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quantum pumps, S-matrix, adiabatic, curvature, random matrix theory
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\title{Geometry, Statistics and Asymptotics of Quantum Pumps}
\begin{document}
%%%%%%%%%%%%%%%%%
\tightenlines
\author{ J.~E.~Avron ${}^{(a)}$, A. Elgart ${}^{(a)}$, G.M. Graf ${}^{(b)}$
and L. Sadun ${}^{(a,c)}$}
\address{${}^{(a)}$ Department of Physics, Technion, 32000 Haifa,
Israel}
\address{${}^{(b)}$ Theoretische Physik, ETH -H\"onggerberg, 8093 Z\"urich,
Switzerland}
\address{${}^{(c)}$ Department of Mathematics, University of Texas, Austin Texas 78712, USA}
\draft
\maketitle
%%%%%%%%%%%
\begin{abstract}
We give a pedestrian interpretation of a formula of B\"uttiker
et.\ al.\ (BPT) relating the adiabatically pumped current to the
$S$ matrix and its (time) derivatives. We describe the geometric
content of BPT and of the corresponding Brouwer pumping formula.
As applications we derive explicit formulas for the joint
probability density of pumping and conductance in the context of
random matrix theory, and derive an asymptotic formula for hard
pumping in case that the $S$ matrix is periodic in the driving
parameters.
\end{abstract}
\pacs {PACS numbers: 72.10.Bg, 73.23.-b}
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}
\narrowtext {\bf Introduction:} Is there a geometric description
of adiabatic quantum transport? For {\em non-dissipating transport
phenomena} there is such a framework
\cite{ThoulessBOOK,avron,thouless94,stone} where transport
coefficients are identified with the adiabatic curvature
\cite{berry}. It has not been clear if there is a geometric
theory for {\em dissipative} transport. Recently, Brouwer
\cite{brouwer}, and Aleiner et.\ al.\ \cite{aleiner}, building on
results of B\"uttiker, Pretre and Thomas (BPT) \cite{bpt},
suggested that a geometric description of dissipative transport
goes via the $S$ matrix and adiabatic scattering theory. Some of
this work, and certainly our own work, was motivated by
experimental results of Switkes et. al. \cite{marcus} on quantum
pumps.
In this article we examine the formula of
BPT \cite{bpt}, which relates adiabatic charge transport to the $S$
matrix and its (time) derivatives, in the special case of single-channel scattering.
We show that the formula is the sum of 3 terms, each referring to an
easily understood process at the Fermi energy. The first two processes are
dissipative and non-quantized. The third is nondissipative, but integrates
to zero for any cyclic variation in the system.
Next, we describe the geometric significance of BPT and the
pumping formula of Brouwer \cite{brouwer}. The latter can be
interpreted as curvature being formally identical to the adiabatic
curvature\cite{berry}. But, whereas non-dissipative transport has
both interesting geometry and interesting topology, dissipative
transport gives rise to an interesting geometry but to trivial
topology. In particular, all Chern numbers
associated to the Brouwer formula are identically zero.
We proceed with two applications. First we give an elementary and
explicit derivation of the main qualitative features of the joint
probability density for pumping and conductance in the framework
of random matrix theory. This problem was studied in
\cite{brouwer}. Brouwer's results go beyond ours as he also
calculates the tails of the distributions and we don't. Finally,
we calculate the asymptotics of hard pumping in systems where the
$S$ matrix depends periodically on two parameters. If the system
traverses a circle of radius $R$ in parameter space, with $R$
large, then the amount of charge transported is order $\sqrt{R}$,
multiplied by a quasi-periodic (oscillatory) function of $R$.
We shall use units where $e=m=\hbar=1$, so the electron charge is
$-1$ and the quantum of conductance is $e^2/h = {1 \over 2\pi}$. For the sake of
simplicity and concreteness we take the the dispersion relation in
all the channels to be quadratic and identical: $E={1\over
2}\,k^2$. The mutual Coulombic interaction of the electrons is
disregarded.
%\section{
{\bf BPT:}
Consider a scatterer connected to
leads that terminate at electron reservoirs. All the reservoirs are
initially at the same chemical potential and at zero temperature.
The scatterer is described by its (on-shell) $S$ matrix, which, in
the case of $n$ channels is an $n\times n$ matrix parameterized
by the energy and other parameters associated with the adiabatic
driving of the system (e.g. gate voltages and magnetic fields).
