Y-junction of Luttinger-liquid wires out of equilibrium
D. N. Aristov,1, 2, 3 I. V. Gornyi,2, 4 D. G. Polyakov,2 and P. W¨lfle2, 5
o
1

NRC “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
3
St.Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia
4
A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
5
Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

arXiv:1701.06886v2 [cond-mat.str-el] 11 Apr 2017

2

We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider
the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron
interactions in one of the arms of the junction (“tunneling tip”) is different from that in the other
two arms (which together form a “main wire”). The scaling dependence of the two independent
conductances on bias voltages is determined within a fermionic renormalization-group approach in
the limit of weak interactions. The solution shows that, in general, the conductances scale with
the bias voltages in an essentially different way compared to their scaling with the temperature
T . Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically
obtained from the linear ones by simply replacing T with the “corresponding” bias voltage or the
largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup
of the main wire into two disconnected pieces in the limit of zero T and a zero source-drain voltage.

I.

INTRODUCTION

Models of one-dimensional quantum wires in the presence of electron-electron interactions have been intensively studied since the 1950s [1, 2], mostly in the framework of the Tomonaga-Luttinger liquid model (TLL) featuring a linear electron dispersion relation [3]. Of particular interest is charge transport through junctions of
TLL wires. It is well recognized that the charge screening
at a two-lead junction, in the limit of zero temperature
T , may block transport completely or may enable ideal
transport, depending on whether the electron-electron interaction is repulsive or attractive [4]. The T dependence
of the conductance obeys, then, power-law scaling with
an exponent that depends on the strength of interactions
in the TLL wires. It is also understood that the nonlinear conductance, at a finite bias voltage V , in the limit
of vanishing T is described by a power law in V whose
exponent is identical to that for the T dependence in the
linear response [5–11]. This implies that the scaling behavior of the nonequilibrium conductance with varying
V may simply be obtained from the linear conductance
by replacing T with V .
In the present paper, we focus on a three-lead junction
(“Y-junction”) to which two bias voltages Va and Vb can
be applied, and derive the scaling behavior of the conductances as functions of Va,b . One of the most relevant experimental setups that correspond to the three-way junction consists of a tunneling tip contacting a quantum wire
(“main wire” below), in which case Va and Vb are the
source-drain and tunneling bias voltages, respectively.
We show that neither of the “simple” prescriptions—
replacing T in the linear conductance with the corresponding voltage (Vb for the tunneling conductance and
Va for the conductance of the main wire) or with the
largest one for each of the conductances—is applicable
in the case of multi-lead junctions, where there are two

or more bias voltages. Moreover, generically, the scaling behavior of the conductances with the bias voltages
is essentially different compared to their scaling with T .
It is worth mentioning that the possibility of an intricate interplay between different cutoffs for the scaling of
currents and noise in a nonequiulibrium Y-junction was
pointed out in Ref. [12], where the question of whether
the tunneling bias stops the renormalization of the junction was raised.
We build on the works on linear transport through Yjunctions [13–16] and nonlinear transport through twolead junctions [17] that provide a general framework for
understanding the nonequilibrium properties of multilead systems. We use a fully fermionic formalism [18]
which explicitly assumes the existence of thermal Fermi
leads and thus avoids, by construction, complications
arising within the conventional bosonization approach
when interactions are only present inside a finite segment
of a wire [19–21]. The perturbative fermionic RG theory, formulated in Ref. [18], was efficiently used for the
problem of a double barrier [22, 23] and of a Y-junction
[13, 24, 25] in TLL. The theory has been extended to
an arbitrary interaction strength by summing up an infinite series in perturbation theory (ladder summation).
The results for the beta-function to two-loop order and
for the scaling exponents at the fixed points within this
approach are in complete agreement with known exact
results [15, 26–28]. This theory was further developed to
calculate the conductance for a nonequilibrium two-lead
junction (biased by a finite source-drain voltage) [17].

II.

