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@xmath2d electron systems exhibit a number of peculiarities which destroy the familiar fermi - liquid behavior known from higher dimensions .
main reason is the geometrical restriction of the motion in @xmath2d where electrons can not avoid each other . as a consequence excitations are plasmons similar to sound waves in solids .
the corresponding phase is called a luttinger liquid ( ll ) @xcite . renewed interest in lls arises from progress in manufacturing narrow quantum wires with a few or a single conducting channel .
examples are carbon nanotubes @xcite , polydiacetylen @xcite , quantum hall edges @xcite and semiconductor cleave edge quantum wires @xcite .    from a theoretical point of view
1d quantum wires allow the investigation of the interplay of interaction and disorder effects since short range interaction can be treated already within a harmonic bosonic theory @xcite .
central quantity is the interaction parameter @xmath3 which plays the role of a dimensionless conductance of a clean ll @xcite .
the effect of disorder on transport in lls has been so far considered in two limiting cases :    \(i ) the effect of a _ single impurity _
was considered in @xcite . here
the conductance depends crucially on @xmath3 .
impurities are irrelevant for attractive ( @xmath4 ) and strongly relevant for repulsive interaction ( @xmath5 ) , respectively . for finite voltage @xmath6 and @xmath5 ,
the conductance is @xmath7 @xcite .
these considerations can be extended to two impurities .
depending on the applied gate voltage , coulomb blockade effects may give rise to resonant tunneling @xcite .
\(ii ) in the opposite case of a _ finite density of _ weak _ impurities _ , ( gaussian ) disorder is a relevant perturbation for @xmath8 leading to the localization of electrons . for weak external electric field
@xmath1 the conductivity is highly nonlinear : @xmath9 @xcite . at low but finite temperatures
@xmath0 this result goes over into the vrh expression for the linear conductivity @xmath10 @xcite . at higher temperatures
there is a crossover to @xmath11 @xcite .    on the contrary , much less
is known in the case of a _
finite density of _ strong _ pinning centers _
@xcite which we will address in the present paper .
in particular we determine both the temperature and electric field dependence of the ( nonlinear ) conductivity for this case in a broad temperature and electric field region .
the main results of the paper are the conductivities ( [ eq : large_e ] ) , ( [ eq : large_t ] ) , ( [ eq : small_e ] ) and ( [ eq : small_t ] ) as well as the crossover behavior summarized in fig .
[ sigma ] .
starting point of our calculation is the action of interacting electrons subject to an external uniform electric field @xmath1 and strong pinning centers . in bosonized form
the action takes the form @xmath12 the phase @xmath13 is related to the electron density @xmath14 .
@xmath15 is the fermi wave vector , @xmath16 and @xmath17 . @xmath18 and @xmath19 denote the plasmon velocity and the thermal de broglie wave length , respectively .    the phase field between the impurities
can now be easily integrated out leaving only its values @xmath20 _ at _ the impurity sites @xmath21 which are assumed to be randomly distributed .
the action can then be expressed in terms of fourier components @xmath22 , @xmath23 .
thus @xmath24\bigg\}\end{aligned}\ ] ] where @xmath25 and @xmath26 .
since @xmath27 below we will assume the @xmath28 to be random phases but keep the impurity distance @xmath29 approximately constant @xmath30 .
next we consider the current resulting from tunneling processes between metastable states , assuming strong pinning and weak quantum fluctuations , i.e. @xmath31 . the tunneling process starts from a classical _ metastable _ configuration @xmath32 which minimizes the impurity potential for all values of @xmath33 , @xmath34 .
hence @xmath35 where @xmath36 is integer . among the many metastable states there is one ( modulo @xmath37 ) zero field _
state @xmath38 where @xmath39_g$ ] @xcite . here
@xmath40 and @xmath41_g$ ] denotes the closest integer to @xmath42 .
a new metastable state follows from the ground state by adding integers @xmath43 to the @xmath44 .
next we consider an _ instanton _
configuration which connects the original state @xmath32 with the new state @xmath45 , @xmath36 depends in general on @xmath33 . to be specific
, we assume a double kink configuration for the instanton at each impurity site : @xmath46 for @xmath47 and @xmath48 for @xmath49 . @xmath50 is the kink / anti - kink position , @xmath51 is the approximate width of the kinks and @xmath52 their distance .
it is plausible that in the saddle point configuration all @xmath53 will be the same , an approximation we will use in the following . with @xmath54 the instanton action can then be rewritten as @xmath55 \right\}\nonumber\ ] ] where the sum goes only over impurities with @xmath56 .
@xmath57 is a constant that includes the core action of a kink and an anti - kink : @xmath58 , where @xmath59 .    for a given initial metastable state @xmath60
, @xmath61 is a function of the variational parameters @xmath62 .
the nucleation rate @xmath63 and hence the current @xmath64 is given by @xmath65 here we employ an approximate treatment in which we assume @xmath66 for @xmath67 and @xmath68 elsewhere , i.e. tunneling is assumed to occur simultaneously through @xmath69 neighboring impurities .
the instanton is then a rectangular object with extension @xmath70 and @xmath71 in @xmath72 and @xmath33 direction , respectively .
the instanton action can then be written as @xmath73 here we introduced the dimensionless field strength @xmath74 where @xmath75 , @xmath76 denotes the compressibility .
