Conductance scaling of junctions of Luttinger-liquid wires out of equilibrium
D. N. Aristov1, 2, 3 and P. W¨lfle2, 4
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
Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

arXiv:1802.03927v2 [cond-mat.str-el] 28 Apr 2018

2

We develop the renormalization group theory of the conductances of N-lead junctions of spinless
Luttinger-liquid wires as functions of bias voltages applied to N independent Fermi-liquid reservoirs.
Based on the perturbative results up to second order in the interaction we demonstrate that the
conductances obey scaling. The corresponding renormalization group β−functions are derived up
to second order.
I.

INTRODUCTION

The charge transport through junctions connecting
quantum wires modeled by the Tomonaga-Luttinger liquid model (TLL) has been studied intensely over the past
several decades. In the linear response regime it has been
shown since the first studies of a two-lead junction1–4 that
the conductance obeys scaling as a function of temperature, at least in the vicinity of certain special values of
the conductance. This behavior is captured in the framework of a renormalization group (RG) formulation, where
the special values are identified as fixed points of the RG
flow. Initially the flow equations were derived within the
bosonization approach. The latter has the difficulty that
the conductance of a wire of finite length depends on the
contact resistances at the links to the external charge
reservoirs (accounting for the transition of bosonic excitations into fermionic quasiparticles), which so far have
not been determined. Alternatively, a purely fermionic
formulation may be used, which avoids the problem of
contact resistance.
The latter approach has been pioneered by5 in the
limit of weak interaction and was later extended to arbitrary coupling strength by6 . In simple words, the standard procedure for how to derive the RG flow equations,
e.g. for the two-lead junction, is to first calculate the
conductance G0 (θ) in the absence of interaction, as a
function of the parameter θ (or of several parameters in
the case of multi-lead junctions) determining the scattering strength of the junction. Then the conductance is
calculated in perturbation theory in the interaction, allowing to identify the linear logarithmic corrections (for
example ∝ ln(ω0 /T ), if the temperature T is cutting
off the infrared singularities of the theory and ω0 is an
ultraviolet cutoff), and extract the RG β−function as
β = −dG/d ln T at ln(ω0 /T ) = 0. The resulting function β depends on the interaction strength α and on the
parameter θ characterizing the junction. Inverting the
functional dependence G0 (θ), the parameter θ may be
expressed by G0 , which in the sense of the RG structure may be replaced by the renormalized G at scale T .
This procedure may be justified within a more rigorous
scheme using ideas first formulated by7,8 , see6 . In order
to explicitly demonstrate the scaling property, it should

be shown that all terms of powers higher than linear in
ln(ω0 /T ), appearing in perturbation theory, are generated by iteration of the RG equations. For the case of
the conductance of a two-lead junction this has been verified up to order (ln(ω0 /T ))36 . The approach sketched
above has also been applied to multilead junctions, such
as the Y-junctions in the weak coupling limit9–11 and at
strong coupling12 , and chiral Y-junctions13–15 , as well as
X-junctions16 .
The results on transport through Y-junctions obtained
in our previous work are generally in good agreement
with results obtained by the bosonization method (BM)
in the linear response regime17–20 , keeping in mind that
the overall magnitude of the current can not be determined accurately by BM, as mentioned above. In the few
cases where discrepancies have arisen, such as in the limit
of very strong attractive interaction,12 the BM calculation employed additional assumptions which we believe
to be incorrect. There are only few works on transport
through Y-junctions out of equilibrium, in which scaling
has been assumed to exist without proof21,22 .
The fermionic transport formulation is general and
physically appealing. It may be extended to systems out
of equilibrium in a natural way. Transport through a
two-lead junction at finite voltage bias V and low temperatures has been considered in23 , with results in agreement with those of other methods, where applicable24–27 .
Not too surprisingly, one finds the conductance following a power law in V , with exponent identical to the
one governing the temperature power law of the linear
response conductance. Recently, transport through a Yjunction out of equilibrium has been studied by assuming
the scaling property to hold in this case as well22 . The
latter assumption is not trivial, if only for the following
reason: the Y-junction is connected to three charge reservoirs held at three different chemical potentials, in general. Consequently, the two independent conductances
depend on two independent bias voltages Va , Vb . There
appears to be scaling of the conductances in both variables, Va and Vb . The question is then how the scaling
in T is expressed in terms of voltage, i.e. which of the
two voltages or both enter the corresponding formulas.
Indeed it has been found in22 that the scaling as a function of Va , Vb may not be obtained from the scaling of the

2
linear response conductance with T by any simple recipe.
In the present paper we demonstrate that scaling in the
case of multi-lead junctions out of equilibrium is valid, by
explicitly calculating the terms of second power in the
scaling variable Λ = ln(ω0 / ) in the conductances of a
symmetric Y-junction in second order in the interaction.
Here ω0 and are ultraviolet and infrared cutoffs in energy. We then show that all of these terms are generated
by the RG equations, proving the validity of scaling, at
least up to this order.