CONDUCTANCES OF A Y-JUNCTION

We start by identifying the scaling terms in the perturbative expansion of the conductances to first order
in the interaction strength. These are the terms that

2
diverge logarithmically as ln with decreasing infrared
cutoff in energy space . Here, we do not attempt to derive a loop expansion for the renormalization group (RG)
at nonequilibrium from an effective low-energy model.
Rather, we rely on the perturbative RG formalism which
assumes that scaling for a given number of the coupling
constants (two in our case) holds at higher orders in their
perturbative expansion in the interaction strength, i.e.,
that the second-order terms in the expansion that diverge
as ln2 are all generated by the RG equations, etc.
Moreover, to demonstrate the peculiar scaling behavior
of the conductances at nonequilibrium, we restrict ourselves here to the case of the lowest-order perturbative
RG, i.e., only keep the terms in the beta-functions (for
two conductances) of the first order in the interaction
strength. This corresponds to the one-loop RG taken
to the limit of weak interaction. Already at this level
the similarity between the scalings with the bias voltages
and T proves to be broken and the fixed points show a
nontrivial dependence of the conductances on Va and Vb .
The renormalizability of the nonequilibirum Y-junction
model to one-loop order will be addressed elsewhere [29].
We consider a Y-junction whose main wire is characterized by the dimensionless interaction constant α across
its whole length. The junction is thus symmetric in the
sense that the interaction constants α1 and α2 in arms 1
and 2, respectively, are equal to each other, α1 = α2 ≡ α.
The main wire is connected to reservoirs with chemical
potentials µ1 and µ2 , as shown in Fig. 1. The third arm
of the junction is a tunneling-tip wire, with the interaction constant α3 , connected to a reservoir at a chemical
potential µ3 = 0. The currents Ji flowing from the reservoirs i = 1, 2, 3 with the chemical potentials µi towards
the junction obey Kirchhof’s law, i Ji = 0.
It is convenient to introduce two independent currents
Ja,b and two independent bias voltages Va,b as follows:
Ja =

1
(J1 − J2 ),
2

Va = µ1 − µ2

(1)

for the main wire and
Jb =

1
(J1 + J2 − 2J3 ) = −J3 ,
3

Vb =

1
(µ1 + µ2 ) (2)
2

µ1

µ3

(3)

For the main wire which is symmetric under interchange
of the ends, the off-diagonal conductances vanish in the
linear-response limit,

Va
Vb
J2

µ2
FIG. 1: Setup of the Y-junction out of equilibrium. The main
wire is shown as a blue vertical line, the tunneling tip as a
red horizontal line. The reservoirs at the chemical potentials
µa,b,c are depicted as gray blocks, with the currents Ja,b,c
flowing out from them in the presence of the bias voltages Va
and Vb .

ends of the main wire, including the chemical potentials.
The reason why the case µ1 = −µ2 is also referred to as
“symmetric” is the presence of the time-reversal invariance which implies that the cases µ1 = µ, µ2 = −µ and
µ1 = −µ, µ2 = µ are completely equivalent. We will see,
though, that Gab , Gba can be expressed in terms of the
diagonal conductances.
We formulate our model in the scattering-state basis.
in
out
The incoming (ψj ) and outgoing (ψj ) electronic waves
at the junction (x → 0) are related by the S matrix
in
in
in
Ψout (x) = S · Ψin (x), where Ψin = (ψ1 , ψ2 , ψ3 )T and
similar for Ψout . The scattering matrix S characterizing
the symmetric junction is given explicitly in Appendix
A. The unitarity of the S-matrix imposes the constraint
0 ≤ Gb ≤ 1 − 4 Ga −

for the tunneling tip. The conductances are then defined
as
Ja = Ga Va + Gab Vb ,
Jb = Gb Vb + Gba Va .

