Tunneling into a Luttinger liquid revisited
D. N. Aristov1,2 , A. P. Dmitriev3,2 , I. V. Gornyi2,3 , V. Yu. Kachorovskii3,2, D. G. Polyakov2, and P. W¨lﬂe4,2
o
1
Petersburg Nuclear Physics Institute, 188300 St.Petersburg, Russia
Institut f¨r Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
u
3
A.F.Ioﬀe Physico-Technical Institute, 194021 St.Petersburg, Russia
4
Institut f¨r Theorie der kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
u
(Dated: February 21, 2018)

arXiv:1006.4844v2 [cond-mat.str-el] 24 Dec 2010

2

We study how electron-electron interactions renormalize tunneling into a Luttinger liquid beyond
the lowest order of perturbation in the tunneling amplitude. We ﬁnd that the conventional ﬁxed
point has a ﬁnite basin of attraction only in the point contact model, but a ﬁnite size of the contact
makes it generically unstable to the tunneling-induced breakup of the liquid into two independent
parts. In the course of renormalization to the nonperturbative-in-tunneling ﬁxed point, the tunneling
conductance may show a nonmonotonic behavior with temperature or bias voltage.
PACS numbers: 71.10.Pm, 73.21.Hb

Electron tunneling into a correlated many-electron system is one of the most essential tools to probe the nature of the correlations. A key concept here is that the
tunneling density of states reﬂects how diﬃcult it is for
electronic states to rearrange themselves to accommodate
the extra charge of the tunneling electron. The hallmark
of strong correlations is the “zero-bias anomaly” (ZBA)
[1]—the nonlinear behavior of the tunneling current as a
function of the bias voltage—resulting from a singularity
in the tunneling density of states at the Fermi energy.
The prototype model and the best-understood example
of a strongly correlated metallic state is the Luttinger liquid (LL) in one-dimensional electron systems [2]. The LL
behavior has been observed in nanowires—of which the
most prominent examples are the carbon nanotubes and
the semiconductor quantum wires—through the powerlaw suppression [3, 4] of the tunneling conductance with
decreasing bias voltage and/or temperature.
When using the term “tunneling”, we often have at the
back of our minds that the tunneling probes the properties of the system into which the electron tunnels “noninvasively”, i.e., the tunneling amplitude is inﬁnitesimally
small. A more subtle and complete understanding of the
interplay of strong correlations and tunneling emerges
when the latter is treated beyond the lowest order of perturbation theory. It is the purpose of this paper to study
the junction between a LL and a tunnel electrode (Fig. 1)
for arbitrary strength of tunneling. Apart from the conceptual interest, the “three-way junction” is considered
a key element for device engineering in nanoelectronics.
Our main result is the phase diagram for the
interaction-induced renormalization of the parameters of
the tunnel junction. We show that a homogeneous LL is
actually generically unstable at low energies to arbitrarily weak tunneling and the breakup into two independent
semi-inﬁnite wires. The ZBA at the true (stable) ﬁxed
point (FP) is strongly enhanced. For suﬃciently weak interaction or suﬃciently strong tunnel coupling, the tunnel conductance actually grows with decreasing energy

x
y

tout

r

tb
t
tin
FIG. 1: Junction between a quantum wire (horizontal) and a
tunnel electrode (vertical). The arrows denote the scattering
processes (and their amplitudes) for incoming electrons.

scale (temperature, bias voltage) before it reaches maximum and starts to renormalize toward zero.
Let us specify the model. The Hamiltonian reads
= Hw + He + Htun , where ( = 1) Hw =
1
†
describes the LL
µ dx −iµvψµ ∂x ψµ + 2 V0 nµ n−µ
wire (for compactness, we focus here on the spinless case),
µ = ± denotes electrons moving to the right (+) and to
the left (−). The electron operator ψ, deﬁned in the complete basis of scattering states of the three-way junction,
is decomposed inside the wire as a sum of chiral components: ψ(x) = µ ψµ (x), and nµ (x) is the density ﬂuctuation in the channel µ. We assume that the interaction
potential between electrons in the wire is screened by a
nearby metallic gate and take it to be point-like with the
zero-momentum component V0 . The interaction between
electrons with the same µ is then fully incorporated in the
renormalization of the velocity v [2]. We focus in this paper on the case of small α = V0 /2πv. We assume that the
interaction is present in a ﬁnite region of length L around
the tunnel contact, which models a LL wire connected
to noninteracting leads. For simplicity, we represent here
the tunnel electrode as a noninteracting semi-inﬁnite wire
0
†
described by He = −ive µ µ −∞ dy ψµ ∂y ψµ , where ψµ
are the chiral components of the electron operator ψ(y)
on the half-axis of y. In the absence of tunneling, the
scattering states at y < 0 are characterized by the phase
H

