Y-junction connecting Luttinger liquids: fixed point structure and conductances
D.N. Aristov1, 2, 3 and P. W¨lfle2, 4
o

arXiv:1105.5883v2 [cond-mat.str-el] 6 Oct 2011

2

1
Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
3
Department of Physics, St.Petersburg State University,
Ulianovskaya 1, Petrodvorets, St.Petersburg 198504, Russia
4
Institute for Condensed Matter Theory, and Center for Functional Nanostructures,
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
(Dated: June 15, 2018)

We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y-junction, within the scattering state formalism. We first
formulate the problem in current algebra language and focus on the case of a symmetric set-up,
for which the scattering matrix and the matrix of conductances is parametrized by two variables.
For these we derive coupled RG equations, first in a ladder summation up to infinite order in the
interaction. The fixed point structure and the implicit solution of these equations is presented. It is
shown that higher order terms beyond the ladder approximation do not change the scaling behavior
near the fixed points. For sufficiently strong attractive interaction a new fixed point with unusual
properties is found.

I.

INTRODUCTION

Electron transport in strictly one-dimensional quantum wires is governed by the Coulomb interaction between electrons. This is spectacularly demonstrated
by the fact that within the Tomonaga-Luttinger liquid
(TLL) model in the limit of temperature T → 0 the conductance of a quantum wire with finite barrier tends to
zero, provided the interaction is repulsive and assumes
the maximum value G = 1 (in units of the conductance
quantum G0 = e2 /h ) for attractive interaction. The
latter behavior appears independent of the strength of
the scattering at the barrier, and can be traced back to
the formation of Friedel oscillations of the charge density around the barrier, leading to an infinitely extended
effective barrier potential in that limit.
TLL behavior has recently been studied experimentally in carbon nano-tubes.1,2 In a future nanoelectronics constructed out of quantum wires, junctions of three
or more wires will necessarily be involved, requiring a
knowledge of the fundamental transport behavior of electrons in such structures. It is known that such systems
exhibit rather rich TLL effects which have been the subject of a number of recent papers.3–17 Much of the work in
this field has used the bosonization method, which gives
rise to the problem of how to preserve the fermionic character of charge carriers. When the number of wires meeting at a junction exceeds two, the Klein factors, which
give Fermi statistics to the bosonized operators of different wires, are more difficult to handle.3 Oshikawa et al.
recently introduced18 a new method to study this problem, mapping it into the dissipative Hofstadter model
(DHM), which describes a single particle moving in a
uniform magnetic field and a periodic potential in two
dimensions and coupled to a bath of harmonic oscillators. When the three quantum wires enclose a magnetic
flux, the mapping to the DHM also allows to identify a
new low energy chiral fixed point with an asymmetric

flow of current that is highly sensitive to the sign of the
flux. The paper by18 is giving a systematic and thorough treatment of the three lead junction problem, and
we shall refer to it later, when we compare our results
with theirs. It should be noted, however, that in spite
of their success, the above mentioned methods do not in
general allow to determine the fixed point (FP) structure
of the theory in an unbiased way, as they require a certain
knowledge of the existence of FPs as a starting point.
In this paper we follow a different route: we describe the transport properties in fermionic language,
thus avoiding the problem of Klein factors and the necessity to assume the existence of FPs. As shown in our
previous work, one may recover the known results on
transport through a TLL wire with barrier and obtain
new results not accessible by other methods within the
fermionic formulation.19 Moreover, as this method is formulated in a scattering wave picture, the connection of
the interacting wire to non-interacting reservoirs is naturally included. While this formulation has previously
been regarded as being restricted to weak coupling6,20,21
we have shown in an earlier work that it may be extended
to the strong coupling regime in a controlled way by using
an infinite resummation of perturbation theory (ladder
summation). The result obtained in this approximation
is universal, i.e. does not depend on the ultraviolet cutoff
chosen.
We restrict ourselves to a symmetric time-reversal invariant setup, which is characterized by two independent
components of the conductance tensor. This setup in case
of one wire without interaction is relevant to the problem
of tunneling into a Luttinger liquid and was studied in22
for weak interaction. There it has been found that the
asymmetric fixed point A (perfectly conducting wire and
a vanishing tunneling amplitude into the tip, see below)
becomes unstable in the case of weak repulsive interaction once the description of the Y-junction is not reduced
to the simple tunneling model. A further remarkable re-

