The Mesoscopic Chiral Metal--Insulator Transition
S. Kettemann$^{(1)}$
B. Kramer$^{(1)}$
T. Ohtsuki$^{(1,2)}$
Sharp localization transitions of chiral edge states in disordered quantum wires, subject to strong magnetic field, are shown to be driven by crossovers from two- to one-dimensional localization of bulk states. % These crossovers are found to cause chiral metal--insulator transitions. % They are characterized by jumps of % the localization length of the edge states from {exponentially large} % to finite % values. As a result, the two-terminal conductance is found to exhibit at zero temperature discontinuous transitions between {exactly} integer plateau values and zero, reminiscent of first order phase transitions. We discuss the corresponding phase diagram. The spin of the electrons is shown to result in a multitude of phases, when the spin degeneracy is lifted by the Zeeman energy. The width of conductance plateaus is found to depend sensitively on the spin flip rate $1/\tau_s$.
mesoqhe03.tex
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\title{The Mesoscopic Chiral Metal--Insulator Transition}  
% 
\author{S. Kettemann$^{(1)}$, B. 
  Kramer$^{(1)}$, T.  Ohtsuki$^{(1,2)}$ }  
% 
\address{ $^{1}$I. Institut f\"ur 
  Theoretische Physik, Universit\" at Hamburg, Jungiusstra\ss{}e 9, 20355 
  Hamburg, Germany\\ 
% 
 $^{2}$Department of Physics, Sophia 
  University, Kioi-Choi 7-1, Chiyoda-Ku, Tokyo 102-8554, Japan} 
\maketitle 
%\date{April 21, 2003} 
%% 
%% 
\begin{abstract} 
  Sharp localization transitions  of  chiral edge states 
 in disordered quantum wires,  subject to  strong magnetic field,  are 
  shown  to be 
  driven by crossovers from two- to one-dimensional localization of  bulk
  states.
% These crossovers are found to cause chiral metal--insulator transitions.
%  They are characterized by  jumps of
%  the localization length of the edge states from  {\it exponentially large}
% to finite
%  values.
 As a result, the two-terminal conductance 
  is found  to 
 exhibit at zero temperature discontinuous transitions between {\it exactly}
  integer
  plateau values and zero, reminiscent of first order phase transitions.
 We discuss the corresponding phase diagram.  The 
  spin of the electrons is shown to result in  a   multitude of phases, when 
  the spin degeneracy is lifted by the Zeeman energy. 
 The width of  conductance plateaus is  found to depend sensitively
 on the spin flip rate $1/\tau_s$. 
\end{abstract} 
% 
\pacs{PACS numbers: 72.10.Fk, 72.15.Rn, 73.20.Fz} 
%\maketitle 
\vskip2pc] 
 
The high precision of the quantization of the Hall conductance of a two
dimensional (2D) electron system (2DES) in  a strong  magnetic field
\cite{klitzing} is known to be due to the binding of electrons to localized
states in the bulk of the 2DES. Thereby,  a change of electron density does not
 change the  Hall conductance \cite{laughlin,aoki,halperin}.
The localization length in  tails of  Landau bands is very small,
of the order of the cyclotron length
$l_{\rm cyc} = v_F/\omega_B = \sqrt{2 n+ 1} l_B$. 
  It increases towards the centers of the Landau
bands, $E_{n0} = \hbar \omega_B (n +1/2)$ ($n=0,1,2,...$), with $\omega_B = e
B/m^{*}$ the cyclotron frequency ($e$ elementary charge, $m^{*}$ effective
mass),  $v_F$ the Fermi velocity, 
and $l_B^2 = \hbar/e B$ defines the magnetic length.
  In an {\em infinite} 2DES in 
perpendicular magnetic field, the localization length at energy $E$ diverges
as $ \xi \sim |E -E_{n0}|^{-\nu}$, reminiscent of 2nd order phase
transitions.  The critical exponent $\nu$ is found numerically for the lowest
two Landau bands, $n=0,1$, to be $\nu = 2.33\pm0.04$ for spin-split Landau
levels \cite{hucke,huckerev}.  Analytical \cite{raikh} and
experimental studies \cite{scalingexp} are consistent with this value. In a
{\em finite} 2DES, a region of extended states should exist in the centers of
 disorder broadened Landau-bands. These states extend beyond the system
size $L$. The width of these regions is given by $\Delta E = (l_{\rm
  cyc}/L)^{1/\nu} \Gamma$, where   $\Gamma = \hbar ( 2
\omega_B/\pi\tau)^{1/2}$ is the band width,
with elastic scattering time $\tau$. 
 
In quantum Hall bars of finite width, there exist in addition 
edge states with energies lifted by the confinement potential above the
energies of  centers of  bulk Landau bands, $E_{n0}$ \cite{halperin}.
Previously, there has been a considerable interest in the study of
mesoscopically narrow quantum Hall bars\cite{haug}, with  emphasis on conductance
fluctuations\cite{timp,ando94}, edge state mixing
\cite{edgemixing,shkledge,ohtsuki,mani},  breakdown of the quantum Hall
effect\cite{breakdown}, and  quenching of the Hall effect due to classical
commensurability effects \cite{roukes}.
%  When  the localization length of bulk states becomes of the 
%   order of the wire width, the electrons can diffuse freely  
%  between the edges, and  opposite  edge states can  mix.   
It is known  that in the presence of white noise disorder
the edge states do mix with the bulk states when the Fermi energy is moved
into the center of a Landau band. It had been suggested that this might result
in localization of edge states \cite{ohtsuki,mani}.
In this paper, we show  that this is
indeed the case. In particular, at zero temperature the
two-terminal conductance  of a quantum wire in a
magnetic field exhibit for uncorrelated disorder and hard wall confinement
discontinuous transitions between integer plateau values and zero,  Fig \ref{loclength}.
\begin{figure}
\begin{center}
\vspace{-4.5cm}
%\scalebox{0.4}{
  \includegraphics[width=.45 \textwidth ]{prlfig1.ps}
\vspace{-0.cm}
\caption{ Solid curve, left axis: Two--terminal conductance
  of  a wire  (width $L_y = 80 a$,  length $L = 5000 a$, hard wall
  confinement)
 in units of $G_0= e^2/h$ as  function of
  energy $E$ in units of $t$ (hopping amplitude in the tight binding
  model). Disorder strength,  $W = .8 t$. There are  $x = .025$ 
 magnetic  flux quanta through an elementary
  cell $a^2$.
% corresponding to a magnetic length $l_B = (1/2 \pi
%  b)^{1/2} a \approx 2.52 a$. 
 Dashed curve, right axis: 
  bulk localization length $\xi$ in units of
   $a$ as  function of
   $E$, from   transfer matrix method for  wire of
  identical properties but periodic boundary conditions and $L_{max} = 100 000
  a$.  Straight dashed  line:   $L_y = 80 a$.
   }
 \label{loclength}
\end{center}
\vspace{-.5cm}
\end{figure}

