Coulomb drag of Luttinger liquids and quantum-Hall edges
Karsten Flensberg

arXiv:cond-mat/9802220v2 [cond-mat.mes-hall] 2 Jun 1998

Danish Institute of Fundamental Metrology,
Building 307, Anker Engelunds Vej 1, DK-2800 Lyngby, Denmark
(February 19, 1998)
We study the transconductance for two coupled one-dimensional wires or edge states described
by Luttinger liquid models. The wires are assumed to interact over a ﬁnite segment. We ﬁnd for
the interaction parameter g = 1/2 that the drag rate is ﬁnite at zero temperature, which cannot
occur in a Fermi-liquid system. The zero temperature drag is, however, cut oﬀ at low temperature
due to the ﬁnite length of the wires. We also consider edge states in the fractional quantum Hall
regime, and we suggest that the low temperature enhancement of the drag eﬀect might be seen in
the fractional quantum Hall regime.

in the linear and the non-linear regimes in Ref. [13]. In
Ref. [14] the non-linear transresistance for two LLs at
zero temperature was studied and it was argued that absolute drag (equal currents in the two wires) is possible.
Here we show a related eﬀect for the linear conductance.

One-dimensional (1D) systems have attracted much attention since the advances in lithographical fabrication
techniques have made it possible to study systems such
as e.g. quantum point contacts, quantum wires, quantum dots, and nano-tubes. Interacting 1D systems are
particularly interesting, because they are believed to be
Luttinger liquids (LLs) and thus exhibit non-Fermi liquid behavior. However, it is still not clear to what extent
the non-Fermi liquid behavior can be seen in a transport experiment for clean systems, i.e. without impurity
scattering. Several authors [1] have shown that the interactions do not inﬂuence the conductance, and the reason
for a possible interaction induced eﬀect observed [2] at
ﬁnite temperature remains unexplained.

We concentrate on the temperature dependence of the
linear response. Utilizing a mapping to two decoupled
LLs, it is shown that a LL description shows zero temperature drag for the case where the coupling parameter
g = 1/2 [15]. Moreover, we ﬁnd an interesting dependence of the length of the interaction region and a regime
for g < 1/2, where the drag current is almost equal to the
drive current. Our results are applied to edge states in
coupled FQH systems with a narrow contriction, which
is diﬀerent from the situation of the Ref. [11].

Another very interesting testing ground for LL behavior is edge states in the fractional quantum Hall (FQH)
regime, where edge states can be described as chiral LLs
[3]. Surprisingly, experiments have shown that the tunneling between two edge states follows the LL behavior
[4] both for compressible and incompressible states. Theoretical descriptions in terms of 1D edge channels have
been developed [5], and several authors [6,7] have recently
extended this idea and calculated tunneling density of
states which oﬀers an explanation of the observed characteristics.

LI

LW

FIG. 1.
Schematic outline of the geometry considered
here. The two wires are of length LW , but they interact only
in a region of length LI .

In this paper, another experiment which measures
the interaction eﬀects is suggested, namely a Coulomb
drag experiment. Coulomb drag has during recent years
proved to be a powerful tool for studying interaction and
screening properties of coupled two-dimensional electron
systems [8], including phonon-mediated interactions [9],
coupled QH systems both in the integer regime [10] and
in the fractional regime where recent experiments show
Coulomb drag that saturates at the lowest temperatures
at ﬁlling factor close to one half [11]. This type of behavior cannot be explained within the present weak coupling
theories for bulk composite-Fermion drag [12].

In our model the two spin-less LLs are coupled by
Coulomb interactions, and interwire tunneling is neglected. They are coupled in a ﬁnite region of length
LI . The wires have lengths LW and it is assumed that
the intrawire interaction is constant in this region, see
Fig. 1. Thus the results are valid for temperatures or
voltages larger than the cut-oﬀ given by the energy of
charge excitations with wavelengths of order LW , i.e.,
T > TW = hvF /LW , where vF is a Fermi velocity.
¯
The Hamiltonians for the separate systems are those
of two LLs (i = 1, 2) (we use h = kB = 1)
¯

1D drag between two Fermi liquids has been studied
1

H0i =

vi
2

dx

[Pi (x)]2 +

1
2
,
2 [∂x φi (x)]
gi

can be neglected. The ﬁnal expression for the interaction
now reads

(1)

where gi , vi are the interaction parameters and Fermi
velocities, respectively, and where the densities, ρL
and ρR , of left and right movers respectively, enter as
√
∂x φi (x) = [ρLi (x) + ρRi (x)] π and P (x) = −[ρRi (x) −
√
ρLi (x)] π. The ﬁelds φi and Pi form conjugate variables:
[φi (x), Pj (x′ )] = iδij δ(x − x′ ).
The Coulomb interaction between the two wires is
given by
Hint =

dxdy U12 (x, y)ρ1 (x)ρ2 (y).