The BPT formula \cite{bpt} says that the charge $dq_\ell$ entering
the scatterer from the $\ell$-th lead due to an adiabatic variation of $S$ is
\be
d{ q}_\ell = {i \over 2 \pi}\, \Tr \left (Q_\ell \,dS
S^\dagger \right ),
\label{dQ} \ee
where $Q_\ell$ is a projection on the channels in the $\ell$-th lead,
and the $S$ matrix is evaluated at the Fermi energy. In the special case of two
leads, each lead carrying a single channel,
\be
S = \pmatrix{r &
t' \cr t & r'},\quad Q_\ell=\pmatrix{1&0\cr 0&0} \label{whatsS}
\ee
where $r,\ (r') $ and $t,\ (t')$ are the reflection and
transmission coefficients from the left (right) and $Q_\ell$
projects on the left lead. In this case Eq.~(\ref{dQ}), for the charge
entering through the left lead, reduces to
\be
2\pi\,d{ q}_\ell={i}\,(\bar r dr + \bar t'd t')= -{|r|^2 d\arg(r)}
- {|t'|^2 d\arg(t')}.\label{dq} \ee We shall present an elementary
derivation of (\ref{dq}).
%\subsection{
{\bf Scattering parameters:}
Every unitary $2 \times 2$ matrix can be expressed in the form:
\be S =
e^{i\gamma} \pmatrix{\cos(\theta) e^{i \alpha} & i \sin(\theta)
e^{-i\phi} \cr i \sin(\theta) e^{i\phi} & \cos(\theta) e^{-i
\alpha}}, \label{S1}
\ee
where $0\le \alpha,\phi<2\pi,\
0\le\gamma<\pi$ and $0\le\theta\le\pi/2$. In terms of these parameters,
the BPT formula reads
\be 2 \pi d{ q}_\ell=-\cos^2(\theta) d\alpha+\sin^2(\theta)d\phi - d\gamma.\label{dq2}
\ee
The basic strategy of
our derivation of Eq.~(\ref{dq2}) is to find processes that vary
each of the parameters in turn, and keep track of how much current
is generated by each process. An underlying assumption is that
current depends only on $S(k_F)$ and $\dot S(k_F)$, so that
processes that give rise to the same change in the $S$
matrix also give rise to the same current. Because we do not prove
this assertion, our derivation cannot be considered a complete proof.
We understand the four parameters as follows. The parameter
$\alpha$ is associated with translations: Translating the
scatterer a distance $\ell=\alpha/2k_F$ to the right multiplies
$r,\ (r')$ by $e^{i\alpha},\ (e^{-i\alpha}) $, and leaves $t$ and
$t'$ unchanged. The parameter $\phi$ is associated with a vector
potentials near the scatterer, with $\phi = -\int A$. This phase
shift across the scatterer multiplies $t,\ (t')$ by $e^{i \phi},\
(e^{-i \phi}) $, and leaves $r$ and $r'$ unchanged. The parameter
$\theta$ determines the conductance of the system: $g= |t|^2/2\pi
= \sin^2(\theta)/2\pi$. Finally, $\gamma = (-i/2)\log\det S$ is
related, by Krein's spectral shift \cite{krein}, to the number of
electrons trapped in the scatterer. As a consequence, for a closed
path in the space of Hamiltonians $\oint d\gamma=0$.
{\bf Changing $\alpha$--The snow plow:} To determine the effect of
changing $\alpha$ we imagine a process that changes $\alpha$ and
leaves the other parameters fixed, namely translating the system a
distance $d\ell = d\alpha/2k_F$. The scatterer passes through a
fraction $|t|^2$ of the $k_F d\ell/\pi = d\alpha/2\pi$ electrons
that occupy the region of size $d\ell$, and pushes the remaining
$|r|^2 d\alpha/2\pi$ electrons forward. Thus \be 2 \pi dq = -
\cos^2(\theta) d\alpha. \label{dalpha} \ee
This result can be
obtained less heuristically, \cite{ap}, by working in the
reference frame of the moving scatterer and integrating the
contribution of each wave number from 0 to $k_F$. In either
approach it is clear that the process is dissipating since energy
exchanging processes with the ``snow plow" take place.
{\bf Changing $\phi$--EMF:} To change $\phi$, we vary the vector potential.
This
induces an EMF of strength $ -\int \dot A = \dot\phi$. The current is
simply the
voltage times the Landauer conductance $|t|^2/2\pi$ \cite{landauer}.
Integrating over time gives
\be
2 \pi dq = \sin^2(\theta) d \phi.
\label{dphi}
\ee
This current is clearly dissipating as well.