II.

THE MODEL

We consider a system of spinless fermions in one dimension, interacting in each of N quantum wires in the
region a < |xj | < L, j = 1, .., N , adiabatically connected to N charge reservoirs at |xj | > L. The N wires
are connected by a junction located in the narrow regime
|xj | < a, which scatters the fermions as described by an
S matrix with elements Sjk , where j, k = 1, ...N .
We assume the interaction to be described by a TLL
model in the form
N

∞

H=

0
int
[Hj (x) + Hj (x)Θ(x; a, L)]

dx
0

(1)

j=1

where
†
†
0
Hj (x) = vj [ψj,in (x)i ψj,in (x) − ψj,out (x)i ψj,out (x)]

and
†
†
int
Hj (x) = 2πvj αj [ψj,in (x)ψj,in (x)ψj,out (x)ψj,out (x)]

Here vj is the Fermi velocity, αj is the interaction constant in lead j and Θ(x; a, L) = 1 in the interval a <
|x| < L and zero elsewhere. The fermionic field operators
†
ψj,ηj (x) create particles at position x in scattering states
|j, ηj ; ω of energy ω, in wire j and with chirality ηj = ±1,
labeling incoming (ηj = −1) and outgoing (ηj = +1)
states. In the following we will put vj = 1 for simplicity.
The outgoing fermion operators are connected with the
incoming ones by the S matrix, ψj,out (0) = Sjk ψk,in (0).
The perturbation theory will be formulated in the language of Keldysh matrix single particle Green’s functions

Gω (l+ , y; j+ , x) = −ieiω(y−x)

∗
θ(y − x)δlj , Slm Sjm hm
,
0,
−θ(x − y)δlj

Gω (l+ , y; j− , x) = −ieiω(y+x)

Slj , Slj hj
,
0,
0

Gω (l− , y; j+ , x) = −ie−iω(y+x)

∗
0, Sjl hl
∗ ,
0, −Sjl

θ(x − y),
hl
,
0,
−θ(y − x)
(3)
where hl (ω) = tanh[(ω −µl )/2T ] is the Keldysh function,
with µl the chemical potential in wire l ; summation over
m is implied in the first line of (3). The functions hl (ω)
carry the information on the out of equilibrium conditions.
Gω (l− , y; j− , x) = −ieiω(x−y) δlj

III.

DERIVATION OF RENORMALIZATION
GROUP EQUATIONS FOR THE
CONDUCTANCES

We consider the conductances Gj , j = 1, ..NG of a
N −lead junction out of equilibrium (here NG < N 2 is the
number of independent conductances). In order to show
that the conductances obey scaling, we follow the reasoning first developed by Callan and Symanczik Ref.7,8
and apply it to the problem at hand. Assume that we
know the perturbation series of the conductances Gj .
The conductances depend on the scattering properties
of the junction, expressed by the S-matrix elements Slj
(which may be expressed in terms of the conductances
G0 of the system in the absence of interactions), and on
j
the interaction (coupling constant α). The possible existence of scaling is signaled by the appearence of powers
of the scaling variable Λ = ln(ω0 / ). The perturbation
series of Gj in terms of the interaction has the general
form
Gj = G0 + αAj ({G0 }) + α2 Bj ({G0 }) + O(α3 )
j
i
i

where Aj , Bj are polynomials of first and second order
in the scaling variable Λ, respectively, polynomials of the
bare conductances G0 , functions of the NV independent
j
bias voltages Vj , and of additional coupling constants αj
in the form of ratios αj /α. Now we invert the series to
express the conductances G0 in the non-interacting limit
j
in terms of the full conductances Gj
G0 = Gj + αAj ({Gi }) + α2 B j ({Gi }) + O(α3 )
j

