arXiv:0807.0194v1 [cond-mat.mes-hall] 1 Jul 2008

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Nattermann

International Journal of Modern Physics B
c World Scientific Publishing Company

TRANSPORT IN LUTTINGER LIQUIDS WITH STRONG
IMPURITIES

SERGEY MALININ
Department of Chemistry, Wayne State University
5101 Cass Avenue, Detroit, Michigan 48202, USA
malinin@chem.wayne.edu
THOMAS NATTERMANN
Institut f¨r Theoretische Physik, Universit¨t zu K¨ln
u
a
o
Z¨lpicherstr. 77, D-50937 K¨ln, Germany
u
o
natter@thp.uni-koeln.de

The tunnel current of a Luttinger liquid with a finite density of strong impurities is
calculated using an instanton approach. For very low temperatures T or electric fields
E 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 T
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.
Keywords: Luttinger liquids; disorder; transport

1. Introduction
1D 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 1D where electrons cannot avoid each other. As
a consequence excitations are plasmons similar to sound waves in solids. The corresponding phase is called a Luttinger liquid (LL) 1,2 . Renewed interest in LLs
arises from progress in manufacturing narrow quantum wires with a few or a single
conducting channel. Examples are carbon nanotubes 3 , polydiacetylen 4 , quantum
Hall edges 5 and semiconductor cleave edge quantum wires 6 .
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 7 . Central quantity is the
interaction parameter K which plays the role of a dimensionless conductance of a
clean LL 8,1 . 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 8,9,10,11. Here the conduc1

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tance depends crucially on K. Impurities are irrelevant for attractive (K > 1) and
strongly relevant for repulsive interaction (K < 1), respectively. For finite voltage V
2
and K < 1, the conductance is ∼ V K −2 8 . These considerations can be extended
to two impurities. Depending on the applied gate voltage, Coulomb blockade effects
may give rise to resonant tunneling 8,10 .
(ii) In the opposite case of a finite density of weak impurities, (Gaussian) disorder is a relevant perturbation for K < 3/2 leading to the localization of electrons. For
√
weak external electric field E the conductivity is highly nonlinear: σ(E) ∼ e−c/ E
12,13,14,15 . At low but finite temperatures T this result goes over into the VRH
√
′
expression for the linear conductivity σ ∼ e−c / T 13,14,15,16,17 . At higher temperatures there is a crossover to σ ∼ T 2−2K 18 .
On the contrary, much less is known in the case of a finite density of strong
pinning centers 19,2 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 (5), (6), (7) and (8) as well as the
crossover behavior summarized in Fig. 1.
2. Model and Instantons
Starting point of our calculation is the action of interacting electrons subject to an
external uniform electric field E and strong pinning centers. In bosonized form the
action takes the form
L λT

S=

2πK
0

0

N

dxdy (∂y ϕ)2 + (∂x ϕ + f x)2 −

i=1

uδ(x − xi ) cos(2ϕ + 2kF xi )

(1)

The phase ϕ(x) is related to the electron density ρ(x) = π −1 (kF +∂x ϕ)(1+2 cos(2ϕ+
2kF x)). kF is the Fermi wave vector, τ = y/v and f = F K/v . v and λT = v/T
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 ϕ(xi , y) ≡ φi (y) at the impurity sites xi which are assumed to
be randomly distributed. The action can then be expressed in terms of Fourier
components φi (y) = λT −1 ωn φi,ωn e−iωn y , ωn = 2πn/λT . Thus
N

S=

2πK

i=0

ωn

ωn
λT

ω n ai
|φi+1,ωn − φi,ωn |2
+ |φi,ωn |2 + |φi+1,ωn |2 tanh
sinh ωn ai
2

−f (ai−1 + ai )φi,0 + ueff

dy 1 − cos 2φi (y) + 2παi

(2)

where ai = xi+1 − xi and αi = kF xi /π. Since kF ai ≫ 1 below we will assume the
αi to be random phases but keep the impurity distance ai approximately constant
ai ≈ a.

