Devil's staircase of incompressible electron states in a nanotube
Dmitry S. Novikov
\nin It is shown that a periodic potential applied to a nanotube can lock electrons into incompressible states. Depending on whether electrons are weakly or tightly bound to the potential, excitation gaps open up either due to the Bragg diffraction enhanced by the Tomonaga -- Luttinger correlations, or via pinning of the Wigner crystal. Incompressible states can be detected in a Thouless pump setup, in which a slowly moving periodic potential induces quantized current, with a possibility to pump on average a fraction of an electron per cycle as a result of interactions.
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\begin{document}
\title{Devil's staircase of incompressible electron states in a nanotube}
%\title{Incompressible states and adiabatic pumping in nanotubes}
\author{Dmitry S. Novikov}
\email{dima@alum.mit.edu}
\affiliation{Department of Electrical Engineering and Department of Physics,
Princeton University, Princeton, NJ 08544}
%\affiliation{Department of Physics, Massachusetts Institute of Technology, 
%Cambridge, MA 02139}
\date{December 16, 2004}


%\setlength\topmargin{0.0cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}

\nin
It is shown that a periodic potential applied to a nanotube can lock electrons into 
incompressible states.
Depending on whether electrons are weakly or tightly bound to the potential,
excitation gaps open up either due to  
the Bragg diffraction enhanced by the Tomonaga -- Luttinger correlations, 
or via pinning of the Wigner crystal.
Incompressible states can be detected in a Thouless pump setup, in which
a slowly moving periodic potential induces  quantized current,
with a possibility to pump on average a fraction of an electron per cycle
as a result of interactions.

\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pacs{71.10.Pm, 85.35.Kt, 64.70.Rh}

\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






\nin
A carbon nanotube (NT) is a strongly interacting electron system
known to be a host of many-body effects \cite{Nygard}, 
including Luttinger liquid behavior \cite{Stone}.
At half-filling the NT is effectively dilute, and a crossover from the
liquid to the 1d Wigner crystal \cite{WC} is expected \cite{Levitov'01}.
 

Here we suggest that an external periodic potential can be a probe 
of both crystallization and Luttinger correlations. 
We show that incompressible electron states arise when the 
electron number density $\bar\rho$ (relative to half--filling) is
{\it commensurate} with the potential period $\lsaw$:
\be \label{rho-alpha}
\bar \rho 
%= \sum_{\alpha=1}^4\rho_{\alpha} 
= {\mtot / \lsaw} \,,  \quad \mtot = 4m \,.
\ee
In Eq.~(\ref{rho-alpha}), $m$ is the number of the NT electrons of each 
of the four polarizations \cite{Dresselhaus} %(called below ``flavors'') 
per period. %potential period. % $\lsaw$. 
To calculate excitation gaps we 
generalize the Pokrovsky--Talapov theory \cite{PT}
for the case of the four %nanotube 
coupled fermion modes.
%and calculate excitation gaps. % via the phase soliton approach. 


In the absence of interactions, Bragg diffraction 
on the potential
opens minigaps 
at integer density (\ref{rho-alpha}), 
$|m|=1,2,...$ \cite{Talyanskii'01}. 
%Electron
Interactions profoundly change the spectrum, yielding 
a devil's staircase of incompressible states at {\it rational} $m=p/q$.
%Semiclassically, 
In such a state, the NT electron system is locked by the %periodic 
potential into a $q\lsaw$--periodic structure.
If detected, {e.g.} in a Thouless pump setup \cite{Talyanskii'01},
corresponding minigaps would provide a direct probe of interactions, 
%of electron interactions between the NT electrons, 
with a possibility to map the devil's staircase by
pumping at fractions of the base frequency. 
%(as described below).


%the Bloch theory only allows 
%for the incompressible states in which the density (\ref{rho-alpha}) 
%corresponds to {\it integer} $m$. 
%These states are the result of the Bragg diffraction of the non--interacting 
%electrons on the periodic external potential. As a result of this diffraction,
%{\it minigaps} in the single electron spectrum open\cite{Talyanskii'01} when 
%$m=0, \, \pm 1, \,\pm 2, \, ...$ . 
%In the present work we will show that electron--electron interactions result in
%additional incompressible states that arise at any {\it rational} $m=p/q$.
%In such a state, the NT electron system is locked by the periodic 
%potential into a $q\lsaw$--periodic structure sketched in 
%Fig.~\ref{fig:soliton-sketch}.  
%Naturally, the states with the lower denominator $q$ will be
%more pronounced. Realistically, due to the finite system size and 
%finite temperature, only a few states with small enough $q$
%could be reliably detected.
%However, these fractional---$m$ states are very important.
%Minigaps that correspond to them
%are interaction induced and vanish in the noninteracting limit.



%Periodic potential is a natural way to detect crystallization since 
%when the density is commensurate with the period, one expects  
%locking of the periodic electron configuration into incompressible states.
%This is a quantum--mechanical manifestation of the Pokrovsky--Talapov transition
%for the case of four nanotube modes,
%with the excitation gaps calculated by the phase soliton approach.


One-dimensional interacting electrons are conventionally described 
%in terms of the density modes of the 
by the Tomonaga - Luttinger liquid \cite{Stone}.
%This is a description in terms of the {hydrodynamic modes} of density,
This hydrodynamic approach 
%implies that the excitations are { delocalized},
is valid 
%Naturally, this picture is valid 
in a small momentum shell near the Fermi points,
with excitations extended over the whole system.
%where the electronic dispersion can be linearized.
%On the other hand, 
%At low density the Coulomb interaction can overcome the Fermi energy, 
%resulting in the semiclassical electrons arranging into a regular
%lattice (Wigner crystal with a quasi--long--range order). 
%\cite{WC}
%\cite{Ruzin,Schulz,Giamarchi}
%that can manifest itself 
%in oscillations of conductivity with density \cite{Ruzin,Scott-Thomas}.
%%Unfortunately, conventional bosonization techniques are not designed to work 
%%at the bottom of the band.
%Treatment of the curvature perturbatively within bosonization (by a gradient 
%expansion) introduces a ``dangerously irrelevant'' perturbation
%that causes ultraviolet dirvergencies.
%
%
%
Adequate description of crystallization  and commensurability 
%is a fundamental problem that 
requires
%with a potential to solve the problem of 
%the Coulomb drag.~\cite{Drag}
%(utilized {\it e.g.} to calculate Coulomb drag) 
%In one dimension, theoretical treatment 
%must 
including
%is perplexed by the problem of including 
the {\it curvature} of the electronic dispersion
that becomes important at low density.
%into the bosonized description \cite{Stone} of 1d fermions.
%~\cite{Pustilnik}
%Indeed, simple Gaussian theory cannot capture these effects since   
%it describes delocalized excitations.
%Indeed, at low density, the Fermi level is close to the bottom of the band,
%in which case one cannot linearize the dispersion near the Fermi points.
%It is %Technically, 
%the curvature that 
The curvature % becomes important at low density and 
can yield crystallization or commensuration by
coupling charge and spin modes %\cite{Levitov'01}
and by introducing a {\it length scale} 
into an otherwise scale-invariant Gaussian theory. 
%Such a length scale, as well as binding spin with charge can produce 
%resulting in crystallization or commensuration. 
%the desired effects.
%Unfortunately, conventional bosonization techniques are not designed to work 
%at the bottom of the band.
%Treatment of the curvature perturbatively within bosonization (by a gradient 
%expansion) introduces a ``dangerously irrelevant'' perturbation
%that causes ultraviolet dirvergencies.
%This curvature can originate from interactions.
In this work we treat both electron interactions and the curvature
of the dispersion non-perturbatively
%even at low density 
%all the way down to the Wigner crystal limit,  
%and treat the effect of curvature exactly, 
by making use of the 
relativistic Dirac spectrum of a half--filled nanotube.
Curvature is controlled by the Dirac gap and is bosonized exactly
by virtue of the massive Thirring - sine-Gordon duality. \cite{Coleman,Haldane}
This enables us to study incompressible states both in the limit
of a narrow--gap Luttinger liquid and in that of the locked Wigner crystal.
%
%in the semi--classical 
%limit where the conventional hydrodynamic approach \cite{Stone} is not applicable.