R

G=

G
0

K

G
GA

(2)

in the non-interacting limit. The Green’s functions depend on energy ω, position x, wire index j and chirality
ηj , G = Gω (lηl , y; jηj , x).
The Green’s functions for each pair of chiralities ηl , ηj
are given by

(4)

(5)

where Aj , B j are again polynomials of first and second
order in Λ. By substituting G0 as given in Eq. (5) into
j
Eq. (4) we find the following relations
Aj ({Gi }) = −Aj ({Gi })
B j ({Gi }) = −Bj ({Gi }) −
l

∂Aj ({Gi })
(6)
Al ({Gi })
∂Gl

3
We now use that G0 ({Gi }) must be independent of Λ
j

µ1
dG0 ({Gi })
j
dΛ

=0=

∂G0 ({Gi })
j
∂Λ

∂G0 ({Gi })
j

+

∂Gl

l

∂Gl
(7)
∂Λ

to find the renormalization group equation

∂Gj
=−
∂Λ

l

∂G0 ({Gi })
j
∂Gl

−1

∂G0 ({Gi })
l
∂Λ

J1

J3
µ3

(8)

Va
Vb

The inversion of the matrix Djl = ∂G0 ({Gi })/∂Gl is
j
obtained from Eq. (5) as
∂G0 ({Gi })
j
∂Gl

−1

= δjl + α

∂Aj ({Gi })
∂Gl

µ2

(9)

Substituting ∂G0 ({Gi })/∂Λ we finally get
l
∂Gj
∂Aj ({Gi })
=α
∂Λ
∂Λ
∂Bj ({Gi })
+ α2
−
∂Λ

∂ 2 Aj ({Gi })
Al ({Gi })
∂Λ∂Gl
l
(10)
In order for scaling to hold, the r.h.s. of Eq. (10) should
not depend on Λ. More specifically, it should be O(Λ0 ),
possibly containing Heaviside step functions (see below),
indicating a stop of the RG flow. Verifying this property
amounts to proving the validity of scaling up to the order
considered.

IV.

J2

SYMMETRIC Y-JUNCTION OUT OF
EQUILIBRIUM

We now apply the above general derivation of RG equations for the conductances to a concrete example. We
consider charge transport through a Y-junction with a
symmetric main wire (labelled 1, 2) contacted by a tip
wire at the center (3) (see Fig.1). The three half-wires
of length L are adiabatically connected with reservoirs
kept at chemical potentials µj , j = 1, 2, 3. We assume
that there is no interaction within the junction of radius
a. The scattering states of each wire are labeled by wire
index j, chirality ηj = +, − (outgoing or ingoing), energy ω, and position x > 0 in the interval [a, L]. The
junction is 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 third arm of the
junction is a tunneling-tip wire, with interaction constant
α3 , which we will assume to vanish in the following. We
define currents Jj flowing from the reservoirs toward the
junction.

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 .

The S matrix of the symmetric Y-junction may be
parametrized as follows


r1 , t1 , t2
S =  t1 , r1 , t2  .
(11)
t2 , t2 , r2
The symmetric form of interaction, α1 = α2 , keeps the
renormalized S matrix in symmetric form (11). We use
the parametrization
r1 = 1 (cos ϑ + e−iψ ),
2
t2 =

i
√
2

sin ϑ,

t1 = 1 (cos ϑ − e−iψ ) ,
2

r2 = cos ϑ .

(12)

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

(13)

for the main wire and
Jb =

1
(J1 + J2 − 2J3 ) = −J3 ,
3

Vb =

1
(µ1 + µ2 ) (14)
2

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

Jb = Gba Va + Gb Vb

(15)

It is found that in the symmetric setup Gab and Gba appear due to asymmetry produced by the voltages, they

4
may be expressed in terms of the diagonal conductances
Ga , Gb and therefore do not flow independently. We
therefore do not consider the off-diagonal conductances
in the following. In terms of parametrization (12) the
conductances are given by Ga = (1 − cos ϑ cos ψ)/2,
Gb = sin2 ϑ.
A.

where for n = a, b

Ga ( ) = G0 + α (a0 Λa + a0 (Λb+ + Λb− )) ,
a
1
2
Gb ( ) = G0 + α a0 (Λb+ + Λb− )
b
3

a1 (Ga , Gb ) = −2Ga (1 − Ga ) + 1 Gb ,
2
a2 (Ga , Gb ) = − 1 [1 − Ga + g3 (1 − 2Ga )]Gb ,
8