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Transport in disordered Luttinger liquids

Next we consider the current resulting from tunneling processes between
metastable states, assuming strong pinning and weak quantum fluctuations, i.e.
K ≪ 1. The tunneling process starts from a classical metastable configuration
˜
φi which minimizes the impurity potential for all values of y, E = 0. Hence
˜
φi = π(ni − αi ) where ni is integer. Among the many metastable states there
˜
is one (modulo π) zero field ground state φ0 where ni = n0 = j≤i [∆αj−1 ]G 13 .
i
i
Here ∆αj = αj+1 − αj and [α]G denotes the closest integer to α. A new metastable
state follows from the ground state by adding integers qi = ±1 to the n0 .
i
Next we consider an instanton configuration which connects the original state
˜
˜
φi with the new state φi + π, ni depends in general on y. To be specific, we assume a
˜
double kink configuration for the instanton at each impurity site: φi (y) = φi +π , for
˜i , for |y − yi | > Di + d, with a linear interpolation
|y − yi | < Di − d, and φi (y) = φ
between the two values at the kink walls in the regions |y − yi | − Di < d. yi ± Di
is the kink/anti-kink position, d ∼ 1/u is the approximate width of the kinks and
2Di their distance. It is plausible that in the saddle point configuration all yi will
be the same, an approximation we will use in the following. With zi = πDi /a the
instanton action can then be rewritten as
SI ≈

2
K

i

˜
∆φi
zi
cosh((zi+1 − zi )/2)
(zi+1 − zi ) − f a2 zi + s + ln
tanh cosh zi
π
cosh((zi+1 + zi )/2)
2

where the sum goes only over impurities with zi > 0. s is a constant that includes
the core action of a kink and an anti-kink: s = ln(Cau) ≫ 1, where ln C/K ≫ 1.
˜
For a given initial metastable state {φi }, SI is a function of the variational
parameters {Di ; i = 1, ..., N }. The nucleation rate Γ and hence the current I
N

is given by I ∝ Γ ∝

i=0

i∞
0

dDk exp(−S/ ). Here we employ an approximate

treatment in which we assume Di ≡ D = az/π for k < i ≤ k + m and Di = 0
elsewhere, i.e. tunneling is assumed to occur simultaneously through m neighboring
impurities. The instanton is then a rectangular object with extension ma and 2D
in x and y direction, respectively. The instanton action can then be written as
Sinst =

2
K

zσm (k) + ln(1 + e−2z ) + m s + ln tanh

z
E
−z
2
Ea

.

(4)

Here we introduced the dimensionless field strength f a2 /π = E/Ea where Ea =
1/(κa2 ), κ = K/π v denotes the compressibility. σm (k) = νk (1) + νk+m (−1) /2
plays the role of a surface tension of the vertical boundaries of the instanton where
νk (q) = q 2 − 2q ∆αk − [∆αk ]G . In the ground state σm (k) is equally distributed
in the interval 0 ≤ σm (k) < 2 14 . The second and the third contribution in (4)
result from the horizontal boundaries of the instanton and include their surface
tension s/a 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 Sbath = 2 mη ln z ,
K
η ≪ 1, in the action in order to allow for energy dissipation 14 . However, we will

(3)

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omit η-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 ∂Sinst /∂z < 0 for z → ∞, i.e. σm (k) <
mEa /E. The tunneling probability follows from the saddle point value of the instanm
E
ton action where z fulfils the condition σm (k) − m Ea + tanh z − 1 + m η + sinh z = 0 .
z
3. Results and Conclusions
We discuss now several special cases: (i) For sufficiently large fields E ≫ Ea
the saddle point is zs ≈ Ea ≪ 1 which gives a tunneling probability Γ ∝
E
2m
(E/Ea ) K −1 e−2ms/K . The exponent −1 results from the integration around the saddle point. Because of small K and correspondingly large kink core action, tunneling
through single impurities (m = 1) is preferred and hence the nonlinear conductivity
is given by
2

σ(E) ∼ (E/Ea ) K

−2 −2s/K

e

,

Ea < E < E1,cr = Ea es ,

(5)

in agreement with previous results for tunneling through a single weak link 8 if we
identify e−s/K ∼ t with the hopping amplitude t through the link. The upper field
strength for the validity of this result can be estimated from Ds u ≡ zs au < 1 since
the instantons loose then their meaning. Using u → ueff ≈ kF (u/kF )1/(1−K) we
find E1,cr ∼ (kF /eκa)(u/kF )1/(1−K) , which can be also read off directly from (5)
as E1,cr ∼ Ea es . Classically (K = 0), E1,cr corresponds to the case when the field
energy Ea the electron gains by moving to the next impurity is smaller than the
pinning energy u/κ.
At finite temperatures there is a crossover to a temperature T ≈ EaK dependent
conductivity
σ(T, E) ∝