1
2

2

,

(5)

such that the region of allowed conductance values in the
Gb -Ga plane is bounded by a parabola [13].
The Hamiltonian of the TLL model reads, then, as:
∞

H=

dx [H0 + Hint θ(x − )θ(L − x)] ,
0

H0 = iv Ψ† Ψin − Ψ†
out Ψout ,
in

Gab (Va → 0, Vb → 0) = Gba (Va → 0, Vb → 0) = 0. (4)
This is no longer true in the nonequilibrium case, where
Gab = Gba , except if |µ1 | = |µ2 |, meaning that the system is symmetric with respect to interchanging the two

J1

J3

(6)

3

Hint = 2πv

αj ρj ρj ,
j=1

with the interaction present only in the interval

<x<

3
x

L. The incoming (ρj ) and outgoing (ρj ) density operators are defined by
ρj = Ψ+ ρj Ψin ,
in

ρj = Ψ+ ρj Ψin ,
in

(7)

where ρj = S + · ρj · S and the matrix ρj is given by
+
(ρj )αβ = δαβ δαj , so that (ρj )αβ = Sαj Sjβ . The shortdistance scale defines the ultraviolet energy cutoff ω0 =
v/ ; we set v = 1 below.
We consider the currents as functions of the infrared
cutoff = ω0 e−Λ . The first-order interaction-induced
corrections to the currents are described by the diagrams
in Fig. 2. The calculation, similar to that in Ref. [17],
is presented for the Y-junction geometry in Appendix A.
(0)
Without interaction, the currents are given by Ja =
(0)
(0)
(0)
Ga Va , Jb = Gb Vb . The scale-dependent corrections
to the currents at T = 0 in the limit L → ∞, are written,
to first order in the interaction, as:
(1)
Ja,b

ω0

=

dω Ra,b (ω) ,
(8)

Ra = A1 F (ω, Va ) + A2 [F (ω, Vb+ ) − F (ω, Vb− )] ,
Rb = B2 [F (ω, Vb+ ) + F (ω, Vb− )] ,

Ω
z

F (ω, V ) = (|ω + V | − |ω − V |)/ω

Gb
,
4
α + (α + 2α3 )(1 − 2Ga )
Gb ,
A2 = −
16
α(1 − 2Ga ) + (α + 2α3 )(1 − Gb )
B2 = −
Gb .
8

dω F (ω, V ) = θ( − |V |) 2V ln

ω

Ω+ω
Ω

γ

x

γ
˜

FIG. 2: Diagrams for the corrections to the currents to first
order in interactions. The current is calculated at point z and
the interaction takes place at point x (both z and x include
the space coordinate and the wire index). The solid and wavy
lines stand for the electron Green’s functions and interaction,
respectively. The fermion-boson vertices in Keldysh space
are denoted by the matrices γ and γ ; for more detail, see
˜
Appendix A and Ref. [17].

(so that its derivative with respect to Λ vanishes exponentially in Λ → ∞). We thus obtain the scaling contributions to the currents as

(9)

(12)

(0)

where we set Ja = Ja (ω0 ), etc. An extension of this
calculation to the case of finite T is presented in Appendix B.
The corrections to the conductances are then given by
Ga ( ) = Ga (ω0 ) + 2A1 Λ θ( − Va ) + A2 Λ θ+ ( ) ,
(13)
Gb ( ) = Gb (ω0 ) + 2B2 Λ θ+ ( ) ,
and

We note that if the tip is detached from the main wire,
so that Gb = 0, we recover the equations for a two-lead
wire from Ref. [17].
The ω integration of F (ω, V ) yields

.

z

Jb ( ) = Jb (ω0 ) + 2B2 [Vb+ θ( − Vb+ )
ω0
,
+ Vb− θ( − |Vb− |)] ln

A1 = −α Ga (1 − Ga ) −

ω0

+

and

We choose, without a loss of generality, Va,b ≥ 0, so that
|Vb− | ≤ Vb+ . The prefactors A1 , A2 , B2 are found in the
form

θ( − |V |) 2V ln

ω

Ω+ω

x

Vb± = Vb ± Va /2.