2
sult of that work is the finding of a nonmonotonic behavior of the conductance (e.g. as a function of temperature), in certain cases. We extend that work here to
arbitrary interaction and find similar behavior at strong
interaction.
The resulting coupled RG equations will first be analyzed with respect to their fixed point structure. We find
four fixed points, labelled N,A,M,Q, where at N (Neumann) the three wires are totally separated, at A (asymmetric) the third wire is separated, while the main wire
is perfectly conducting, at M (mystery) and Q (quaint)
the conductances take an intermediate value, depending
on the interaction strength. All fixed points are located
on the boundary curve marking the allowed area in the
plane of the two conductances. For repulsive interaction
only N is stable, A being a saddle point. For attractive
interaction N is unstable and M is stable. For stronger attractive interaction, beyond a threshold value A becomes
an additional stable fixed point. The conductances are
calculated analytically for several special cases of interest. We present and discuss the power law exponents
appearing in the various regimes.
In order to check the reliability of the ladder summation result, we calculate all scale dependent terms (in the
limit of T = 0 ) up to third order in the interaction. We
find several terms additional to the ladder terms. These
terms are nonuniversal and subleading, in the sense that
they disappear for the case of repulsive interaction, when
one of the stable fixed points is approached. For attractive interaction, the properties of the new fixed point M
appear to be affected by the three-loop terms beyond the
ladder approximation.
The paper is organized as follows. In section II we
define the model of a Y- junction considered here. We
introduce the currents in the SU(3)-representation and
define the Hamiltonian in the chiral representation. Section III is devoted to an analysis of the scattering matrix
and the derivation of the conductance tensor in terms
of the S-matrix components. We consider a simple tunneling case and the totally symmetric case, both characterized by only a single free parameter in the S-matrix
and the general Y-junction case with two parameters and
therefore two independent conductance components. In
section IV we show how the RG equation for the simple
tunneling case is extracted from perturbation theory in
the ladder approximation, and discuss the ensuing conductance. The RG equation for the general Y-junction
case is presented in section IV B. We further present the
results on the nonuniversal terms in third order perturbation theory in section V. Finally, the RG phase diagram
is presented and discussed in section VI.

II.

MODEL OF A Y-JUNCTION

We consider the setup of a Y-junction: a quantum wire
of finite length 2L connected to noninteracting leads, and
a tip with attached wire forming a contact (junction) in

the center of the wire, which for simplicity is assumed to
have the same length L, also connected to a noninteracting lead. We choose the origin of the coordinate system
at the junction and denote the two halves of the main
wire by indices 1 and 2 and the tip by 3. We consider
spinless fermions. In the following we will refer to the
so defined symmetric (leads 1, 2) set up as a Y-junction.
A particularly simple version to be considered below is a
simple tunneling junction (no tunneling barriers, only on
site tunneling). Below we will also briefly consider a totally symmetric junction (with respect to interchanging
any pair of wires).
The electrons are assumed to interact via a shortranged interaction of arbitrary strength and sign between
incoming and outgoing fermions. In the present case, instead of defining right- and left-movers, it is more appropriate to speak of incoming and outgoing waves, with
respect to the junction, or “origin”.
The short-range interaction is characterized by the amplitude g in the main wire ( 1 and 2) and a different amplitude g3 in the third wire. This interaction takes place
at x < L, measuring from the origin. In order to spatially
separate effects of the potential scattering at the origin
and the interaction, we also assume that the interaction
takes place at x > a0 , whereas the potential scattering
happens at x < a0 . The scale a0 appears below as shortrange cutoff of the theory.
The Hamiltonian of the system is
3

0

H =

Hj

dx
−∞

(1)

j=1

†
†
Hj = vF ψj,in i ψj,in − vF ψj,out i ψj,out
†
†
+2πvF gj ψj,in ψj,in ψj,out ψj,out