The localization length $\xi$ in a 2DES with broken time reversal symmetry is
expected from  scaling theory\cite{ab,weg,hikami,elk} and
numerical scaling studies \cite{mk81,transfer} to be 
\begin{equation} \label{2db} 
\xi_{\rm 2D}  = l_0\, e^{\pi^2 g^2}. 
\end{equation} 
 depending exponentially on 
 $g$,  the 2D conductance parameter per spin channel.  
 $l_0$ is the short distance cutoff, which
is the elastic mean free path $l = 2 g(B=0)/k_{{\rm F}}$ ($k_{\rm F}$ Fermi
wave number) at moderate magnetic fields, $b\equiv\omega_B \tau < 1$.
 For stronger magnetic fields, $b>1$,   $l_0$ crosses
over to the cyclotron length $l_{\rm cyc}$.
   $g$ exhibits Shubnikov-de-Haas oscillations as function
of  magnetic field for $b>1$. Maxima occur when the Fermi energy is in
the center of  Landau bands. Thus, the localization length 
increases strongly from  band tails to  band centers, even 
 when  the wire  width $L_y$  is too narrow to allow delocalisation 
of bulk states. For uncorrelated impurities, within self
consistent Born approximation \cite{ando},  the maxima 
 are given by $g(E = E_{n 0}) =  (2  n + 1)/\pi = g_n
$.  Thus, $ \xi_{\rm 2D} (E_{n 0}) = l_{\rm cyc} \exp{( \pi^2 g_n^2 )} $ are macroscopically
large in  centers of higher Landau bands, $n > 1 $
\cite{huckerev,levitation}.  However, when the width of the system $L_y$ is
smaller than  $\xi_{\rm 2D}$, localization is expected to
behave quasi-1D. In other words,  electrons in  centers of  Landau
bands can diffuse between the edges of the system, but are localized parallel
to the  edges if $L_y < \xi_{\rm 2D}$. 
 The quasi-1D localization length is known to
depend only linearly on $g$. In a magnetic field, when
time reversal symmetry is broken, it is \cite{ef,larkin,dorokhov}
\begin{equation} \label{1db} 
\xi_{\rm 1D} = 2 g (B)  L_y.   
\end{equation} 
There is a crossover from 2D to 1D
localization as the Fermi energy is moved from  tails to  centers of 
Landau bands. 
 Performing  a    renormalization  of the wire conductance, 
 one obtains for the  localization length \cite{crossover},
%   Performing  this  renormalization for wires of 
% finite width $L_y$ in the unitary regime, we obtain 
%  in appendix B,  that the  
%  localization length $\xi$  satisfies in the unitary regime the equation,  
\begin{equation}  
\label{lcb} 
\xi^2  = 4  L_y^2 g^2 - \frac{2 L_y^2 }{ \pi^2} 
 \ln  \left[ \frac{1+ (L_y/ l_0 )^2}{ 1 + ( L_y/  \xi )^2} \right].   
\end{equation}  
 Its solution shows   a crossover between the quasi-1-D--
 and the 2D--behaviour,
 Eq.  (\ref{1db}) and  
 Eq. (\ref{2db}), respectively. 

% \begin{figure}[b]
%\begin{center} 
%\vspace{-0.5cm} 
%\includegraphics[width=.4\textwidth]{bild3.ps} 
%\vspace{-4cm} 
%\caption[]{The localization 
%  length $\xi$ as function of conductance $g$, in the unitary limit, $\xi >
%  L_B$, where
%  $L_B =l_B$, the magnetic length, in the 2D limit \cite{prbr}.
%  For comparison, the quasi-2D behavior $\xi_{\rm 2D} = (2 g/ k_{\rm F})
%  \exp(\pi^2 g^2)$ (dotted) and the quasi-1D limit $2 L_y g$ (dashed-dotted)
%  are shown. The dashed horizontal line indicates $L_y$.}
% \label{lcbg} 
%\vspace{-.5cm} 
%\end{center} 
%\end{figure}  
%Here, $\Sigma = 1$, for spin degenerate energy levels,
% and $\Sigma = 2$, when
%the spin degeneracy is broken, and energy levels are mixed by spin flip.  We
%assume $\Sigma=1$ in the following.  
%Eq. (\ref{lcblambert}) is solved in
%two limiting cases.  For $L_y \ll\xi$, there is a logarithmic correction to
%the quasi-1D result,
%\begin{equation} 
%\label{lcb1d} 
%\xi \mid_{\xi \gg L_y} = 2 L_y g \left[ 1 - \frac{1}{\pi^2 g^2} \ln 
%\sqrt{1+ \left( \frac{L_y}{ l_0} \right)^2} \right]^{1/2}. 
%\end{equation} 
%For $\xi < L_y$, Eq. (\ref{lcblambert}) can be simplified to
%\begin{equation} 
%\xi \mid_{\xi \ll L_y} = l_0 \exp\left[ \frac{\pi^2}{4}  
%\left(4 g^2 -\frac{\xi^2}{L_y^2} \right)\right].  
%\end{equation} 
%The solution of this equation can be written in terms of the
%Lambert-W-function $W_{0}$ \cite{lambert},
%\begin{equation}  
%\label{lcb2d}  
%\xi  \mid_{\xi \ll L_y} = \frac{\sqrt{2} L_y}{  \pi } \,W_0^{1/2} 
%\left[\frac{l_0^2 }{2 L_y^2} \,e^{2 \pi^2g^2 } \right]. 
%\end{equation}  
%%For fixed width $L_y$, the localization length $\xi$ is plotted in Fig.
%%\ref{lcbg} as function of conductance parameter $g$, using Eq.~(\ref{lcb1d})
%%for $\xi>L_y$, and Eq. (\ref{lcb2d}) for $\xi < L_y$.
 

%Quantum wires of disordered electrons in a strong magnetic field exhibit the
%quantum Hall effect, when their width $L_y$ and the mean free path $l$ do
%exceed the cyclotron length $l_{\rm cyc}$.  We consider quantum Hall wires of
%diffusive widths, where the mean free path $l$ is smaller than $L_{y}$.  In
%the opposite case, $l> L_y$, the ballistic motion between the edges of the
%wire results in anomalous magneto-resistance phenomena due to classical
%commensurability effects of the cyclotron orbits\cite{roukes}.

