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\begin{document}

\title{The problem of Macroscopic Charge Quantization in the Coulomb Blockade}

\classification{PACS 73.23Hk, 73.43Hq}
\keywords      {renormalization group, theta-vacuum, coulomb blockade}

\author{I.S. Burmistrov}{
  address={L.D. Landau Institute for Theoretical
Physics, Kosygina street 2, 117940 Moscow, Russia},
altaddress={Department
of Theoretical Physics, Moscow Institute of Physics and
Technology, 141700 Moscow, Russia}
}

\author{A.M.M Pruisken}{
  address={Institute for Theoretical Physics, University of
Amsterdam, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands}
}



\begin{abstract}
Based on the Ambegaokar-Eckern-Sch\"{o}n approach to the Coulomb
blockade, we develop a complete quantum theory of the single
electron transistor. We identify a previously unrecognized
physical observable in the problem that, unlike the usual average
charge on the island, is robustly quantized for any {\em finite}
value of the tunneling conductance as the temperature goes to
absolute zero. This novel quantity is fundamentally related to the
non-symmetrized current noise of the system. Our results display
all of the superuniversal topological features of the $\theta$ angle
concept that previously arose in the theory of the quantum Hall
effect.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Coulomb blockade has traditionally been viewed as an
experimental demonstration of ``macroscopic charge
quantization"~\cite{QExpFirst}. The standard experimental set-up
is the single electron transistor (SET)~\cite{SETExp} which is a
mesoscopic metallic island coupled to a gate and connected to two
metallic reservoirs by means of tunnelling contacts with a total
conductance $g$ (see Fig.~\ref{FIGURE1}).

Even though in the absence of tunneling the problem is generally
well understood, it has nevertheless been shown that for any
finite value of $g$, no matter how small, the averaged charge $Q$
on the island is {\em un-}quantized as the temperature ($T$) goes to
absolute zero~\cite{Matveev}. This raises fundamental questions
about the exact significance of the experiments and the physical
quantities in which the Coulomb blockade is usually expressed.

In a recent investigation~\cite{ShortPaper} the authors reported
complete quantum theory of the SET. This theory is motivated by
the formal analogies that exist between the
Ambegaokar-Eckern-Sch\"{o}n (AES) theory on the one hand, and the
theory of the quantum Hall effect ~\cite{PruiskenBurmistrov2} on
the other.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%\sidecaption
\includegraphics[width=110mm]{Figure1.eps}
\caption{a) Sketch of the SET device. b) Equivalent circuit of the
SET.} \label{FIGURE1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
By studying the sensitivity of the system to changes in the
boundary conditions it was shown that the AES theory generally
defines two physical observables, the SET conductance
$g^{\,\prime}$ and a novel quantity termed $q^{\,\prime}$ that is
fundamentally related to the current noise in the SET. The
$q^{\,\prime}$ is in all respects same as the Hall conductance in
the quantum Hall effect and, unlike the averaged charge $Q$ on the
island, it is robustly quantized in the limit $T \rightarrow 0$
independent of the value of $g$.

The physical parameters $g^{\,\prime}$ and $q^{\,\prime}$ set the
stage for a unifying renormalization theory of the SET within
which the various disconnected pieces of existing computational
knowledge of the AES theory can in general be understood. In this
paper we review some of the main ingredients of this theory and
provide some more detail on the development of the scaling diagram
in the $g^\prime$- $q^\prime$ plane.
%%
\section{AES Model}
%%
The action involves a single abelian phase $\phi(\tau)$ describing
the potential fluctuations on the island
$V(\tau)=i\dot{\phi}(\tau)$ with $\tau$ denoting the imaginary
time~\cite{AES}. The theory is defined by
\begin{equation}\label{Zstart}
 Z = \int \mathcal{D}[\phi] e^{-S [\phi]}, \quad S [\phi] =
 S_d +S_t + S_c.
\end{equation}
The action $S_d$ describes the tunneling between the island and
the reservoirs
\begin{equation}\label{SdStart}
 S_d[\phi] = \frac{g}{4}\int_{0}^{\beta} d\tau_1
 d\tau_2\,
 \alpha(\tau_{12})e^{-i[ \phi(\tau_1)-\phi(\tau_2)]}.
\end{equation}
Here, $\beta=1/T$, $\tau_{12}=\tau_{1}-\tau_2$ and $g=g_l+g_r$,
where $g_{l,r}$ denotes the dimensionless bare tunneling
conductance between the island and left/right reservoir (see
Fig.~\ref{FIGURE1}). The kernel $\alpha(\tau)$ is usually
expressed as
$\alpha(\tau)=(T/\pi)\sum_{n}|\omega_n|e^{-i\omega_n\tau}$ with
$\omega_n=2\pi T n$. The part $S_t$ describes the coupling between
the island and the gate and $S_c$ is the effect of the Coulomb
interaction between the electrons
\begin{equation}\label{StcStart}
 S_t[\phi] = - 2\pi i q W [\phi] , \quad
 S_c[\phi] = \frac{1}{4 E_c} \int_{0}^{\beta} d \tau\, \dot{\phi}^2 .
\end{equation}
Here, $q$ is the external charge and $W[\phi] =
1/(2\pi)\int_{0}^{\beta}d\tau \dot{\phi}$ is the winding number or
{\em topological charge} of the $\phi$ field. For the system in
equilibrium the winding number is strictly an integer~\cite{BEAH}
which means that Eq.~\eqref{StcStart} is only sensitive to the
{\em fractional} part $k-1/2 < q \leqslant k+1/2$ of the external
charge $q$.

