Content-Type: multipart/mixed; boundary="-------------0002140421579" This is a multi-part message in MIME format. ---------------0002140421579 Content-Type: text/plain; name="00-72.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-72.comments" 4 pages, no figures ---------------0002140421579 Content-Type: text/plain; name="00-72.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-72.keywords" quantum pumps, S-matrix, adiabatic, curvature, random matrix theory ---------------0002140421579 Content-Type: application/x-tex; name="aegs.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aegs.tex" \documentstyle[aps,prl,multicol]{revtex} %%%%%%%%%%%%%%%%%% %\documentstyle[preprint,aps]{revtex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\documentclass {article} %\usepackage{amsfonts} \DeclareSymbolFont{AMSb}{U}{msb}{m}{n} \DeclareSymbolFontAlphabet{\mathbb}{AMSb} \newcommand{\complex}{\kern.1em{\raise.47ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \newcommand{\ket}[3]{\left\vert #1 , #2;#3 \right\rangle} \newcommand{\kernel}[3]{\left\langle #1\left\vert #2\right\vert#3 \right\rangle} \newcommand{\half}{\frac{1}{2}} \newcommand{\braket}[2]{\left\langle #1\vert#2 \right\rangle} \newcommand{\C}{\complex} \newcommand{\CP}{{{\mathbb{CP}}^1}} \newcommand{\Tr}{{\hbox{Tr}}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bearray}{\begin{eqnarray}} \newcommand{\eearray}{\end{eqnarray}} \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|