2
(a)

c

1

FIG. 2: Renormalization group ﬂows (schematic) for the point
tunnel junction. Two stable FPs at ρ = 0 and ρ = 1 describe
the homogeneous wire and the breakup of the wire into two
independent semi-inﬁnite pieces, respectively. Tunneling into
the wire is blocked at both points.

of the reﬂection coeﬃcient r = eiφr .
The term Htun describes the tunnel junction. We start
by considering the simplest and commonly used model
(“point contact”) for the tunneling Hamiltonian: Htun =
t0 ψ † (y = −0)ψ(x = 0) + H.c., take t0 to be real and
put the phase φr = 0. In the absence of interaction,
the scattering amplitudes (Fig. 1) can be shown to obey
tb = −ρ, tin = 1 − ρ, tout = t = −i sgn(t0 )[2ρ(1 − ρ)]1/2 ,
r = 1 − 2ρ, where ρ = 2|t0 |2 /(vve + 2|t0 |2 ). We see
that the virtual transitions from the wire into the tunnel
contact and back lead to backscattering in the wire.
As is commonly known [5], a weak backscattering amplitude in a LL is renormalized by interaction and behaves as (Λ/|ǫ|)1−K , where the energy ǫ is counted from
the Fermi level, Λ is the ultraviolet cutoﬀ, and the Luttinger parameter K = (1 − α)1/2 (1 + α)−1/2 ≃ 1 − α
for small α. At ﬁrst glance, one might expect that the
amplitude tb is renormalized similarly. However, the fact
that unitarity in the tunnel junction is imposed on the
3 × 3 (and not 2 × 2) scattering matrix has dramatic consequences for the renormalization. We ﬁnd (see below)
that, to second order in α, the exact-in-ρ beta-function
β(ρ) = ∂ρ/∂L, where L = ln(Λ/|ǫ|) for |ǫ| ≫ v/L, for
the point tunnel contact reads:
β(ρ) = αρ(1 − ρ)[ ρ − α(1 − 2ρ)(1 − ρ + ρ2 )/2 ] .

(1)

For ρ ≪ 1 it reduces to
β(ρ) ≃ −α2 ρ/2 + αρ2 ,

(b)

ρ≪1.

(2)

It is most important that β(ρ) in Eq. (2) does not contain
the term αρ linear in both α and ρ, which would give
the renormalization of tb similar to that for the weak
impurity in a LL wire. To better understand this, recall
that the renormalization can be described [6] in terms
of scattering oﬀ the Friedel oscillations produced by the
scatterer. A subtlety of the point tunnel junction is that
both tb and tin —in contrast to the impurity—are real
and, as a result, the contributions to β(ρ) at order O(αρ)
from the Friedel oscillations that “dress” the junction at
x < 0 and x > 0 exactly cancel each other.
The ﬁrst term in Eq. (2) means the usual suppression
2
[2] of the tunneling density of states ∝ (|ǫ|/Λ)(1−K) /2K
in a homogeneous LL. The second term comes from the

0

x

0

x

FIG. 3: Lowest-order scattering processes leading to the
renormalization of the reﬂection amplitude r. Dots: tunneling amplitude t0 . Wavy lines: bare interaction α between the
right and left movers. The straight single and double lines denote the electron Green function in the wire and in the tunnel
electrode, respectively.

renormalization of backscattering along the wire at ﬁrst
order in α, as found earlier in Ref. [7]. What seems to
have not been discussed in the literature is that the two
terms have opposite signs (for α > 0), so that β(ρ) for
small α vanishes at ρ = ρc ≃ α/2, which signiﬁes an
unstable FP (Fig. 2) leading to the phase transition separating two phases with ρ = 0 and ρ = 1 [8]. For ρ, α ≪ 1
the solution of Eq. (2) gives
ρ≃

ρ0
,
2ρ0 /α + (1 − 2ρ0 /α)(Λ/|ǫ|)α2 /2

(3)

where ρ0 is the bare value of ρ.
We see that the tunneling is suppressed by interaction
only if its bare amplitude is suﬃciently small (or, equivalently, if the interaction is suﬃciently strong). Otherwise,
according to Eq. (1), the tunneling constant t0 monotonically increases in the course of renormalization. The
tunneling transparency Gt = 2|t|2 , however, shows a nonmonotonic behavior with increasing t0 :
Gt = 4ρ(1 − ρ) = 8vve |t0 |2 /(vve + 2|t0 |2 )2 .

