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\markboth{{\bf 561 F 2005 Lecture 5}}{{\bf 561 F 2005 Lecture 5}}
\begin{document}

{\bf 561 Fall 2005 Lecture 5  - modified 9/15/05 to correct small errors
and typos}

{\bf Linear Response Theory: Response Functions - Retarded Green's
Functions}

References:   P. C. Martin; Mahan 2.9 and 3.3; Fetter 5; Doniach;
nice short summary of quantum formulation in Phillips, Ch. 8.

{\bf 1.  Classical Oscillator with thermal damping force}

Consider an oscillator driven by external force (Following P. C. Martin, Ch. 1),
\begin{equation}
m \ddot{x}(t) + m\omega_0^2 x(t) = F^{int}(t) + F^{ext}(t),
\end{equation}
where $F^{int}$ represents internal forces from parts of the
system not explicitly taken into account - random forces that act
to bring the oscillator into equilibrium. The oscillator is driven
from equilibrium by $F^{ext}(t)$ leading to $\langle F^{int}(t)
\rangle \neq 0$. The forces always tend to restore equilibrium on
the average and an approximate form is a friction term $\langle
F^{int}(t) \rangle = - m \gamma \dot{x}(t)$ with $\gamma >0$. Then
 the time evolution of the average of $x$ is given by
\begin{equation}
m \left[ \frac{d^2}{dt^2} + \omega_0^2 + \gamma \frac{d}{dt}
\right]\langle x\rangle(t) =
 F^{ext}(t).
\end{equation}

A Green's function for this system is defined by the response to a
delta function source (We will use the same symbol $\chi$ for both
the in time and in frequency, with the arguments shown
( $\chi(t)$ or $\chi(\omega)$ when we need to differentiate the
functions.)
\begin{equation}
m \left[ \frac{d^2}{dt^2} + \omega_0^2 + \gamma \frac{d}{dt}
\right] \chi(t-t^{\prime})  = \delta(t-t^{\prime}),
\label{eq:G}
\end{equation}
and
\begin{equation}
\langle x\rangle(t)= \int_{-\infty}^{\infty} dt^{\prime} \;
\chi(t-t^{\prime}) F^{ext}(t^{\prime})
\end{equation}

Eq. \ref{eq:G} does not uniquely specify the Green's function;
depending on boundary conditions, one can specify the different
Green's functions:
\begin{itemize}
\item Retarded - Needed for causal response functions -
$\chi^A(t-t^{\prime}) = 0$ for $t-t^{\prime} < 0$
\item
Advanced -
$\chi(t-t^{\prime}) = 0$ for $t-t^{\prime} > 0$
\item Time-ordered - Useful for treating particle
interactions in perturbation expansions (more on this later)
\end{itemize}

The response to a single frequency driving force $F^{ext}(t) =
F^{ext}(0)exp(i\omega t)$ is given by the Fourier transform
\begin{equation}
\chi(\omega) = \int_{-\infty}^{\infty} dt \; \chi(t)
e^{i\omega t} ; \;\;\; \chi(t)  = \int_{-\infty}^{\infty}
\frac{d\omega}{2\pi} \chi(\omega)e^{-i\omega t}  \label{eq:FT}
\end{equation}

For the damped oscillator example,
\begin{equation}
\chi(\omega) = \frac{1}{m}  \left[\frac{-1}{\omega^2 - \omega_0^2
+ i \gamma\omega}  \right] \label{eq:G-osc2}
\end{equation}
We show below that this is an example of a causal retarded
function.

\medskip
{\bf 2.  Retarded Causal Green's Functions  (Response Functions) - complex
analysis}

(Note: Complex analysis holds for both classical and
quantum response functions.  The description of the classical
oscillator follows P. C. Martin, Ch. 1; it gives a useful physical picture
but is not essential to derive the quantum expressions. )

Consider $\chi(z)$ as a complex function of a complex frequency $z$.
A retarded function satisfies $\chi(t-t^{\prime}) = 0$
for $t-t^{\prime} < 0$,
\begin{equation}
\chi(\omega) = \int_{-\infty}^{\infty} dt \; \chi(t)
e^{i\omega t}.
\label{eq:FT2}
\end{equation}
This equation can be considered the definition of $\chi(\omega)$ for complex
$\omega$, which we sometimes write as $\chi(z)$ to avoid confusion
with teh cases where $\omega$ is a real frequency. The integral is
well-defined for $Im \omega > 0$, since we
only need to consider $t>0$ and the exponential factor converges for
large $Im \omega > 0$.  Thus a physical, causal response function
is analytic in the upper plane ($Im \omega > 0$).