%How does the transition from a liquid to a crystal occur?
%It has been shown by Schulz \cite{Schulz} that the density correlation
%function in both the Luttinger liquid
%regime and in the Wigner crystal regime develops a singularity at $4k_F$ 
%that corresponds to separation between electrons of different spin polarizations.
%Onset of crystallization has been observed in the exact diagonalization of 
%the fermions with long--range intraction on a ring \cite{Giamarchi}, 
%as well as by Hartree--Fock calculation \cite{Mueller}.
%However, so far the clear analytical description of the crossover into 
%a Wigner crystal regime has not been achieved.


%A partial resolution to this question has been given 
%in the insightful paper by Schulz \cite{Schulz}.
%There the case of spin--$\frac12$ electrons was considered. 
%It was shown that the density correlation function in both the Luttinger liquid
%regime and in the Wigner crystal regime develops a singularity at $4k_F$ 
%that corresponds to separation between electrons of different spin polarizations.
%It was pointed out that due to absense of a phase transition in one dimension, 
%a plasmon mode of a 
%semi--classical Wigner crystal would have the same dispersion as that given by 
%the Luttinger liquid Hamiltonian.

%Glazman, Ruzin, and Shklovskii \cite{Ruzin}.
%Matveev \cite{Matveev0.7}.


Our course of action is to introduce the bosonized description for the NT electrons,
develop the phase soliton method %integrate over quantum fluctuations
and find excitation gaps from the renormalized 
sine--Gordon action, % of the four modes,
draw the phase diagram in the semiclassical limit, and comment on 
the experimental means to detect incompressible states.




{\it The model.---} 
%Nanotube as an ideal Dirac system.
Nanotube electrons in the forward scattering approximation \cite{fsa}
%According to the latter, even in the presence of electron interactions,
%the two inequivalent Dirac points in the Carbon Brilloin zone
%remain approximately decoupled.
%As a result, near half--filling, electron properties of 
%nanotubes can be approximately 
%described in terms of the four fermion modes
are described by the four flavors of Dirac fermions
%\be \label{psi-alpha}
$\psi_{\alpha} = \lp {\psi^R_{\alpha} \ \psi^L_{\alpha}} \rp^T$, 
$\alpha=1-4$,
%\ee
%that are the Weyl spinors,
%$\psi_{\alpha}, \ 1\leq \alpha \leq 4$, 
whose interaction is written in terms of the smooth envelope
%\be \label{rho}
$\rho(x)=\sum_{\alpha = 1}^4\psi^+_{\alpha}(x)\psi_{\alpha}(x) $
%\ee
of the total charge density.
% in which we neglected the 
%strongly oscillating components $\sim e^{\pm 2iK_{\rm BZ}x}$, with 
%$K_{\rm BZ}\sim 1/a_{\rm cc}$ 
%defining the position of the Dirac points in the Brilloin zone of graphene.
The second-quantized Hamiltonian
$\H=\H_0 + \H_{\rm bs} + \H_{\rm ext}$,
%The Hamiltonian in the second--quantized form is 
%\be \label{H}
%\H=\H_0 + \H_{\rm bs} + \H_{\rm ext} \ .
%\ee
%The first term 
where $\H_0$ is the massless Dirac Hamiltonian
%in the forward scattering approximation \cite{Kane'97,Egger'97}, 
%discussed below, 
\be \label{H0}
\H_0 = -i\hbar v\int 
\sum_{\alpha = 1}^4\psi^+_{\alpha}\sigma_3\partial_x\psi_{\alpha} dx 
+ \frac{1}{2}\sum_k\rho_{-k} V(k)\rho_{k} 
\ee
%
with the Coulomb interaction 
%$V(x) = \frac{e'^2}{|x|}$,
%$V(k) = e'^2\ln\lb 1+(ka)^{-2}\rb$, $e'^2 = \frac{2e^2}{\varepsilon+1}$
%$V(x) = \frac{2e^2}{\varepsilon+1} \frac1{|x|}$,
$V(k) = \frac{2e^2}{\varepsilon+1}\ln\lb 1+(ka)^{-2}\rb$
for a tube of radius $a$ 
placed on a substrate with the dielectric constant $\varepsilon$.
%the 1D Coulomb interaction 
%\be \label{V}
%V(k) = \textstyle{\frac{2}{\varepsilon+1} V_0(k)} \ ,
%\ee
%where
%\be \label{V0}
%V_0(k) \simeq e^2\ln\lb 1+(ka)^{-2}\rb \ .
%\ee
%The second term in the Hamiltonian (\ref{H}) describes the effect of 
The curvature of the electron dispersion controlled by
the gap $2\Delta_0$ at half-filling introduces backscattering
at each NT Dirac point:
%between the left and right moving fermions {\it within} each Dirac point:
\be \label{Hbs}
\H_{\rm bs}  =  
\Delta_0 \int 
{\textstyle \sum_{\alpha = 1}^4} \psi^+_{\alpha}\sigma_1\psi_{\alpha} dx \ .
\ee
%This gap controls the curvature of the electronic dispersion and it is
%essential for the subsequent analysis.
We emphasize that the backscattering  $\Delta_0$ is {\it not} the usual
interaction-induced $V(2k_F)$ term (which is undetectably small in metallic NTs), 
but rather is present at the single-particle level
\cite{Dresselhaus,curvature}:
Depending on the tube chirality, the bare gap $\Delta_0$ 
can be in the range $\Delta_0\lesssim$10\,meV to $\sim$0.5\,eV;
it
%$0 \leq \Delta_0 \lesssim 0.5$\,eV. 
%Although gaps can open due to interactions even in the metallic tube 
%$\Delta_0=0$, we will rely on the gap from the bandstructure.
can also be controlled by the parallel magnetic field \cite{B-paral}.
Adding the periodic potential $U(x)$ and the chemical potential $\mu$
[$\mu=0$ at half-filling] results in
\bea
\label{Hext}
\H_{\rm ext} = \int \! dx \, \rho(x) \lf U(x)-\mu \rf \ ,\\
\label{SAW}
U(x) = A \cos \ksaw x \ , \quad \ksaw={2\pi / \lsaw} \ .
\eea
Qualitatively, our findings will be valid for any realistic potential
%realized by the means described in the Introduction,
which justifies the simplest choice (\ref{SAW});
typically, $\lsaw \sim 0.1-1\,\mu$m.
%In the present work we consider 
%the nanotube subject to the external periodic potential
%which for the purpose of simpler algebra we take to be a harmonic function,
The Hamiltonian ${\cal H}$
%(\ref{H}) %written in the forward scattering approximation
is U(4) invariant with respect to rotations in the fermion flavor space. 
%the fermion flavors 
%$\psi_{\alpha}$.