Eq.~\eqref{SdStart} has classical finite action solutions $\phi_W
(\tau)$ with a non-zero winding number that are completely
analogous to Yang-Mills instantons. The general expression for
winding number $W$ is given by ~\cite{Instantons1,Instantons2}
\begin{equation}
 e^{i\phi_{W}(\tau)} = e^{-i 2\pi T\tau} \sum \limits_{a=1}^{|W|}\frac{e^{i 2\pi
 T\tau}-z_a}{e^{-i 2\pi T\tau}-{z}_a^*}.\label{InstSolG}
\end{equation}
For instantons ($W>0$) the complex parameters $z_a$ are all inside
the unit circle and for anti-instantons ($W<0$) they are outside.
The classical action is finite $ S_d[\phi_W]+S_t[\phi_W] =  g|W|/2
- 2\pi q W i$ leaving the set of parameters
$\{z_1,\dots,z_{|W|}\}$ as $2|W|$ zero modes in the problem.

On the weak coupling side (large $g$) the dominant contribution
comes from widely separated single instantons with $W=\pm 1$. This
leads to the dilute instanton gas approximation where
$\textrm{arg}\, z /2\pi T$ is recognized as the {\em position} of
the single instanton and $\lambda=(1-|z|^2)\beta$ equals the {\em
scale size} or the duration of the potential pulse
$i\dot{\phi}_{W}(\tau)$. The main effect of $S_c$ in
Eq.~\eqref{StcStart} is to provide a cut-off for large frequencies
or small scale sizes $\lambda$.
%%
\section{Kubo Formulae for the Observable Parameters\label{KuboFormulae}}
%%
To develop a general quantum theory of the SET that encompasses
both the weak and strong coupling aspects of the AES model we make
use of the fact that $\tilde{\phi}(\tau)=\omega_n\tau$ satisfies
the classical equation of motion of Eq.~\eqref{Zstart}. By
employing $\tilde{\phi}(\tau)$ as a background field then the
effective action $S^{\,\prime}[\tilde{\phi}]$ is properly defined
in terms of a series expansion in powers of $\omega_n$. Retaining
only the lowest order terms in the series we can write
\begin{equation}\label{Seff}
S^{\,\prime} [\tilde{\phi} ]=\beta \left [
\frac{g^{\,\prime}}{4\pi}
 |\omega_n| - i q^{\,\prime} \omega_n + O(\omega_n^2) \right ].
 %+\frac{\omega_n^2}{4 E_c^\prime}\right ).
 \end{equation}
The quantities of physical interest are the parameters $g^\prime$
and $q^\prime$ with $k-1/2< q^{\,\prime} \leqslant k+1/2$. They
are formally given in terms of the linear response expressions
according to ~\cite{ShortPaper,Det}
\begin{equation}
g^{\,\prime} = 4\pi \Im \frac{\partial
K^R(\omega)}{\partial\omega}\Biggr |_{\omega=0},\qquad
 q^{\,\prime} = Q + \Re
\frac{\partial K^R(\omega)}{\partial\omega}\Biggr |_{\omega=0} .
\label{Back}
\end{equation}
Here, $Q=q + i\langle\dot{\phi}\rangle/(2E_c)$ denotes the average
charge on the island. The function $K^R(\omega)$ is defined as the
analytic continuation $i\omega_n\to \omega+i0^+$ of the following
expression
\begin{equation}
K(i\omega_n)=-\frac{g}{4\beta}\int_0^\beta {d\tau_1
 d\tau_2} e^{i\omega_n\tau_{12}} \alpha(\tau_{12})
D(\tau_{21}) \label{KMdef}
\end{equation}
with $D(\tau_{12})= \langle \exp i [{\phi(\tau_2)-\phi(\tau_1)}]
\rangle$. The function $K^R(\omega)$ is written more explicitly in
terms of the retarded propagator $D^R(E)$ ~\cite{Det}
\begin{equation}
K^R(\omega) = g  \int \frac{d E d E^\prime}{4\pi^3}  E^\prime
\frac{n_B(E^\prime)-n_B(E)}{E-E^\prime+\omega+i 0^+} \Im D^R(E) \nonumber\\
\label{KRdef}
\end{equation}
where $n_B(E) = [\exp(\beta E)-1]^{-1}$ is the Bose-Einstein
distribution.