T
Ta

2
K

−2

e−2s/K sinh(

Ea T
)
,
T Ea

EKa < T < T1,cr = Ta es

(6)

when the instanton extension 2Ds reaches λT , i.e. for E < Ea T /Ta. For temperatures higher than T1,cr isolated impurities are weak. Following the arguments of 19
one expects in this region σ ∼ T 2−2K .
(ii) In the opposite case of weak fields, E ≪ Ea , tunneling happens simultaneously through many impurities and the saddle point is zs ≫ 1. In this case we can
estimate the typical surface tension as σk (m) ≈ 1/m for a chosen pair of sites k and
k + m, respectively 14 . For very large values of m we can treat m as continuous
and the saddle point condition gives ms ≈ Ea /E ≫ 1 and zs ≈ sEa /(2E). The
tunneling probability and hence the current is proportional to
√
2s
I ∼ σ(E) ∼ e− K Ea /E , E < Ea .
(7)
If we write the result in the VRH form 16 I ∼ e−2ma/ξloc we can identify the
localization length ξloc ≈ aK/s of the tunneling charges. There is a crossover to

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Transport in disordered Luttinger liquids

a temperature dependent conductivity if λT < 2Ds , i.e. for E < sT Ea /Ta < Ea
where
√
2
Ea T
)
,
EKa/s < T < Ta /s
(8)
σ(T ) ∼ e− K csTa /T ) sinh(
T Ea
Results (7) and (8) are in agreement with those obtained for weak pinning 13,14 . (iii)
T
T 1,cr

u

T

2
K

Ta /s
T

−2

2
-2
K

strong
pinning

kF
T1,cr

Ta

uc

E

2
K

c
K

T 2-2K
Tloc

T as
T

e

2s E a
K E

Ea

T

−2

~T a s

e

e -c

E 1,cr

E

weak
pinning
T 2-2K(T)
Ta

F

T

Fig. 1. Left: Field and temperature dependence of the conductivity in the various regions of the
T −E plane. Ta , T1,cr , s, Ea and E1,cr are explained in the text. The region Ta > T > Ta /s, E < Ea
T
is characterized by activated behavior σ ∼ e−Ta /T sinh( Ea ) Ea Right: u − T phase diagram of
T
the linear conductivity of disordered LLs. For strong pinning, u > uc ∼ kF (kF a)K−1 and T <
T1,cr ∼ Ta (u/uc )1/(1−K) , Ta /s separates the VRH from the single impurity hopping regime. For
T > T1,cr impurities become weak. For weak pinning, u < uc , Tloc ∼ Ta (u/uc )2/(3−2K) separates
VRH from renormalized power law behavior. For T > Ta the power law is unrenormalized.

If m is not too large (e.g. for large a) we have to take into account the discreteness
Ea /E. Since Sinst (z(m), m)
of m. An instanton solution exists only for m >
has always a negative derivative with respect to m at m → Ea /E + 0, but for
reasonably large values of s the interval of m with negative derivative is much shorter
than 1 and hence the optimal hopping length ms (E) is the smallest integer exceeding
Ea /E, which we denote as
Ea /E G+ . To be more realistic we have to take
into account the randomness of the impurity distances ai such that decreasing the
field (or the temperature), the current jumps by a factor ∼ e−2am /ξloc . Clearly, for
long wires these jumps will average out.
Finally, we briefly compare the present case of Poissonian strong disorder,
ueff a ≫ 1 with the Gaussian weak disorder, ueff a ≪ 1 considered in 10,13,14,18 .
In the latter case u and a are sent simultaneously to zero but the quantity
−3
3
u2 /a ∼ ξ0 ≪ kF is assumed to be finite, ξ0 denotes the bare correlation length.
−1
Fluctuations on scales smaller than ξ0 renormalize ξ0 → ξ ∼ kF (ξ0 kF )3/(3−2K) . At
low T the conductivity is of variable range hopping type (8) up to a temperature
Tloc = v/ξ = Ta (u/uc )2/(3−2K) where uc ≈ kF (akF )K−1 . For higher T there is a
direct crossover to σ ∼ T 2−2K(T ) where K is now renormalized by disorder fluctuations 18,19 . This renormalization disappears only at much higher Ta ∼ v/a. Both

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weak and strong pinning theories should roughly coincide for u → uc ≈ kF (akF )K−1
where Ta ≈ T1,cr ≈ Tloc which is indeed the case since ξ ≈ a. In the strong pinning
region ξ continues as ξ ∼ a/s.
Experimentally, a linear variable range hopping conductivity has been seen in
carbon-nanotubes 3 and polydiacetylen 4 .
Acknowledgements
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.
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