+ 2θ(|V | − ) V − sgn(V ) + V ln

Ω

Ω

and

ω0

γ

Ja ( ) = Ja (ω0 ) + 2 [A1 Va θ( − Va ) + A2 Vb+ θ( − Vb+ )
ω0
,
(11)
−A2 Vb− θ( − |Vb− |)] ln

where we defined

I(ω0 , , V ) =

x

γ
˜

ω0

Gab ( ) = −2A2 Λ θ− ( ) ,
Gba ( ) = −B2 Λ θ− ( ) ,

(14)

where
θ± ( ) = θ( − |Vb− |) ± θ( − Vb+ )

ω0
,
V
(10)

In the last line of Eq. (10), we neglected the contribution
of < |V |. This is because it is not an infrared-divergent
scaling contribution, being only linearly dependent on

takes the value of 2 for > Vb+ and 0 for < |Vb− |, and
1 in between. The off-diagonal conductances are only
nonzero in the energy window |Vb− | < < Vb+ . It is
worth noting that, since A2 and B2 depend only on the
diagonal conductances, Gab and Gba are not independent
quantities.

4
III.

IV.

RG EQUATIONS TO FIRST ORDER IN
THE INTERACTION

We now assume that the conductances obey scaling
behavior (to be verified in Ref. [29]), as expressed by
Ga,b ( ; Va , Vb , ω0 ) = Φa,b

Va Vb
, ,
ω0 ω0 ω0

,

Near the fixed point N , we linearize the beta-function
in Ga , Gb . By introducing the combination
1
Gc = Ga − Gb ,
4

(15)

where Φa,b (x, y, z) is the scaling function. Differentiating
this relation with respect to Λ, with the use of Eqs. (13)
and (14) we get the perturbatrive RG equations
dGa
≡ βa (Ga , Gb ) = 2A1 θ( − Va ) + A2 θ+ ( ) ,
dΛ
dGb
≡ βb (Ga , Gb ) = 2B2 θ+ ( ) ,
dΛ

(16)

we eliminate the terms in the RG equation for Gc that
are proportional to θ+ ( ) and get
dGc
= −2αGc θ( − Va ) ,
dΛ
α + α3
dGb
=−
Gb θ+ ( ) .
dΛ
2

Gc ( )
=
Gc (ω0 )
(17)

2
3 Vb

< Va < 2Vb ,

2Vb < Va .

( /ω0 )

2α

,

> Va ,

2α

(Va /ω0 )

,

(20)

< Va ,

where
1
Gc (ω0 ) = Ga (ω0 ) − 4 Gb (ω0 ) .

with the initial values Ga,b (ω0 ). The three quantities
A1 , A2 , B2 will be taken to depend on the flowing conductances determined by the RG method. Since they
do not depend on the off-diagonal conductances, the RG
equations Eq. (16) form a closed system determining the
diagonal conductances.
It is important to emphasize that the dependence of
the right-hand side of the RG equations (16) on through
the step functions only means that the renormalization
occurs in several steps with different beta-functions at
each step. Specifically, Eqs. (16) and (17) lead to different scaling behavior of the conductances in three distinct
cases:
Va < 2 Vb ,
3

(19)

The solution for Gc reads:

and
dGab
= −2A2 θ− ( ) ,
dΛ
dGba
= −B2 θ− ( ) ,
dΛ

STABLE FIXED POINT

(21)

Recall that we have assumed positive voltages Va,b > 0,
such that Vb+ > |Vb− |. For large > Vb+ , we find
α+α3

Gb ( ) = Gb (ω0 )

,

ω0

> Vb+ .