The quantities $g^{\,\prime}$ and $q^{\,\prime}$ probe the low
energy dynamics of the SET since they are, by construction, a
measure for the sensitivity of the system to infinitesimal changes
in the boundary conditions. Eq.~\eqref{Back} defines in fact
exactly the same quantities that one normally would obtain in
ordinary linear response theory~\cite{ShortPaper,Det}. For
example, $g^{\,\prime}$ is same as the Kubo
formula~\cite{Cur,GG2,Cond} relating a small potential difference
$V$ between the reservoirs to the current $\langle I \rangle$
across the island: $\langle I \rangle = e^2 G V/h$ where the SET
conductance $G = g_l g_r g^\prime/(g_l+g_r)^2$ and $h$ is Planck's
constant. The new quantity $q^{\,\prime}$ is more transparently
written in terms of the quantum current noise according to
\begin{equation}
q^\prime = Q - \frac{(g_l+g_r)^2}{2g_l g_r}  i
\frac{\partial}{\partial V} \int_{-\infty}^0 dt \langle  [I(0),
I(t)] \rangle  \label{CorI}
\end{equation}
in the limit $V \rightarrow 0$.
%%
\section{Weak Coupling Regime, $g^{\,\prime}\gg 1$\label{WCRegime}}
%%

By evaluating Eq.~\eqref{Back} in a series expansion in powers of
$1/g$ one obtains the well-known perturbative results for
$g^{\,\prime}(T)$~\cite{perturb,HZ,Beloborodov}. The new quantity
$q^{\,\prime}$ is unaffected by the quantum fluctuations. To
establish the renormalization of $q^{\,\prime}$ it is necessary to
include the effect of instantons. Extending the methodology of
Ref.~\cite{PruiskenBurmistrov2} we find~\cite{Det}
\begin{eqnarray}
\Im D^R(\omega) &=& \pi \beta \omega \delta(\omega)\left [1-
\frac{2}{g}\ln \frac{gE_ce^\gamma}{2\pi^2 T} \right ] \nonumber
\\&+& \Im \left (\frac{2\pi i/g}{\omega+i0^+} - \frac{2\pi
i/g}{\omega+ig E_c/\pi}\right) \nonumber \\ &-& \frac{g^2
E_c}{\pi^2 T} e^{-g/2} \Re \Biggl( \Biggl [ \pi \beta \omega
\delta(\omega)\nonumber\\
&-& \frac{1}{\omega+i0^+} + \frac{1}{\omega+i2\pi T} \Biggr ]
e^{-i2\pi q} \Biggr) \label{EqImDR}
\end{eqnarray}
where $\gamma= 0.577\dots$ denotes the Euler constant. The first
two lines in Eq.~\eqref{EqImDR} are the one-loop perturbative
results whereas the terms proportional to $\exp(-g/2)$] are
typically instanton terms with $W=\pm 1$. Using Eq.
~\eqref{EqImDR} we obtain Eq. ~\eqref{KRdef} as follows
\begin{eqnarray}
K^R(\omega) &=& \frac{i\omega g}{4\pi} \left [ 1- \frac{2}{g}\ln
\frac{e g E_c}{2\pi^2 T} +\frac{2}{g}\psi \left
(1-\frac{i\omega}{2\pi T}\right ) \right ] \nonumber \\
&-& \frac{g^3 E_c}{2\pi^2} e^{-g/2} e^{i 2\pi q} \Bigl [\psi \left
(1\right ) -\psi \left
(1-\frac{i\omega}{2\pi T}\right )
\Bigr ] \nonumber \\
%
 &-&\frac{g^3 E_c}{2\pi^2}e^{-g/2}\cos 2\pi q \sum_{n>1}^{n_{max}} \frac{1}{n} \nonumber\\