ω0

d
ω0

∂βn (G0 , G0 )
a
b
βa (G0 , G0 )
a
b
∂G0
a

(19)

The expression in square brackets here reads
∂a0 0
∂a0 0
2
1
a θ ( )θa ( ) +
a θ+ ( )θa ( )
0 1 a
∂Ga
∂G0 1
a
∂a0 0
∂a0 0
1
1
+
a +
a
θa ( )θ+ ( )
0 2
∂Ga
∂G0 3
b

Ga =

+

∂a0 0
∂a0 0
2
2
a +
a
0 2
∂Ga
∂G0 3
b

(20)

θ+ ( )θ+ ( )

and
∂a0 0
3
a θ+ ( )θa ( )
∂G0 1
a
∂a0 0
∂a0 0
3
3
+
a2 +
a θ+ ( )θ+ ( )
∂G0
∂G0 3
a
b

Gb =
(17)

a3 (Ga , Gb ) = − 1 [1 − Ga − 1 Gb + g3 (1 − Gb )]Gb .
2
2
and g3 = α3 /α.
Differentiating (16) with respect to Λ = ln(ω0 / ) and
replacing a0 → aj , we obtain the RG β−functions as
j
∂Ga
= βa (Ga , Gb )
∂Λ
= α[a1 (Ga , Gb ) θa ( ) + a2 (Ga , Gb ) θ+ ( )] ,
∂Gb
= βb (Ga , Gb ) = α a3 (Ga , Gb ) θ+ ( ) .
∂Λ

d

∂βn (G0 , G0 )
a
b
+
βb (G0 , G0 )
a
b
∂G0
b

(16)

where G0 = Ga,b (ω0 ) and we use a shorthand notation
a,b
Λa = ln(ω0 / max[ , Va ]), Λb± = ln(ω0 / max[ , |Vb± |]),
with Vb± = Vb ± Va /2 . Here we defined a0 = aj (G0 , G0 )
a
j
b
with

βn (G0 , G0 )
a
b

ω0

Gn = 2

Perturbation theory results and RG-equation in
first order

As shown in22 the diagonal conductances in first order
are given by

d

Gn =

(18)

with θa ( ) = θ( − Va ) and

(21)

Eqs. (20), (21) contain θ functions in four different
combinations. One of them, θa ( )θa ( ) results simply in
Λ2 /2. Others, e.g. θ+ ( )θa ( ) lead to more complicated
a
expressions and depend on the relation between Va , Vb .
As was shown in22 , there are two most interesting cases.
In one of them, with Va Vb , we can let θ+ ( ) 2θ( −
Va ), which results in unique value of logarithm, Λa . In
another regime, Va
Vb , we let θ+ ( ) 2θ( − Vb ) and
obtain two values, Va and Vb .
In this second regime we can express the integrals appearing in (19) as

θ+ ( ) = θ( − |Vb− |) + θ( − Vb+ ) ,
The effect of the θ−functions is to define different
forms of the functions Aa,b in different intervals of , and
hence Λ. For example, for given Va > 2Vb > 0 we have
θ+ ( ) = 2 for > Vb + Va /2 and θa ( ) = 1 for > Va
(interval I), > Vb + Va /2 and θa ( ) = 0 for < Va
(interval II), θ+ ( ) = 1 for |Vb − Va /2| < < Vb + Va /2
(interval III) and θ+ ( ) = 0 for |Vb − Va /2| > (interval
IV ).
If the differential equations (18) are valid, then the calculated second order corrections, ∼ α2 Λ2 , are cancelled
in the above procedure, leading to Eq. (10). Alternatively, we may compare these corrections with the predicted form stemming from equation (18).
The conductances Ga,b up to second order in Λ are determined by solving the RG-equations iteratively. We
substitute the first order results, Eq. (16), into the
β−functions and integrate. We get
Ga,b =

G0
a,b

+ αGa,b +

1 2
2 α Ga,b ,

1
dΛ dΛ θa ( )θa ( ) = 2 Λ2
a

dΛ dΛ θa ( )θ+ ( ) = 2Λa Λb − Λ2
b
(22)
dΛ dΛ θ+ ( )θa ( ) = Λ2
b
dΛ dΛ θ+ ( )θ+ ( ) = 2Λ2
b
which leads to some simplification of the second-order
corrections as predicted by the RG equations
∂(a0 + a0 )
1
2
∂G0
a
0
2 0 ∂a2
+ (2Λa Λb − Λb )a3
∂G0
b

Ga = Λ2 (a0 + a0 )
a
1
2

Gb = Λ2 (a0 + a0 )
b
1
2

∂a0
∂a0
3
+ a0 3
3
0
∂Ga
∂G0
b

(23)

5
We find that the similar expressions applicable for the
first regime Va
Vb are obtained from (23) by the replacement Λb → Λa . These results may now be compared with the explicit calculation of the second order
perturbative corrections, undertaken in the next section.