{\bf Changing $\theta$ and $\gamma$--Krein's spectral shift:}
First suppose our scatterer is right-left symmetric, so $r=r'$ and
$t=t'$. Then changes in $\theta$ and $\gamma$ would draw equal
amounts of charge to the scatterer from the left and right leads.
The charge that accumulates on the scatterer is given by Krein
spectral shift \cite{krein}. The charge coming from the left is
half this, namely \cite{ap}:
\be
2 \pi dq={2 \pi i \over 4\pi} \, d\log\det S = - d \gamma.
\label{dgamma}
\ee
The current through the two leads is unaffected by a (fixed)
translation, or by a (fixed) vector potential, i.e., by
nonzero values of $\alpha$ and $\phi$. Since every $S$ matrix
can be obtained by translating and adding a vector potential to
a right-left symmetric scatterer, the formula (\ref{dgamma})
applies to all possible $S$-matrices. Since, by assumption, the
current depends only on the $S$ matrix, the formula (\ref{dgamma})
applies to all possible scatterers, symmetric or not.
Changing $\gamma$ gives non-dissipative transport which vanishes
for a closed loop. Thus, only dissipating processes contribute to
the transport of a quantum pump (contrary to an assertion made in
\cite{aleiner}).
Combining (\ref{dalpha}),
(\ref{dphi}) and (\ref{dgamma}), gives BPT, Eq.~(\ref{dq2}).
{\bf Geometrical interpretation:} ${\cal A}=-2\pi \,dq$ is the
1-form associated with Berry's phase\cite{berry}. If we define the
spinor $| \psi \rangle = \pmatrix{r \cr t'}$ then \be{\cal A} =
{-i} \langle \psi | d \psi \rangle. \label{BerrydQ} \ee $\cal A$
is the global angular form of $S^3$, the three sphere associated with
$\{r,t'\}$ (since $|r|^2+|t'|^2=1$). ${\cal A}$
reduces to the angular form on the circles $t=0$ and $r=0$, and is
invariant under unitary transformations of $S^3\subset {\C}^2$.
This makes ${\cal
A}$ a natural geometric object. It is remarkable that geometry and
transport coincide.
To compute the charge transported by a closed cycle $C$
in parameter space, we can either integrate the 1-form $\cal A$
around $C$, or (by Stokes' theorem) integrate the exterior
derivative of $\Omega= d \cal A$ over a disk $D$ whose boundary is
$C$. $\Omega$ is the curvature 2-form of Brouwer
\cite{brouwer}: \bearray \Omega & = & d\,(|r|^2)\wedge d\, \arg
(r)+ d\,(|t'|^2)\wedge d\, \arg (t')\cr &=&{-i} \langle d\psi | d
\psi\rangle = -i{ d\bar z\wedge dz \over (1+|z|^2)^2},
\label{Omega} \eearray where $z=r/t'$. The second expression is
formally identical to the adiabatic (Berry's) curvature that
appears in adiabatic quantum transport in closed systems.
In the third expression one sees that the curvature sees only the
ratio $r/t'$, and not $r$ and $t'$ separately. The $z$ plane
includes the point at infinity ($t'=0$) and should be viewed as
$\CP\cong S^2$. The curvature $\Omega$ is
the $U(2)$-invariant area form on $\CP$, and its integral
over all of $\CP$ is $2\pi$. $\Omega$ is also the curvature of the
Hopf fibration $\pi:\,S^3 \to \CP$ that sends a unit vector
$(r,t') \in \C^2$ to the point $r/t' \in \CP$.
In the study of non-dissipative quantum transport, Chern numbers,
topological invariants that equal the integral of the curvature
over closed surfaces in parameter space, play a role. In the
context of adiabatic scattering, however, all Chern numbers are
zero, since the vector bundle over parameter space is
topologically trivial: The vector $(r, t')$ gives a section of
this bundle.
% Put another way, quantum pumps have nontrivial
%geometry but trivial topology.
These geometrical constructions generalize to systems with many
channels, \cite{ap}.
{\bf Integrality and Duality:} Imagine a loop $C$ in parameter
space, bounding a disk $D$, that is mapped by $\pi\circ S$ to a
loop $\tilde C$ in $\CP$. A special, yet interesting, case is when
$\tilde C$ is a single point $z_0$ in $\CP$. At first, one might
think that such a loop can not transport charge, for an integral
of the curvature would seem to give zero since a point has no
area. This is false. $z_0$ can also be viewed as the boundary of
all of $\CP$ with one point removed, in which case the charge
transport is $\pm 1$, and also as the boundary of a multiple cover
of $\CP$ with one point removed so that charge transport could be
any {\em integer}. To determine the integer one needs to know not
only what is the image under $\pi\circ S$ of $C$ but also what is
the image of $D$: The integer is the (signed) number of preimages
(in $D$) of a generic point $z\in\CP$. From this follows a duality
property: The (signed) number of points in $D$, in which the
scatterer is a perfect insulator, is equal to the number of points
where it is a perfect conductor. Examples exhibiting this duality
are described in \cite{ap}.