The conductance  per spin channel, $g(b) =
\sigma_{xx}(B)/\sigma_0$, is given by the Drude formula $ g(b) =
g_0/(1+b^2)$, ($g_0 = E \tau/\hbar$, $b=\omega_B \tau$) for weak magnetic field, $b
<1$.  For $b>1$, when the cyclotron length $ l_{\rm cyc}$ is smaller than the
mean free path $l$, disregarding the overlap between  Landau bands, $g$ is
obtained in SCBA \cite{ando},
\begin{equation} \label{scba} 
g(B)  = \frac{1}{\pi}  \left(2 n+1 \right) \left( 1 - \frac{(E_{\rm F}- 
 E_n)^2}{\Gamma^2}\right),  
\end{equation} 
for $ \mid E - E_n \mid < \Gamma$.  One obtains the localization length for $b>1$ and
$\mid \epsilon/b -n -1/2 \mid < 1$ by inserting $g$,  Eq.~(\ref{scba}) into Eq.~(\ref{lcb}).
% when $\xi>L_y$, and into
%Eq.~(\ref{lcb2d}) if $\xi < L_y$.
 It oscillates between maximal values in
 centers of  Landau bands, and minimal values in  band tails  (Fig.~\ref{lcbosc}).
  For $n>1$, one finds in  band centers,
%$ \xi ( L_y) > L_y$ for uncorrelated disorder,
\begin{equation} \label{middle} 
\xi_n =  \frac{2}{\pi} \left( 2 n + 1 \right) L_y   
\left[ 1- \frac{\ln 
\sqrt{ 1+ \left(L_y/l_{\rm cyc} \right)^2}}{(n+1/2)^2} \right]^{1/2}.    
\end{equation}

In the center of the lowest  Landau band ($n=0$), Eq.~(\ref{1db})
gives a value $\xi_0 ( B) \approx (2/\pi) L_y$, smaller
than $L_y$. There, the localization is 2D and the topological term
\cite{pruiskenrev} is  effective, leading to  criticality, and  diverging
localization lengths.  In a wire of finite width, $L_y$ the localization length $\xi$
saturates to the critical value $\xi_{\rm crit} \approx 1.2 L_y$
\cite{hucke,huckerev} larger than    $L_y$. Comparison with
$\xi_{n}$, Eq. (\ref{middle}), shows  that the  noncritical  quasi-1D localization
length exceeds   $\xi_{\rm crit}$
in all but the lowest Landau bands.
 \begin{figure}[htbp]
\begin{center} 
\vspace{-1cm} 
\includegraphics[width=.4 \textwidth]{prlfig2.ps} 
\vspace{-4.3cm} 
\caption[]{  The 
  localization length, $\xi$, as obtained by inserting
  $g(B)$ in 2nd order Born approximation in Eq.~(\ref{lcb}), including a
  summation over all Landau levels (full line). Broken line:
 $\xi$, with   $g(B)$ in 
 SCBA,  neglecting overlap between  Landau bands, Eq.
  (\ref{scba}).  The result of 
  transfer matrix calculations 
  is plotted  (dot size: numerical error $\approx 1\%$) for periodic boundary
  conditions \cite{kwaz93} as function of magnetic flux
 through  unit cell 
  $x = a^2/2 \pi l_B^2$. Dimensionless conductance
parameter  per spin channel  is $g(B=0) = 5.1$.  At
  weak magnetic fields  doubling of $\xi$ due to breaking of 
  time reversal symmetry is seen, in good agreement  with 
 the  analytical  crossover   formula (full line) \cite{crossover,prbr}. 
 Horizontal dashed  line: $L_y= 30 a$. }
 \label{lcbosc} 
\vspace{-.5cm} 
\end{center} 
\end{figure} 
%  We note that we have assumed in the
%derivation of Eq.~(\ref{middle}), that the conductivity is homogenous. In a
%strong magnetic field, the formation of edge states results in strongly
%inhomogenous and anisotropic conductivity, thereby reducing the mixing of
%these edge states with the bulk states \cite{shkledge}.  However, for
%uncorrelated disorder potential it has been shown previously, that the edge
%states do mix with the bulk states in the middle of the Landau bands
%\cite{ohtsuki}.
If all states in  centers of  higher Landau bands are localized along
the wire, the question arises if  there exist extended
states in the quantum Hall wire at all.  
% We therefore {\it consider}
% an argument 
% introduced  by Halperin
% in order to prove the existence of extended
%states in the Landau bands
% \cite{halperin}.
 Consider  an annulus, with a
circumference larger than the localization length in the center of a Landau
band.  When a magnetic flux pierces  the annulus, localized states are
unaffected. Guiding centers of states which extend around the
annulus do shift in position and energy, however\cite{halperin}. 
 In a confined wire there exist chiral edge states
\index{chiral edge states}, which  extend around the annulus.  A magnetic
flux change  moves these states  down and up  inner and outer edges,
respectively. As shown above,  in the middle of the Landau band, the  electrons can
diffuse freely from edge to edge, but are localized along the annulus
 with $\xi > L_y$.  As a
consequence, when adiabatically changing the magnetic flux, an ``edge state'' 
has to  move from the inner to the outer edge, since it cannot enter the band of
localized states. 
 This fact has  been interpreted as  proof for the existence of an
extended bulk state, extending around the annulus and between  edges, at the
energy $ E_{m}$, with $\xi (E_m) = L_y$ \cite{halperin}.
 In the following, we   show that  rather 
  a transition from 
  extended chiral edge states to localised states occurs at these energies, $E_m$. 
%  Then, one is forced to the
%conclusion that the transition from localized bulk states to an extended state
%occurs not in the band center, but at energies where the dimensional crossover
%happens, namely $E_{m}$.
 