%By virtue of the massive Thirring -- quantum sine--Gordon duality,
Bosonization of the nanotube electrons %\cite{Levitov'01} 
%\be \label{Theta}
%$\psi_{\alpha} = (2\pi a})^{-\frac12} e^{i\Theta_{\alpha}}$
$\psi_{\alpha} = {1\over \sqrt{2\pi a}}\, e^{i\Theta_{\alpha}}$
%\lp \matrix{
%e^{i(\Phi_{\alpha}-\Theta_{\alpha})}  \cr
%e^{i(\Phi_{\alpha}+\Theta_{\alpha})} 
%}\rp \ .
%\ee
is exact even in the presence of (\ref{Hbs}). %\cite{Coleman,Haldane}
It maps the problem of the four interacting Dirac fermion modes onto 
the sine-Gordon model of the four coupled bose fields $\Theta_{\alpha}$.
We rotate \cite{Levitov'01,Talyanskii'01} 
to the basis of the charge mode $\theta^0$ and three neutral modes $\theta^a$,
in which case the charge density 
$\rho(x) = \textstyle{\frac{2}{\pi}}\, \partial_x \theta^0$, and 
%the Gaussian part $\L_0$ [bosonized $\H_0$] of the total action 
%$\L = \L_0 + \L_{\rm bs}$ is diagonal 
the Gaussian action (bosonized $\H_0$) is diagonal
[$\hbar = v =1$]:
\be
%\matrix{
\L_0 =  {1\over 2\pi} \int\! dx  
\lp
%\frac1{v^2}
(\partial_{t} \theta^0)^2 - K\, (\partial_{x}\theta^0)^2 
%\rp
%\cr
%& + & 
+
%\ds{\hbar v\over 2\pi} \int\! dx \, 
%\ds
\ts{\sum_{a=1}^3 \lp 
%\frac1{v^2}(\partial_t\theta^a)^2 - (\partial_x\theta^a)^2
\partial_{\mu}\theta^a
\rp^2 }  \rp \, .
%}  
\label{L0}
\ee
The slow momentum dependence of the charge stiffness 
%\be \label{K}
$K_k = 1 + 4\nu V(k)$, $\nu^{-1} = {\pi \hbar v}$,
%\ee
is irrelevant, and we take it as constant 
%Below we drop the (irrelevant) logarithmic dependence of the 
%stiffness $K_q$ on the momentum, assuming a constant value
$K\!\equiv\! K_{k\sim 1/\lch}$,  
$\lch$ being the size of the charged-mode soliton
% of the charge mode 
(described below).
% that is found self--consistently. 
%which is the screening length %beyond which the Coulomb interaction is screened, 
%for the Coulomb interaction,
Assuming $\lch \sim l_s$, with the screening length $l_s \sim 1\, \mu$m,  
%assuming $l_s\sim 1\,\mu$m,
%it depends on the distance to the metallic gate and 
%at least cannot exceed the tube length $L$.
%Since the dependence of the stiffness $K$ on $\lch$ is only logarithmic,
and using $e^2/\hbar v\simeq 2.7$, % with $l_{\rm ch} \sim 1\,\mu$m,
one estimates $K\simeq 40$ for 
for the stand-alone tube; $K\simeq 10$ 
if the tube is placed on a substrate
with %realistic 
dielectric constant $\varepsilon\simeq 10$.   
The logarithmic behavior of 
%Coulomb form factor 
$V(k)$ 
underlies the fact 
that the Coulomb interaction $V(x)\propto 1/|x|$
is essentially {\it local} in one dimension, 
$\frac1{2} \int dx dx' \rho(x)V(x-x')\rho(x') 
\simeq \frac{\hbar v}{2\pi}\int dx K (\partial_x\theta^0)^2$.


The nonlinear part of the sine-Gordon Lagrangian [coming from $\H_{\rm bs}$]
reads 
\cite{Talyanskii'01,Levitov'01,fracnt} 
\bea
\L_{\rm bs} = -  \int\! dx \, g_0 %\frac{g_0}{\pi}
\F \lp
\theta^0 + 2\tilde\mu\ksaw x - 2\tilde A \sin \ksaw x ,\ \theta^a\rp 
%\vphantom{1^{2^3}}
%\rf  
\ ,
\label{LF}
%\eea
%Here 
%\be
\\
\label{f} 
\F \lp\theta^0, \,
\theta^a\rp
%\!=\! {\frac14} \ts{\sum} \cos 2\Theta_{\alpha}
= \cos\theta^0 \cdot \ts{\prod_{a=1}^3 \cos\theta^a +  
\sin\theta^0 \cdot \prod_{a=1}^3 \sin\theta^a}  \ , 
\eea
where %(bare backscattering)
%\be \label{def-lambda0}
$g_0= {4 \Delta_0/\pi D a^2}$, 
%$g_0={4\Delta_0/\hbar v a} = {4 \Delta_0/D a^2}$, 
%\ee
and $D\simeq \hbar v/a$ is the 1d bandwidth. 
%$\mu$ is the chemical potential calculated from the half filling, 
In Eq.~(\ref{LF}) we included the 
coupling (\ref{Hext}) to external fields
%Completing the square % \cite{Talyanskii'01,fracnt} 
%removes 
by shifting the charge mode 
%leaves $\theta^a$ intact, and shifts 
%\be \label{theta-gauge}
$\theta^0 \to \theta^0 - \textstyle\frac2{\hbar v}\int^x K^{-1} 
(U-\mu)\, dx'$, 
with $\tilde\mu = \mu/(K\epsilon_0)$, $\tilde A = A/(K\epsilon_0)$,
$\epsilon_0=\hbar \ksaw v$.
%with $\tilde \mu = \mu/K\hbar\ksaw v$, $\tilde A = A /K\hbar\ksaw v$.