@xmath77 plays the role of a surface tension of the vertical boundaries of the instanton where @xmath78_g\big)$ ] . in the ground state @xmath79
is equally distributed in the interval @xmath80 @xcite .
the second and the third contribution in ( [ eq : m - instanton ] ) result from the horizontal boundaries of the instanton and include their surface tension @xmath81 and their attractive interaction .
the last term describes the volume contribution resulting from the external field .
in addition , we have to include a small dissipative term @xmath82 , @xmath83 , in the action in order to allow for energy dissipation @xcite .
however , we will omit @xmath84-dependent terms in all results where they give only small corrections ( apart from possible pre - exponential factors which we do not consider ) .
a necessary condition for tunneling is @xmath85 for @xmath86 , i.e. @xmath87 . the tunneling probability follows from the saddle point value of the instanton action where @xmath88 fulfils the condition @xmath89
we discuss now several special cases : ( i ) for sufficiently large fields @xmath90 the saddle point is @xmath91 which gives a tunneling probability @xmath92 .
the exponent @xmath93 results from the integration around the saddle point .
because of small @xmath3 and correspondingly large kink core action , tunneling through single impurities ( @xmath94 ) is preferred and hence the nonlinear conductivity is given by @xmath95 in agreement with previous results for tunneling through a single weak link @xcite if we identify @xmath96 with the hopping amplitude @xmath97 through the link .
the upper field strength for the validity of this result can be estimated from @xmath98 since the instantons loose then their meaning .
using @xmath99 we find @xmath100 , which can be also read off directly from ( [ eq : large_e ] ) as @xmath101 .
classically ( @xmath102 ) , @xmath103 corresponds to the case when the field energy @xmath104 the electron gains by moving to the next impurity is smaller than the pinning energy @xmath105 .    at finite temperatures
there is a crossover to a temperature @xmath106 dependent conductivity @xmath107 when the instanton extension @xmath108 reaches @xmath109 , i.e. for @xmath110 . for temperatures higher than @xmath111 isolated impurities are weak . following the arguments of @xcite
one expects in this region @xmath112 .
\(ii ) in the opposite case of weak fields , @xmath113 , tunneling happens simultaneously through many impurities and the saddle point is @xmath114 . in this case
we can estimate the typical surface tension as @xmath115 for a chosen pair of sites @xmath116 and @xmath117 , respectively @xcite .
for very large values of @xmath69 we can treat @xmath69 as continuous and the saddle point condition gives @xmath118 and @xmath119 . the tunneling probability and
hence the current is proportional to @xmath120 if we write the result in the vrh form @xcite @xmath121 we can identify the localization length @xmath122 of the tunneling charges .
there is a crossover to a temperature dependent conductivity if @xmath123 , i.e. for @xmath124 where @xmath125 results ( [ eq : small_e ] ) and ( [ eq : small_t ] ) are in agreement with those obtained for weak pinning @xcite
.     plane . @xmath126 and @xmath103 are explained in the text .
the region @xmath127 , @xmath128 is characterized by activated behavior @xmath129 right : @xmath130 phase diagram of the linear conductivity of disordered lls . for strong
pinning , @xmath131 and @xmath132 , @xmath133 separates the vrh from the single impurity hopping regime . for @xmath134 impurities
become weak . for weak
pinning , @xmath135 , @xmath136 separates vrh from renormalized power law behavior .
for @xmath137 the power law is unrenormalized . ]
\(iii ) if @xmath69 is not too large ( e.g. for large @xmath138 ) we have to take into account the discreteness of @xmath69 .
an instanton solution exists only for @xmath139 .
since @xmath140 has always a negative derivative with respect to @xmath69 at @xmath141 , but for reasonably large values of @xmath57 the interval of @xmath69 with negative derivative is much shorter than @xmath2 and hence the optimal hopping length @xmath142 is the smallest integer exceeding @xmath143 , which we denote as @xmath144_{g+}$ ] . to be more realistic we have to take into account the randomness of the impurity distances @xmath145 such that decreasing the field ( or the temperature ) , the current jumps by a factor @xmath146 .
clearly , for long wires these jumps will average out .
finally , we briefly compare the present case of poissonian strong disorder , @xmath147 with the gaussian weak disorder , @xmath148 considered in @xcite . in the latter case @xmath149 and @xmath138
are sent simultaneously to zero but the quantity @xmath150 is assumed to be finite , @xmath151 denotes the bare correlation length .
fluctuations on scales smaller than @xmath151 renormalize @xmath152 . at low @xmath0
the conductivity is of variable range hopping type ( [ eq : small_t ] ) up to a temperature @xmath153 where @xmath154 . for higher @xmath0
there is a direct crossover to @xmath155 where @xmath3 is now renormalized by disorder fluctuations @xcite .
this renormalization disappears only at much higher @xmath156 . both weak and strong pinning theories
should roughly coincide for @xmath157 where @xmath158 which is indeed the case since @xmath159 . in
the strong pinning region @xmath160 continues as @xmath161 .
experimentally , a linear variable range hopping conductivity has been seen in carbon - nanotubes @xcite and polydiacetylen @xcite .
the authors thank a. altland , t. giamarchi , b. rosenow , s. scheidl for useful discussions .
this work is supported by the sfb 608 of dfg . s.m .
acknowledges financial support of rfbr under grant no .
03 - 02 - 16173 .
t. giamarchi and h. maurey , in _ correlated fermions and transport in mesoscopic systems _ , ed .
s. montambaux and j. tran than van , ed .
frontiere , gif sur yvette , ( 1996 ) , ( cond - mat/9608006 ) . in their notation @xmath163 .
the tunnel current of a luttinger liquid with a finite density of strong impurities is calculated using an instanton approach . 
 for very low temperatures @xmath0 or electric fields @xmath1 the ( nonlinear ) conductivity is of variable range hopping ( vrh ) type as for weak pinning . for higher temperatures or fields 
 the conductivity shows power law behavior corresponding to a crossover from multi- to single - impurity tunneling . for even higher @xmath0 and 
 not too strong pinning there is a second crossover to weak pinning . the determination of the position of the various crossover lines both for strong and weak pinning allows the construction of the global regime diagram .