Using the transfer matrix method \cite{transfer}, we have calculated the
localization length as function of energy $E$ in a tight binding model of a
disordered quantum wire in  perpendicular magnetic field, with periodic
boundary conditions, Fig. 1 (dashed curve, right axis).
  Indeed, its
maxima are seen to increase linearly with energy $E$ in
  agreement with  Eq. (\ref{middle}).  The transfer
matrix result for the   2-terminal conductance $G$ \cite{pmr92}
is shown in Fig. 1 (solid curve, left axis) for a wire of identical
properties, but  hard wall boundary
conditions and finite length $L$.
% \begin{figure} 
% \begin{center} 
% \vspace{-3.5cm} 
% \includegraphics[width=.5 \textwidth]{bildloce.ps} 
% \vspace{-4cm} 
% \caption{ The localization length $\xi$, in units of  
%   the lattice spacing $a$, 
%   as a function of energy $E$ in units of $t$, the hopping 
%   amplitude of the tight binding model, 
%   as calculated with the transfer matrix 
%   method  for  $L_{max} = 100 000 a$.  The straight line denotes the width of the wire $L_y = 100 a$. 
%   Periodic boundary conditions are assumed. The disorder strength is $W = .8 
%   t$. The perpendicular magnetic 
%   field is $x = .025$  in units of flux per flux quantum through the 
%   elementary cell of the size $a^2$, corresponding to  
%  a magnetic length $l_B = (1/2 
%   \pi b)^{1/2} a \approx 2.52 a$.  } 
%  \label{loclength} 
% \end{center} 
% \end{figure} 
% \vspace{-1.5cm} 
% \begin{figure} 
% \begin{center} 
% \vspace{-2cm} 
% \hspace{-1cm}\includegraphics[width=.5 \textwidth]{bildmeso2.ps} 
% \vspace{-3.5cm} 
% \caption{  The two-terminal conductance  
%   of a wire identical to the one studied in Fig. 3, but of  length $L = 
%   5000 a$ with hard wall confinement of the width $L_y =100 a$.  Sharp steps 
%   in the conductance are seen whenever $\xi$ crosses $L_y$ in Fig. 3.} 
%  \label{loclengthW} 
% \end{center} 
% \end{figure}

We verify that the condition $\xi(E_{m,p}) = L_y$, yields 
the energies $E_{m,p}$, $p = \pm$, at which $m$ edge states mix, and
transitions from the quasi-1D chiral metal to an insulator occur, as signaled by sharp 
jumps of $G$ in Fig. 1. Here, $m=n$
when this energy is above the bulk energy of the $n$-th Landau band,
$p=+$, and $m= n-1$ when it is below it, $p=-$.
  For $\xi  < L_y$,  backscattering between edges is exponentially 
suppressed and the localisation length of edge states 
 increases exponentially as $\xi_{\rm edge} = \xi \exp (L_y/\xi)$.  
  These results are  summarised in the phase diagram, Fig. 3 a, 
 where the value of  $G$, in units of $e^2/h$, 
 is given  as function of wire width $L_y$ and energy $E$ in units of $\hbar \omega_B$. 
 We find that  $G= m$, where $m$ is the number of extended edge states. 
 When  $L_y \le l_{cyc} \propto \sqrt{2 n+1}$,  all edge states are localised, 
  and all conductance plateaus collapse, $G=0$, as seen in Fig. 1, close to the middle of 
  the band, $E=0$. 

 Taking into account the spin, the  Zeeman splitting $E_n^+-E_n^- = g_Z \mu_B B$,
 lifts the spin degeneracy,  where  $g_Z$ is the material dependent Zeeman-g-factor, 
 and $\mu_B$  Bohr's magneton. 
Without spin flip scattering  edge states of different spins are not mixed.  
 The 
 phase diagram,  Fig. 3 b, is then a  
   superposition of   two   phase diagrams, Fig. 3 a, for each spin.
  There are phases where the conductance is
   equal to  the total number of edge channels, $G=m^+ + m^-$,
 when the bulk localisation length of electrons  $\xi^{\sigma}$,
  does not   exceed the wire width,  $\xi^{\sigma} < L_y$, for either spin $\sigma =\pm$.
 When,  $\xi^{\sigma} > L_y$ for 
 one spin $\sigma$ only,  the conductance is carried by the number of edge states 
 with  opposite spin, $G=m^{-\sigma}$, only.
 If $\xi^{\sigma} > L_y$ for both $\sigma =\pm$,
  the conductance vanishes,   $G=0$.
  With  strong  spin flip scattering,  
 all edge state mix with  bulk states for   $\xi^{\sigma} > L_y$, yielding $G=0$.  
 There are  conductance plateaus
  equal to the total number of edge states,
 $G=m^+ + m^-$, only if 
  $\xi^{\sigma} < L_y$ is fullfilled for both spins,   $\sigma =\pm$ (Fig. 3 c).
 Thus,  both the  sequence  
  and   width of  conductance plateaus are  
  sensitive measures of  the spin flip scattering rate $1/\tau_s$,  due to 
 electron-electron interaction, spin-orbit interaction, 
 or scattering from  nuclear spins\cite{komi}. 
 Furthermore,   the  enhancement of the
 $g_Z$--factor above its bulk value due to the exchange interaction 
   depends on the  Fermi energy\cite{fang}.
 Accordingly, the 
 width of the conductance plateaus  changes with  energy. 
 In a real sample there exists a slowly
varying potential disorder which can stabilize  edge states against
mixing with bulk states  \cite{ohtsuki}. When the
confinement potential is varying slowly on the magnetic length scale $l_B$,
the   energies $E_{m,p}$ are expected to split into $m$ energies,
 at which  edge
states mix one by one with  bulk states,  accompanied by steps of  heights $1$ in 
 conductance $G$. Both potentials can be 
 effectively modified by the Coulomb interaction as obtained
  by a selfconsistent solution of the Poisson equation. 
 \begin{figure}[]
\begin{center} 
\vspace{-2.6cm} 
\hspace{-1cm}
\includegraphics[width=.8 \textwidth]{prlfig3.ps} 
\vspace{-10.4cm} 
\caption[]{  Schematic phase diagram of a quantum Hall wire, 
 for  $L \gg L_y$.  The 
full lines indicate    jumps between integer
 plateau values of  conductance $G$,  in units of $e^2/h$,
as denoted by integers. 
(a) Spinless electrons (b) Electrons with spin and  Zeeman--Splitting 
 $(E_n^+ - E_n^-)/\hbar \omega_B = g_Z/2 $ without spin flip, and 
(c) with strong spin flip rate.
 Dotted line:  inverse cyclotron length, $1/l_{cyc}$. 
 For a particular value of inverse width $1/L_y$ (dashed line) 
 a sequence of conductance plateaus  similar to Fig. 1 is obtained. 
 In the limit of $1/L_y \rightarrow 0$ there are  delocalisation critical points
  of bulk states (circles). 
 }
 \label{phase} 
\vspace{-.5cm} 
\end{center} 
\end{figure} 
 In the presence of long range interactions,  interacting edge states  
 form a correlated Luttinger liquid. Renormalisation due to edge plasmon 
 excitations   enhances
 the interedge scattering amplitude\cite{kane}.
 This results in  a decrease of  the localisation length of the edge states. 
 Accordingly, the width of the plateaus of lower Landau bands is ecpected to be reduced
 due to the Luttinger liquid correlations.
 Similarly, in the fractional quantum Hall regime, 
 where the edge state excitations are strongly correlated even
 without long range interactions\cite{kane}, 
 a reduction of the localisation length of edge states is expetced as function of the 
 filling factor $\nu$. 