{\bf Statistics of weak pumping:}
In this section
we consider how well a random scatterer transports charge when two
parameters are varied gently and cyclically. More precisely, we
consider the charge transported by moving along the circle $X_1 =
\epsilon \cos(\tau)$, $X_2 =\epsilon \sin(\tau)$ in parameter
space. If $\epsilon$ is small, then the charge transport is close
to $- \frac{\pi \epsilon^2}{2\pi}\,\Omega(\partial_1, \partial_2)$,
evaluated at the origin, where $\partial_j$ are the tangent vectors
associated with the parameters $X_j$. The vectors $\partial_j$
map to random vectors on $U(2)$, which we assume to be Gaussian
with covariance $C$. The problem is then to
understand the possible values of the curvature $\Omega$ applied
to two random vectors.
To do this this, we first need to understand the
statistics of 2-forms applied to pairs of random vectors, and
to understand the geometry of the group $U(2)$.
{\bf Areas of random vectors:} Take two random vectors in
${\mathbb R}^2$, and see how much area they span. By random
vectors we mean independent, identically distributed Gaussian
random vectors whose components $X_j$ have the covariance $\langle
X_i X_j \rangle = C \delta_{ij}$. The area $A$ turns out to be
distributed as a 2-sided exponential:
\be dP(A) = {1\over 2C}\,
e^{-|A|/C}\, dA.\label{exp}
\ee
This is seen as follows. If the two
vectors are $X$ and $Y$, then the area is $X_1 Y_2 - X_2 Y_1$.
Clearly $X_1 Y_2$ and $-X_2 Y_1$ are independent random variables,
and a calculation shows that their characteristic function is
$1/\sqrt{k^2 C^2+1}$, so their sum is a random variable with
characteristic function $(k^2C^2 + 1)^{-1}$, and so exponentially
distributed.
{\bf Geometry of $U(2)$:} We parametrize the group $U(2)$ by the
angles $(\alpha, \gamma, \phi, \theta)$, as in (\ref{S1}). A
standard, bi-invariant metric on $U(2)$ is
\bearray\frac{1}{2}\,Tr(dS^*\otimes dS)&=&(d\gamma)^2+
\cos^2\theta\, (d\alpha)^2\cr &+&\sin^2\theta\,
(d\phi)^2+(d\theta)^2. \eearray
In this metric the vectors
$\partial_i$ are orthogonal but not orthonormal. Unit tangent
vectors are \be e_\gamma=\partial_\gamma, \ e_\alpha=
\frac{1}{\cos\theta}\,
\partial_\alpha,\ e_\phi=\frac{1}{\sin\theta}\,
\partial_\phi,\ e_\theta=\partial_\theta. \ee The volume form is
$\sin(\theta)\cos(\theta)\, d\alpha\wedge d\gamma\wedge d\phi\wedge
d\theta.$ The curvature 2-form, from
Eq.~(\ref{Omega}), is
\be
\Omega = {-2} \sin(\theta)\cos(\theta)\, d\theta \wedge(d\alpha +
d\phi) \label{Omega2}. \ee
A scattering matrix is time reversal invariant if and only if $t =
t'$. The space of time-reversal matrices is parametrized exactly
as before, only now with $\phi$ identically zero. The volume form
for the metric inherited from $U(2)$ is $\cos(\theta)
d\alpha\wedge d\gamma\wedge d\theta, \label{volume2}$ and the
curvature form is now $\Omega = {-2} \sin(\theta)\cos(\theta)
d\theta \wedge d\alpha$.
{\bf Weak pumping:} We are now prepared to compute the statistics
of weak pumping, assuming that our system is described by random
matrix theory.
This problem was studied by Brouwer \cite{brouwer}, assuming that
the Hamiltonians are Gaussian random variables. While our results
generally agree with Brouwer's, the tails of the probability
distributions are different. The source of this discrepancy is
explained below.