\begin{itemize}
\item There are poles in $\chi(\omega)$ {\it only} in lower half of
complex plane ($Im \; \omega < 0$) no matter how complicated the
system and the functional form of  $\chi(\omega)$.  For example, the
response function Eq. \ref{eq:G-osc2} for the damped oscillator
can readily be shown to
have two poles at $\pm \omega_0 - i \eta$ where $Im \eta > 0$.
Show they this is related to the causal assumptions in the
formulation of the damping.  in be worked out easily

\item Using Cauchy Relations and closing the contour in upper plane (where $\chi$
is analytic) leads to
\begin{equation}
\chi(z) = \frac{1}{2\pi i}\int_{-\infty}^{\infty} d\omega \; \frac{\chi(\omega)}
{\omega - z}, \;\;\; Im z > 0,
\end{equation}
where the integral is for $\omega$ on the real axis (i.e.,
involves the physically measurable $\chi(\omega)$ for real frequencies $\omega$.
The analytic continuation of Eq. \ref{eq:FT2} to $Im z < 0$
gives $\chi(z) = 0$ since $z$ is outside the contour which
is closed in the upper plane.

 \item The fact that $\chi(z) = 0$ for $z$ approaching the real axis from
 below is one of the ways to derive the famous Kramers-Kronig relations for real
 frequency $\omega$ valid for all causal response functions
 (see many texts for details)
\begin{equation}
 Re\chi(\omega) = -\frac{1}{\pi}\int_{-\infty}^{\infty} d\omega^{\prime} \;
\frac{Im \chi(\omega^{\prime})}
{\omega - \omega^{\prime}}, \;\;\; Im\chi(\omega) =
\frac{1}{\pi}\int_{-\infty}^{\infty} d\omega^{\prime} \;
\frac{Re \chi(\omega^{\prime})}
{\omega - \omega^{\prime}}.
\end{equation}

\item Interpretation of the complex response for $\chi(\omega)$ for real $\omega$:
\begin{itemize}
\item $Re\chi(\omega)$ is the real response to a perturbation - in-phase
- no energy loss
\item $Im\chi(\omega)$ is the imaginary response to a perturbation - out-of-phase
- represents energy loss
\end{itemize}

\item Sum rules: Since particles always act as free particles at high
enough frequency, it follows that $\chi(z) \rightarrow - \frac{1}{m z^2}$.
Using the first KK relation for $Re \chi$ it follows that (straightforward to
show by expanding energy denominator in KK integrand)
\begin{equation}
\frac{1}{m} = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega^{\prime} \;
Im \chi(\omega^{\prime})\omega^{\prime}.
\end{equation}

\item Static Response Function: for a static perturbation $\langle x \rangle =
\chi(\omega = 0) F^{ext}$ to linear order.  Using the KK relation we find
\begin{equation}
\chi(\omega = 0)  = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega^{\prime} \;
\frac{Im \chi(\omega^{\prime})}
{\omega^{\prime}}.
\end{equation}

\end{itemize}

{\bf 3.  Quantum Theory for Retarded Causal Green's Functions }

(Following Phillips Sec 8.3; more details are given in Fetter, Ch. 5, sect 13,
 Mahan 2.9 and 3.3)

Consider a system described by a hamiltonian $H = H_0 + W(t)$ where
$W(t)$ is an external perturbation turned on at some time.
In the Schrodinger representation, the operators are independent of time and
the wavefunction $\Psi(t)$ evolves. For ensembles one can work with
the density matrix $\rho(t)$ and the expectation value of an observable $Y$
is given by
\begin{equation}
\langle Y \rangle (t)  = Tr ( \rho(t) Y),
\end{equation}
with $\rho$ the density matrix in the Schrodinger representation, which satisfies the
equation
\begin{equation}
i\hbar\frac{d \rho(t)}{dt} = [H,\rho(t)] = [H_0 + W(t),\rho(t)],
\end{equation}
where $[A,B] = AB - BA$ is the commutator.