(4)

If ρc < 1/2, Gt grows until it reaches maximum Gt =
1 (which means a completely transparent contact with
r = 0) at ρ = 1/2 and decreases at larger ρ, eventually
vanishing at ρ = 1. By contrast, Gt for ρ < ρc decreases
monotonically in the course of renormalization.
It is instructive to analyze the scattering processes diagrammatically in the energy-space representation; in particular, this makes it easy to count powers of t0 and α.
The ﬁrst and second terms in Eq. (2) correspond, respectively, to diagrams (b) and (a) in Fig. 3 for the amplitude
r. In Fig. 3(a), tunneling with the amplitude t0 into the
right-moving state is followed by scattering at ﬁrst order in α oﬀ the Friedel oscillation, whose amplitude is
proportional to tb ∼ O(t2 ). Returning to the tunnel elec0
trode costs one more power of t0 . Altogether, this gives
α|t0 |4 , which explains the origin of the second term in
Eq. (2) (and—since the process of ﬁrst order in α necessarily includes backscattering at the contact—it also
explains why the term of order αρ is absent). The other

3
(a)

(b)

(e)

Gt

(d)

1

(c)

FIG. 4: Skeleton diagrams (a) and (b) contribute to the exactin-ρ β-function at second order in α, diagrams (c)-(e) do not.
The thick straight lines denote the noninteracting electron
Green function fully dressed by the tunneling vertices.

Gt

c

0
0

1

1 - Gw

scattering process [Fig. 3(b)] is of second order in α because of the creation of an electron-hole pair, but requires
only two powers of t0 (only to get into the wire and come
back): this gives α2 |t0 |2 .
More complicated diagrams of higher orders in ρ are
constructed similarly: to second order in α they are compactly represented in Fig. 4, where the thick lines denote
the noninteracting Green function dressed in all possible ways by the tunnel vertices. Importantly, only oneloop diagrams (a) and (b) in Fig. 4 contribute to the
β-function to second order in α. Diagrams (c)-(e) cancel
all singular interaction-induced terms in the S-matrix except those given by the renormalization group equation.
In particular, the perturbative corrections to r which
come from Figs. 4(a) and (b) read δr(4a) → −αttout t∗ L
b
and δr(4b) → −α2 ttout (t∗ |tb |2 +t∗ |tin |2 )L/2. Calculating
in
b
diagrams (a) and (b) for other amplitudes gives similar
expressions, which—when written for the point tunnel
contact in terms of ρ—all reduce to the single equation,
Eq. (1).
As follows from Eq. (1), tunneling into the wire is
blocked at both FPs ρ = 0 and ρ = 1: Gt vanishes as
2
|ǫ|α /2 at ρ → 0 and as |ǫ|α at ρ → 1. We see that the
ZBA for small α is strongly enhanced in the latter case.
The diﬀerence in the exponents reﬂects the diﬀerence in
the state of the wire: although the tunnel contact is decoupled from the wire at both FPs, the wire at ρ = 0 is
homogeneous, while at ρ = 1 it is broken up into two disconnected pieces (the exponent coincides then with that
for tunneling into the end of a semi-inﬁnite wire [2]).
We now turn to a more general form of the tunnel coupling and show that the FP at ρ = 0, obtained above for
the commonly used model of the point contact, is actually generically unstable. The general parametrization of
the 3 × 3 S-matrix obeying time reversal symmetry and
mirror symmetry x ↔ −x (in particular, at an extended
tunnel contact) gives for the moduli (which we focus on
here) of the scattering amplitudes:
|tb |2 = ρ2 ,

|t|2 = |tout |2 = 2ρ(cos ϕ − ρ) ,

(5)

where ρ and ϕ are constrained by the condition 0 < ρ <
cos ϕ < 1 and the unitarity condition reads 2|t|2 + |r|2 =
|tout |2 + |tin |2 + |tb |2 = 1. The point contact with real
t0 corresponds to ϕ = 0. The wire decoupled from the
tunnel electrode corresponds to ρ = cos ϕ.

FIG. 5: Renormalization group ﬂows for the tunnel junction
in the Gt –Gw plane for α = 0.15. The only stable FP is
at Gt = Gw = 0 (breakup of the junction into three parts).
The unstable FP at Gt = Gc corresponds to the point ρc in
t
Fig. 2. The dashed line separates the regions in which Gt
grows (above) and falls oﬀ (below) with decreasing energy.