(22)

In the intermediate range |Vb− | < < Vb+ , the scaling
exponent is reduced by a factor of 2:
(α+α3 )/2

Gb ( ) = Gb (Vb+ )

,

Vb+

For the lowest running energies,
stops and

|Vb− | < < Vb+ .
(23)
< |Vb− |, the scaling

(18)

At high energies,
> max{Va , Vb+ }, the scaling follows the behavior found in the linear-response regime
[13, 24] in all these cases. On the other hand, below
max{Va , Vb+ }, the scaling depends on which of the cases
specified in Eq. (18) is realized.
It follows from the expressions for A1 , A2 , B2 in terms
of Ga , Gb that in the limit Va,b → 0, the system of RG
equations (16) possesses three fixed points, i.e., there are
three solutions of the pair of equations βa (Ga , Gb ) =
βb (Ga , Gb ) = 0 (we restrict our discussion to repulsive
interactions; see the discussion in Ref. [28]). There is
one stable fixed point, termed N , at Ga = Gb = 0, and
there are two unstable fixed points, A at Ga = 1, Gb = 0
and M at GM , GM . All the fixed points belong to the
a
b
boundary of the range of allowed conductances in the
Gb -Ga plane, specified above. Below, we discuss scaling
of the conductances in the vicinity of the stable fixed
point. The analysis of the nonlinear conductances near
the unstable fixed points is given in Appendix C.

Gb ( ) = Gb (|Vb− |) ,

< |Vb− |.

Summarizing, the diagonal conductances near the N
point are found as
Gb (0) = Gb (ω0 )
Ga (0) =

2
|Vb2 − Va /4|
ω0

1
Ga (ω0 ) − Gb (ω0 )
4

Va
ω0

α+α3

,
2α

(24)

1
+ Gb (0) .
4
(25)

Note that the renormalized conductance Gb (0) exhibits
a split zero-bias anomaly (i.e., vanishes to zero) at Vb =
±Va /2 (or, equivalently, µ1 = µ3 or µ2 = µ3 ). This is
similar to the splitting of the tunneling density of states
in a nonequilibrium TLL wire with a double-step distribution function supplied by the leads [30]. In our case,
however, the effect of a double-step distribution is produced by scattering off the junction. Remarkably, it is

5

1
1
Gab (0) = Gba (0) = [Gb (Vb+ ) − Gb (0)] = Gb (ω0 )
2
2


α+α3
α+α3
2 − V 2 /4|
|Vb
Vb + Va /2
a
.
×
−
ω0
ω0

The currents are obtained by means of Eqs. (3) that combine the diagonal and off-diagonal conductances. Differentiating the currents with respect to the bias voltages
yields differential conductances
∂Jb
˜
Gb =
,
∂Vb

∂Ja
˜
Gab =
,
∂Vb

∂Jb
˜
Gba =
.
∂Va

Note that such a differentiation might potentially give
rise to divergent terms in the differential conductances.
However, the singular contributions coming from the diagonal conductances are cancelled by the terms produced
by differentiation of the off-diagonal conductances. As a
result, in the limit of weak interaction, α, α3
1, we
obtain for the renormalized conductances related to the
current in the main wire:
˜
Ga (0)

Ga (0),

˜
Gab (0)

Gab (0)

(27)

at arbitrary voltages, including the singular points. At
the same time, in the differential tunneling conductance
we have to keep the small (in α+α3 ) correction stemming
from the off-diagonal conductance Gba ,
˜
Gb (0)

(α + α3 )Va
Gb (0) + Gb (ω0 )
2Vb + Va

Vb + Va /2
ω0

α+α3

,

(28)
near the zero-bias anomaly at Vb = Va /2 = 0, where
Gb (0) = 0. This means that the differential conductance
˜
Gb for tunneling into the biased wire does not completely
vanish,
˜
Gb (0)

Vb =Va /2

α + α3
Gb (ω0 )
2

Va
ω0

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(26)

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sufficient to align the chemical potential in the tunneling
tip with the chemical potential only in one arm of the
main wire to make Gb vanish. One more difference is that
the critical exponent for the split zero-bias anomaly in
our case (fixed point N ) is linear in interactions, whereas
in the case of noninvasive tunneling (fixed point A, see
Appendix C, with α3 = 0) it is quadratic. It is also worth
noticing that the tunneling zero-bias anomaly also shows
up in the conductance of the main wire, Eq. (25).
For the off-diagonal conductances, we get