%Raising the chemical potential 
%just above the (renormalized) charge gap $\Delta_{\rm ch}$ (estimated below),
%one adds a single electron to the finite--length nanotube.
In what follows, it is useful to 
first describe elementary excitations of the stand-alone tube, $U\equiv 0$.
As shown by Levitov and Tsvelik \cite{Levitov'01}, in the bosonized language
adding one electron corresponds to a  
{\it composite soliton} of both the charge and the flavor modes. 
In such a composite object, the charge mode changes by $\pi/2$
(adding unit charge) over the length $\lch$, %$\sim l_s$, 
whereas the neutral sector adds a particular SU(4) flavor to the
electron by means of the solitons of $\theta^a$
``switching'' by $\pm \pi/2$ 
on a shorter scale %$\lfl \ll \lch$
$\lfl \sim K^{-1/2}\lch$,
right in the middle of the charge soliton.
The composite soliton is a unit charge configuration
of the minimal energy, obtained by optimizing the 
action $\L_0 + \L_{\rm bs}$ in the limit of large Coulomb repulsion $K\gg 1$.
Technically, such an optimization results in the soft neutral modes $\theta^a$
adjusting to create the effective potential 
$\bar \F(\theta^0) = \min_{\{ \theta^a\} } \F (\theta^a , \ \theta^0)
\sim \cos 4\theta^0$ for the stiff charge mode $\theta^0$.
%In particular, the neutral modes ``switch'' by 
%when the chemical potential is just above the charge gap
%$\mu \gtrsim \Delta_{\rm ch}$, $\Delta_{\rm ch} \simeq K^{1/2}\Delta_{\rm fl}$,
%$\Delta_{\rm fl} \simeq D^{1/5} \Delta_0^{4/5}$,
%This crossover is determined by the 




In the absence of external potential, $U\equiv 0$,
the system $\L_0 + \L_{\rm bs}$ 
describes the Wigner crystal - Luttinger liquid crossover.
%describes a crossover from the low density limit of localized electrons
%(Wigner crystal) to the scale--invariant Tomonaga--Luttinger liquid.
%Let us briefly describe this crossover starting from the half--filled tube
%($\mu=0$) which is a band insulator (``Dirac vacuum''). 
In particular, raising the chemical potential from
half-filling %($\mu=0$) 
[band insulator, or the ``Dirac vacuum'']
to just above the charge gap %$\mu > \Delta_{\rm ch}$, %\cite{Levitov'01} 
produces the train of weakly overlapping
composite solitons described above [$\bar\rho\lfl \ll 1$, ``Wigner crystal''],
in which overlapping charge solitons %($\bar\rho\lch > 1$) 
maintain a quasi-long-range order. 
Further increase of %$\mu\gg \Delta_{\rm ch}$ corresponds to 
$\mu$ leads to 
strongly overlapping solitons rendering the nonlinear term (\ref{LF}) irrelevant
[$\bar\rho\lfl\gg 1$, Luttinger liquid].
%Depending on the value of the chemical potential $\mu$, 
%the saddle point of the action $\L_0 + \L_{\rm bs}$ determines at what point in the
%crossover between the Wigner crystal and the Luttinger liquid the system is.
%In the Wigner crystal regime, low density results in a small overlap of the 
%flavor solitons (small exchange), whereas the overlap between the ``fat''
%charge solitons is nothing else but the Coulomb interaction 
%preserving quasi--long--range order. 
%The logarithmic behavior of the Coulomb
%form factor $V(q)$ underlies the fact that Coulomb force is essentially {\it local}
%in one dimension, 
%$\frac{e^2}{2} \int dx dx' \rho(x) \rho(x')/|x-x'| 
%\simeq \frac1{2\pi}\int dx K (\partial_x\theta^0)^2$.
%Conversely, when the electron density $\bar \rho$ is high, $\bar \rho \lfl \gg 1$,
%system enters the scale--invariant Luttinger liquid regime 
%of delocalized excitations, the flavor solitons strongly overlapping.
%Technically, at large $\mu\gg \Delta_{\rm ch}$,
%the nonlinear term (\ref{LF}) becomes irrelevant, and the system is described 
%by the conventional Gaussian bosonized Lagrangian (\ref{L0}).
We emphasize that the crystal - liquid crossover occurs due to the
{\it finite} soliton size $\lfl\propto \Delta_0^{-\zeta}$ 
that scales inversely with the {curvature} of the dispersion.
Curvature is also responsible for binding flavor to charge through the term 
(\ref{LF}).
%Technically, when $\Delta_0=0$, the chemical potential can be gauged away,
%which explains why it is practically impossible to dope the truly metallic tube.
The $U\equiv 0$ system is compressible \cite{Levitov'01}.



Periodic potential locks electrons into incompressible states.
%Adding periodic potential $U(x)$ %(\ref{SAW}) 
%generates {\it incompressible states}.
Technically, the term (\ref{LF}) becomes relevant whenever the density 
$2\tilde\mu = m$ [Eq.~(\ref{rho-alpha})] is integer \cite{Talyanskii'01}.
When $m$ is a simple fraction, $2\tilde\mu = p/q$, to identify incompressible states 
one utilizes the phase soliton method \cite{PT}.
We generalize it for the case of the four nanotube modes   
by expanding in powers of the coupling $g_0$ as 
$\theta^j = 
\bar \theta^j + g_0{\theta^j}^{(1)} + ... + g_0^n{\theta^j}^{(n)} + ... $,
$j=0, a$, %$a=1-3$,
and finding the effective Lagrangian 
$\L_{m}[\bar \theta^0, \bar \theta^a]$ 
for the {\it phase} modes $\bar\theta^0$ and $\bar\theta^a$, $a=1-3$, 
to the lowest order in $g_0$. %\cite{fracnt}
The phase modes $\bar \theta^0, \bar \theta^a$ 
are constant in the commensurate phase, whereas an
excitation is a {\it composite phase soliton}, in which the
phase fields $\bar \theta^0(x), \bar \theta^a(x)$ describe a slow 
deformation of the regular commensurate configuration. 
Excitation gap is given by the energy of the 
composite soliton, renormalized by quantum fluctuations. 


We note that it is the curvature $\propto\Delta_0$ that yields %commensuration.
incompressible states.
Same is true for the non-interacting electrons: 
When $\Delta_0=0$, the external potential is gauged away from 
the Hamiltonian $-i\hbar v \sigma_3\partial_x + U(x)$.


The composite phase soliton is a result of optimization of the 
corresponding effective action $\L_{m}[\bar \theta^0, \bar \theta^a]$
(examples of which are given below). % \cite{fracnt}
This problem will be similar 
to the $U\equiv 0$ case described above: %\cite{Levitov'01}
When interaction is strong, $K\gg 1$, the neutral phase modes 
$\bar\theta^a$ adjust [on the scale $\bar\lfl$]
to create an effective potential for the charged 
phase mode $\bar\theta^0$.
The crucial difference from the former case is that 
this optimization %of $\L_m$ 
will qualitatively depend
on whether the system is in the regime of 
the Luttinger liquid or in that of the Wigner crystal. In this sense,
periodic potential naturally distinguishes between the opposite sides of the 
crystal - liquid crossover, by bringing about the additional
{\it length scale}, its period $\lsaw$.
%, that needs to be compared with the ``size of the electron'', $\lfl$. 
Technically, the saddle point of $\L_m$ will depend on   
whether the neutral modes %[that ``switch'' on the scale $\lfl$]
are adiabatic or fast on the length scale on 
which the effective potential (\ref{LF}) changes appreciably.
Below we consider both cases separately.