(2)

where we did not write explicitly the range function of
the interaction. In the chosen parametrization the amplitudes gj are dimensionless and enter the subsequent
expressions in the most convenient form.
We unfold the setup in the usual way,23,24 by putting
the incoming fermions on the negative x-axis and the
fermions, which have passed through the junction, at
positive |x|. As a result, we have a three-fold multiplet
of right-going fermions with non-local interaction. This
procedure is depicted in Fig. 1
The boundary condition at the origin is described by
the scattering S-matrix as follows. For elastic scattering by the central dot, the outgoing fermions at the origin are connected to the incoming ones by the relation
†
†
∗
ψk,out = Skm ψm,in . We choose S in the symmetric way
as described at length in the next subsection.
In the scattered states representation, the right- and
left- going fermionic densities acquire the form
ψj,in (x) = ψj (x),
†
ψj,in (x)ψj,in (x)

ψj,out (x) = (S.ψ)j (−x)

†
ψj,out (x)ψj,out (x) = ρj (−x)
˜
(3)
˜
here and below we use the notation A = S † .A.S.

= ρj (x),

3
may now be written.

S

g

ρ1 (−x)˜1 (x) + ρ2 (−x)˜2 (x)
ρ
ρ

1

2

1
= 2 Ψ† (−x)

2
3 λ0

+

1
√ λ8
3

Ψ(−x)
(5)

× Ψ† (x)

g3

(a)

+

1 ˜
√ λ8
3

Ψ(x)

˜
+ 1 Ψ† (−x)λ3 Ψ(−x)Ψ† (x)λ3 Ψ(x)
2

3
g
1

2
3 λ0

˜
where λj = S † .λj .S, or simply
1

2

S

2

3

1
1
1
1 ˜
ρ1 ρ1 + ρ2 ρ2 = √ λ0 + √ λ8 √ λ0 + √ λ8
˜
˜
3
6
3
6
1 ˜
+ λ3 λ3 ≡ ρ+ ρ+ + ρ− ρ−
˜
˜
2

3

(b)

g3

FIG. 1: (Color online) (a) The geometry of a Y -junction
is shown together with currents of incoming and outgoing
fermions, the type of fermions in the scattered states representation is indicated by its color. The local short-range
interaction between the fermions is shown by a wavy line.
(b) The equivalent representation in the chiral fermion basis
after the unfolding procedure; the initially local interaction
becomes non-local, as described by Eq. (8).

1
1
√ λ0 − √ λ8
6
3

ρ3 ρ3 =
˜


1
ρ1 = 0
ˆ
0

0
ρ2 = 0
ˆ
0

0
ρ3 = 0
ˆ
0


0 0
0 0 =
0 0

0 0
1 0 =
0 0

0 0
0 0 =
0 1

1
2

1
2
λ0 + √ λ8 + λ3
3
3

1
2

1
2
λ0 + √ λ8 − λ3
3
3

(4)

1
1
√ λ0 − √ λ8
6
3

H=

addition to these, we use also the matrix λ0 =
which is proportional to the unit matrix, 1.

2
3

1,

The outgoing densities are given by ρj
˜
=
†
∗
δjk Skl Skm ψl ψm = Ψ† .S † .ˆj .S.Ψ, so that in the
ρ
ˆ
matrix representation ρj = S † .ˆj .S. We will mostly
˜
ρ
omit the hat sign over ρ below.
In terms of the above densities the interaction terms

†
ψj i ψj

dx
−∞

(8)

j=1,...3

+ 2πΘ(a < x < L) (gρ+ ρ+ + gρ− ρ− + g3 ρ3 ρ3 )
˜
˜
˜
where ρ± , ρ3 refer to −x and ρ± , ρ3 refer to x, and Θ(a0 <
˜ ˜
x < L) is equal to 1 within the interval specified and zero
elsewhere.
For simplicity we do not consider here the so-called
g4 part of the fermionic interaction, i.e. the terms
†
†
πvF gj [(ψj,in ψj,in )2 + (ψj,out ψj,out )2 ]. It is known that
¯
the g4 -interaction can be absorbed into the redefinition of the group velocity inside the interacting region,
vF j = vF (1 + gj ). For finite L one can show that g4 does
˜
¯
not lead to a renormalization of d.c. conductance, which
is the quantity of our interest below. The effect of g4 on
the a.c. conductance can be analyzed, e.g., following the
guidelines in25 .
III.