T (c) =

TrK [γ ext GΩ (j, +, z; l, −ηl , x)
µ,µ =1,2 ηl ,ηl =+,−
µ

· γ GΩ+ω (l, −ηl , x; l , −ηl , x )¯ µ
¯
γ
B.

· GΩ+ω+ω (l , −ηl , x ; l, ηl , x)γ µ

Perturbation theory in second order

The contribution to the current in the outgoing channel
j at position z in second order in the interaction may be
expressed as
dΩ dωdω
(2π)2

(2)

Jj (z) = −2

L

dxdx
a

(24)

· GΩ+ω (l, ηl , x; l , ηl , x )γ µ GΩ (l , ηl , x ; j, +, z)]
(27)
Here GΩ are 2 × 2 matrices of Green’s functions in
Keldysh space in the absence of interaction, but in the
presence of the scattering effect of the junction, which is
expressed in terms of the S−matrix elements Sij , as presented above. The dependence on the coordinates may
be split off:

αl αl Tjll (x, x ; Ω, ω, ω )

×
l,l

(a,b,c)

Tjll
(a)

(b)

(a,b,c)

(x, x ; Ω, ω, ω ) = e−2i(ωx+ω x ) Tjll

(c)

(Ω, ω, ω )
(28)

The trace TrK is over the lower (fermionic) Keldysh
µ
¯µ
indices; the fermion-boson vertices, γij → γ µ , γij → γ µ ,
¯
tensors of rank 3 defined in Keldysh space, are given by
FIG. 2: Three skeleton diagrams showing the leading contribution to currents in second order of perturbation

There are three diagrams (see Fig. 2 ) contributing
in second order, each one with arrows both forming a
right-handed or a left-handed loop, which amounts to
letting Ω → −Ω, giving the identical result, thus leading
to a prefactor of 2. The three diagrams give rise to the
combination Tjll (x, x ; Ω, ω, ω ) = T (a) + T (b) + T (c) ,
where

1
γij = γij =
¯2

1
√ δij ,
2

2
γij = γij =
¯1

1
√ τ1 ,
2 ij

(29)

with τ 1 the first Pauli matrix. The external vertex is
given by

ext
γij =

i
2

1 1
−1 −1

=

i
2

1
−1

1 1 ,

(30)

TrK [γ ext GΩ (j, +, z; l, −ηl , x)

T (a) =
µ,µ =1,2 ηl ,ηl =+,−
µ

γ
· γ GΩ+ω (l, −ηl , x; l , −ηl , x )¯
¯

µ

· GΩ+ω+ω (l , −ηl , x ; l , ηl , x )γ µ
· GΩ+ω (l , ηl , x ; l, ηl , x)γ µ GΩ (l, ηl , x; j, +, z)]
(25)
is a rainbow diagram with nested self-energy insertions,
T (b) =

TrK [γ ext GΩ (j, +, z; l, −ηl , x)¯ µ
γ

which suggests to interpret the trace in Keldysh space
as operating with the Keldysh matrices on the vector
T
1 −1
and forming the inner product of the resulting
i
vector with the vector 2 1 1 .
The calculation of second order corrections to the currents Ja , Jb is a tedious procedure and is discussed in
more detail in the Appendix. Here we provide the summary of this calculation. We find the corrections in the
form

µ,µ =1,2 ηl ,ηl =+,−

γ
· GΩ+ω (l, −ηl , x; l, ηl , x)γ µ GΩ (l, η, x; l , −ηl , x )¯ µ
·GΩ+ω (l , −ηl , x ; l , ηl , x )γ µ GΩ (l , ηl , x ; j, +, z)]
(26)
is a chain diagram with two self-energy insertions in series, and the third diagram has crossed self-energy insertions

