Nonequilibrium transport for crossed Luttinger liquids
Andrei Komnik and Reinhold Egger

arXiv:cond-mat/9802116v1 [cond-mat.str-el] 11 Feb 1998

Fakult¨t f¨r Physik, Albert-Ludwigs-Universit¨t, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
a u
a
(February 11, 1998)

Transport through two one-dimensional interacting metals (Luttinger liquids) coupled together at a single point is
analyzed. The dominant coupling mechanism is shown to
be of electrostatic nature. Describing the voltage sources by
boundary conditions then allows for the full solution of the
transport problem. For weak Coulomb interactions, transport
is unperturbed by the coupling. In contrast, for strong interactions, unusual nonlinear conductance laws characteristic for
the correlated system can be observed.

U2 2

U1 2

I2

I1

PACS numbers: 71.10.Pm, 72.10.-d, 73.40.Gk

I1
The physics of one-dimensional (1D) conductors has
received much attention lately, chieﬂy due to fabrication
advances and the discovery of novel 1D materials such as
carbon nanotubes [1]. From the theoretical point of view,
these systems are of interest since Coulomb interactions
invalidate the ubiquitous Fermi liquid description. The
resulting state is often of Luttinger liquid (LL) [2,3] type
characterized by, e.g., spin-charge separation, suppression of the tunneling density of states, and interactiondependent power laws in the transport behavior. However, so far the unambiguous experimental observation of
LL behavior has been diﬃcult to achieve.
In this paper, we study two correlated 1D metals coupled in a point-like manner (“crossed Luttinger liquids”).
For the standard two-chain problem, where two Luttinger
liquids are connected all along the conductors, the coupling normally destroys the LL phase [4]. In the case
of a point-like coupling, however, the LL characteristics
can survive and lead to the unusual transport features
reported below. The most promising candidates for their
experimental observation are carbon nanotubes. At not
exceedingly low temperatures, metallic single-wall nanotubes exhibit LL behavior (with an additional ﬂavor
index) [5]. In a remarkable recent experiment, Tans et
al. [6] were able to attach leads to a single nanotube.
So far, transport measurements have been dominated by
Coulomb charging eﬀects due to rather large contact resistances between the leads and the nanotube, thereby
masking any possible deviation from Fermi liquid theory. In the near future this problem might be overcome,
and non-Fermi liquid laws should indeed emerge. Other
realizations of crossed Luttinger liquids could be based
on, e.g., 1D quantum wires in semiconductor heterostructures [7], or edge states in a fractional quantum Hall bar
[8].

I2

U1 2

U2 2

FIG. 1. Two Luttinger liquids coupled together at one
point (x = 0) and connected to external reservoirs held at
constant voltages ±U1 /2 and ±U2 /2. In the absence of sin′
gle-particle tunneling, the currents obey Ii = Ii .

The geometry of our system is shown in Fig. 1. We
shall consider two spinless Luttinger liquids characterized
by the same interaction constant g [9]. Here the noninteracting value is g = 1, and externally screened Coulomb
interactions imply 0 < g < 1 [2,3]. For the nanotube
experiment of Ref. [6], one has an externally unscreened
e2 /|x−x′ | interaction potential and therefore very strong
correlations. Strictly speaking, this interaction leads to
g → 0 in an inﬁnite system, but the ﬁnite length of the
nanotube in Ref. [6] implies g ≈ 0.2. A natural and
quite simple description of Luttinger liquids is oﬀered by
the standard bosonization method [3]. External reservoirs (voltage sources) can be incorporated by imposing
boundary conditions [10] for the phase ﬁelds employed in
the bosonization scheme. This approach oﬀers a general
and powerful route to studying multi-terminal LandauerB¨ttiker geometries [11] for strongly correlated electrons.
u
The crossed Luttinger liquids depicted in Fig. 1 may be
the simplest example for such a problem.
We start by expressing the right- and left-moving
(p = ±) component of the electron operator ψpi (x) in
conductor i = 1 or 2 in terms of the dual bosonic phase
ﬁelds θi (x) and φi (x) obeying the algebra
[φi (x), θj (y)]− = −(i/2)δij sgn(x − y) .