 In 3--D layered systems in 
 perpendicular  magnetic field,  surface states  form 
 2D chiral metals in  plateau regions where  bulk states are localised\cite{chalker}. 
% These chiral metals have very peculiar properties. Furthermore, 
% there 
% exist  broad bands of metals  in the middle of  Landau bands, due to
% the insulator--metal transition of  bulk states.
 There
are  transitions between these
 2D chiral metals and  insulating
  states in long quasi-1D wires of  layered electron systems. 
%   similar to the 1D-chiral--metal to insulator transition studied here.
% A detailed analysis of the 
% corresponding phase diagram is  subject of further research. 

We conclude, that in quantum Hall bars of finite width $L_y \ll \xi_n$ at low
temperatures  quantum phase transitions occur between
 extended chiral  edge states and a quasi-1D insulator. 
These are driven by the  crossover from 2D to 1D localization of  bulk states. These
metal-insulator transitions
% can be deliberately classified as 
{\it resemble } first-order
phase transitions in the sense that the localization length
 abruptly jumps between exponentially large  and finite values.
 In the thermodynamic limit, {\it fixing the aspect ratio $c = L/L_y$, 
 when sending $L \rightarrow \infty$, then $c \rightarrow \infty$, 
  the two--terminal  conductance 
 jumps between exactly integer values and     zero}. 
 The transitions occur at energies where the localization
length of  bulk states is equal to the geometrical wire width. Then,  $m$ edge
states  mix and  electrons are free to diffuse between
the wire boundaries but become Anderson localized along the wire.
% The
%transitions are accompanied by peaks in the longitudinal conductance with the
%heights $m e^2/h$ ($m$ integer), indicating the existence of 
% $m$ extended states at these
% energies.
 At finite temperature, this
phenomenon can be observed, when the phase coherence length exceeds the
quasi-1D localization length in  centers of  Landau bands, $L_{\varphi}
> \xi_n$. It may accordingly be called {\it mesoscopic
 Chiral Metal--Insulator Transition}. 
 In Hall bars of large aspect ratios at low temperatures one should observe 
transitions of the two-terminal resistance from integer quantized plateaus,
$R_n = h/n e^2$ to a Mott variable-range hopping regime of exponentially
diverging resistance. Such experiments would yield new information about edge
states in quantum Hall bars.
At higher temperature, when $L_{\varphi}< \xi_n$, the conventional form of the
integer quantum Hall effect is recovered \cite{klitzing}.
   
{\bf Acknowledgments} The authors gratefully acknowledge useful discussions
with Bodo Huckestein,  Mikhail Raikh,   Isa Zarekeshev, and  
Grzegorz Michalek.
 This research was
supported by  German Research Council (DFG), Grant No. Kr 627/10,  
 Schwerpunkt "Quanten-Hall-Effekt",
 and  by EU TMR-network Grant. No.  HPRN-CT2000-0144.  
 
\begin{references} 
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{\bf 45}, 494 (1980). 
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   Phys. Rev. B {\bf 23}, 5632 (1981). 
\bibitem{aoki} T. Ando and H. Aoki, Solid State Commun. {\bf 38}, 1079 (1981).  
R. B. Laughlin, Phys. Rev. Lett. {\bf 50}, 1395(1983). 
\bibitem{halperin} B. I. Halperin,  
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\bibitem{hucke} J. T. Chalker and G. J. Daniell,  
Phys. Rev. Lett. {\bf  61}, 593 (1988); 
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\bibitem{huckerev} 
 B. Huckestein,  
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% \bibitem{milnikov} G. V. Mil'nikov and I. M. Sokolov, JETP Lett. {\bf 48} , 
%  536 (1988).     
\bibitem{raikh} A. G. Galstyan and M. E. Raikh,  Phys. Rev.  {\bf B 56}, 
  1422 (1997); D. P. Arovas, M. Janssen, and B. Shapiro, Phys. Rev.
{\bf  B 56}, 4751 (1997).
\bibitem{scalingexp} 
H. P. Wei, et al.  
% D. C. Tsui, M. A. Paalanen, and A. M. M. Pruisken, 
  Phys. Rev. Lett. {\bf 61}, 1294 (1988); 
 S. Koch, et al.,  
%R. J. Haug, K. v. Klitzing, and K. Ploog,  
 Phys. Rev. Lett. {\bf 67}, 883 (1991); 
L. W. Engel, et al., 
%D. Shahar, C. Kurdak, and D. C. Tsui, 
 Phys. Rev. Lett. {\bf 71}, 2638 (1993); 
M. Furlan, Phys. Rev. B {\bf 57}, 14818 (1998); 
 F. Hohls, et al.,  
% U. Zeitler, and R. J. Haug, 
Phys. Rev. Lett. {\bf 88}, 036802 (2002). 
%\bibitem{nonscaling} 
%    N. Q. Balaban, U. Meirav, and I. Bar--Joseph, 
%Phys. Rev. Lett. {\bf 81}, 4967 (1998). 
%\bibitem{aokiando} H. Aoki and T. Ando, 
%Phys. Rev. Lett. {\bf 54}, 831 (1985).