Ga = Ba1 Fa1 + Ba2 Fa2 + Ba3 Fa3 ,
Gb = Bb1 Fb1 + Bb2 Fb2 + Bb3 Fb3
with

(31)

6

1
Gb (Ga − 1 + g3 (2Ga − 1))
16
× (4Ga + 3Gb − 4 + g3 (6Gb − 4)) ,

Ba1 =

Ba2 = − (2Ga − 1) 4G2 − 4Ga + Gb ,
a
1
Ba3 = − (1 + 2g3 ) Gb 4G2 − 4Ga + Gb ,
a
8
1
Gb (2Ga + Gb − 2 + 2g3 (Gb − 1))
Bb1 =
16
× (4Ga + 3Gb − 4 + g3 (6Gb − 4)) ,
1
Bb2 = Gb (1 + 2g3 ) Ga Gb − 4(Ga − G2 ) − g3 Gb ,
a
4
1 2
G (Gb + 4g3 (1 + g3 ) (Gb − 1)) .
Bb3 =
16 b
(32)
Coefficients Fj are defined as integrals over energy,
they are independent of Ga,b and are discussed in the
Appendix.
We distinguish again between two regimes: i) Va Vb
and ii) Va
Vb . In second regime we find, using the
results given in the Appendix
Fa1 =Λ2 ,
b

Fb1 = 2Λ2 ,
b

Fa2 = − Λ2 ,
a
Fa3 =Λ2
b

Fb2 = 2Λ2 ,
b

− 4Λa Λb ,

Fb3 =

(33)
2Λ2 .
b

which gives
Ga = Ba1 Λ2 − Ba2 Λ2 + Ba3 (Λ2 − 4Λa Λb ),
b
a
b
Gb = 2(Bb1 + Bb2 + Bb3 )Λ2 ,
b

(34)

in the first regime we should merely replace Λb
Λa
in these expressions. In both cases one can check the
equivalence of (23) and (34), which proves that the second order corrections are indeed exactly generated by the
RG equations (18).
Alternatively, we checked the validity of (18) by application of Eq. (10). In the non-trivial second regime Va
Vb , we use the above expressions, (34) and find that the
terms ∼ α2 in (10) are proportional to (Λa − Λb ) dΛb /dΛ,
which is identically zero.

C.

RG-equations to second order

As shown previously,23 there are also contributions linear in Λ to the conductances in second order of the interaction , i.e. subleading terms ∼ α2 Λ. These contributions arise from the diagram shown in Fig. 3, featuring
two fermion loops. Terms linear in Λ generate contributions to the RG beta functions. They give rise to α2 corrections to the scaling exponents. The modification
of the subleading terms in multi-lead junctions out of
equilibrium was not previously analyzed.
Performing calculations similar to the one described in
the Appendix we arrive at the following results. The beta

FIG. 3: A skeleton diagram showing the subleading contribution to currents in second order of perturbation

functions in Eq. (18) retain their general structure, with
updated coefficient functions aj . We have to replace
1
a1 → (1 − α(Ga − 2 )) a1 ,
1
a2 → a2 + α Gb (Ga − 2 )
8

× (1 − 1 Gb − g3 (Gb + g3 (Gb − 1))),
4
a3 → a3 −

α
4 Gb

1 + 2Ga (Ga − 1) −

(35)

3
4 Gb

1
+ 4 (Gb + 2g3 (Gb − 1))2 .

For the detached third wire, Gb = 0, we obtain
a2 (Ga , 0) = a3 (Ga , 0) = 0 and the modification of
a1 (Ga , 0) is in accordance with the second order expansion of Eq. (47) in Ref.23 . Notice that for the pure tunneling case,10 when Gb = 4Ga (1 − Ga ), the part with
θa ( ) disappears, a1 = 0, and the two RG equations
with θ+ ( ) become linearly dependent, since in this case
d
22
dΛ (Gb − 4Ga (1 − Ga )) = 0.
At the same time there is not much simplification in
(35) in the case of Va ∼ Vb , when we can let θ+ ( )
2θa ( ). The remaining expressions are complicated, as
can be seen, e.g. by expanding Eq. (12) in14 in powers
of α at c = 0.
V.