Hint =

1
dxdy U12 (x, y) ×
2π 2 α2
√
cos[2(kF 1 x − kF 2 y) + 2 π(φ1 (0) − φ2 (0))]
√
+ cos[2(kF 1 x + kF 2 y) + 2 π(φ1 (0) + φ2 (0))] . (6)

Now we transform the Hamiltonian to that of two (interacting) LLs scattering against a single impurity potential. Deﬁne the new ﬁeld operators
Φ = φ1 + φ2 ,

Note that the interwire interaction U12 (x, y) only acts
within a region of length LI . Here the subsystem densities are given by

P = (P1 + P2 )/2,

(7a)

Θ = φ1 − φ2 ,

(2)

Π = (P1 − P2 )/2,

(7b)

deﬁned such that Φ, P and Θ, Π are conjugate pairs. The
interwire interaction term becomes particularly simple in
this basis and the Hamiltonian transforms to

ρi (x) = ρLi (x) + ρRi (x) + Ψ† (x)ΨLi (x) + Ψ† (x)ΨRi (x),
Li
Ri

H = H0 + H ′ + Hint ,

(3)

where

where ΨL(R)i are Fermion operators corresponding to the
left(right) movers. In the bosonization language these
are given by [16] (one set for each wire)ΨL(R) (x) =
√
x
exp ±ikF x ∓ 2π −∞ dyρL(R) (y) / 2πα, α is a cut-oﬀ
parameter [17].
We separate the interwire interaction in 4 terms describing the diﬀerent possibilities of forward and/or backward scattering. Note that since the interaction region is
ﬁnite, momentum need not be conserved in the scattering
process. We have
Hint =

H0 =

H′ = v
¯

(5a)
(5b)
(5c)

√
B d (x, y) = φ1 (x) A2 (y) + A† (y) / π + 1 ↔ 2.
2

1
[∂x Φ(x)]2
g2
¯

1
[∂x Θ(x)]2 ,
g2
¯

(9)

dx aP (x)Π(x) +

b
∂x Φ(x)∂x Θ(x) ,
g2
¯

(10)

2
2
where v = (v1 + v2 ), 1/¯2 = (v1 /g1 + v2 /g2 )/4¯, a =
¯
g
v
2
2
2
2
(v1 − v2 )/¯, and b = (v1 /g1 − v2 /g2 )/(v1 /g1 + v2 /g2 ).
v
The current operator for the LL model is given by
√
ji = vF Pi / π and through the continuity equation, the
√
current is expressed as jj (x) = −∂t φj (x, t)/ π. Using
the Kubo formula, we obtain the transconductance in
terms of the new ﬁelds deﬁned in Eq. (7) as

where

B (x, y) = A1 (x)A2 (y) + h.c.,
B c (x, y) = [∂x φ1 (x)∂y φ2 (y)] /π,

dx [P (x)]2 +

and where H ′ describes the interaction between the new
ﬁeld operators

B a (x, y) + B b (x, y) + B c (x, y) + B d (x, y) , (4)

b

v
¯
2

+ [Π(x)]2 +

dxdy U12 (x, y) ×

B a (x, y) = A1 (x)A† (y) + h.c.,
2

(8)

(5d)

G21 (ω) =

iωe2 r
r
[DΦ (x, x′ ; ω) − DΘ (x, x′ ; ω)] ,
4π

(11)

where the Green’s functions
DΦ (t − t′ ) = −iΘ(t − t′ ) [Φ(x, t′ ), Φ(x′ , t)] ,

Here A(x) is a ”backscattering operator” deﬁned as
√
A(x) = exp (i2kF x + i2 πφ(x)) /(2πα).
The two terms B b and B d correspond to nonmomentum-conserving scatterings processes. The terms
B c and B d do not provide a mechanism for Coulomb drag
(to any order in perturbation theory) and, furthermore,
since the renormalization due to these terms is not important at low energies, they are omitted. We will make
one further approximation which is valid when the relevant energy scale is smaller than vF /LI . In this limit, the
spatial dependence of φ’s in the backscattering operator