{\it Bragg diffraction in a four-flavor Luttinger liquid.---}
In the adiabatic limit $\bar\lfl \gg \lsaw$ of extended flavor excitations,
{\it exchange} is important:
%the flavor excitation is extended, which results in importance 
%of {\it exchange}:
%This means that effects of exchange are important: 
The system
(correlated over many $\lsaw$) ``knows'' that it is comprised of particles
of the four different flavors. This limit is 
%Qualitatively,
%this limit describes the Bragg diffraction in a 
%narrow--gap Luttinger liquid; it is also 
connected to the non-interacting case, where particles repel 
only due to the Pauli principle. 
For integer density $m$, averaging (\ref{LF}) over the period $\lsaw$ 
reduces the problem to the $U\equiv 0$ case 
with $\L_m = \L_0 + \L_{\rm bs}$, %$[\bar\theta^0, \bar\theta^a]$,
$g_0 \to g_0 J_m(2\tilde A)$, 
%\cite{Levitov'01}, 
yielding the $K\gg 1$ minigaps 
$\Delta_m \simeq K^{1/2} D^{1/5} | \Delta_m^{(0)}|^{4/5}$
that are enhanced by the quantum fluctuations compared to the noninteracting
values $\Delta_m^{(0)} = 2\Delta_0 J_m(2\tilde A)$ \cite{Talyanskii'01}.
 

The fractional $m=\frac12$ case is considered in detail in Ref.~\onlinecite{fracnt}.
A somewhat lengthy calculation yields the effective action of the form
$\L_{1/2} = \L_0 
+ 
%\int \! dx (
v(\tilde A) \sum_{a} \cos 2\bar\theta^a \cos 2\bar\theta^0 
- u(\tilde A) \sum_{a>b} \cos 2\bar\theta^a \cos 2\bar\theta^b
$. %\cite{fracnt} 
We find the {\it charge} excitation gap 
$\Delta_{1/2} \propto 
{K-1\over \sqrt{K}} \, D %\, |v_{1/2}(2\tilde A)| 
\lp {\Delta_0 \over \epsilon_0}\rp^2$
(that vanishes in the noninteracting limit $K\to 1$ 
in accord with the Bloch theory),
whereas {\it flavor} 
excitations are governed by the SU(4)$\simeq$O(6) Gross - Neveu
Lagrangian \cite{GN} that can be written in terms of six Majorana fermions
$\chi \sim e^{i\theta}$,
$\L_{\rm GN} 
%= {1\over 2\pi} \int \! dx 
%\lf (\partial_{\mu}\theta^a)^2  
%- g_{\rm GN} \sum_{a>b} \cos2\theta^a\cos2\theta^b \rf
%= \int \! dx 
=  i\bar\chi_j \gamma_{\mu}\partial_{\mu}\chi_j - 
u(\tilde A) (\bar \chi_j \chi_j)(\bar \chi_{j'} \chi_{j'})$.
Although the charge gap $\Delta_{1/2}$ vanishes for certain values of the potential
amplitude (zeroes of $v(\tilde A)$ given in Ref.~\onlinecite{fracnt}), 
the excitation gap never closes due to flavor,
since the Gross - Neveu coupling $u(\tilde A) \neq 0$.
%that has a {\it gap} $\Delta_{\rm fl}$.
%Gaps $\Delta_{p/q}$ for higher fractions could vanish due to fluctuations.





{\it Locked Wigner crystal.---}
%In the remainder of this work 
Below we focus on the 
opposite limit $\bar\lfl \ll \lsaw$, in which the charge
mode dominates, whereas exchange (overlap of the flavor solitons) is
relatively unimportant. 
We will show that the effective description in this regime is that 
of a {\it single} mode of fermions with the density (\ref{rho-alpha}),
locked by the external potential.
%In this regime the system of the four fermion flavors effectively behaves 
%as a {\it single} mode system with the density (\ref{rho-alpha}).
Physically this happens since when the electron wavefunctions (represented 
by solitons) are {localized} on the scale $\lsaw$, 
Coulomb repulsion wins over exchange.
Naturally, in this case 
the appropriate saddle point of $\L_0 + \L_{\rm bs}$ %gives 
is equivalent to
the semi-classical 
description, with the WKB condition $\bar\lfl \ll \lsaw$. 
Technically, % in this limit 
the neutral mode ``switching'' produces the effective charge mode potential
with a quarter-period, % $\sim \cos 4\theta^0$,
similarly to the $U\equiv 0$ case above:
$g_0 \F ( \theta^0 + m\ksaw x - 2\tilde A \sin \ksaw x ,\ \theta^a)
\sim g \cos ( 4\theta^0 + \mtot\ksaw x -8\tilde A \sin \ksaw x)$. %+higher harmonics.
This potential  now depends
on the {\it total density} $\mtot$, Eq.~(\ref{rho-alpha}). 
%$\mtot=4m$. 


To obtain the renormalized coupling $g$ we utilize adiabaticity of $U(x)$,
$\bar\lfl \ll \lsaw$.  
%the potential is adiabatic on the scale $l<\lfl$.
The renormalization group 
produces the effective coupling $g$ 
on the scales $a < l < \lfl$ that appear
``microscopic'' for the external potential.
Thus the neutral modes can be integrated out, yielding 
both $\lfl$ and $g$ 
%in the strong coupling limit become
independent of the shape of the potential $U(x)$.
%, renormalizing $g$.
% (as long as it is adiabatic). 
The problem 
%of estimating $g$ 
becomes similar to the $U\equiv 0$ case.
%The soliton sizes $\lch$ and $\lfl$ follow from the variational estimate
%$K/\lch^2 \sim 1/\lfl^2 \sim g(\lch,\lfl)$. In the limit of $K\gg 1$
%one can neglect the fluctuations of the charge sector, 
The coupling $g$ %and the flavor soliton size $\lfl$ 
follows from the 
scaling $g \simeq g_0 (\lfl/a)^{-3/4}$ and 
self-consistency $g(\lfl)\sim 1/\lfl^2$. 
%due to quantum fluctuations of the 
%neutral sector.
%obtaining $\lfl \sim K^{-1/2}\lch$, %and $g\simeq {\Delta'/\hbar v' a'}$,  
%the charge and flavor gaps $\Delta_{\rm ch} \simeq K^{1/2}\Delta_{\rm fl}$,
%$\Delta_{\rm fl} \simeq D^{1/5} \Delta_0^{4/5}$.
\cite{Talyanskii'01,Levitov'01,fracnt}
%where the spatially oscillating terms are discarded and 
%Here the Fourier coefficients $f_n$ are defined in Eq.~(\ref{f-fourier}),
%and in the last line we used the approximation (\ref{F=cos4}).