(1)

The bosonization formula then reads [3]
√
ηpi
exp[−ipkF x − ip πg θi (x) − i π/g φi (x)] ,
ψpi (x) = √
2πa
(2)
1

g < 1/2. In contrast, a static potential scatterer in one
of the conductors would be relevant already for g < 1
[13]. In our case, electrons in conductor 1 experience
the ﬂuctuating potential scattering V1 due to electrons in
conductor 2, implying a doubled scaling dimension.
The second potentially important process is singleparticle hopping from one conductor into the other. It is
helpful to distinguish processes that do (do not) preserve
the p = R, L = ± index, yielding the two perturbations
†
V2 = λ2 p ψp1 (0)ψp2 (0)+ H.c. (preserving the p index)
†
and V3 = λ3 p ψp1 (0)ψ−p2 (0)+ H.c. (not preserving the
p index). They have the bosonized form

where the same average density kF /π is assumed for both
conductors. The short-distance cutoﬀ (lattice spacing) in
Eq. (2) is taken as a = 1/kF . To ensure anticommutation
relations among diﬀerent branches (p, i), we use (real)
Majorana fermions ηpi fulﬁlling [ηpi , ηp′ i′ ]+ = 2δpp′ δii′ .
In the following, only products of Majorana fermions will
appear. A valid choice for these products employs the
standard Pauli matrices [5],
ηp1 ηp2 = iσx ,
ηp1 η−p1 = ipσz ,

ηp1 η−p,2 = −ipσy ,

(3)

ηp2 η−p2 = −ipσz .

Assuming that the conductors do not contain impurities,
the Hamiltonian of the uncoupled system is
1
H0 =
2

(∂x φi )2 + (∂x θi )2

dx

,

2λ2
√
σx cos{ πg [θ1 (0) − θ2 (0)]}
πa
× sin{ π/g [φ1 (0) − φ2 (0)]}
2λ3
√
σy sin{ πg [θ1 (0) + θ2 (0)]}
V3 =
πa
× cos{ π/g [φ1 (0) − φ2 (0)]} .
V2 = −

(4)

i=1,2

where we have put h = 1 and the sound velocity v =
¯
vF /g = 1. Adiabatically connected voltage sources can
then be taken into account by Sommerfeld-like boundary
conditions. Applying the voltage U1 along conductor 1,
and U2 along conductor 2, see Fig. 1, they read [10]
ρp=±,i (x → ∓∞) = ±

eUi
,
4πg

ρi (x) =

(5)

dλ
= [1 − 2g] λ + [g − 1/g] t2 ,
dℓ
dt
= [1 − (g + 1/g)/2] t ,
dℓ

4πg θi (x)] , (6)

where the ﬁrst (“slow”) term is due to the sum of rightand left-moving densities ρRi + ρLi , and the second
(“fast”) term arises from mixing right- and left-movers.
The ∓ signs correspond to i = 1, 2, respectively. One
checks easily that most contributions to V1 are irrelevant for g ≤ 1, i.e., they have scaling dimension η > 1.
Keeping the fast component in Eq. (6) yields the only
important term,
V1 = −

λ1
sin[ 4πg θ1 (0)] sin[ 4πg θ2 (0)] ,
(πa)2

(9)

For the standard two-chain problem, a (bulk) coupling
term formally identical to Eq. (8) has been discussed
in Ref. [14]. Both V2 and V3 have scaling dimension
η2 = η3 = (g + 1/g)/2 > 1, from which one might
naively conclude that they are irrelevant [15]. However,
this conclusion is premature because V2 and V3 have
conformal spin S = 1 [16]. For an operator with nonzero conformal spin, the standard criterion for relevance
η < 1 does not apply, since relevant perturbations may be
generated in higher orders of the renormalization group
(RG). This phenomenon indeed occurs in the standard
two-chain problem [17], where the (bulk) coupling term
corresponding to Eq. (8) generates relevant particle-hole
and/or particle-particle excitation operators. Similarly,
we ﬁnd that V2 and V3 together generate the electrostatic coupling V1 given in Eq. (7), but no other relevant
terms. Omitting irrelevant operators, the resulting RG
equations take the closed form

