prime}(\omega)\). Here we use the real cosine drive (A.13); the result (A.16) follows directly.

A.4.2 Physical interpretation

• \(\chi^{\prime}(\omega)\) : the dispersive (reactive) part of the response - describes energy storage, analogous to the elastic modulus or inductance.
• \(\chi^{\prime \prime}(\omega)\) : the absorptive (dissipative) part - describes irreversible energy transfer from the field to the system (viscosity, resistance, damping). This is the directly measurable part in spectroscopy.

A.6.1 Derivation of the KMS condition

Define the two-time correlation function \(C_{A B}(t)=\langle A(t) B(0)\rangle_{0}=\operatorname{Tr}\left(\rho_{0} A(t) B\right)\). Using \(\rho_{0}=\) \(e^{-\beta H_{0}} / Z_{0}\) and the cyclic property of the trace:

\[ \begin{align*} \operatorname{Tr}\left(e^{-\beta H_{0}} A(t) B\right) & =\operatorname{Tr}\left(e^{-\beta H_{0}} e^{i H_{0} t / \hbar} A e^{-i H_{0} t / \hbar} B\right) \\[4pt] & =\operatorname{Tr}\left(B e^{-\beta H_{0}} e^{i H_{0} t / \hbar} A e^{-i H_{0} t / \hbar}\right) \\[4pt] & =\operatorname{Tr}\left(e^{-\beta H_{0}} e^{i H_{0}(t+i \hbar \beta) / \hbar} A e^{-i H_{0}(t+i \hbar \beta) / \hbar} B\right) . \tag{A.19} \label{A.19} \end{align*} \]

The third line follows because \(e^{\beta H_{0}} B e^{-\beta H_{0}}\) can be absorbed by shifting the time argument \(t \rightarrow t+i \hbar \beta\) in the evolution operator. This gives the KMS condition:

\[ \langle A(t) B(0)\rangle_{0}=\langle B(0) A(t+i \hbar \beta)\rangle_{0} . \tag{A.20} \label{A.20} \]

The KMS condition states that a thermal state can be distinguished from an arbitrary state by a universal periodicity in imaginary time of period \(\hbar \beta\). It serves as the rigorous mathematical definition of thermal equilibrium in both quantum mechanics and quantum field theory, even in situations (e.g., infinite-volume limits) where a density matrix may not exist.

A.6.2 Role of the KMS condition

The KMS condition is the algebraic engine of the entire formalism:

• 1.Operator reordering. It allows the two orderings \(\langle A(t) B\rangle_{0}\) and \(\langle B A(t)\rangle_{0}\) to be related via an imaginary time shift, which in Fourier space becomes multiplication by \(e^{\beta \hbar \omega}\).

• 2.Detailed balance. Fourier-transforming Eq. (A.20) gives the detailed balance relation (see next subsection).
• 3.Fluctuation-dissipation theorem. Combining detailed balance with the definition of the susceptibility yields the FDT (Section A.7).

A.6.3 Detailed balance

Define \(C_{A B}(\omega)=\int_{-\infty}^{\infty} d t e^{i \omega t}\langle A(t) B(0)\rangle_{0}\). Taking the Fourier transform of Eq. (A.20):

\[ C_{A B}(\omega)=e^{\beta \hbar \omega} C_{B A}(\omega), \tag{A.21} \label{A.21} \]

since the imaginary time shift \(t \rightarrow t+i \hbar \beta\) generates a factor \(e^{-\omega \cdot \hbar \beta}=e^{-\beta \hbar \omega}\) upon Fourier transformation, or equivalently:

\[ C_{A B}(\omega)=e^{\beta \hbar \omega} C_{B A}(\omega) . \tag{A.22} \label{A.22} \]

This is the quantum detailed balance condition: the spectral weight for a transition that emits a quantum of energy \(\hbar \omega\) is Boltzmann-enhanced relative to the reverse process, consistent with the system preferring lower-energy configurations at finite temperature.

A.7.1 Connecting the commutator to the correlation functions

The retarded susceptibility (A.10) involves the commutator \(\langle[A(t), B(0)]\rangle_{0}\). Decomposing into the two orderings:

\[ \langle[A(t), B(0)]\rangle_{0}=C_{A B}(t)-C_{B A}(-t) . \tag{A.23} \label{A.23} \]

Fourier-transforming:

\[ \chi_{A B}^{\prime \prime}(\omega)=\frac{1}{2 \hbar}\left[C_{A B}(\omega)-C_{B A}(-\omega)\right] . \tag{A.24} \label{A.24} \]

(The factor \(1 / 2 \hbar\) comes from the definition (A.10) and the relation between the full susceptibility and its imaginary part via the Heaviside function.)