\bibitem{haug} R. J. Haug, and K. von Klitzing, Europhys. Lett. {\bf 10}, 489 (1989).  
\bibitem{timp} G. Timp, et al.,  
%A. M. Chang, P. Mankiewich, R. Behringer, J. E. Cunningham, T. Y. Chang, and R. E. Howard, 
    Phys. Rev. Lett. {\bf 59}, 732 (1987); 
% Fractional quantum Hall effect in one dimension 
%    G. Timp, et al.,  
% R. Behringer, J. E. Cunningham, and R. E. Howard, 
    ibid. {\bf 63}, 2268 (1989).   
\bibitem{ando94} 
%} Localization and fluctuations in the quantum Hall regime 
    T. Ando 
    Phys. Rev. B {\bf 49}, 4679 (1994). 
\bibitem{edgemixing} 
% Quantum ballistic and adiabatic electron transport studied with quantum point contacts 
    B. J. van Wees, et al.,  
% L. P. Kouwenhoven, E. M. M. Willems, C. J. P. M. Harmans, J. E. Mooij, H. van Houten, C. W. J. Beenakker, J. G. Williamson, and 
%    C. T. Foxon, 
    Phys. Rev. B {\bf 43}, 12431 (1991).  
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A. V. Khaetskii and B. I. Shklovskii, 
 Zh. Eksp. Teor. Fiz. {\bf 85}, 721 (1983) 
 [Sov. Phys. JETP {\bf 58}, 421 (1983)]; 
M. E. Raikh and T. V. Shahbazyan,  
Phys. Rev. B {\bf 51}, 9682 (1995). 
\bibitem{ohtsuki} 
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    (1988). 
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    (1989); T. Ohtsuki and Y. Ono, J. Phys. Soc. Jpn. 58, 956 (1989); 
 T. Ando, Phys. Rev. B {\bf 42}, 5626 (1990).  
% Y. Ono, T. Ohtsuki, and B. Kramer in:  
% {\it High Magnetic Fields in Semicondoctor Physics}, ed. by G. Landwehr, 
% Springer Ser. Sol. St. Sci. {\bf 101}, Springer, Berlin, 60 (1992). 
%  see also  B. Kramer, {\it Quantization of transport},  
% in {\it Quantum transport and dissipation}, T. Dittrich, P. H\" anggi,  
%  G.--L. Ingold, B. Kramer, F. Sch\" on, and W. Zwerger,  
%  Wiley--VCH, Weinheim, (1998). 
\bibitem{mani} R. G. Mani and K. v. Klitzing,  
 Phys. Rev. B {\bf 46}, 9877 (1992); 
R. G. Mani, K. von Klitzing, and K. Ploog,  
 Phys. Rev. B {\bf 51}, 2584 (1995).  
\bibitem{breakdown}  
% Critical currents of the quantum Hall effect in the mesoscopic regime 
    P. G. N. d. Vegvar, et al.,  
% A. M. Chang, G. Timp, P. M. Mankiewich, J. E. Cunningham, R. Behringer, and R. E. Howard, 
    Phys. Rev. B {\bf 36}, 9366 (1987). 
\bibitem{roukes} M. L. Roukes, et al.,  
%A. Scherer, S. J. Allen, Jr., H. G. Craighead, R. M. Ruthen, E. D. Beebe, and 
%J. P. Harbison, 
 Phys. Rev. Lett. 
{\bf  59}, 3011 (1987). 
%T. Park et al., 
% and G. Gumbs, and   M. Pepper, 
% Phys. Rev. B {\bf 56}, 6758 (1997).
  \bibitem{chalker}  J. T. Chalker and A. Dohmen
    Phys. Rev. Lett. 75, 4496 (1995).
\bibitem{ab} E. Abrahams, et al. 
% P. W. Anderson, D. C. Licciardello, V. Ramakrishnan, 
 Phys. Rev. Lett { \bf 42} 673(1979). 
\bibitem{weg} F. Wegner, Z. Physik { \bf B 36},l209(1979);  
Nucl. Phys. B {\bf 316}, 663(1989) 
\bibitem{hikami}  S. Hikami, Prog. Theor. Phys. {\bf 64}, 1466 (1980). 
\bibitem{elk} K. B. Efetov, A. I. Larkin, D. E. Khmel'nitskii, 
 Zh. Eksp. Teor. Fiz. {\bf 79}, 
1120(1980)  
[Sov. Phys. JETP {\bf 52}, 568(1980)]. 
\bibitem{mk81} A. MacKinnon, and B. Kramer, Phys. Rev. Lett. {\bf 47}, 1546 
 (1981); Z. Phys. {\bf B  53}, 1 (1983). 
\bibitem{transfer}
{B. Kramer} and A.~MacKinnon, Rep. Prog.Phys. {\bf 56}, 1469 (1993).
\bibitem{ando}  
 T. Ando and Y. Uemura, 
 J. Phys. Soc. Jpn. {\bf 36}, 959 (1974); 
T. Ando,  
          J.  Phys.  Soc.  Jpn.   
                       {\bf 36} 1521 (1974); T. Ando, J. Phys. Soc. Japan {\bf 37}, 622 (1974). 
\bibitem{levitation} 
%D. Khmelnitskii, Phys. Lett. 106A, 182 (1984); 
% R. B. Laughlin,  
%     Phys. Rev. Lett. {\bf  52}, 2304 (1984); 
 B. Huckestein,  
   Phys. Rev. Lett. 84, 3141-3144 (2000). 
\bibitem{ef} 
K. B. Efetov,  
{\it Supersymmetry in Disorder and Chaos} Cambridge University Press,  
Cambridge (1997). 
\bibitem{larkin} K. B. Efetov, A. I. Larkin, Zh. Eksp Teor. 
 Fiz. {\bf 85}, 764(1983) ( Sov. Phys. JETP {\bf 58}, 444 ).  
\bibitem{dorokhov} 
O. N. Dorokhov,  Pis'ma Zh. Eksp. Teor. Fiz. 36, 259 (1982)) 
[JETP Lett. {\bf 36}, 318 (1983)];,  Zh. Eksp. Teor. Fiz. 85, 1040 (1983) 
[ Sov. Phys. JETP {\bf 58}, 606 (1983)];   Solid State Commun. {\bf 51}, 381 
    (1984).       
\bibitem{crossover} S. Kettemann,  
                     J. Phys. Soc. Jpn. Suppl. A   {\bf 72},  197 (2003); 
 S. Kettemann, acc. for publ. in  Phys. Rev. {\bf B} (15)  xx,xxxx  (2003).  
\bibitem{kwaz93} V. Kalmeyer, D. Wei, D.P. Arovas and S.-C. Zhang, 
Phys. Rev. B{\bf 48}, 11095 (1993).  
 \bibitem{prbr} S. Kettemann,   Phys. Rev. {\bf B 62  }, R13282
 (2000);S. Kettemann, R. Mazzarello,   Phys. Rev.  {\bf B 65   }
  085318 (2002).
\bibitem{lambert} R. M. Corless, et al., 
%G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and 
%  D. E. Knuth, 
 Adv. Comput. Maths {\bf 5}, 329 (1996).
% \bibitem{mazza} S. Kettemann, R. Mazzarello,   Phys. Rev.  {\bf B 65   } 
%  085318 (2002). 
% \bibitem{khavin} J.-L. Pichard, M. Sanquer, K. Slevin, and P. Debray, 
% Phys. Rev. Lett. 65, 1812 
% (1990); 
%  M. E. Gershenson et al.,  
% Phys. Rev. Lett. {\bf 79}, 725(1997); 
%   Phys. Rev. {\bf B 58}, 8009 (1998). 
% \bibitem{prbr} S. Kettemann,   Phys. Rev. Rapid Commun.  {\bf B 62  }, R13282 
%   (2000). 
\bibitem{pruiskenrev} 
  A. M. M. Pruisken in {\it The Quantum Hall effect},  
ed. by R. E. Prange, S. M. Girvin, Springer New York ( 1990). 
 