Here and below λj , with j = 1, . . . 8 are the traceless
Gell-Mann matrices, discussed in the Appendix A. In

(7)

1
1
1
1
Here we defined ρ+ = √3 λ0 + √6 λ8 , ρ3 = √6 λ0 − √3 λ8 ,
1
and ρ− = √2 λ3 .
In terms of these quantities, the Hamiltonian takes the
following form (from now on we set vF = 1)
∞

Let us explain the meaning of these quantities.
We introduce a multiplet of incoming fermions Ψ =
(ψ1 , ψ2 , ψ3 ). The incoming density ρj = Ψ† .ˆj .Ψ is given
ρ
by a diagonal matrix, i.e.

1
1 ˜
√ λ0 − √ λ8
6
3

(6)

S-MATRIX AND CONDUCTANCES
A.

S-matrix

The most general S-matrix is defined as follows


r1 , t12 , t13
S = t21 , r2 , t23 
t31 t32 r3

(9)

where rj is the reflection amplitude for wire j, and tjk is
the transmission amplitude between wires j and k. The

4
matrix S is unitary S † S = 1, which allows its param8
eterization via the exponential, S = exp i j=0 θj λj .
Obviously, there is a redundancy in the description by
the nine real-valued parameters θj in the exponent, as
only the densities and not the fermion amplitudes enter
the observables.
The number of physically relevant parameters for the
description of the S-matrix may be determined as follows. One may fix the relative U (1) phase between the
ingoing and outgoing electrons in each wire, by demanding that the reflection coefficients are real valued. This
excludes the λ3 and λ8 components. It turns out that it
is more convenient to keep the λ8 component, at the cost
of introducing some redundancy, see below. In addition
the overall phase described by the λ0 term may be set to
zero.
A further reduction in the number of independent parameters may be derived from the following consideration. In the limit of three almost detached wires, we
have S
1 + i 8 θj λj , and θj
1. This limiting
j=0
case elucidates the meaning of the θj as the tunneling
amplitudes between the corresponding wires. For example, the infinitesimal hopping between wires 1 and 3 of
†
the form (t13 ψ3 ψ1 + h.c.)|x=0 leads to θ4 = Re(t13 )/vF ,
θ5 = Im(t13 )/vF with vF the Fermi velocity, cf.18 . It
should be noted that this correspondence between the
Hamiltonian and the S-matrix holds only in the limiting
cases, and starts to depend on the definition of the regularization procedure in higher orders of tkl , i.e. beyond
the Born approximation.19
By using the above correspondence of the (small) θj
and the tunneling amplitudes it becomes clear that without loss of generality one may require real-valued hopping
amplitudes t13 and t23 , i.e. coinciding phases for hopping
from the third wire into wires 1 or 2 (in the absence of
magnetic flux). This makes θ5 = θ7 = 0. After that
the phase of the complex valued amplitude t12 between
the wires 1 and 2 is fixed. In the absence of magnetic
fields it must be zero18 anyway, which further reduces
the number of components, since it requires θ2 = 0.
Thus in the time-reversal invariant situation, the Smatrix can be parametrized by three angles, θ1 , θ4 , θ6 .
In the presence of a magnetic field a fourth component,
θ2 , appears. We do not consider this case here.
In this paper we concentrate on the case of tunneling
into the center of a Luttinger liquid wire. This amounts
to full symmetry between the wires 1 and 2, and it reduces the number of independent parameters further to
only two, θ1 and θ4 = θ6 .
As mentioned above, we will in addition keep the angle
θ8 . The reason for the introduction of the λ8 component
is two-fold. First, the explicit analytic expressions for S
below are somewhat simplified. Second, even if we choose
to start without λ8 component, it will be generated during the renormalization process, as we show below in Sec.
IV B. This means that the three angles θj , j = 1, 4, 8, are
not independent. Below we will identify proper combi-

nations of these variables forming a minimal set of two
independent variables, in terms of which all the other
quantities may be expressed. We then parametrize the
˜
˜
three angles in terms of a set of new angles θ, φ, ψ as
follows
8