where ρ±,i is the density of right/left-moving particles
injected into conductor i. Outgoing particles are assumed
to enter the reservoirs without reﬂection.
Let us now consider a point-like coupling of both 1D
conductors at, say, x = 0. For example, in a nanotube
setup, two nanotubes could be stacked on top of each
other. Such a contact causes (at least) two diﬀerent coupling mechanisms [12].
First, there arises a (density-density) electrostatic interaction of the form V1 = λ1 ρ1 (0)ρ2 (0). Using Eqs. (2)
and (3), and omitting the mean density kF /π which is
supposedly neutralized by positive background charges,
the bosonized representation of the density operator is
σz
g
∂x θi (x) ∓
sin[2kF x +
π
πa

(8)

(10)

where λ2 = λ3 ≡ t is the hopping amplitude and λ1 ≡ λ
the electrostatic coupling. The standard ﬂow parameter
is deﬁned by dℓ = −d ln ωc , where ωc is a high-frequency
cutoﬀ that is reduced under the RG transformation.
Let us ﬁrst discuss the case g = 1. The electrostatic
coupling is irrelevant, i.e., we may eﬀectively put λ = 0,
but the hopping term stays marginal. By refermionizing the Hamiltonian H0 + V2 + V3 , and employing the
boundary conditions (5), one arrives at the familiar results for uncorrelated electrons in the geometry of Fig. 1,
see Ref. [11]. We therefore recover the usual LandauerB¨ttiker formalism. Second, for g < 1, the hopping amu
plitude t(ℓ) always scales to zero as ℓ → ∞, and the

(7)

with scaling dimension η1 = 2g. Clearly, this coupling becomes relevant for suﬃciently strong interactions,
2

eﬀects of single-particle tunneling can be captured by a
renormalization of the bare electrostatic coupling λ, see
Eq. (10). We shall assume henceforth that this renormalization has been carried out, and only the electrostatic
interaction V1 will be kept. In that case, the currents
ﬂowing through conductor i = 1 or 2 satisfy Ii = Ii′ , see
Fig. 1, and can be computed from the bosonized current
operator [3]
Ii = e

g/π ∂t θi (x = 0, t) .

the problem of an elastic potential scatterer embedded
into a spinless LL [13]. However, this LL now has the
doubled interaction strength parameter g = 2g. The
¯
boundary conditions (15) specify the eﬀective voltages
√
¯
Ur = (U1 + rU2 )/ 2 applied to channel r = ±. In analogy to Eq. (11), currents in channel r = ± are deﬁned by
¯
Ir = e g/π ∂t ϑr (0), and from Eq. (13), we then ﬁnd the
√
¯
¯
currents Ii = (I+ ± I− )/ 2 ﬂowing in conductor i = 1, 2.
The Hamiltonian (14) has been discussed in detail before, see, e.g., Refs. [13,18–21]. For arbitrary g, the exact
¯
solution of the transport problem has been given in Ref.
[19]. This solution exploits the integrability of Eq. (14)
and employs the thermodynamic Bethe ansatz. Simpler
exact solutions are possible by means of refermionization
techniques for g = 1 [see Eq. (12)] and g = 1/2. The case
¯
¯
g = 1/2 thus corresponds to an uncorrelated situation in
the new basis (13), and g = 1/4 is the Toulouse point
[20]. Progress can also be made by expanding in |ǫ| ≪ 1
for g = 1 − ǫ [18] or g = 1/2 − ǫ [21].
¯
¯
Employing the exact results of, e.g., Ref. [19], at zero
temperature we ﬁnd the asymptotic low-voltage behavior

(11)

For weak interactions, 1/2 < g < 1, the electrostatic
coupling λ(ℓ) also ﬂows to zero as ℓ → ∞. In that case,
at low energy scales, crossed Luttinger liquids are basically insensitive to the coupling considered here. At
asymptotically low energy scales, the currents are then
Ii = (e2 /h) Ui . The ﬁnite-temperature or low-voltage
corrections due to the irrelevant operators Vi can be computed by perturbation theory in the respective coupling
strengths λi . Since the ﬂuctuating potential scattering is
irrelevant for 1/2 < g < 1, the corrections due to V2,3 are
governed by the standard exponent α = (g + g −1 − 2)/4
for tunneling into a bulk LL [2,3]. This is in contrast to
a static potential scatterer, where tunneling into the end
of a LL matters at low energy scales [13].
Directly at g = 1/2, the operator V1 is marginal, and
straightforward refermionization yields
1
e2
Ui .
Ii =
h 1 + (λ/2πa)2