\bibitem{pmr92}J. B. Pendry, A. MacKinnon, and P. J. Roberts, Proc. R. 
Soc. London A{\bf 437}, 67 (1992).  
\bibitem{komi} S. Komiyama, O. Astafiev, T. Machida, Physica {\bf E 20}, 43 (2003). 
\bibitem{takashima} K. Takashima, et al., Physica {\bf E 20}, 160 (2003). 
\bibitem{fang} F. F. Fang and P. J. Stiles, Phys. Rev. {\bf 174}, 823 (1968).
\bibitem{kane} C. L. Kane, and M. P. A. Fisher, in {\it Perspectives in Quantum Hall Effects}, 
 ed. by S. Das Sarma and A. Pinczuk, Wiley New York (1997).  
\end{references} 
\end{document} 
 





cond-mat0307044.extracted_bib
\bibitem{klitzing} K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett.  
{\bf 45}, 494 (1980). 

\bibitem{laughlin} R. B. Laughlin 
   Phys. Rev. B {\bf 23}, 5632 (1981). 

\bibitem{aoki} T. Ando and H. Aoki, Solid State Commun. {\bf 38}, 1079 (1981).  
R. B. Laughlin, Phys. Rev. Lett. {\bf 50}, 1395(1983). 

\bibitem{halperin} B. I. Halperin,  
 Phys. Rev. B {\bf 25}, 2185 (1982). 

\bibitem{hucke} J. T. Chalker and G. J. Daniell,  
Phys. Rev. Lett. {\bf  61}, 593 (1988); 
 B. Huckestein and B. Kramer,  Phys. Rev. Lett. {\bf 64}, 1437 (1990) 

\bibitem{huckerev} 
 B. Huckestein,  
 Rev. of Mod. Phys. {\bf 67}, 357 (1995). 
% 
\bibitem{milnikov} G. V. Mil'nikov and I. M. Sokolov, JETP Lett. {\bf 48} , 
%  536 (1988).     

\bibitem{raikh} A. G. Galstyan and M. E. Raikh,  Phys. Rev.  {\bf B 56}, 
  1422 (1997); D. P. Arovas, M. Janssen, and B. Shapiro, Phys. Rev.
{\bf  B 56}, 4751 (1997).

\bibitem{scalingexp} 
H. P. Wei, et al.  
% D. C. Tsui, M. A. Paalanen, and A. M. M. Pruisken, 
  Phys. Rev. Lett. {\bf 61}, 1294 (1988); 
 S. Koch, et al.,  
%R. J. Haug, K. v. Klitzing, and K. Ploog,  
 Phys. Rev. Lett. {\bf 67}, 883 (1991); 
L. W. Engel, et al., 
%D. Shahar, C. Kurdak, and D. C. Tsui, 
 Phys. Rev. Lett. {\bf 71}, 2638 (1993); 
M. Furlan, Phys. Rev. B {\bf 57}, 14818 (1998); 
 F. Hohls, et al.,  
% U. Zeitler, and R. J. Haug, 
Phys. Rev. Lett. {\bf 88}, 036802 (2002). 
%
\bibitem{nonscaling} 
%    N. Q. Balaban, U. Meirav, and I. Bar--Joseph, 
%Phys. Rev. Lett. {\bf 81}, 4967 (1998). 
%
\bibitem{aokiando} H. Aoki and T. Ando, 
%Phys. Rev. Lett. {\bf 54}, 831 (1985).


\bibitem{haug} R. J. Haug, and K. von Klitzing, Europhys. Lett. {\bf 10}, 489 (1989).  

\bibitem{timp} G. Timp, et al.,  
%A. M. Chang, P. Mankiewich, R. Behringer, J. E. Cunningham, T. Y. Chang, and R. E. Howard, 
    Phys. Rev. Lett. {\bf 59}, 732 (1987); 
% Fractional quantum Hall effect in one dimension 
%    G. Timp, et al.,  
% R. Behringer, J. E. Cunningham, and R. E. Howard, 
    ibid. {\bf 63}, 2268 (1989).   

\bibitem{ando94} 
%} Localization and fluctuations in the quantum Hall regime 
    T. Ando 
    Phys. Rev. B {\bf 49}, 4679 (1994). 

\bibitem{edgemixing} 
% Quantum ballistic and adiabatic electron transport studied with quantum point contacts 
    B. J. van Wees, et al.,  
% L. P. Kouwenhoven, E. M. M. Willems, C. J. P. M. Harmans, J. E. Mooij, H. van Houten, C. W. J. Beenakker, J. G. Williamson, and 
%    C. T. Foxon, 
    Phys. Rev. B {\bf 43}, 12431 (1991).  

\bibitem{shkledge} B. I. Shklovskii,  Pis'ma Zh. Eksp. Teor. Fiz. {\bf 36}, 43 
 (1982) [ JETP Lett. {\bf 36}, 51 (1982)];  
A. V. Khaetskii and B. I. Shklovskii, 
 Zh. Eksp. Teor. Fiz. {\bf 85}, 721 (1983) 
 [Sov. Phys. JETP {\bf 58}, 421 (1983)]; 
M. E. Raikh and T. V. Shahbazyan,  
Phys. Rev. B {\bf 51}, 9682 (1995). 

\bibitem{ohtsuki} 
 T. Ohtsuki and Y. Ono, Solid State Commun. 65, 403 (1988); and, 68, 787 
    (1988). 
  Y. Ono, T. Ohtsuki and B. Kramer, J. Phys. Soc. Jpn. 58, 1705 
    (1989); T. Ohtsuki and Y. Ono, J. Phys. Soc. Jpn. 58, 956 (1989); 
 T. Ando, Phys. Rev. B {\bf 42}, 5626 (1990).  
% Y. Ono, T. Ohtsuki, and B. Kramer in:  
% {\it High Magnetic Fields in Semicondoctor Physics}, ed. by G. Landwehr, 
% Springer Ser. Sol. St. Sci. {\bf 101}, Springer, Berlin, 60 (1992). 
%  see also  B. Kramer, {\it Quantization of transport},  
% in {\it Quantum transport and dissipation}, T. Dittrich, P. H\" anggi,  
%  G.--L. Ingold, B. Kramer, F. Sch\" on, and W. Zwerger,  
%  Wiley--VCH, Weinheim, (1998). 