 &-&  \sum_{n>0}^{n_{max}} \frac{g E_c T}{2\pi^2 T n+ g E_c}
\end{eqnarray}
where $\psi(z)$ denotes the Euler di-gamma function and $n_{max}
\approx g E_c/T$. The final results for the observable parameters
of Eq.~\eqref{Back} can be written as follows
\begin{eqnarray}
g^{\,\prime}(T) &=& g  - 2 \ln \frac{g E_c e^{\gamma+1}}{2\pi^2 T}
-\frac{g^3 E_c}{6T} e^{-g/2} \cos 2\pi q \label{gPrT} \nonumber \\ \\
%
q^{\,\prime}(T) &=& q - \frac{g^3 E_c}{24\pi T}  e^{-g/2} \sin 2\pi
q . \label{qPrT}
\end{eqnarray}
These results describe the Coulomb blockade in the weak coupling
regime $g^{\,\prime}\gg 1$ or at high temperatures $T \gg g^3 E_c
e^{-g/2}$ such that the amplitude of oscillations with varying
external charge $q$ is small. These oscillations are nevertheless
much stronger than those appearing in the average charge $Q$ on
the island ~\cite{Instantons21,Instantons3,Beloborodov}
\begin{equation}
Q(T) = q - \frac{g^2}{\pi} e^{-g/2} \ln \left (\frac{E_c}{2\pi^2
e^\gamma T}\right ) \sin 2\pi q . \label{QresInst}
\end{equation}
Eqs~\eqref{gPrT} and \eqref{qPrT} are completely analogous to the
instanton results originally obtained in the theory of the quantum
Hall effect~\cite{Inst-87,Comment} and recently studied
experimentally ~\cite{Murzin}. It should be mentioned that the
result for the SET conductance Eq.~\eqref{gPrT} coincides with
that obtained in Ref.~\cite{Cond}.


%%
\section{Strong Coupling Regime, $g^{\,\prime}\ll 1$\label{SCRegime}}
%%
Near the point $g=0$ and $q=k+1/2$ one can project the theory onto
the low energy states with $Q=k$ and $Q=k+1$
respectively~\cite{Matveev}. The AES theory is then mostly
elegantly described by the spin $1/2$ effective action~\cite{Det}
\begin{eqnarray}
S_{pf} &=& \beta E_c q^2 + \beta \frac{\Delta}{2}+\int_0^\beta d\tau
\overline{\psi} \left (\partial_\tau -\eta +\frac{\Delta}{2}
\sigma_z\right )\psi \nonumber
\\ &+& \frac{g}{4}\int_0^\beta d\tau_1 d\tau_2
\alpha(\tau_{12})[\overline{\psi}(\tau_1)\sigma_-\psi(\tau_1)]
\nonumber \\
&& \hspace{2.5cm}\times [\overline{\psi}(\tau_2)\sigma_+\psi(\tau_2)].
\label{Scrit}
\end{eqnarray}
Here, $\psi$ and $\overline{\psi}$ are the Abrikosov's two-component
pseudofermion fields~\cite{Abrikosov,IzyumovSkryabin}. The
quantity $\Delta= E_c(2k+1-2q)>0$ is the energy gap between the
charging levels with $Q=k$ and $Q=k+1$ respectively and $\sigma_j$
with $j=x,y,z$ stands for the Pauli matrices and $\sigma_{\pm} =
(\sigma_{x}\pm i\sigma_y)/2$.

We have introduced a chemical potential $\eta$ in Eq.
\eqref{Scrit} in order to eliminate the contributions from the
non-physical states with pseudofermion number $N_{pf}\neq 1$. This
is accomplished by taking the limit $\eta\to -\infty$ at the end
of all computations.