SUMMARY

In this paper we established the validity of the RG
equations for the conductances of multilead junctions of
Tomonaga-Luttinger liquid wires in a situation out of
equilibrium. Comparing to the equilibrium case, when
the RG flow stops at some unique cutoff, which characterizes the low-energy scale of the whole system of wires,
the out-of-equilibrium situation can be characterized by
several such scales, referring to (N − 1) relative voltages
between the N wires. In this situation it is not clear
which of these scales should be used as a cutoff in the
corresponding expressions. Previously we found22 that
the RG equations contained several functions, describing
partial stops of the RG flow, so that the direction of the
flow could alter during the renormalization process. In
this paper we formulate the statement about the scaling property for the set of conductances, characterizing
a general setup with N wires. Then we consider the particular example of the Y junction (N = 3) with different

7
strength of interaction in the main wire and the tunneling
tip. We focus on the two most interesting regimes, when
i) all voltages are of the same order and ii) the voltage
Va in the main wire is much smaller than the voltage Vb
at the tip.
The second order corrections are calculated in two
ways. One way is the iteration of the RG equations to second order, which is less trivial in the presence of several
cutoffs. A second way is the direct calculation of second
order corrections by means of computer algebra, which
requires considering a large number of partly canceling
contributions. We find that both ways of calculation lead
to identical results in both regimes. As a by-product we
derived the corrections to the beta functions of second
order in the interaction.
We believe that our results may be useful for a generalization of ideas of scaling in the presence of several low energy cutoffs, appearing particularly in out-of-equilibrium
situations.
VI.

ACKNOWLEDGEMENTS

It is convenient to symmetrize the appearing expressions with respect to ω → −ω, picking the odd-in-ω part
of the integrand, and then to consider a positive interval
of energies in subsequent integrations :
ω0
0

(A1)

with ω0 ultraviolet cutoff.
Let us now discuss the integration over Ω. In general,
we find terms, linear in hl (ω) = tanh[(ω − µl )/2T ] ≡
h0 (ω − µl ), and cubic in this quantity, ∼ hl hm hj . The
quadratic terms, hl hm , disappear.
Every
cubic
combination
has
the
form
h0 (Ω1 )h0 (Ω2 )h0 (Ω3 ), with Ωj = Ω + . . ., (e.g.
Ωj = Ω − µ1 + ω or Ωj = Ω − µ2 + ω − ω ). In order to
regularize the integral over Ω, we subtract a term h0 (Ω3 )
so that the combination (h0 (Ω1 )h0 (Ω2 ) − 1)h0 (Ω3 ) is
convergent at Ω → ±∞. All regularization terms h0 (Ω3 )
are combined with the other terms of the first power in
h0 (. . .).
In the so regularized terms we may shift the argument
and write

The work of D.A. was partly supported by RFBR grant
No. 15-52-06009 .
Appendix A: Details of calculation

dω dω
4 ωω

dΩ(h0 (Ω + ω1 )h0 (Ω + ω2 ) − 1)h0 (Ω + ω3 ),
˜
˜
˜
=

dΩ(h0 (Ω + ω1 − ω3 )h0 (Ω + ω2 − ω3 ) − 1)h0 (Ω),
˜
˜
˜
˜

≡ f3 (˜ 1 − ω3 , ω2 − ω3 )
ω
˜ ˜
˜
The expression for the corrections (24) requires five
integrations.
Let us first discuss the integration over x, x . The dependence of each Green’s function in (3) on the coordinates comes from two factors: the step functions, θ(x),
and the oscillatory exponentials, eiωx . The outgoing current is determined at a point z in the lead, which is outside the interacting region. In our terms this means that
the coordinate z is greater that any other of the coordinates, x, x . This allows to simplify the step functions by
replacing θ(z − x) = 1, θ(x − z) = 0, etc. The corrections
to the incoming currents are zero, which is verified by
putting θ(z − x) = 0, θ(x − z) = 1. The exponents e±iωx
do not contain Ω, and after appropriate change of sign
in ω , ω can be reduced to unique form e−2i(ωx+ω x ) .
The remaining expressions may still contain θ(x − x ),
θ(x −x), however, after symmetrization, x ↔ x , ω ↔ ω ,
these stepwise functions combine to unity.
The integration over x, x is now simple, since the dependence on the coordinates in each term is reduced to
e−2i(ωx+ω x ) . We have
L

dx1 e−2iωx =
a

e−2iωa − e−2iωL
1
→
2iω
2iω

where the last equality is obtained because the rapidly
oscillating factor e−2iωL is only important as an infrared
cutoff at the smallest ω, and in our case this cutoff is
provided by the voltages. The integration over x,x hence
leads to the overall factor, −1/(4ω1 ω2 ).