\bibitem{mani} R. G. Mani and K. v. Klitzing,  
 Phys. Rev. B {\bf 46}, 9877 (1992); 
R. G. Mani, K. von Klitzing, and K. Ploog,  
 Phys. Rev. B {\bf 51}, 2584 (1995).  

\bibitem{breakdown}  
% Critical currents of the quantum Hall effect in the mesoscopic regime 
    P. G. N. d. Vegvar, et al.,  
% A. M. Chang, G. Timp, P. M. Mankiewich, J. E. Cunningham, R. Behringer, and R. E. Howard, 
    Phys. Rev. B {\bf 36}, 9366 (1987). 

\bibitem{roukes} M. L. Roukes, et al.,  
%A. Scherer, S. J. Allen, Jr., H. G. Craighead, R. M. Ruthen, E. D. Beebe, and 
%J. P. Harbison, 
 Phys. Rev. Lett. 
{\bf  59}, 3011 (1987). 
%T. Park et al., 
% and G. Gumbs, and   M. Pepper, 
% Phys. Rev. B {\bf 56}, 6758 (1997).
  
\bibitem{chalker}  J. T. Chalker and A. Dohmen
    Phys. Rev. Lett. 75, 4496 (1995).

\bibitem{ab} E. Abrahams, et al. 
% P. W. Anderson, D. C. Licciardello, V. Ramakrishnan, 
 Phys. Rev. Lett { \bf 42} 673(1979). 

\bibitem{weg} F. Wegner, Z. Physik { \bf B 36},l209(1979);  
Nucl. Phys. B {\bf 316}, 663(1989) 

\bibitem{hikami}  S. Hikami, Prog. Theor. Phys. {\bf 64}, 1466 (1980). 

\bibitem{elk} K. B. Efetov, A. I. Larkin, D. E. Khmel'nitskii, 
 Zh. Eksp. Teor. Fiz. {\bf 79}, 
1120(1980)  
[Sov. Phys. JETP {\bf 52}, 568(1980)]. 

\bibitem{mk81} A. MacKinnon, and B. Kramer, Phys. Rev. Lett. {\bf 47}, 1546 
 (1981); Z. Phys. {\bf B  53}, 1 (1983). 

\bibitem{transfer}
{B. Kramer} and A.~MacKinnon, Rep. Prog.Phys. {\bf 56}, 1469 (1993).

\bibitem{ando}  
 T. Ando and Y. Uemura, 
 J. Phys. Soc. Jpn. {\bf 36}, 959 (1974); 
T. Ando,  
          J.  Phys.  Soc.  Jpn.   
                       {\bf 36} 1521 (1974); T. Ando, J. Phys. Soc. Japan {\bf 37}, 622 (1974). 

\bibitem{levitation} 
%D. Khmelnitskii, Phys. Lett. 106A, 182 (1984); 
% R. B. Laughlin,  
%     Phys. Rev. Lett. {\bf  52}, 2304 (1984); 
 B. Huckestein,  
   Phys. Rev. Lett. 84, 3141-3144 (2000). 

\bibitem{ef} 
K. B. Efetov,  
{\it Supersymmetry in Disorder and Chaos} Cambridge University Press,  
Cambridge (1997). 

\bibitem{larkin} K. B. Efetov, A. I. Larkin, Zh. Eksp Teor. 
 Fiz. {\bf 85}, 764(1983) ( Sov. Phys. JETP {\bf 58}, 444 ).  

\bibitem{dorokhov} 
O. N. Dorokhov,  Pis'ma Zh. Eksp. Teor. Fiz. 36, 259 (1982)) 
[JETP Lett. {\bf 36}, 318 (1983)];,  Zh. Eksp. Teor. Fiz. 85, 1040 (1983) 
[ Sov. Phys. JETP {\bf 58}, 606 (1983)];   Solid State Commun. {\bf 51}, 381 
    (1984).       

\bibitem{crossover} S. Kettemann,  
                     J. Phys. Soc. Jpn. Suppl. A   {\bf 72},  197 (2003); 
 S. Kettemann, acc. for publ. in  Phys. Rev. {\bf B} (15)  xx,xxxx  (2003).  

\bibitem{kwaz93} V. Kalmeyer, D. Wei, D.P. Arovas and S.-C. Zhang, 
Phys. Rev. B{\bf 48}, 11095 (1993).  
 
\bibitem{prbr} S. Kettemann,   Phys. Rev. {\bf B 62  }, R13282
 (2000);S. Kettemann, R. Mazzarello,   Phys. Rev.  {\bf B 65   }
  085318 (2002).

\bibitem{lambert} R. M. Corless, et al., 
%G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and 
%  D. E. Knuth, 
 Adv. Comput. Maths {\bf 5}, 329 (1996).
% 
\bibitem{mazza} S. Kettemann, R. Mazzarello,   Phys. Rev.  {\bf B 65   } 
%  085318 (2002). 
% 
\bibitem{khavin} J.-L. Pichard, M. Sanquer, K. Slevin, and P. Debray, 
% Phys. Rev. Lett. 65, 1812 
% (1990); 
%  M. E. Gershenson et al.,  
% Phys. Rev. Lett. {\bf 79}, 725(1997); 
%   Phys. Rev. {\bf B 58}, 8009 (1998). 
% 
\bibitem{prbr} S. Kettemann,   Phys. Rev. Rapid Commun.  {\bf B 62  }, R13282 
%   (2000). 

\bibitem{pruiskenrev} 
  A. M. M. Pruisken in {\it The Quantum Hall effect},  
ed. by R. E. Prange, S. M. Girvin, Springer New York ( 1990). 
 

\bibitem{pmr92}J. B. Pendry, A. MacKinnon, and P. J. Roberts, Proc. R. 
Soc. London A{\bf 437}, 67 (1992).  

\bibitem{komi} S. Komiyama, O. Astafiev, T. Machida, Physica {\bf E 20}, 43 (2003). 

\bibitem{takashima} K. Takashima, et al., Physica {\bf E 20}, 160 (2003). 

\bibitem{fang} F. F. Fang and P. J. Stiles, Phys. Rev. {\bf 174}, 823 (1968).

\bibitem{kane} C. L. Kane, and M. P. A. Fisher, in {\it Perspectives in Quantum Hall Effects}, 
 ed. by S. Das Sarma and A. Pinczuk, Wiley New York (1997).  