Diagrams (a) and (b) in Fig. 4, where the thick lines denote now the “dressing” by the tunnel vertices described
by Eqs. (5), yield a set of two closed [9] ﬂow equations
for arbitrary ρ and ϕ. In the limit of small ρ ≪ α, cos ϕ,
the β-functions read [10]: ∂ρ/∂L ≃ αρ(sin2 ϕ − α/2) and
∂ϕ/∂L ≃ (α/2) sin 2ϕ. It follows that the FP (ρ = 0, ϕ =
0) is actually a saddle point in (ρ, ϕ) space. The key difference from Eq. (1) is that the function ∂ρ/∂L for small
ρ now acquires the term αρ sin2 ϕ. This means that at
ϕ = 0 the cancellation of scattering on the Friedel oscillations at order O(αρ) [cf. the discussion below Eq. (2)]
is no longer exact. Written in terms of the dimensionless
conductances Gt = 2|t|2 and Gw = |tin |2 + |t|2 /2 [11], the
ﬂow equations are:
∂Gt /∂L = αGt (f1 + αf2 ) ,
∂Gw /∂L = α(f3 + αf4 ) ,

(6)

where f1 = −1 + Gt /2 + Gw , f2 = −1/2 + Gt(3 − Gt )/8 +
Gw (1 − Gw ), f3 = Gt (1 + Gw )/4 − 2Gw (1 − Gw ), f4 =
(1 − 2Gw )[Gt (4 + Gt )/32 − Gw (1 − Gw )]. The ﬂow stops
at ǫ given by T , bias, or v/L, whichever is larger.
The ﬂow diagram for Gt and Gw is shown in Fig. 5.
Two limiting curves are the line Gt = 4Gw (1−Gw ) which
describes the point tunnel junction, with the unstable FP
from Fig. 2 at Gt = Gc ≃ 2α, and the line Gt = 0 which
t
describes a decoupled tunnel electrode. The FP (Gt =
0, Gw = 1) is seen to be unstable to the decrease of Gw .
It is this point that describes the commonly expected
outcome of the renormalization: a homogeneous LL with
tunneling blocked by the ZBA. In fact, this point has a
ﬁnite basin of attraction (the line 0 < Gt < Gc ) only in
t
the model of the point tunnel contact (which, albeit being
widely used, is not generic in this sense). Generically, all
ﬂows are toward the point Gt = Gw = 0 which describes
the breakup of the junction into three disconnected parts.
For 1 − Gw ≪ 1 and the bare values Gt0 , 1 − Gw0 ≪ α,

4

FIG. 6: Left panel: schematics of the renormalization of the
tunnel conductance Gt on the log-log scale for the generic
tunnel contact with ϕ = 0 (solid lines) and the point contact
(dashed). Curves I and IV (II and III) correspond to the bare
conductance Gt0 below (above) the dashed line in Fig. 5. The
arrow indicates the point at which the contact is reﬂectionless. The ﬂow of Gt stops at the largest scale of ǫ among T ,
v/L, and the voltage between the tunnel electrode and the
wire. Right panel: the tunnel conductance of the point contact for α = 0.4 and Gt0 above (upper curve) and just below
(lower) the critical conductance Gc ≃ 0.45 (dashed line). The
t
criticality retards the development of the ZBA.

Eqs. (6) can be linearized, which gives:
κ

1 − Gw − Gt /4 = (Gt0 /Gt ) (1 − Gw0 − Gt0 /4) , (7)
where κ ≃ 2(2 − α)/α. We see that if the ﬂow is initially
close to the limiting curve 1 − Gw ≃ Gt /4 predicted by
the point contact model, it shows eventually a kink, after which the “deviation” from the point contact model
begins to grow sharply (κ ∼ α−1 for small α), see Fig. 5.
The critical exponents change at the kink: before it both
2
Gt and 1 − Gw decrease as |ǫ|α /2 [12], after it Gt continues to follow this power law but 1 − Gw increases as
|ǫ|−2α . Near the FP at Gw = 0, Eqs. (6) predict that the
exponent of Gt also changes and both Gt and Gw vanish
at the FP as |ǫ|α [13]. The scaling of Gt,w ∝ |ǫ|α describes tunneling into the end of a semi-inﬁnite LL wire
[12], which means enhancement (curve I in Fig. 6) of the
ZBA as compared to the conventional picture of tunneling into a homogeneous LL (curve IV). If the bare values
Gt0 , 1 − Gw0 lie above the dashed line in Fig. 5, interactions ﬁrst make the tunnel contact more transparent
(curves II and III in Fig. 6), so that both Gt and 1 − Gw
ﬁrst grow, but eventually the ﬂow is attracted to the
same FP Gt = Gw = 0. Note that the development of
the ZBA can be strongly hindered in the vicinity of Gc ,
t
as shown in Fig. 6 (right).
To summarize, we have shown that the picture of tunneling into a LL is qualitatively modiﬁed when the tunneling amplitude is not treated as inﬁnitesimally small.
The conventional FP has a ﬁnite basin of attraction only
in the model of the point tunnel contact, but taking a
ﬁnite size of the contact (or any perturbation induced by
the contact in the wire) into account makes it unstable.