Ii =

dx (∂x ϕ± )2 + (∂x ϑ± )2
±

λ
cos
2(πa)2

(12)

λB = (cg /a) (λ/a)1/[1−2g] ,

(17)

where cg is a numerical constant of order unity [19]. The
result (16) holds under the condition
e|U1 ± U2 | ≪ λB .

(18)

If both voltages approach zero, the linear conductance vanishes in both 1D conductors. We thus ﬁnd
a pronounced zero-bias anomaly, with characteristic
interaction-dependent power laws for small voltages.
Let us now discuss the full current-voltage characteristics. A particularly simple solution emerges at the
Toulouse point g = 1/4 by refermionization [20] of
Eq. (14) under the boundary conditions (15). At zero
temperature, the result is

(14)

¯
I± = (e2 /h) [U± − V± ] ,

8πg ϑ± (0) ,

(19)

where V± is the four-terminal voltage [10] subject to the
self-consistency equation

and the boundary conditions (5) determining the density
ρp,r of p = ± movers injected into channel r = ± take
¯
the form
p e(U1 + rU2 )
ρp,r (x → −p∞) = √
¯
.
4πg
2

(16)

where the ± sign corresponds to i = 1, 2, respectively.
The energy scale λB generated by the bare electrostatic
coupling λ is given by

which again obey the algebra (1). Remarkably, the
Hamiltonian H0 + V1 decouples into the sum H+ + H−
with
1
2

sgn(U1 + U2 ) [e|U1 + U2 |/λB ]1/g−1

± sgn(U1 − U2 ) [e|U1 − U2 |/λB ]1/g−1 ,

Each conductor exhibits a response only to the voltage
applied to itself, with the conductance now explicitly depending on the electrostatic coupling strength λ.
For suﬃciently strong interactions, g < 1/2, the electrostatic coupling λ(ℓ) ﬂows to strong coupling. To proceed, we switch to the linear combinations
√
ϑ± (x) = {θ1 (x) ± θ2 (x)}/ 2 ,
(13)
√
ϕ± (x) = {φ1 (x) ± φ2 (x)}/ 2 ,

H± =

e2 λB
h e

eV± = 2λB tan−1 {[2eU± − (3/2)eV± ]/λB } ,

(15)

(20)

where λB = λ2 /4(πa)3 in accordance with Eq. (17). Under the condition (18), the exact result (19) reproduces
Eq. (16) again. In the absence of a coupling, λB = 0, one
ﬁnds the correct unperturbed currents Ii = (e2 /h)Ui .

Therefore we are left with two completely decoupled
systems r = ±, each of which is formally identical to
3

for useful discussions. This work was supported by the
Deutsche Forschungsgemeinschaft (Bonn).