It is easiest to work in the interaction representation in which an
operator $\widehat{O}(t)$ is defined by
\begin{equation}
\widehat{O}(t) = e^{\frac{+iH_0 t}{\hbar}} O  e^{\frac{-iH_0 t}{\hbar}},
\end{equation}
where $O$ is a time-independent operator in the Schrodinger picture. Then
\begin{equation}
\langle \widehat{Y}(t) \rangle  = Tr ( \widehat{\rho}(t) \; \widehat{Y}(t)),
\end{equation}
where the evolution is given by
\begin{equation}
i\hbar\frac{d \widehat{\rho}(t)}{dt} = [\widehat{W}(t), \widehat{\rho}(t)]
\end{equation}
or the formal solution
\begin{equation}
\widehat{\rho}(t) = T \exp\left(-\frac{i}{\hbar} \int^t_{-\infty}
[\widehat{W}(t^{\prime}), \widehat{\rho}(t^{\prime})] dt^{\prime}\right).
\end{equation}

The time ordering operator $T$ specifies how the exponential is
to be interpreted. (See Mahan, Fetter and other references for a careful
exposition.)  To lowest order it is simply
\begin{equation}
\widehat{\rho}(t) = \widehat{\rho}_0(t) -\frac{i}{\hbar} \int^t_{-\infty}
[\widehat{W}(t^{\prime}), \widehat{\rho}_0(t^{\prime})] dt^{\prime}.
\end{equation}

The expectation value is given by
\begin{equation}
\langle \widehat{Y}(t) \rangle = \langle \widehat{Y}(t) \rangle_0
-\frac{i}{\hbar} \int^t_{-\infty}
Tr (  \widehat{Y}(t)[\widehat{W}(t^{\prime}), \widehat{\rho}_0(t^{\prime})]
dt^{\prime} + \ldots.
\end{equation}
Cyclic permutation leads to the desired result:
\begin{equation}
\delta \langle \widehat{Y}(t) \rangle = \langle \widehat{Y}(t) \rangle - \langle \widehat{Y}(t) \rangle_0
= \int^t_{-\infty} \chi_{YW}(t,t^{\prime}) dt^{\prime},
\end{equation}
where the response function for $Y$ due to $W$ is given by

\begin{equation}
\chi_{YW}(t,t^{\prime}) = -\frac{i}{\hbar} \langle [\widehat{Y}(t),\widehat{W}(t^{\prime})] \rangle_0.
\end{equation}
(Note: Phillips defines $\chi$ without the factor $-\frac{i}{\hbar}$,
 but it is only with
 the factor that that $\chi$ has the expected properties, e.g., that
 $Im \; \chi(\omega)$ is the energy loss function.)
 Here $\langle \ldots \rangle_0$ denotes expectation value with the unperturbed
$\widehat{\rho}_0$, and the response function is a commutator of the
response $\widehat{Y}(t)$ and the perturbation $\widehat{W}(t^{\prime})$ in
the interaction representation.

{\bf 4.  Example:  density-density response function}

Consider the problem of charged particles in an external potential
\begin{equation}
W(t) = \int d^3 r \hat{n}(r) e \phi^{ext}(r,t),
\end{equation}
where $\hat{n}(r)$ is the density operator, $e$ the charge, and $\phi^{ext}(r,t)$
the perturbing potential that is explicitly time-dependent.

To linear order the response of the ground state
to the perturbation is
\begin{equation}
\delta \langle \widehat{n}(r,t) \rangle
= -\frac{i}{\hbar} \int^t_{-\infty} dt^{\prime} \; \int d^3 r^{\prime}
e \phi^{ext}(r^{\prime},t^{\prime})
\langle [\widehat{n}(r^{\prime},t^{\prime}), \widehat{n}(r,t)] \rangle_0.
\end{equation}
If the density-density response function is defined by (here $\Theta$ is
the step function)
\begin{equation}
\chi_{nn}(rt,r^{\prime} t^{\prime}) =
= -\frac{i}{\hbar} \Theta(t-t^{\prime})
\langle [\widehat{n}(r^{\prime},t^{\prime}), \widehat{n}(r,t)] \rangle_0,
\end{equation}
then the response of the system to linear order is
\begin{equation}
\delta \langle \widehat{n}(r,t) \rangle
= -\frac{i}{\hbar} \int^{\infty}_{-\infty} dt^{\prime} \; \int d^3 r^{\prime} \;
\chi_{nn}(rt,r^{\prime} t^{\prime}) e \phi^{ext}(r^{\prime},t^{\prime}).
\end{equation}

We will return to the density-density correlation function in the analysis
of the dielectric function.

{\bf Linear Response Theory -  continued next time:}\\
Example of phonons - also see Homework 2\\
General properties of Green's functions\\
Fluctuation-dissipation theorem\\
Scattering spectra and experiments

\end{document}