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\title{Universal Conductance Fluctuations by Carlo Beenakker}

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\maketitle

{\small This definition has been adapted from~\cite{0904.1432}.}

In the 1960's, Wigner, Dyson, Mehta, and others discovered that the fluctuations in the energy level density are governed by level repulsion and therefore take a universal form \cite{Por65}.
The universality of the level fluctuations is expressed by the
Dyson-Mehta formula \cite{Dys63} for the variance of a linear statistic \footnote{The quantity $A$ is
called a linear statistic because products of different $E_{n}$'s do
not appear, but the function $a(E)$ may well depend non-linearly on
$E$.}
$A=\sum_{n}a(E_{n})$ on the energy levels $E_{n}$.  The Dyson-Mehta formula reads 
\begin{equation}
{\rm Var}\,A=\frac{1}{\beta}\,\frac{1}
{\pi^{2}}\int_{0}^{\infty}dk\,|a(k)|^{2}k,\label{DysonMehta}
\end{equation}
where $a(k)=\int_{-\infty}^{\infty}\!dE\,{\rm e}^{{\rm i}kE}a(E)$ is
the Fourier transform of $a(E)$.  Eq.\ \eqref{DysonMehta} shows that:
1.~The variance is independent of microscopic parameters; 2.~The
variance has a universal $1/\beta$-dependence on the symmetry index.

\begin{figure}[!tb]
\centerline{
\includegraphics[width=20pc]{UCFAu.png}
}
\caption{ Fluctuations as a function of perpendicular magnetic field of the
  conductance of a 310~nm long and 25~nm wide Au wire at 10~mK. The trace
  appears random, but is completely reproducible from one measurement to the
  next. The root-mean-square of the fluctuations is $0.3\,e^{2}/h$, which is
  not far from the theoretical result $\sqrt{1/15}\,e^{2}/h$ [See Sec.~II of~\cite{0904.1432}]. Adapted from
  Ref.\ \cite{Was86}.
\label{fig_UCFAu}}
\end{figure}

In a pair of seminal 1986-papers \cite{Imr86,Alt86}, Imry and Altshuler and Shklovkski\u{\i} proposed to apply random matrix theory to the
phenomenon of universal conductance fluctuations (UCF) in metals, which was discovered
using diagrammatic perturbation theory by Altshuler \cite{Alt85} and
Lee and Stone \cite{Lee85}. UCF is the occurrence of sample-to-sample
fluctuations in the conductance which are of order $e^{2}/h$ at zero
temperature, \textit{independent} of the size of the sample or the
degree of disorder --- as long as the conductor remains in the diffusive
metallic regime (size $L$ large compared to the mean free path $l$, but small compared to the localization length $\xi$). An example is shown in Fig.~\ref{fig_UCFAu}. 

The similarity between the statistics of energy
levels measured in nuclear reactions on the one hand, and the
statistics of conductance fluctuations measured in transport
experiments on the other hand, was used by Stone \textit{et al.} \cite{Mut87,Sto91} to construct a \cref{random matrix theory} of quantum
transport in metal wires. The random matrix is now not the Hamiltonian $H$, but the transmission matrix $t$, which determines the conductance through the \cref{Landauer formula}
\begin{equation}
G=G_{0}{\rm Tr}\,tt^{\dagger}=G_{0}\sum_{n}T_{n}.\label{Landauer}
\end{equation}
The conductance quantum is $G_{0}=2e^{2}/h$, with a factor of two to account for spin degeneracy. Instead of repulsion of energy levels, one now has repulsion of the transmission eigenvalues $T_{n}$, which are the eigenvalues of the transmission matrix product $tt^{\dagger}$. In a wire of cross-sectional area ${\cal A}$ and Fermi wave length $\lambda_{F}$, there are of order $N\simeq A/\lambda_{F}^{2}$ propagating modes, so $t$ has dimension $N\times N$ and there are $N$ transmission eigenvalues. The phenomenon of UCF applies to the regime $N\gg 1$, typical for metal wires.