Generically, at the only stable FP the junction breaks
up into three disconnected parts. Flowing toward this
FP, the tunnel conductance may behave nonmonotonically with bias voltage or temperature. Our predictions
can be veriﬁed by systematically varying the distance to
the tunnel electrode in experiments on carbon nanotubes
or semiconductor nanowires.
We thank S. Das, Y. Oreg, I. Saﬁ, and O. Yevtushenko
for interesting discussions. The work was supported by
the DFG/CFN, the EuroHORCs/ESF, the RAS, GIF
Grant No. 965, the RFBR, and the DFG-RFBR.

[1] Yu.V. Nazarov and Ya.M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University
Press, Cambridge, 2009).
[2] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004).
[3] M. Bockrath et al., Nature 397, 598 (1999); Z. Yao et
al., ibid. 402, 273 (1999).
[4] O.M. Auslaender et al., Science 295, 825 (2002); E. Levy
et al., Phys. Rev. Lett. 97, 196802 (2006); Y. Jompol et
al., Science 325, 597 (2009).
[5] C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46, 15233
(1992).
[6] D. Yue, L.I. Glazman, and K.A. Matveev, Phys. Rev. B
49, 1966 (1994).
[7] S. Lal, S. Rao, and D. Sen, Phys. Rev. B 66, 165327
(2002); S. Das, S. Rao, and D. Sen, ibid. 70, 085318
(2004).
[8] In Ref. [7], this FP was missed because the calculation of
the β-function was restricted to the ﬁrst order in α. The
FP at ρ = 0, 1 obtained in Ref. [7] was entirely due to the
presence of interaction in the LL tunnel electrode. The
latter FP was also described in X. Barnab´-Th´riault et
e
e
al., Phys. Rev. B 71, 205327 (2005)—whose method is
restricted to the same level of accuracy—for the case of
tunneling between identical LLs. For a discussion of the
junction of identical LLs and the stable (breakup) FP
at ρ = 1, see also R. Egger et al., New J. Phys. 5, 117
(2003); M. Oshikawa, C. Chamon, and I. Aﬄeck, J. Stat.
Mech. (2006) P02008; A. Agarwal et al., Phys. Rev. Lett.
103, 026401 (2009).
[9] The evolution of the phases of the S-matrix is completely
determined by the ﬂow of the moduli of its elements.
[10] In the limit of ρ → 0, the linearized-in-ρ ﬂow equations were also proposed in I. Saﬁ, arXiv:0906.2363 [see
Eqs. (127) there]—in a form which depends crucially on
an unspeciﬁed “nonuniversal” coeﬃcient c (in contrast,
all coeﬃcients in our calculation are well-deﬁned and give
c = α/2). Note that the linearized-in-ρ equations yield
the growth of ρ for ϕ = 0 but this does not guarantee the
ﬂow to the breakup FP at ρ = 1. To obtain the breakup,
starting from the vicinity of the FP at ρ = 0, one should
go beyond the linear-in-ρ approximation [see Eqs. (6)].
[11] The “wire conductance” Gw gives the current in a biased
wire under the condition that the applied potentials are
such that no current ﬂows through the tunnel electrode.
[12] Following the method of D.N. Aristov and P. W¨lﬂe,
o
Phys. Rev. B 80, 045109 (2009), the one-loop contributions can be summed up to inﬁnite order in α to give a
phase portrait similar to that in Fig. 5 and the critical ex-

5
ponents of Gt,w equal to (1 − K)2 /2K and (1 − K)/K for
the FPs at Gw = 1 and Gw = 0, respectively (D.N. Aristov and P. W¨lﬂe, unpublished).
o

[13] Note that the wire transparency |tin |2 vanishes at this
FP with the doubled exponent as |ǫ|2α .