Comparing Eqs.~\eqref{Zstart} and \eqref{Scrit} we identify the
spin operators $\overline{\psi}(\tau)\sigma_{\pm}\psi(\tau)$ with the
AES operators $\exp(\pm i \phi(\tau))$ projected onto the states
with $Q=k$ and $Q=k+1$ respectively. It is interesting to notice
that the effective action~\eqref{Scrit} is similar to the $XY$
case of the Bose-Kondo model for spin
$s=1/2$~\cite{LarkinMelnikov,SiDemler}. Similar to the one loop
renormalization group procedure described in Ref.~\cite{SiDemler}
we sum the leading logarithms and find
\begin{eqnarray}
\Im D^R(\omega) &=& \frac{\pi}{\gamma^2}\delta(\omega-
\Delta^{\,\prime}) \tanh \frac{\beta \Delta^{\,\prime}}{2} ,\nonumber\\
\gamma^2 &=&  1+\frac{g}{2\pi^2} \ln
\frac{\Lambda}{\max\{\Delta^{\,\prime},T\}}. \label{DRSC}
\end{eqnarray}
Here, $\Delta^\prime=\Delta/\gamma^2$ denotes the renormalized
energy gap and $\Lambda$ is a high energy cut-off of order $E_c$.
We obtain the following result for Eq.~\eqref{KRdef}
\begin{eqnarray}
K^R(\omega) &=& \frac{g}{\gamma^2}
\frac{\omega+\Delta^{\,\prime}}{4\pi^2}\tanh
\frac{\beta\Delta^{\,\prime}}{2} \Bigl [ \psi\left (1-i
\frac{\omega+\Delta^{\,\prime}}{2\pi T}\right )
\nonumber \\
&-&\psi\left (1- \frac{i\Delta^{\,\prime}}{2\pi T}\right
)-\frac{2\pi}{\Delta^{\,\prime}}Y(\Delta^{\,\prime}) \Bigr
]\label{KRSCres}
\end{eqnarray}
with $Y(\Delta^{\,\prime}) = T\sum_{\omega_n>0}
\omega_n\Delta^{\,\prime}/(\omega_n^2+\Delta^{\,\prime 2})$. Based
on Eq.~\eqref{KRSCres} we obtain the following expressions for the
physical observables in Eq.~\eqref{Back}
\begin{eqnarray}
g^{\,\prime}(T) &=& \frac{g}{2\gamma^2}\frac{\beta
\Delta^\prime}{\sinh \beta \Delta^\prime}, \label{PrSCres1} \\
 q^{\,\prime}(T)
&=& Q(T) - \frac{\gamma^2-1}{2\gamma^2} \tanh \frac{\beta
\Delta^\prime}{2}. \label{PrSCres2}
\end{eqnarray}
Here, Eq.~\eqref{PrSCres1} and $Q(T)$ are the same as the results
obtained in Ref.~\cite{Schon}
\begin{equation}
Q(T) = \frac{1}{2} \left (1 - \frac{1}{\gamma^2} \tanh \frac{\beta
\Delta^\prime}{2} \right ) .\label{QSCres}
\end{equation}
In the limit $T=0$ we find
%
\begin{equation}
 Q(T=0) = \frac{g}{4\pi^2} \ln
 \frac{\Lambda}{\Delta^{\,\prime}}\Bigr / \left (1+\frac{g}{2\pi^2}
 \ln\frac{\Lambda}{\Delta^{\,\prime}}\right )
\end{equation}
%
which is the result obtained by Matveev~\cite{Matveev}. It says
that in the presence of tunneling $g\neq 0$ the average charge
$Q(T)$ on the island is {\em un-} quantized. On the other hand,
Eqs.~\eqref{PrSCres2} and ~\eqref{QSCres} imply
\begin{equation}
q^{\,\prime}(T) = k+\frac{1}{1+e^{\beta \Delta^\prime}}
\label{qPrSCres}
\end{equation}
indicating that the novel physical quantity $q^{\,\prime}$ is
\emph{robustly quantized} as one moves away from quantum
criticality.

%Summation of the leading logarithms yields the %renormalization of the energy gap, $\Delta \to \Delta^\prime$,
%but preserves the  zero width for the resonance at $\omega=%\Delta^\prime$ in $D^R(\omega)$.