α+α3

= 0,

in contrast to Gb .
We now turn to the discussion of the general result for
the N point, Eqs. (24) and (25), in two limiting cases. In
particular, one may wonder whether in the case that one
of the bias voltages is much larger than the other, the
smaller of the voltages may be neglected. If true, this

FIG. 3:
Zero-T RG flows for different voltages Va /ω0 =
0.3, 0.1, 0.03, 0.01 (from top to bottom) for the interaction
constants α = 0.3, α3 = 0.01, the tunneling bias Vb = 0.3 ω0 ,
and the bare conductances Ga (ω0 ) = 0.26, Gb (ω0 ) = 0.42.
The position of fixed points N , A, M at equilibrium is depicted by blue dots. The limiting RG values of the conductances out of equilibrium lie on the parabolic fixed line.

would make it possible for the linear-response scaling to
carry over to the nonlinear regime by simply replacing T
with the larger of the voltages. As we show now, this is
not always the case.

1.

Vb

Va

Consider first the case of small bias on the tunneling
tip, Vb
Va , i.e., Vb+
|Vb− |
Va /2
Vb . From
Eqs. (24) and (25), we have

Gb (0)

Gb (ω0 )

Va
ω0

α+α3

,

(29)
Va
ω0

1
Ga (0) = Ga (ω0 ) − Gb (ω0 )
4
1
+ Gb (ω0 )
4

Va
ω0

2α

α+α3

.

(30)

These results are obtainable from the linear-response
conductances as a function of T , Gb (T ) ∝ T α+α3 and
1
Ga (T ) − 4 Gb (T ) ∝ T 2α , by substituting Va (the larger
of the voltages) for T . Note that, despite the deceitfully
simple appearance, this is rather nontrivial. Indeed, Gb
here is controlled by Va , which is to say that the bias
voltage Vb driving the current Jb does not enter the nonlinear dependence of Gb . Thus, one of the naive “general”
prescriptions mentioned in Sec. I, which would imply the
replacement of T by the corresponding voltage (in the
present case this would be Vb ), fails.

6
2.

Vb

Va

In the opposite limiting case of small bias in the main
wire, Vb
Va , we have
Gb (0)
Ga (0)

Gb (ω0 )

Vb
ω0

α+α3

,

(31)
Va
ω0

1
Ga (ω0 ) − Gb (ω0 )
4
1
+ Gb (ω0 )
4

Vb
ω0

2α

α+α3

.

(32)

Now we see that Gb (0) is found to scale as (Vb /ω0 )α+α3 ,
which is obtainable by replacing T with Vb in the linearresponse expression for Gb (T ), i.e., both “naive” prescriptions from Sec. I do work. However, Ga (0) depends
now on both voltages, which cannot be obtained from the
linear-response scaling of Ga (T ) by the replacement of T
with one of the voltages. Thus, we find the remarkable result that, depending on the initial conductances Ga,b (ω0 )
and the interaction constants α and α3 , the conductance
Ga (0) of the main wire is controlled by either the corresponding voltage Va applied to the main wire or by the
larger voltage Vb applied to the tunneling tip. We, therefore, conclude that none of the prescriptions translating
the linear and nonlinear conductances into each other is
general.
It is also interesting to look at the situation from another perspective: by observing that the renormalization
of the conductances may continue for energy scales below
the larger voltage. In particular, the larger tunneling bias
does not completely stop the RG flow for the zero-T conductance Ga of the main wire. One might expect, then,
that Ga at Va → 0 would renormalize to zero, similarly
to a two-lead junction. This is not the case, as seen from
Eq. (32), which can be traced back to the condition (5)
following from the unitarity of the S-matrix. That is, remarkably, a finite tunneling bias, although not stopping
the RG flow for Ga , still prevents the interaction-induced
breakup of the main wire into two disconnected pieces,
as illustrated in Fig. 3.