For systems without time reversal symmetry, random matrix theory
posits that the $S$ matrix is distributed on $U(2)$ with a uniform
measure. Since the
conductance $g$ is $g\propto |t|^2 = \sin^2\theta$ we have
that $dg\propto \sin\theta\cos\theta d\theta$, proportional to the
volume form: The conductance $g$ is therefore uniformly
distributed.
A random tangent vector to $U(2)$ is
$X= X_\theta e_\theta+
X_\alpha e_\alpha+X_\phi e_\phi+X_\gamma e_\gamma,$
where $X_j$
are Gaussians with $\langle X_jX_k\rangle=C \delta_{jk}$. The
curvature associated with two random tangent vectors $X,\ Y$ is
\be \Omega(X,Y)= -2\Big( X_\theta \, W_\theta
-Y_\theta\,Z_\theta\Big),
\ee
where
$W_\theta=\sin\theta\,
Y_\alpha+\cos\theta \,Y_\phi$ and
$Z_\theta=\sin\theta\,X_\alpha+\cos\theta\, X_\phi.$
%
The variables $W_\theta$ and $Z_\theta$ are independent,
each with variance $C$. From
Eq.~(\ref{exp}), the distribution of the curvature is exponential
and independent of $|t|$. The joint distribution of curvature,
$\omega$, and conductance, $g= {1\over 2\pi}\, |t|^2$ is given by
the probability density
\be {\pi \over 2 C}\ e^{-|\omega|/2C}
d\omega\, dg \label{NoTRI2}
\ee
with $\omega$ ranging from
$-\infty$ to $\infty$ and $g$ from 0 to $1\over 2\pi$.
For systems with time reversal symmetry, random matrix theory says
that the $S$ matrix is uniformly distributed on the $t=t'$
submanifold, with the metric inherited from $U(2)$. The tangent
vectors are now Gaussian random variables of the form $ X=
X_\theta e_\theta+ X_\alpha e_\alpha+X_\gamma e_\gamma,$ and the
curvature is now
\be
\Omega(X,Y)= -2\sin\theta\,\Big( X_\theta \,
Y_\alpha-Y_\theta\,X_\alpha\Big).\ee Since the curvature depends
on $\theta$ the curvature and the conductance are correlated. The
volume form indicates that $\sqrt g$, and not $g$, is uniformly
distributed. This favors insulators. The joint distribution for
curvature and conductance is
\be {1 \over 4 \sqrt{g} C}\,
e^{-|\omega|/2C\sqrt{2\pi g}}\, d\omega\, d(\sqrt g).
\label{TRI2}
\ee
This formula says that, statistically, good pumps tend to be
good conductors; $\omega/\sqrt{g}$, rather than $\omega$ itself,
is independent of $g$.
We have assumed, so far, that the variance $C$ is a constant.
There is no reason for this and it is natural to let $C$ itself be
a random variable. Given a probability distribution for the
covariance, $d\mu(C)$, one integrates the formulas (\ref{NoTRI2})
and (\ref{TRI2}) over $C$. One sees, by inspection, that in the
absence (presence) of time reversal symmetry, $\omega$
($\omega/\sqrt{g}$) is independent of $g$. Furthermore, the
distribution of $\omega$ after integrating over $g$ is smooth away
from $\omega=0$, but has a discontinuity in derivative (log
divergence) at $\omega=0$. In these qualitative features, our
results agree with Brouwer's. However, the tails of the
distribution for large pumping may depend on the tail of
$d\mu(C)$; While (\ref{NoTRI2}) and (\ref{TRI2}) have
exponentially small tails, power law tails in $d\mu(C)$ will lead to
power law tails in $\omega$. Since we do not determine
$d\mu(C)$ we cannot determine the tails. Using random matrix
theory Brouwer determined the power decay in $\omega$
\cite{brouwer}.
{\bf Hard Pumping:} Finally, we consider what happens for hard
pumping. Here one can no longer evaluate the curvature at a point
and multiply by the area. One needs to honestly integrate the
curvature. Hard pumping was addressed by \cite{aleiner} who
studied it in the context of random matrix theory and showed,
using rather involved diagrammatic techniques, that pumping scales
like the root of the perimeter. Here we shall describe a
complementary, elementary result that holds provided the $S$ matrix
is a periodic function of the parameters. This is the case, for
example, when the pumping is driven by two Aharonov-Bohm fluxes.
With $S(x,y)$ periodic in the driving parameters $x$ and $y$, so
is the curvature $\Omega(x,y)=\sum
\hat\Omega_{mn}\,e^{i(mx+ny)}.$ Since the global angular form is
also periodic, $\hat\Omega_{00}=0$.
The integral $\int_{|x|