\cref{Random matrix theory} is based on the fundamental
assumption that all correlations between the eigenvalues are due to the
Jacobian  $J=\prod_{i<j}|T_{i}-T_{j}|^{\beta}$ from matrix elements to eigenvalues. If all correlations are due to the Jacobian, then the
probability distribution $P(T_{1},T_{2},\ldots T_{N})$
of the $T_{n}$'s should have the form $P\propto
J\prod_{i}p(T_{i})$, or equivalently,
\begin{align}
P(\{T_{n}\})\propto{}&\exp\Bigl[-\beta\Bigl(\sum_{i<j}
u(T_{i},T_{j})+\sum_{i}V(T_{i})\Bigr)\Bigr],
\label{Pglobala}\\
u(T_{i},T_{j})={}&-\ln|T_{j}-T_{i}|,
\label{Pglobalb}
\end{align}
with $V=-\beta^{-1}\ln p$.
Eq.\ \eqref{Pglobala} has the form of a Gibbs distribution at
temperature $\beta^{-1}$ for a fictitious system of classical particles
on a line in an external potential $V$, with a logarithmically
repulsive interaction $u$. All microscopic parameters are contained in
the single function $V(T)$. The logarithmic repulsion is
independent of microscopic parameters, because of its geometric
origin.

Unlike the RMT of energy levels, the correlation function of the $T_{n}$'s is not
translationally invariant, due to the
constraint $0\leq T_{n}\leq 1$ imposed by unitarity of the scattering matrix. Because of this
constraint, the Dyson-Mehta formula \eqref{DysonMehta} needs to be modified, as shown in Ref.\ \cite{Bee93a}. In the large-$N$ limit, the variance of a linear
statistic $A=\sum_{n}f(T_{n})$ on the transmission eigenvalues is given by
\begin{equation}
{\rm Var}\,A=\frac{1}{\beta}\,\frac{1}{\pi^{2}}\int_{0}^{\infty}
dk\,|F(k)|^{2}k\tanh(\pi k).\label{CB}
\end{equation}
The function $F(k)$ is defined in terms of the function $f(T)$ by the
transform
\begin{equation}
F(k)=\int_{-\infty}^{\infty}\!dx\,{\rm e}^{{\rm i}kx}f
\left(\frac{1}{1+{\rm e}^{x}}\right).\label{Fkdef}
\end{equation}
The formula \eqref{CB} demonstrates that the universality which was the
hallmark of UCF is generic for a whole class of transport properties,
viz.\ those which are linear statistics on the transmission
eigenvalues. Examples, reviewed in Ref.\ \cite{Bee97}, are the critical-current fluctuations in Josephson
junctions, conductance fluctuations at normal-superconductor interfaces, and fluctuations in the
shot-noise power of metals.

{\small For more details see~\cite{0904.1432}, Sec.~II.}

\begin{thebibliography}{aaaaaa}
\bibitem{0904.1432} C.W.J. Beenakker \url{0904.1432} [cond-mat.mes-hall]
\bibitem{Por65} C.E. Porter, ed., \textit{Statistical Theories of Spectra: Fluctuations}, Academic Press, New York 1965.
\bibitem{Dys63} F.J. Dyson and M.L. Mehta, J. Math. Phys. \textbf{4} (1963) 701.
\bibitem{Was86} S. Washburn and R.A. Webb, Adv. Phys. \textbf{35} (1986) 375.
\bibitem{Imr86} Y. Imry, Europhys. Lett. \textbf{1} (1986) 249.
\bibitem{Alt86} B.L. Altshuler and B.I. Shklovski\u{\i}, Sov. Phys. JETP \textbf{64} (1986) 127.
\bibitem{Alt85} B.L. Altshuler, JETP Lett. \textbf{41} (1985) 648.
\bibitem{Lee85} P.A. Lee and A.D. Stone, Phys. Rev. Lett.  \textbf{55} (1985) 1622.
\bibitem{Mut87} K.A. Muttalib, J.-L. Pichard, and A.D. Stone, Phys. Rev. Lett. \textbf{59} (1987) 2475.
\bibitem{Sto91} A.D. Stone, P.A. Mello, K.A. Muttalib, and J.-L. Pichard, in: \textit{Mesoscopic Phenomena in Solids}, ed.\ by B.L. Altshuler, P.A. Lee, and R.A. Webb, North-Holland, Amsterdam 1991.
\bibitem{Bee93a} C.W.J. Beenakker, Phys. Rev. Lett. \textbf{70} (1993) 1155.
\bibitem{Bee97} C.W.J. Beenakker, Rev. Mod. Phys. \textbf{69} (1997) 731.

\end{thebibliography}
\end{document}