A.7.2 Applying detailed balance

Substituting \(C_{B A}(-\omega)=e^{-\beta \hbar \omega} C_{A B}(\omega)\) (which follows from Eq. (A.21) applied with \(\omega \rightarrow-\omega\) and swapped indices):

\[ \chi_{A B}^{\prime \prime}(\omega)=\frac{1}{2 \hbar}\left(1-e^{-\beta \hbar \omega}\right) C_{A B}(\omega) . \tag{A.25} \label{A.25} \]

The symmetric (anticommutator) correlation function is:

\[ S_{A B}(\omega)=\int_{-\infty}^{\infty} d t e^{i \omega t}\left\langle\frac{1}{2}\{A(t), B(0)\}\right\rangle_{0}=\frac{1}{2}\left[C_{A B}(\omega)+C_{B A}(-\omega)\right]=\frac{1}{2}\left(1+e^{-\beta \hbar \omega}\right) C_{A B}(\omega) . \tag{A.26} \label{A.26} \]

Therefore the FDT can equivalently be written in terms of \(S_{A B}(\omega)\), but the cleanest form directly connects \(\chi^{\prime \prime}\) to \(C_{A B}\) :

\[ \begin{align*} \chi_{A B}^{\prime \prime}(\omega) & =\frac{1}{2 \hbar}\left(1-e^{-\beta \hbar \omega}\right) C_{A B}(\omega), \tag{A.27} \label{A.27}\\[4pt] \text { equivalently: } & C_{A B}(\omega)=\hbar\left[n_{B}(\omega)+1\right] \cdot 2 \chi_{A B}^{\prime \prime}(\omega), \tag{A.28} \end{align*} \]

where \(n_{B}(\omega)=\left(e^{\beta \hbar \omega}-1\right)^{-1}\) is the Bose-Einstein distribution.

The FDT equates a non-equilibrium quantity ( \(\chi^{\prime \prime}\), measured by driving the system) to an equilibrium fluctuation ( \(C_{A B}\), measured without any drive). No perturbative assumptions are required; the result is exact. The Bose factor \(n_{B}(\omega)+1\) reflects stimulated plus spontaneous emission in the quantum language.

A.7.3 Classical limit

For \(\hbar \omega \ll k_{B} T\) (high temperature / low frequency): \(\left(1-e^{-\beta \hbar \omega}\right) \approx \beta \hbar \omega\), so:

\[ \chi_{A B}^{\prime \prime}(\omega) \approx \frac{\beta \omega}{2} S_{A B}(\omega), \tag{A.29} \label{A.29} \]

where we used \(S_{A B} \approx C_{A B}\) in the classical limit (the two orderings commute for \(\hbar \rightarrow 0\) ). This is the classical FDT, equivalent to the Einstein-Nyquist relation.

A.9.1 Minimal coupling and the current operator

For a system of charges \(q\) coupled to an electromagnetic field, the minimal coupling Hamiltonian (to lowest order in the vector potential \(\mathbf{A}\) ) is

\[ H=\sum_{n} \frac{\left(\mathbf{p}_{n}-q \mathbf{A}\right)^{2}}{2 m} \approx H_{0}-\frac{q}{m} \sum_{n} \mathbf{p}_{n} \cdot \mathbf{A}=H_{0}-\mathbf{J} \cdot \mathbf{A} V, \tag{A.35} \label{A.35} \]

where \(V\) is the volume and \(\mathbf{J}\) is the current-density operator. The electric field is related to the vector potential in the temporal gauge by \(\mathbf{E}=-\partial \mathbf{A} / \partial t\), so for a monochromatic field \(\mathbf{E}(\omega) e^{-i \omega t}\) :

\[ \mathbf{A}(\omega)=\frac{i}{\omega} \mathbf{E}(\omega) . \tag{A.36} \label{A.36} \]

A.9.2 Kubo formula for conductivity

The Kubo formula gives the induced current as \(J_{i}(\omega)=\chi_{J_{i} J_{j}}(\omega) A_{j}(\omega) V\). Using Eq. (A.36) and \(\sigma_{i j}(\omega)=J_{i}(\omega) / E_{j}(\omega)\) :

\[ \sigma_{i j}(\omega)=\frac{\chi_{J_{i} J_{j}}(\omega)}{i \omega}+\sigma_{i j}^{\mathrm{dia}}, \tag{A.37} \label{A.37} \]

where \(\sigma_{i j}^{\mathrm{dia}}=n q^{2} /(i m \omega)\) is the diamagnetic contribution from the \(\mathbf{A}^{2}\) term. This term ensures that \(\sigma_{i j}(\omega)\) is finite as \(\omega \rightarrow 0\) (it cancels a spurious pole in \(\chi / i \omega\) ).