{\it Re-fermionization.---}
Let us now complete our single-mode description by 
mapping the system in the limit $\bar\lfl \ll \lsaw$ onto the problem 
of spinless Dirac fermions in the external potential (\ref{SAW}).
We introduce the displacement field 
$\Theta=2\theta^0$  % $ \sum_{\alpha=1}^4 \Theta_{\alpha}$
for the total density
$\rho\equiv \frac1{\pi} \partial_x \Theta$.
%Changing of variables:
%\be \label{def-Theta}
%by rescaling the charge mode. 
To preserve commutation relations,
we rescale the canonical momentum 
%$\Pi_{\Theta}$ 
%should be 
%half of that for the field $\theta^0$,
$\Pi_{\Theta}=\frac12 \Pi_{\theta^0}$. 
%with $\Pi_{\alpha}$ defined in Eq.~(\ref{Pi-alpha}).
Changing variables in the Lagrangian 
$\frac{1}{2\pi}\lf K(\partial_x\theta^0)^2 + g\cos(4\theta^0 + ...)\rf$, 
we obtain the effective Lagrangian for the charge mode $\Theta$,
%\begin{widetext}
\be
\matrix{
%\nonumber
\L_{\rm eff}[\Theta] = \ds{\hbar v'\over\pi} 
\int\! dx  
\lf
%\vphantom{\int^c}
\frac1{2v'^2} \lp \partial_t  \Theta\rp^2 
- \frac{K'}2\, \lp\partial_{x}\Theta\rp^2 
%\right. \\ \left.
%\ts{\sum_{a} (\partial_{\mu}\theta^a)^2} 
%\rf   \\
\right. \cr \left.
%\int \!dx\, 
- g \cos(2\Theta+\mtot\ksaw x - 2\tilde A'\sin\ksaw x)
\vphantom{\ds{\int {1\over v^2}}}\rf 
}
\label{L-Theta}
\ee
%The Lagrangian (\ref{L-Theta})
%one needs to rescale 
with the 
rescaled parameters 
$v' \equiv 4v$, $K' \equiv {K/16}$, and 
$\tilde A' \equiv {A/ K'\hbar \ksaw v'} = 4\tilde A$.
%\ee
%This rescaling  %described by Eq.~(\ref{v'K'}) 
%has the following meaning.
The velocity quadrupling corresponds to counting incoming fermions
regardless of their (four) flavors. 
With the same accuracy that allowed us to neglect 
%the energy of 
the flavor sector, 
%${\cal O}(1/K)$)
the rescaled value $K'\approx 1 + {(K-1)/16} \equiv 1 + \nu' V(q)$
%can be understood by noting that 
%looks at first a bit puzzling. However,  
%with the same accuracy that allowed us to discard the flavor modes,
%in the limit of $K\gg 1$,
is by definition the charge stiffness for the spinless Dirac fermions
of velocity $v'$, with the density of states $\nu'={1/\pi\hbar v'}$.
%The chemical potential and the external potential are not rescaled.
The external fields $U$ and $\mu$ are not rescaled.
%
%Re--fermionizing the Lagrangian (\ref{L-Theta}) by 
%
Introducing the Dirac spinors 
$\Psi={1\over \sqrt{2\pi a'}} e^{i\Theta}$ relative to the new 
cutoff $a'\sim \lfl$, 
we obtain the 
effective Hamiltonian 
\be 
\matrix{
\H_{\rm eff}[\Psi] = \ds{\int}\! dx \, 
\Psi^+ \lf -i\hbar v'\sigma_3 \partial_x  + \Delta' \sigma_1 
+ U(x) \right. \cr \left.
- \mu \rf \Psi 
+ \ds{\frac12}\sum_k\rho_{-k}V(k)\rho_{k}
%+ \Psi^+ \rp U(x) - \mu \rp \Psi \ . 
}
\label{H-strong}
\ee
%\end{widetext}
for the fictitious 
spinless Dirac fermions of unit charge, %$e$,
with the number density %(\ref{rho}) 
%in the new variables 
$\rho(x) = \Psi^+\Psi$,
coupled to the external fields in the natural way, Eq.~(\ref{Hext}). 
The fermions $\Psi$ represent the original NT electrons 
traced over the flavor states.
The effective gap $\Delta'$ in Eq.~(\ref{H-strong}) 
corresponds
%is chosen in such a way that it corresponds
to the renormalized coupling $g$
%(\ref{g-ren}) 
entering the Lagrangian 
(\ref{L-Theta})
%\be \label{g-Delta'}
by $g\simeq {\Delta' \over \hbar v' a'}$, %$ \ , \quad a'\simeq \lfl $
%\ee
%similarly to the definition (\ref{def-lambda0}).
%In Eq.~(\ref{g-Delta'}) the new length scale cutoff $a'$ is assumed since
%with regard to the new length scale cutoff $\lfl$;
%the present approach is valid only at the lenth scales $l > \lfl$
%beyond the correlation length of the neutral sector.\cite{log}
%Eq.~(\ref{g-Delta'}) together with Eqs.~(\ref{g-ren}) and 
%(\ref{l0-self-consist}) yields
%\be \label{Delta'}
%Therefore we obtain
yielding
$\Delta' \simeq {\hbar v' \over \lfl} \simeq D^{1/5} \Delta_0^{4/5}$.
%\ee
 
The (charge) excitation gaps are now estimated via the 
phase soliton approach \cite{PT}
for the single-mode system (\ref{L-Theta}) and (\ref{H-strong}),
yielding the devil's staircase.
For integer $\mtot=\pm1, \pm2, ...$, 
%neglecting charge fluctuations,
in the classical limit $K\to \infty$, we find
%$g^{\rm strong} = g J_{\mtot}(2\tilde A')$. 
%\cite{fracnt}
\be
\label{minigaps-ren-4}
\Delta_{m} \simeq K^{1/2} %\left| 
|J_{\mtot}(2\tilde A')|^{1/2}
%\right|^{1/2} 
D^{1/5}\Delta_0^{4/5} \ .
%\quad 
%\mtot=\pm1,  \pm2, ... \ .
\ee
The effective density quadrupling $m\to \mtot$ in (\ref{minigaps-ren-4})
is due to strong interactions: Electrons of all flavors avoid each other,
their ground state being a Slater determinant of the quadrupled size.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{
%\begin{flushright}
%\begin{minipage}[t]{2.0in}
%\vspace{0pt}
%\centering
\includegraphics[width=1.8in]{fig1a.eps}
%\end{minipage}
%\end{flushright}
%\hspace{-2.3in}
%\vspace{-2in}
%\begin{minipage}[t]{1.8in}
%\vspace{.01in}
%\centering 
\includegraphics[width=1.7in,height=1.5in]{fig1b.eps}
%\end{minipage}
}
%\includegraphics[width=3.5in,height=3in]{fig1.eps}
\caption[]{
{\it Left:} 
Phase diagram in the $(A, \mu)$ plane 
according to the model (\ref{hamiltonian-cl}).
%The minimum width of the gap for the incompressible state with any 
%density $n_e-n_h$ is given by the NT charging energy (\ref{gap-min}).
In the limit  $n_e, n_h \gg 1$ 
the period in $A$ becomes $e^2/C_1$.
%all the domains of the 
%phase diagram are identical, with their period in $A$
%given by $e^2/C_1$. 
%Only the $\mu>0$ half of the diagram is shown due to 
%the Dirac symmetry.
%{\it Inset:} 
{\it Right:} 
Mapping the devil's staircase in the Thouless pump setup.
In addition to the integer-$m$ plateaus, the ones with fractional $m$ appear
due to electron interactions. 
}
\label{fig:phdiag}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