DΘ (t − t′ ) = −iΘ(t − t′ ) [Θ(x, t), Θ(x′ , t′ )] ,

(12a)
(12b)

have been deﬁned (note that D(x, x′ ; ω) is independent
of x, x′ in the dc-limit).
For identical wires the part of the Hamiltonian which
couples the Φ and Θ sector in Eq. (10) is equal to zero.
The remaining Hamiltonian is equivalent to two LLs
models scattering against single impurities, but with new
interaction parameters. We can therefore use the results
from this well-studied problem [18,19]. The transconductance simpliﬁes to
2

G21 =

1
(GLutt (V1 , 2g) − GLutt (V2 , 2g)) ,
4

For a long interaction region, i.e. W1 ≪ 1, G21 approaches the value e2 /2h for small temperatures (but
still larger than T ∗ = D|W1 |−2/γ ), see inset of Fig. 2.
In this regime, the diagonal conductance [21] also tends
to e2 /2h and thus the currents in two wires become the
same, which is similar to the absolute drag eﬀects found
in Ref. [14]. This interesting eﬀect occurs because the
momentum conserving backscattering increases with decreasing temperature (second term of Eq. (13) goes to
zero). It saturates when the two currents are the same
and the net momentum exchange hence is zero.

(13)

where g ≡ g1 = g2 and where GLutt (V, 2g) [20] is the conductance of a LL with interaction parameter, 2g, scattering against a single impurity with a backscattering amplitude V = V (2kF ). (Below it is shown that Eq. (13)
is valid even when the velocities are diﬀerent.) Here we
have deﬁned
V1,2 =

D
2πvF

dxdy U12 (x, y) exp [2ikF (x ± y)] , (14)

where the small momentum cut-oﬀ is parametrized in
terms of a high energy cut-oﬀ: D = vF /α.
Now several conclusions follow. Firstly, it is seen that
for a short interaction region LI kF ≪ 1, which implies
V1 ≈ V2 , all momentum transfers have equal weights
and hence there is no Coulomb drag [21]. More importantly, the scaling properties of the model can be read
out [18,19]:
For g > 1/2, GLutt goes to a constant as T → 0, which
means that G21 goes to zero at zero temperature. The
corrections to the low temperature limit give the power
law: G21 ∼ T 4g−2 . For non-interacting wires the drag effect is thus quadratic in temperature; in contrast to the
case where they interact throughout the wires, in which
case the drag scales linearly with temperature [13].
For g < 1/2, both terms in Eq. (13) go to zero, because the model scales toward strong backscattering. The
power laws are however the same but the prefactors will
be diﬀerent and we can conclude that G21 ∼ T 1/g−2 .
For g = 1/2 the problem maps to Fermi-liquids and
we get a temperature independent G21 and hence only
for g = 1/2 does the transconductance remain ﬁnite as
temperature goes to zero.
Consider again the case of identical wires in the vicinity of g = 1/2, where the problem maps to that of two
Fermi-liquids. We may use the perturbative renormalization method developed in Ref. [22] rather than the exact
solution based on the Bethe ansatz [19], and we obtain
for the transconductance [20]
G21

(|W2 |2 − |W1 |2 )tγ
e2
,
=
4π [1 + |W1 |2 tγ ][1 + |W2 |2 tγ ]

FIG. 2.
Transconductance for two identical wires using the expression derived for g close to one half. The full
(dashed) curves are for T /D = 0.1 (0.001). For a short interaction region (left panel, where kF LI = 1) the transconductance peaks near g = 1/2. The peak moves toward g = 1/2 for
smaller temperature. For long interaction region (right panel,
where kF LI = 20), the peak is not developed and instead G21
approaches e2 /2h at low temperature for g < 1/2, which is
shown in the inset for g = 0.4. This regime corresponds to the
limit where only momentum conserving scattering is possible
and where the currents in two wires become almost the same.