completely determined by the diagonal ones, Ga and Gb
for the main wire and the tunneling tip, respectively.
We have calculated the interaction-induced corrections
to the currents to first order in the interaction strength
and identified the scale-dependent terms. This has allowed us to derive two coupled renormalization group
equations for Ga and Gb . We have solved these equation analytically in the vicinity of three fixed points
known to exist in the linear-response regime. For the
case of repulsive interaction, there exists one stable fixed
point at Ga = Gb = 0, around which the conduc1
2α
tances obey power laws Ga − 4 Gb ∝ Va and Gb ∝
2
|Vb2 − Va /4|(α+α3 )/2 with the interaction dependent exponents. This is to say that neither is Ga following a
pure power law in Va , nor is Gb given by a pure power
law in Vb , as one might naively expect. Remarkably, it
is also seen that the RG flow is not generically stopped
by the largest of the two voltages. Notably, the singularity related to the split zero-bias anomaly at Vb = ±Va /2
shows up in both Ga and Gb .
We have thus demonstrated that, in general, the conductances scale with the bias voltages in a way that
is essentially different from their scaling with temperature, unlike in the two-lead junctions. Interestingly,
even though not stopping the renormalization of Ga , a
finite tunneling voltage Vb precludes transport through
the main wire from being blocked in the limit of zero T
and a zero source-drain voltage Va . At the same time, a
finite source-drain voltage Va in the main wire precludes
the differential tunneling conductance from vanishing to
zero at the zero-bias anomaly.

VI.

ACKNOWLEDGEMENTS

The work of D.A. was partly supported by RFBR
grant No. 15-52-06009 and by GIF grant No. 1167165.14/2011.

Appendix A: Derivation of the first-order corrections
V.

SUMMARY

We have developed a framework to study nonlinear
charge transport through a Y -junction, i.e., through
a junction connecting three Luttinger-liquid quantum
wires. The particular example of nonequilibrium on
which we have focused in the present paper is the setup
with the tunneling tip and the main wire biased by the
tunneling and source-drain voltages, respectively. This
setup is characterized by four nonlinear conductances
connecting the two currents with two (source-drain, Va ,
and tunneling, Vb ) voltages. We have shown that the
off-diagonal components of the conductance matrix are

Here we describe how the first-order corrections to the
nonequilibrium currents through a Y-junction were calculated. First of all, we define the Green’s function as
a quantity dependent on two continuum variables, the
space coordinate x and energy ω, the Keldysh index i =
1, 2, the “in-out” label l = 1, 2 and the wire index, which
can, in general, take values j = 1, . . . N ; in our situation
N = 3. This is done according to Eq. (5) in Ref. [17]. The
Keldysh weight functions hj (Ω) = tanh[(Ω−2µj )/2T ] depend on the chemical potential µj and the temperature
T.
As defined in Eq. (5) of [17], the Green’s functions
depend on the elements of the S-matrix, which we assume

7
The quantity I(ω0 , , V ) then reads

to be symmetric:

r1 , t1 , t2
S =  t1 , r1 , t2  .
t 2 , t 2 , r2

ω0



I(ω0 , T, V ) =
(A1)

All the Fermi velocities in the leads are assumed to be
the same.
The calculated corrections to the currents are shown
by the skeleton graphs in Fig. 2. The correction to
the incoming current is zero. Evaluation of the correction to the outgoing current yields three types of terms,
containing quadratic forms in the Keldysh weights, e.g.
hj (Ω)hl (ω + Ω), first powers hj (Ω), hj (ω + Ω), and terms
independent of hj . The integration over Ω leads to
the disappearance of the linear terms, while the terms
quadratic in hj and those independent of hj may be combined into an integrand, convergent at Ω → ±∞, of the
form

ω0
= 2V ln 2πT + 1 − Re ψ 1 +

 ln ω0 , T
|V |,

2πT
2V ×
 ln ω0 e , T

|V |.
V

ωc

Λ(V, T, L) =
0

dω
[f2 (ω + V ) − f2 (ω − V )] sin2 ωL ,
ω

and it provides the linear response to the voltage V . For
T = 0 we have f2 (x) = |x| and the corresponding expressions (8) are given in the main text.