(A2)
with
f3 (A, B) =

Ω
dΩ tanh 2T (tanh Ω+A tanh Ω+B − 1),
2T
2T

A
B
= 2 coth A−B A coth 2T − B coth 2T ,
2T

→ 2(|A| − |B|) sign|A − B| ,
(A3)
the last line obtained in the limit T → 0, which we are
mostly interested in.
The terms linear in h0 (. . .) cancel each other. This can
be proved in two steps. First we subtract one and the
same term h0 (Ω) from each term, h0 (Ω + ω1 ) → h0 (Ω +
˜
ω1 ) − h0 (Ω), making the integral convergent:
˜
dΩ(h0 (Ω + ω1 ) − h0 (Ω)) = −2˜ 1 .
˜
ω

(A4)

The combination of all such terms does not contain ω, ω
and hence vanishes when performing the symmetrization
ω → −ω, leading to above expression (A1). As a second
step we sum all terms with h0 (Ω) and verify that they
cancel each other.
Let us discuss further simplifications. When obtaining
the terms, f3 (. . .), the third argument Ω + ω3 in Eq. (A2)
˜
was chosen by computer, i.e. almost randomly from the
human viewpoint. It means that many appearing terms
may look differently but lead to the same result after
subsequent integrations. To get rid of this ambiguity we

8
use the symmetry properties
(i) f3 (a, b) = f3 (b, a), (ii) f3 (a, b) = −f3 (−a, −b),
(iii) f3 (a, b) ≈ f3 (a − b, −b),
(A5)
where the last approximate equality means generation of
the linear in h0 terms which eventually disappear. The
last equivalence property is (iv) the symmetry with respect to ω ↔ ω .
We find that the number of expressions to be considered is strongly reduced, when we take each correction
term of the form Af3 (a, b), strip its prefactor A and perform the symmetry operations, (i) – (iv), for f3 (a, b). We
thus form an equivalence list of length 24 = 16 which is
ordered according to the computer’s internal rules. We
choose a first element f3 (a∗ , b∗ ) of this ordered list as
a representative. Such operation leaves only the factors
f3 (a∗ , b∗ ) which are not related by symmetry operations,
i.e. are essentially different.
Note that at this step of our analysis we may find terms
of the form f3 (ω − ω + a, b) with a, b dependent only on
the µj . Recalling the initial expression e−2i(ωx+ω x ) we
see that a shift ω = ω + ω and integration over ω leads
¯
to δ(x + x ). In combination with the condition x, x > 0
this means that such terms should be discarded.
In the intermediate expressions for the three types of
diagrams in Fig. 2 we found terms containing neither µ1
nor µ2 . Such unphysical terms cancel in the combination
of all three types of diagrams.
To condense the expressions further we introduce the
symmetric combination
fs (A, B) = −f3 (−ω − A, ω − B) + f3 (−ω + A, ω + B)
+ f3 (ω − A, ω − B) − f3 (ω + A, ω + B)
(A6)

1

2
3

4

5

6

7
8
9
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with the properties
fs (A, B)
fs (A, B)|ω=0

8A , ω
ω,
8B , ω
ω,
= fs (A, B)|ω =0 = 0

(A7)

From the piecewise linear form of (A3) it is clear that
the transition between the values fs 8A and fs 8B
takes place at |ω − ω | |A − B|.
The corrections to the currents are then expressed
through integrals

{Fa1 , Fa2 , Fa3 , Fb1 , Fb2 , Fb3 }
ω0
dωdω
fs (µ1 , 0) − fs (µ2 , 0) ,
=−
4ωω
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fs (µ2 − µ1 , µ2 ) + fs (µ2 − µ1 , −µ1 ) ,
fs (µ1 , 0) + fs (µ2 , 0) , fs (µ1 , µ2 ) ,
fs (µ2 − µ1 , µ2 ) − fs (µ2 − µ1 , −µ1 )}
We have here a generic integral

(A8)

ω0

dωdω
fs (A, B)
4ωω
0
ω0
ω0
+ B ln2
, |A| ∼ |B|,
= A ln2
|A|
|B|
ω0
ω0
ω0
= (A − B) ln2
+ 2B ln
ln
,
|A|
|A| |B|

|A|

|B|,

(A9)
we use these estimates for the calculation of the results
(33) in the main text.

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