In the general case, where the wires have diﬀerent gvalues and velocities, there is a coupling term in the
transformed Hamiltonian, Eq. (10). Since the conductance involves only the ﬁelds at x = 0, we integrate out
all the x = 0 ﬁelds. The resulting action reads
S = S0 + Sint ,
1
|ωn |
∗
S0 =
(Φω Θω )
β ω
g
¯

(15)

n

where t = (T /D), Wi = Vi /vF , and γ = 4g − 2.
In Fig. 2, we show G21 expressed in Eq. (15) for different parameters. The interwire interaction U12 is calculated for typical parameters for GaAs quantum wires
and the distance between the wires is chosen to be
2/kF . In accordance with the arguments given above, the
transconductance peaks when g = 1/2 at low temperatures and the peak moves to smaller g values for higher
temperatures. (Furthermore, the stronger the interwire
interaction the closer is the peak position to g = 1/2.)
Therefore for g < 1/2, G21 shows non-monotonic behavior as a function of temperature.

β

Sint =

√
πΘ(τ )

dτ V2 e2i

(16a)
k+ k−
k− k+
√

+ V1 e2i

Φω
Θω
πΦ(τ )

,

(16b)

+ c.c. , (16c)

0
1/2

1
2
2
.
where k± = 2 (v2 + v1 )(g1 ± g2 )2 /(g1 v2 + g2 v1 )
For the special case, when the two wires have the same
interaction parameter (but unequal velocities) the two
sectors separate and the solution is again given by Eq.
(13). For the general situation, we perform a perturbative renormalization group calculation and ﬁnd

dVi
= (1 − (g1 + g2 ))Vi .
d ln D
3

(17)

The ﬂow is marginal when the sum of the g-constants
is equal to one and the zero temperature drag predicted
above occurs at this line and for g1 + g2 < 1 a strong
enhancement of the Coulomb drag occurs.
Finally, we consider Coulomb drag of edge excitations
in the FQH regime. Recently, there has been a large
activity trying to understand the low energy edge excitations and the tunneling experiments [4] in FQH states
[6,7]. These works show that the edge excitations can
be described as unidirectional bosonic edge excitations,
which were then used to form an electron charge creation
operator as input in an independent boson model for the
tunneling density of states.

[1] I. Saﬁ and H. J. Schulz, Phys. Rev. B 52, R17040 (1996)
D. I. Maslov and M. Stone, Phys. Rev. B 52, R5539
(1995); V. Ponomarenko, Phys. Rev. B 52, R8666 (1995);
A. Kawabata, Phys. Soc. Jap. 65, 30 (1996); Y. Oreg and
A. Finkel’stein, Phys. Rev. B 54, R14265 (1996). I. Saﬁ,
Phys. Rev. B 55 R7331 (1997).
[2] S. Tarucha, T. Honda, and T. Saku, Solid State Comm.,
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135 (1996); A. Yacoby et al., Phys. Rev. Lett. 77, 4612
(1996).
[3] X. Wen, Int. J. Mod. Phys. B 6, 1711 (1992).
[4] M. Grayson et al., Phys. Rev. Lett. 80, 1062 (1998), and
references therein.
[5] I. Aleiner and L. Glazman, Phys. Rev. Lett. 72, 2935
(1995).
[6] S. Conti and G. Vignale, Phys. Rev. B 54, R14309 (1996),
cond-mat/9709055, cond-mat/9801318; J. Han and D.
Thouless, Phys. Rev. B 55, R1926 (1997); J. Han, Phys.
Rev. B 56, 15806 (1997); U. Z¨licke and A. H. MacDonu
ald, cond-mat/9802019.
[7] D.V. Khveshchenko, cond-mat/9710137.
[8] K. Flensberg and B. Y.-K. Hu, Phys. Rev. Lett. 73, 3572
(1994); N. P. R. Hill et al., Phys. Rev. Lett. 78, 2204
(1997).
[9] T. J. Gramila et al., Phys. Rev. Lett. 66, 1216 (1991),
Phys. Rev. B 47, 12957 (1993), Physica B 197, 442
(1994); M.C. Bønsager et al., Phys. Rev. B, in press.
[10] N.P.R. Hill et al., J. Phys.: Condens. Matter 5, 5009
(1993); H. Rubel et al., Phys. Rev. Lett. 78, 1763 (1997);
E. Shimshoni and S.L. Sondhi, Phys. Rev. B 49, 11484
(1994); M.C. Bønsager et al., Phys. Rev. Lett. 77, 1366
(1996), Phys. Rev. B 56, 10314 (1997).
[11] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiﬀer and K.W. West,
Phys. Rev. Lett. 80, 1714 (1998).
[12] S. Sakhi, Phys. Rev. B 56, 4098 (1997); I. Ussishkin and
A. Stern, Phys. Rev. B 56, 4013 (1997); Y.B. Kim and
A.J. Millis, cond-mat/9611125.
[13] B. Y.-K. Hu and K. Flensberg, in Hot Carriers in Semicondoctors, edited by K. Hess (Plenum Press, New York,
1996), p. 943.
[14] Y. V. Nazarov and D. V. Averin, cond-mat/9705158.
These authors consider the ﬁrst term of Eq. (6) valid
for long wires.
[15] Note that since our eﬀect evolves a transport experiment
the zero temperature drag considered here is diﬀerent
from that of a closed system in studied in A. G. Rojo
and G. D. Mahan, Phys. Rev. Lett. 68, 2074 (1992).
[16] J. S´lyom, Advances in Physics 28, 201 (1970).
o
[17] The Fermion operator should include a phase factor
which ensures that the electron operators corresponding
to diﬀerent branches anticommute. However, they are not
important for the present problem.
[18] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68,
1220 (1992); Phys. Rev. B 46, 15233 (1992).
[19] P. Fendley, A. Ludwig, and H. Saleur, Phys. Rev. B 52,
8934 (1995).