{\it Phase diagram.---}
%Discarding zero--point motion in 
%the effective Hamiltonian (\ref{H-strong}), 
%admits a transparent
%classical limit. When the kinetic energy is negligible compared to 
%the interaction, one may discard the zero--point motion of fermions
%
The Hamiltonian (\ref{H-strong}) in the limit $K\gg 1$,
coarse-grained beyond $\lfl$, describes the classical 
chain of $n_e$ electrons and $n_h$ holes per period, $\mtot=n_e-n_h$.
Their positions $x_i$ in the minima and $y_j$ in the maxima of $U(x)$,
as well as the characteristic Coulomb blockade hexagons
labeled by ($n_e$, $n_h$) in Fig.~\ref{fig:phdiag},
are obtained by minimizing $E_{\rm cl}(A)/N - \mu(n_e-n_h)$, where
%their size controlled by the NT capacitance per period, 
%$C_0\simeq {\lsaw\over 2\ln(l_s/\lsaw)}$. 
%We label the domains by the numbers ($n_e$, $n_h$) of electrons and holes
%per period.
%of $N_e$ electrons (with positions $x_i$)
%and $N_h$ holes ($y_j$), with their interaction energy 
\begin{eqnarray}
\matrix{\ds
E_{\rm cl} = 
\sum%_{i=1}^{N_e} 
\lb \Delta' + U(x_i) \rb 
\ + \ 
\sum%_{j=1}^{N_h} 
\lb \Delta' - U(y_j) \rb 
\cr \ds
+ \sum_{i>i'} V(x_i-x_{i'})
+ \sum_{j>j'} V(y_j-y_{j'})
- \sum_{i,j} V(x_i-y_j) \ \ %\,, 
%\cr \ds
%+  {\mathop{\sum_{i,j=1}^{N_e}}_{i> j} 
%{e^2 \over |x_i-x_j|}
%+ \mathop{\sum_{i,j=1}^{N_h}}_{i> j}
%{e^2 \over |y_i-y_j|}
%- \! \mathop{\sum_{i,j=1}}_{i>j}^{N_e, N_h} {e^2 \over |x_i-y_j|}} \ . \quad
}
\label{hamiltonian-cl}
\end{eqnarray}
%Minimizing ${E_{\rm cl}(A) } - \mu(N_e-N_h)$, one obtains the 
%characteristic Coulomb blockade hexagons, Fig.~\ref{fig:phdiag}(a),
for a finite tube of the length $L=N\lsaw$.
In Fig.~\ref{fig:phdiag}, 
the hexagon size is controlled by the NT capacitance per period, 
$C_0\simeq {\lsaw\over 2\ln(l_s/\lsaw)}$. 
%We label the domains by the numbers ($n_e$, $n_h$) of electrons and holes
%per period.
For $n_e, n_h \gg 1$, the Thomas - Fermi approximation
$\rho(x)\approx \rho_{\rm TF}(x) \propto U(x)$
estimates the interaction energy 
%$V_{n_e}$ or $V_{n_h}$ 
inside each half-period (``quantum dot'') as
${e^2\over 2} \int%_{\lsaw/4}^{3\lsaw/4} 
\! dx dx'\, 
{\rho_{\rm TF}(x)\rho_{\rm TF}(x') \over |x-x'|} 
\equiv {n_e^2 e^2\over 2C_1}$,
where the ``dot capacitance'' 
$C_1 \simeq {\lsaw \over 2\pi \ln (\lsaw/a')}$. 
In this limit, all the hexagon domains of 
the phase diagram % in Fig.~\ref{fig:phdiag}
become identical with the width
%period in $A$ being 
${e^2/C_1}$. 
Remarkably, this period asymptotically follows from Eq.~(\ref{minigaps-ren-4})
with $\tilde A' 
%= 4\tilde A = {4A \over K \epsilon_0} 
\approx
{\pi\over 2} \cdot {A\over e^2/C_1}$ [$K\equiv K_{1/\lsaw} \approx K-1$],
using the Bessel function zeroes 
$2\tilde {A'}_{\mtot}^{(n)} \approx {3\pi\over 4} + {\pi \mtot \over 2} + \pi n$, 
% using the correspondence
%where we took the value of the charge stiffness (\ref{K})
%at the momentum scale $\lsaw^{-1}$ and substituted $K\to K-1$ at $K\gg 1$. 
%Utilizing the asymptotic behavior of the $s$--th zero of the Bessel function
%$J_n$,
%[cf. Eqs.~(\ref{minigaps-ren-4}), (\ref{minigaps-ren}), and 
%(\ref{minigaps-nonint})], 
%we obtain the values of $A\simeq (n_e+n_h)e^2/2C_1$ in accord with
%Eq.~(\ref{As}). 
%and 
%made obvious identifications 
%$n\equiv \mtot=n_e-n_h$, 
$n={\rm min}\{ n_e, n_h \}$.
%Away from these critical values the gap increases 
%linearly in $A$ similarly to what was stated above
%[after Eqs.~(\ref{A-gap}) and (\ref{gap-width})].
The borders between the domains correspond to the incompressible states
with {fractional} $n_e$ and $n_h$ (not shown). %(devil's staircase). %\cite{PT}








{\it Quantized current.---}
The devil's staircase can be mapped in 
the Thouless pump setup \cite{Talyanskii'01}.
%in which the adiabatically moving potential wave $U(x-st)$ 
With the Fermi level in the gap,
a slowly moving wave %$U(x-\lsaw ft)$ 
$U(x-st)$ with the frequency  $f=s/\lsaw$ 
will generate the quantized current %\cite{Thouless} 
%\be \label{j-quantized}
$j = \mtot ef$. 
%\ee
The moving potential can be created by
gating, optical methods, or acoustic field.
Estimated gap values $\Delta_m$ in the meV range ensure adiabaticity 
leading to current quantization and possible metrological applications. 
%$k_B T, hf \ll \Delta_m$ 
Novel incompressible states with {fractional} $m$ would correspond  
%can be revealed in a setup of Ref.~\onlinecite{Talyanskii'01}, 
to the current quantized in 
the  {\it fractions} of $4ef$, 
as illustrated in Fig.~\ref{fig:phdiag}.  
Current changes sign at half-filling
due to the Dirac symmetry. %of the NT.
%, where $f$ is the frequency of the adiabatic periodic potential modulation, 




{\it Conclusions.---}
We demonstrated that coupling to a periodic potential
results in the devil's staircase of incompressible electron states.
Excitation gaps are found
in the limit of the narrow-gap Luttinger liquid and in that of the  
Wigner crystal, by selecting the 
saddle point of the nonlinear action of the four bosonic modes.
When the Coulomb interaction dominates, 
%electrons avoid each other due to interaction rather than the Pauli principle, 
the system behaves as a single 
%Dirac 
fermion mode with renormalized mass and velocity. 
Control over the NT gap, the screening length, and the parameters of the  
potential makes this setup
%the adiabatic pumping setup \cite{Talyanskii'01} 
a unique probe of the Luttinger liquid - Wigner crystal crossover and 
%of the electron interactions, 
of commensuration effects in 1d.
Novel effect of adiabatic pumping at fractions of base frequency is linked to
the interaction-induced incompressible states.
%Applied to the Coulomb drag problem, \cite{Drag} 
%the developed approach yields the devil's staircase 
%of the absolute drag states. % in the Coulomb drag context. 
The effective single-mode description could also rationalize
recent manifestations of Wigner crystallization in transport,
such as the $e^2/h$ steps in conductance \cite{Matveev0.7}.