Substituting the Kubo formula (A.10):

Kubo-Greenwood formula:

\[ \sigma_{i j}(\omega)=\frac{1}{i \omega V} \int_{0}^{\infty} d t e^{i \omega t}\left\langle\left[J_{i}(t), J_{j}(0)\right]\right\rangle_{0}+\sigma_{i j}^{\mathrm{dia}} \tag{A.38} \label{A.38} \]

Common omission. Many textbook presentations drop the diamagnetic term \(\sigma_{i j}^{\text {dia }}\). This is only valid for insulators (where it cancels exactly with the \(\omega \rightarrow 0\) limit of the paramagnetic term) or when the \(f\)-sum rule is invoked. For metals or when working at finite \(\omega\), the diamagnetic term must be retained to obtain gauge-invariant results.

A.9.3 DC conductivity and the Einstein relation

In the \(\omega \rightarrow 0\) limit (DC conductivity), the real part of Eq. (A.38) gives:

\[ \sigma_{i j}^{\mathrm{DC}}=\frac{1}{V} \int_{0}^{\infty} d t\left\langle J_{i}(t) J_{j}(0)\right\rangle_{0} \tag{A.39} \label{A.39} \]

which is the standard Green-Kubo relation for conductivity: the DC conductance is the time integral of the equilibrium current-current correlation function.

A.10.1 Non-local susceptibility

When the external perturbation varies in space,

\[ H(t)=H_{0}-\int d^{3} r f(\mathbf{r}, t) B(\mathbf{r}) \tag{A.40} \label{A.40} \]

the linear response is governed by a non-local susceptibility \(\chi_{A B}\left(\mathbf{r}-\mathbf{r}^{\prime}, t-t^{\prime}\right)\) (assuming translational invariance). Transforming to momentum space:

\[ \delta\langle A(\mathbf{k}, \omega)\rangle=\chi_{A B}(\mathbf{k}, \omega) f(\mathbf{k}, \omega), \tag{A.41} \label{A.41} \]

with

\[ \chi_{A B}(\mathbf{k}, t)=\frac{i}{\hbar} \theta(t)\langle[A(\mathbf{k}, t), B(-\mathbf{k}, 0)]\rangle_{0}, \tag{A.42} \label{A.42} \]

where \(A(\mathbf{k}, t)=\int d^{3} r e^{-i \mathbf{k} \cdot \mathbf{r}} A(\mathbf{r}, t)\).

A.10.2 The dynamic structure factor

The dynamic structure factor is the power spectrum of density fluctuations:

\[ S(\mathbf{k}, \omega)=\int_{-\infty}^{\infty} d t e^{i \omega t}\langle\rho(\mathbf{k}, t) \rho(-\mathbf{k}, 0)\rangle_{0} \tag{A.43} \label{A.43} \]

where \(\rho(\mathbf{k})=\sum_{n} e^{-i \mathbf{k} \cdot \mathbf{r}_{n}}\) is the Fourier-transformed density. The FDT (A.27) applied to \(A=\) \(B=\rho_{\mathbf{k}}\) gives:

\[ \chi_{\rho \rho}^{\prime \prime}(\mathbf{k}, \omega)=\frac{1}{2 \hbar}\left(1-e^{-\beta \hbar \omega}\right) S(\mathbf{k}, \omega) . \tag{A.44} \label{A.44} \]

Physical significance of \(S(\mathbf{k}, \omega)\).

• Scattering experiments: \(S(\mathbf{k}, \omega)\) is directly measured in inelastic neutron and X-ray scattering, providing a microscopic probe of the system's dynamics at wavevector \(|\mathbf{k}|\) and energy \(\hbar \omega\).
• Hydrodynamic limit: As \(\mathbf{k} \rightarrow 0\) and \(\omega \rightarrow 0, S(\mathbf{k}, \omega)\) exhibits peaks corresponding to hydrodynamic modes (sound, diffusion), from which macroscopic transport coefficients can be extracted.
• Sum rules: Integrating \(\chi^{\prime \prime}(\mathbf{k}, \omega)\) over \(\omega\) yields the \(f\)-sum rule: \(\int_{-\infty}^{\infty} \frac{d \omega}{\pi} \omega \chi^{\prime \prime}(\mathbf{k}, \omega)=\frac{\hbar k^{2} n}{2 m}\), where \(n\) is the particle number density. This is an exact, model-independent result.