Finally, at lower temperatures $T \lesssim \Delta^\prime$ the {\it
inelastic cotunneling} processes become important~\cite{Averin}.
The lowest order correction to $D^R(\omega)$ due to inelastic
cotunneling is obtained as follows~\cite{Det}
\begin{equation}
\delta D^R_{\rm inel}(\omega) =  \frac{g}{4\pi \gamma^4} \frac{i\omega}{(\omega-\Delta^\prime+i0^+)^2} .
\label{DR_IC}
\end{equation}
Based on this result we compute the corrections to the physical
observables using Eq.~\eqref{KRdef} and Eq.~\eqref{Back} and the
result is
\begin{eqnarray}
\delta g_{\rm inel}^\prime(T) &=& \frac{g^2}{4\pi^2
\gamma^4}\frac{\partial }{\partial\Delta^\prime} \left [
\frac{\beta\Delta^{\prime 2}}{2\pi}\psi^\prime\left (1+\frac{i
\beta\Delta^\prime}{2\pi}\right ) -\Delta^\prime \right ]
\nonumber \\
\delta q^\prime_{\rm inel}(T) &=& \frac{g^2}{4\pi \gamma^4}
\Delta^\prime \frac{\partial^2}{\partial \Delta^{\prime 2}}
\left ( \Delta^\prime \coth \frac{\beta\Delta^\prime}{2}\right ).
\end{eqnarray}
In the regime $T \lesssim \Delta^\prime$ we find
\begin{eqnarray}
\delta g^\prime_{\rm inel}(T) &=&  \frac{g^2}{48 \gamma^4} \frac{T^2}{\Delta^{\prime 2}}
\label{Inel1}\\
\delta q^\prime_{\rm inel}(T) &=& \frac{g^2}{2\pi \gamma^4}
\frac{\Delta^{\prime 2}}{T^2} e^{-\beta \Delta^\prime}.\label{Inel2}
 \end{eqnarray}
Eq.~\eqref{Inel1} coincides with the result found in
Ref.~\cite{Schon}. As a final remark we emphasize that the results
for cotunneling cannot be extrapolated all the way down to the
regime of exponential localization $T \ll \Delta^\prime$ which is
clearly beyond the range of validity of the strong coupling
expansion.
%%
\section{Summary and Conclusions\label{Sum}}
%%
The temperature dependence of the physical observables
$g^{\,\prime}$ and $q^{\,\prime}$ can in general be expressed in
terms of renormalization group $\beta$ functions
%
\begin{equation}
 \frac{d g^{\,\prime}}{d\ln\beta} = \beta_{g}(g^\prime,q^\prime) ~,~~
 \frac{d q^{\,\prime}}{d\ln\beta} = \beta_{q}(g^\prime,q^\prime) .
\end{equation}
%
From Eqs.~\eqref{gPrT}, \eqref{qPrT} we extract for the weak
coupling regime
\begin{eqnarray}
\beta_g &=& -2 - \frac{\pi^2e^{-\gamma-1}}{3}  g^{\,\prime 2}
e^{-g^{\,\prime}/2} \cos 2\pi q^{\,\prime},\\ \beta_q &=&
\frac{\pi e^{-\gamma-1}}{12} g^{\,\prime 2} e^{-g^{\,\prime}/2}
\sin 2\pi q^{\,\prime} .\label{WCRG}
\end{eqnarray}
From Eqs.~\eqref{PrSCres1} and \eqref{qPrSCres} we obtain the
following strong coupling behavior near $q^{\,\prime}=k+1/2$
\begin{equation}
\beta_g = -\frac{g^{\,\prime 2}}{\pi^2},\,\qquad \beta_q =
\Bigl (q^{\,\prime}-k-\frac{1}{2}\Bigr )\left (1-\frac{g^{\,\prime}}{\pi^2}\right
).\label{SCRG}
\end{equation}
This result shows that $q^{\,\prime} =k+1/2$ and $g^{\,\prime}=0$
is the {\em critical} fixed point of the AES theory with
$g^\prime$ a marginally irrelevant scaling variable.