1.0
0.8
0.6
0.4

[1] S. Iijima, Nature 354, 56 (1991); A. Thess et al., Science
273, 483 (1996).
[2] J. Voit, Rep. Prog. Phys. 57, 977 (1995).
[3] A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik,
Bosonization and Strongly Correlated Systems (Cambridge University Press, 1998).
[4] See, e.g., M. Fabrizio, Phys. Rev. B 48, 15 838 (1993).
[5] R. Egger and A.O. Gogolin, Phys. Rev. Lett. 79, 5082
(1997).
[6] S.J. Tans, M.H. Devoret, H. Dai, A. Thess, R.E. Smalley,
L.J. Geerligs, and C. Dekker, Nature 386, 474 (1997).
[7] A.O. Gogolin, Ann. Phys. (Paris) 19, 411 (1994).
[8] Tunneling from three chiral Luttinger liquids onto a dot
is treated by C. Nayak, M.P.A. Fisher, A.W.W. Ludwig,
and H.-H. Lin, preprint cond-mat/9710305.
[9] The same eﬀects as discussed here for spinless electrons
1
are also found for spin- 2 electrons or in carbon nanotubes. The minor modiﬁcations required in these two
cases and the extension to diﬀerent g parameters for both
conductors will be presented elsewhere.
[10] R. Egger and H. Grabert, Phys. Rev. Lett. 77, 538
(1996).
[11] S. Datta, Electronic Transport in Mesoscopic Systems,
Chapter 2 (Cambridge University Press, 1995).
[12] If the conductors are brought into mechanical contact,
local deformations might cause impurity scattering in one
or both conductors. For simplicity, we assume that no
such impurity scattering processes are present.
[13] C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46, 15 233
(1992).
[14] F.V. Kusmartsev, A. Luther, and A.A. Nersesyan, JETP
Lett. 55, 724 (1992).
[15] For a bulk coupling such as in Refs. [4,14], a perturbation
is relevant already for η < 2.
[16] If the operators V2 and V3 are taken in a strictly local sense, the concept of non-zero conformal spin does
not apply and both operators are simply irrelevant. In
practice, however, they might act over some ﬁnite (but
small) distance around x = 0. Thereby one generates the
second-order contribution in Eq. (10).
[17] V.M. Yakovenko, JETP Lett. 56, 510 (1992).
[18] D. Yue, L.I. Glazman, and K.A. Matveev, Phys. Rev. B
49, 1966 (1994).
[19] P. Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev.
B 52, 8934 (1995).
[20] F. Guinea, Phys. Rev. B 32, 7518 (1985); C. de C. Chamon, D.E. Freed, and X. Wen, ibid. 53, 4033 (1996).
[21] U. Weiss, R. Egger, and M. Sassetti, Phys. Rev. B 52,
16 707 (1995); U. Weiss, Solid State Comm. 100, 281
(1996).

0.2
0

-40

-20

0

20

40

FIG. 2. Transport current I1 normalized to the unper0
turbed value I1 = (e2 /h) U1 for g = 1/4, λB /e = 1, T = 0,
and several values of the cross voltage U2 .

The transport current I1 is plotted as a function of
U1 in Fig. 2. Contrary to what is found in the uncorrelated system [11], the current I1 is extremely sensitive
to the applied cross voltage U2 . For U2 = 0, transport
becomes fully suppressed for U1 → 0, with g-dependent
nonlinear low-voltage corrections given by Eq. (16). Increasing U2 for some ﬁxed U1 then leads to an increase
in the current I1 . Eventually, the linear conductance behavior is restored for very large cross voltage. In fact, for
e|U1 ± U2 | ≫ λB , one always recovers the unperturbed
currents Ii = (e2 /h) Ui . The generic correlation eﬀects
are most important under the conditions (18).
Remarkably, there is a suppression of the current if
U1 = ±U2 , which is observed as a “dip” in the normalized current displayed in Fig. 2. This eﬀect can be
rationalized in terms of a partial dynamical pinning of
charge density waves in conductor 1 due to commensurate charge density waves in conductor 2. As can be
checked from Eq. (19), while the nonlinear conductance
G11 ≡ ∂I1 /∂U1 stays positive, one can have a negative value for G12 ≡ ∂I1 /∂U2 . In fact, the latter (oﬀdiagonal) conductance is within the bounds −e2 /2h ≤
G12 ≤ e2 /2h [we note that G12 = 0 for g ≥ 1/2], while
the diagonal conductance fulﬁlls 0 ≤ G11 ≤ e2 /h. The
pronounced and nonlinear sensitivity of the current I1
to the applied cross voltage U2 is a distinct ﬁngerprint
for Luttinger liquid behavior. Parenthetically, anomalous power laws can also be found in the temperature
dependence of the current.
To conclude, we have examined nonlinear transport
through two Luttinger liquids coupled together at one
point. The only relevant coupling is of electrostatic origin and leads to distinct correlation eﬀects for strong
Coulomb interactions, g < 1/2. The theoretical ﬁndings reported here could be of use for the experimental identiﬁcation of non-Fermi-liquid behavior in carbon
nanotubes and other one-dimensional materials.
We thank A.O. Gogolin, H. Grabert, and C.A. Staﬀord
4