iV
2πT

(B1)
We are looking for scaling contributions to I(ω0 , T, V ) as
functions of T . These can be written as
I(ω0 , T, V ) ≈ 2V θ(T − c∗ |V |) ln

ω0
,
2πT

where c∗ is a number of the order of unity. We see that
all the steps in the derivation for T = 0 are applicable
to the case of finite temperature, with the replacement
→ T /c∗ .

dΩ[1 − hj (ω + Ω)hl (Ω)] = 2f2 (ω + µl − µj ) ,
with f2 (x) = x coth(x/2T ). The corrections to the currents ja and jb are obtained by first integrating over
x ∈ [0, L], which leads to the combination of the form
(1 − e2iωL )/ω. At this step, we obtain intermediate expressions as six linear combinations of f2 (ω +µl −µj ) and
three linear combinations of f2 (µl − µj ) with all possible
j = l. The terms with j = l are cancelled.
The last integration is over ω ∈ [−ω0 , ω0 ]. Using the
property f2 (x) = f2 (−x), we simplify these expressions
further and reduce the last integration to the one over the
positive interval, ω ∈ [0, ω0 ]. The resulting corrections
contain terms with large logarithmic factors stemming
from the integral [see Eq. (21) in Ref. [17]]

dωF (ω, V ) =
0

Appendix C: Unstable fixed points

In this Appendix, we analyze the nonequilibrium scaling near the two unstable fixed points. We first consider the solution of the RG equations near the fixed
point A, where Ga = 1 and Gb = 0. Linearizing
¯
in Ga = 1 − Ga
1, we introduce the combination
¯ c = Ga − 1 Gb and express the RG equations as
¯
G
4
¯
dGc
¯
= 2αGc θ( − Va ) ,
dΛ
dGb
α3
= − Gb θ+ ( ) .
dΛ
2

(C1)

The solution to this set of equations reads:
1
¯
¯
Gc (0) = Ga (ω0 ) − 4 Gb (ω0 ) (ω0 /Va )2α ,

(C2)

Gb (0) = Gb (ω0 )(|Vb − Va /2|/ω0 )α3 .

(C3)

and

Notice that the scaling exponent of Gb vanishes in the
absence of interaction in the tip, α3 = 0. It is nonzero
in this case in the second order as α2 , in agreement with
Refs. [13, 30]. The conductance of the main wire is given
by

Appendix B: Extension to finite temperatures
1
¯
Ga (0) = Gc (0) + 4 Gb (0).

Consider now the case of finite T . The contributions to
the current at the lowest order in the interaction strength
are still given by Eq. (8), but now we put = 0 and
the low-energy cutoff is provided by the function F (ω, V )
taken at finite T
F (ω, V ) =

ω+V
ω+V
ω−V
ω−V
coth
−
coth
.
ω
2T
ω
2T

(C4)

¯
We see that Gc (0) is increasing upon renormalization and
thus the RG flow leads away from the fixed point A,
whereas Gb → 0 in the limit Va,b → 0.
Another unstable fixed point is located at
GM =
a

α + α3
,
α + 2α3

GM = 1 −
b

α
α + 2α3

2

.

8
ˇ
ˇ
The RG equations for Ga = Ga −GM and Gb = Gb −GM
a
b
have the form

All coefficients are positive, signaling a runaway flow of
ˇ
ˇ
both Ga and Gb .

ˇ
α
4α
dGa
ˇ
ˇ
=
Ga + Gb θ( − Va )
dΛ
2 α + 2α3
α3 α + α3 ˇ
+
Ga θ+ ( ) ,
2 α + 2α3
ˇ
dGb
α3 (α + α3 )
ˇ
ˇ
=
2αGa + (α + 2α3 )Gb θ+ ( ) .
dΛ
(α + 2α3 )2

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