Eqs ~\eqref{WCRG} and ~\eqref{SCRG} together determine a unifying
scaling diagram of the SET in the $g^{\,\prime}$ and
$q^{\,\prime}$ plane as illustrated in Fig.~\ref{FIGURE2}. The
stable strong coupling fixed points $g^\prime=0$ and $q^\prime=k$
clearly indicate that the AES theory generally displays the
Coulomb blockade as $T$ goes to absolute zero. This scaling
phenomenon is fundamentally different from semiclassical picture
of the Coulomb blockade since it elucidates the discrete nature of
the electronic charge which is independent of tunneling.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\includegraphics[height=65mm]{Figure2}
\caption{Unified scaling diagram of the Coulomb blockade in terms
of the SET conductance $g^{\,\prime}$ and the $q^{\,\prime}$. The
arrows indicate the scaling toward $T = 0$ (see text).}
\label{FIGURE2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In conclusion, the AES model is an extremely interesting and
exactly solvable example of a $\theta$ angle concept that displays
all the {\em super universal} topological features that have
arisen before in the context of the quantum Hall
liquids~\cite{PruiskenBurmistrov2,CPN} as well as quantum spin
liquids~\cite{Spin}. These include not only the existence of {\em
gapless} or critical excitations at $q^\prime=k+1/2$ (or $\theta =
\pi$) but also the {\em robust} topological quantum numbers that
explain the ``macroscopic charge quantization" of the SET at zero
temperature and finite values of $g$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BACKMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theacknowledgments}
  The authors are indebted to A. Abanov for bringing the AES model
to their attention and for valuable conversations. One of us
(I.S.B.) is grateful to O. Astafiev, A. Lebedev and Yu. Makhlin
for helpful discussions. The research was funded in part by the
Dutch National Science Foundations \textit{NWO} and \textit{FOM},
the EU-Transnational Access program (RITA-CT-2003-506095), CRDF,
the Russian Ministry of Education and Science, Council for Grant
of the President of Russian Federation, RFBR, Dynasty Foundation
and the Program of RAS ``Quantum Macrophysics''. One of us (ISB)
acknowledges the hospitality of the Institute for Theoretical
Physics of University of Amsterdam where a part of this work was
performed.
\end{theacknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The bibliography can be prepared using the BibTeX program or
%% manually.
%%
%% The code below assumes that BibTeX is used.  If the bibliography is
%% produced without BibTeX comment out the following lines and see the
%% aipguide.pdf for further information.
%%
%% For your convenience a manually coded example is appended
%% after the \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% You may have to change the BibTeX style below, depending on your
%% setup or preferences.
%%
%%
%% For The AIP proceedings layouts use either
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibliographystyle{aipproc}   % if natbib is available
%\bibliographystyle{aipprocl} % if natbib is missing

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% You probably want to use your own bibtex database here
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliography{sample}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Just a reminder that you may have to run bibtex
%% All of it up to \end{document} can be removed
%% if you don't like the warning.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\IfFileExists{\jobname.bbl}{}
 %{\typeout{}
 % \typeout{******************************************}
  %\typeout{** Please run "bibtex \jobname" to optain}
  %\typeout{** the bibliography and then re-run LaTeX}
  %\typeout{** twice to fix the references!}
  %\typeout{******************************************}
  %\typeout{}
 %}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The following lines show an example how to produce a bibliography
%% without the help of the BibTeX program. This could be used instead
%% of the above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}

\bibitem{QExpFirst} P.\,Lafarge, H.\,Pothier, E.R.\,Williams, D.\,Esteve,
C.\,Urbina, and M.H.\,Devoret, Z. Phys. B \textbf{85}, 327 (1991)

\bibitem{SETExp} T.A.\,Fulton and G.J.\,Dolan, Phys. Rev. Lett.
\textbf{59}, 109 (1987)

\bibitem{Matveev} K.A.\,Matveev, Sov. Phys. JETP \textbf{72}, 892 (1991)

\bibitem{ShortPaper} I.S.\,Burmistrov, and A.M.M.\,Pruisken, Phys. Rev. Lett. {\bf 101}, 056801 (2008)

\bibitem{PruiskenBurmistrov2} A.M.M.\,Pruisken and I.S.\,Burmistrov,
Ann. of Phys. {\bf 316}, 285 (2005)

\bibitem{AES} V.\,Ambegaokar, U.\,Eckern, and G.\,Sch\"{o}n, Phys.
Rev. Lett. \textbf{48}, 1745 (1982)

\bibitem{BEAH} I.S.\,Beloborodov, K.B.\,Efetov, A.\,Altland, and F.W.J.\,Hekking,
Phys. Rev. B \textbf{63}, 115109 (2001); K.B.\,Efetov, and
A.\,Tschersich, Phys. Rev. B \textbf{67}, 174205 (2003)