Following these works, we write the Hamiltonian that
governs the dynamics for (say) the left edge branch as
Hedge,L = 2π vD k>0 ρL (k)ρL (−k), where ρL is the
L
left branch 1D charge operator and [ρL (−k), ρL (k)] =
νkL/2π. Here ν is the ﬁlling factor, vD is the drift velocity at the edge and L a normalization length. Similar expression is obtained for the right branch, similar to the Tomonaga-Luttinger model. The operator which creates a single charge e at point x, moving with velocity vD , in the left channel is Ψ† ∼
L
2π
e−ikD x exp νL k ρL (k)eikx /k . Notice that this operator does not fulﬁll the correct anticommutation relations for fermions. However, the backscattering operator
A = Ψ† ΨL does obey the correct commutation relations
R
and below it is utilized to describe interedge tunnelings
in a model Hamiltonian for coupled FQH systems.
The following experiment is proposed: the coupled
QH systems are narrowed by a constriction such that
edge states moving in opposite directions are coupled
and backscattering can occur. This system can be modeled by writing the backscattering interaction in terms of
the backscattering operator, A. After some simple algebra, we arrive at the Hamiltonian for two coupled LLs,
Eqs. (1) and (6) , with g = ν [23] and conclusions similar to above therefore immediately follow from Eq. (13):
In particular, we predict non-zero drag at zero temperature for half-ﬁlled Landau levels and furthermore near
its maxium value the drag eﬀect increases with decreasing
temperature (or voltage). For long interaction region, the
transconductance goes to a universal value νe2 /2h at low
temperature for ν < 1/2.
In conclusion, we have calculated Coulomb drag for
LLs and found interesting behavior near g=1/2 such as
zero temperature drag and non-monotonic temperature
dependence. These predictions can be tested for edge
states in FQH systems.
The author acknowledges valuable discussions with
Ben Hu and Antti-Pekka Jauho.
Recently, a related paper appeared [24]. These authors
discuss the conductance of crossed LLs coupled at a single point and ﬁnd very similar results to ours.
4

[20] Be aware that in absence of scattering GLutt (2g) = 2e2 /h,
because in this case g should be set to one[1]. This is not
the case for the FQH edge states.
[21] The diagonal conductance is G11 given by Eq. (13) with
a plus instead of the minus and G11 of course remains
ﬁnite for V1 = V2 .
[22] D. Yue, L. Glazman, and K. Matveev, Phys. Rev. B 49,
1966 (1994).

[23] These result will change when including interedge interactions, which lead to small logarithmic corrections. See
Ref. [7] and K. Moon and S. M. Girvin, Phys. Rev. B 54,
4448 (1996); U. Z¨licke and A. H. MacDonald, ibid 54,
u
R8349 (1996).
[24] A. Komnik and R. Egger, Phys. Rev. Lett. 80, 2881
(1998).

5