%How pertinent are the results of this work to the case of the Dirac fermions?
%Indeed, the Dirac fermion is special, since there is a definite 
%``relativistic'' relation 
%between the fermion mass (defininig the curvature of the dispersion) 
%and the Dirac gap, $\Delta = mv^2$. 
%Indeed, the relation $\psi\sim e^{i\theta}$ is asymptotically exact
%when the Fermi energy is large and the dispersion is practically linear.
%There are however reasons to believe that the recipe to include the curvature
%as a sine--Gordon term holds beyond the Dirac model. 
%Remember that the ``Compton'' wavelength $\lambda_C = \hbar v/\Delta = \hbar /mv$
%has a meaning of the minimal size one can confine the Dirac electron.


This work was supported by NSF MRSEC grant DMR 02-13706.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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A.~Odintsov and H.~Yoshioka, Phys. Rev. Lett. {\bf 82}, 374 (1999)



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C. L. Kane and E. J. Mele,
%Size, Shape, and Low Energy Electronic Structure of Carbon Nanotubes, 
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%
% Curvature effect in zig-zag NT, experiment:
%\bibitem{exp-curvature}
C. Zhou, J. Kong, and H. Dai,
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H. Ajiki and T. Ando, 
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%
% Parallel B-field, experiment:
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J. -O. Lee,
%%% {\em et al.}, 
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\bibitem{fracnt}
D.S.~Novikov, 
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preprint {\tt cond-mat/0407498}



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Phys. Rev. D {\bf 10}, 3235 (1974)

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\bibitem{Matveev0.7}

S.~Frank, P.~Poncharal, Z.L.~Wang, and W.A.~de~Heer, 
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M.J.~Biercuk, N.~Mason, J.~Martin, A.~Yacoby, C.M.~Marcus,
Phys. Rev. Lett. {\bf 94}, 026801 (2005); 
%{\tt cond-mat/0406652};
K.A.~Matveev, 
Phys. Rev. Lett. {\bf 92}, 106801 (2004)



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\end{document}


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% applications
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To be precise, this statement is true in 
the forward scattering approximation that holds 
down to undetectably small energies where interaction--induced gaps
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\bibitem{Polyakov}
The author thanks A.M.~Polyakov for a discussion on renormalization

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Similar expression for renormalized gap $\Delta$
has been obtained earlier in Ref.~\onlinecite{Levitov'01}. 
Whereas we agree on scaling exponents of $D$ and $\Delta_0$, 
the $K$--dependence is different. The procedure of Ref.~\onlinecite{Levitov'01}
yields the prefactor $K^{1/5}$ instead of the more natural result 
$K^{1/2 + {\cal O}(\eta)}$ that matches the classical limit.
We believe this discrepancy is a consequence 
of oversimplifying the renormalization group treatment
by introducing a {\it single} soliton scale under the RG 
in Ref.~\onlinecite{Levitov'01}.



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Here we also do not distinguish between $\ln(l_s/a)$ and $\ln(l_s/a')$
in the Coulomb formfactor $V(q)$, with large $K\gg 1$ implying 
large screening length $l_s\sim \lch \gg a, a'$.
This corresponds to treating the tube {\it capacitance} with logarithmic 
accuracy.





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D.S.~Novikov, 
Ph.D. Thesis, Massachusetts Institute of Technology (2003),
unpublished



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\end{document}




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J.~Nygard, D.H.~Cobden, M.~Bockrath, P.L.~McEuen, P.E.~Lindelof,
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%
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L. S. Levitov, A. M. Tsvelik, 
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%
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Y.A.~Krotov, D.-H.~Lee, S.G.~Louie,
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%
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R. Egger, A. O. Gogolin,
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%% cond-mat/9708065
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%%% cond-mat/9708054
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A.~Odintsov and H.~Yoshioka, Phys. Rev. Lett. {\bf 82}, 374 (1999)



\bibitem{curvature} 
C. L. Kane and E. J. Mele,
%Size, Shape, and Low Energy Electronic Structure of Carbon Nanotubes, 
Phys. Rev. Lett. {\bf 78}, 1932 (1997);
%
% Curvature effect in zig-zag NT, experiment:
%\bibitem{exp-curvature}
C. Zhou, J. Kong, and H. Dai,
%Intrinsic electrical properties of individual single-walled Carbon
%nanotubes with small band gaps,
Phys. Rev. Lett. {\bf 84}, 5604 (2000);
M.~Ouyang, J.L.~Huang, C.L.~Cheung, C.M.~Lieber, 
Science {\bf 292}, 5517 (2001)


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H. Ajiki and T. Ando, 
J. Phys. Soc. Jpn. {\bf 65}, 505 (1996);
%
% Parallel B-field, experiment:
%\bibitem{exp-B-parallel}
J. -O. Lee,
%%% {\em et al.}, 
J. R. Kim, J. J. Kim, J. Kim, N. Kim, J. W. Park, and K. H. Yoo,
%Observation of magnetic-field-modulated energy gap in Carbon nanotubes,
Sol. Stat. Comm. {\bf 115}, 467 (2000);
U.C.~Coskun, T.-C.~Wei, S.~Vishveshwara, P.M.~Goldbart, A.~Bezryadin,
%$h/e$ Magnetic flux modulation of the energy gap in nanotube quantum dots,
Science {\bf 304}, 1132 (2004)


\bibitem{fracnt}
D.S.~Novikov, 
% Electron properties of carbon nanotubes in a periodic potential,
preprint {\tt cond-mat/0407498}



\bibitem{GN}
D.J.~Gross and A.~Neveu, 
Phys. Rev. D {\bf 10}, 3235 (1974)

%\bibitem{altland-glazman}
%A.~Altland, L.I.~Glazman, and A.~Kamenev,
%%Electron transport in granular metals,
%Phys. Rev. Lett. {\bf 92}, 026801 (2004)


\bibitem{Matveev0.7}

S.~Frank, P.~Poncharal, Z.L.~Wang, and W.A.~de~Heer, 
Science {\bf 280}, 1744 (1998); 
M.J.~Biercuk, N.~Mason, J.~Martin, A.~Yacoby, C.M.~Marcus,
Phys. Rev. Lett. {\bf 94}, 026801 (2005); 
%{\tt cond-mat/0406652};
K.A.~Matveev, 
Phys. Rev. Lett. {\bf 92}, 106801 (2004)



\end{thebibliography}