\bibitem{Instantons1} S.E.\,Korshunov, JETP Lett. \textbf{45}, 434
(1987)

\bibitem{Instantons2} S.A.\, Bulgadaev, Phys. Lett. A
\textbf{125}, 299  (1987)

\bibitem{Det} I.S.\,Burmistrov, and A.M.M.\,Pruisken,
in preparation

\bibitem{Cur} E.\,Ben-Jacob,
E.\,Mottola, and G.\,Sch\"{o}n, Phys. Rev. Lett. \textbf{51}, 2064
(1983)

\bibitem{GG2} C.\,Wallisser, B.\,Limbach, P.\,vom Stein, R.\,Sch\"{a}fer,
C.\,Theis, G.\,G\"{o}ppert, and H.\,Grabert, Phys. Rev. B
\textbf{66}, 125314 (2002)

\bibitem{Cond} A.\,Altland, L.I.\,Glazman, A.\,Kamenev, and
J.S.\,Meyer, Ann. Phys. (N.Y.) \textbf{321}, 2566 (2006)

\bibitem{perturb} F.\,Guinea and G.\,Sch\"{o}n, Europhys. Lett.
\textbf{1},585 (1986)

\bibitem{HZ} W.\,Hofstetter and W.\,Zwerger, Phys. Rev. Lett.
\textbf{78}, 3737 (1997)

\bibitem{Beloborodov} I.S.\,Beloborodov, A.V.\,Andreev, and A.I.\,Larkin, Phys. Rev.
B \textbf{68}, 024204 (2003)

\bibitem{Instantons21} S.V.\, Panyukov and A.D.\, Zaikin, Phys. Rev. Lett. \textbf{67}, 3168
(1991)

\bibitem{Instantons3} X.\,Wang and H.\,Grabert, Phys. Rev. B
\textbf{53}, 12621 (1996)

\bibitem{Inst-87} A.\,M.\,M.\,Pruisken, Nucl. Phys. B {\bf 285}, 719 (1987);
{\bf 290}, 61 (1987).

\bibitem{Comment} A.\,M.\,M.\,Pruisken and I.\,S.\,Burmistrov,
Phys. Rev. Lett. \textbf{95}, 189701 (2005)

\bibitem{Murzin} S.\,S.\,Murzin, A.\,G.\,M.\,Jansen, and I.\,Claus,
Phys. Rev. Lett. \textbf{92}, 016802 (2004); S.\,S.\,Murzin and
A.\,G.\,M.\,Jansen, Phys. Rev. Lett. \textbf{95}, 189702 (2005)

\bibitem{Abrikosov} A.\,A.\,Abrikosov, Physics \textbf{2}, 21
(1965)

\bibitem{IzyumovSkryabin} see, e.g., Yu.\,A.\,Izyumov and Yu.\,N.\,Skryabin,
\textit{Statistical mechanics of magnetoordered systems}, (Moskva,
Nauka, 1987) (in russian)


\bibitem{LarkinMelnikov} A.I.\,Larkin and V.I.\,Melnikov, Sov. Phys. JETP \textbf{34}, 656
(1972)
%[Zh. \'{E}ksp. Teor. Fiz. \textbf{61}, 1232 (1971)].

\bibitem{SiDemler} L.\,Zhu and Q.\,Si, Phys. Rev. B \textbf{66}, 024426 (2002);
G.\,Zar\'{a}nd and E.\,Demler, Phys. Rev. B \textbf{66}, 024427
(2002)

\bibitem{Schon} H.\,Schoeller and G.\,Sch\"{o}n, Phys. Rev. B
\textbf{50}, 18436 (1994)

\bibitem{Averin} D.V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. {\bf 65} 2446 (1990)


\bibitem{CPN} A.M.M.\,Pruisken, I.S.\,Burmistrov and R. Shankar,
cond-mat/0602653 (unpublished)

\bibitem{Spin} A.M.M.\,Pruisken, R. Shankar and N. Surendran, Phys. Rev. B \textbf{72}, 035329 (2005); Europhys. Lett. {\bf 82}, 47005 (2008).


\end{thebibliography}

\end{document}

\endinput
%%
%% End of file `template-8d.tex'.