√
˜
θj λj = θ cos φ(λ4 + λ6 )/ 2
(10)

j=1

+

1 ˜
2 (θ sin φ

˜
+ ψ)λ1 +

2

1
√

˜
˜
(3θ sin φ − ψ)λ8
3

Explicitly we have the representation of the S-matrix
˜
˜
in terms of three angles θ, φ, ψ


r1 t1 t2
˜
(11)
S = eiψ/3  t1 r1 t2 
t2 t2 r2
with
˜
˜
˜
r1 = 1 (e−iψ + cos θ + i sin θ sin φ)
2
˜
˜
˜
t1 = 1 (−e−iψ + cos θ + i sin θ sin φ)
2
i
˜
t2 = √ sin θ cos φ

(12)

2

˜
˜
r2 = cos θ − i sin θ sin φ
To identify one of the variables as redundant, we use
˜
˜
the transformation (θ, φ, ψ) → (θ, ψ, γ), with
˜
tan γ = sin φ tan θ,
˜
cos θ
cos θ =
.
cos γ

˜
ψ = ψ + γ,
(13)

After some calculation we find the components of the
S-matrix (up to an overall phase factor ei(ψ−γ)/3 ) as
1
r1 = 2 (e−iψ + cos θ)eiγ

t1 = 1 (−e−iψ + cos θ)eiγ
2
t2 =

i
√
2

sin θ

(14)

r2 = cos θe−iγ
We show in the next subsection that the new third
variable γ does not appear in any of the components of
the conductance. In addition, we will see later in Sec.
IV B that γ does not take part in the renormalization
process.
It will turn out to be useful to consider the most
elementary case of tunneling from the third wire into
the main wire (no next neighbor tunneling or tunneling barrier) separately. In that case the S-matrix is
√
√
characterized by a single angle, θ = 2θ4 = 2θ6 ,
whereas θ1 = 0. In terms of the above variables we have
˜
˜
(θ, φ, ψ) = (θ, 0, 0) or else (θ, ψ, γ) = (θ, 0, 0).
In the fully symmetric case of a junction of three
identical wires (identical interaction strength and reflection coefficients) we have only one independent angle

5
characterizing the S-matrix, θ = θ1 = θ4 = θ6 . The
˜
˜
connection to the above (θ, φ, ψ) , (θ, ψ, γ) is not very
transparent, and up to an overall phase factor we have
r1 = r2 = (2 + e3iθ )/3, t1 = t2 = (−1 + e3iθ )/3.

in the general form
Gaa Gab
Gba Gbb

ˆ
G=

=

B.

Conductances

The observables we concentrate on in this work are
the linear conductances. We first discuss the question
of the number of independent linear conductances. The
matrix of conductances, Gjk , is defined through the relation connecting the current Ij in a given wire with the
electric potentials Vk in all the leads as Ij = k Gjk Vk .
The current in the jth wire is given by Ij (x) =
evF ( ρj,in (x) − ρj,out (−x) ). The electric potentials
Vk give rise to the following source term in the Hamil0
tonian, HV = e −∞ dx k Vk (ρk,in (x) + ρk,out (−x)).
In linear response theory19 the currents are given by
0
Ij (x) = −∞ dy( ρj (x)ρk (y) − ρj (−x)ρk (y) )Vk In the
˜
static limit and in the absence of interaction the response
functions may be evaluated to give
ˆ ˆ
Gjk = δjk − T r(ρj ρk ) = δjk − |Sjk |2
˜

(15)

One easily verifies that the charge is conserved, j Gjk =
0, and that applying equal potentials to all wires produces no current, k Gjk = 0.
In view of these conservation laws, it is more instructive to discuss the current response to certain bias voltages. Let us define (Ia , Ib , I0 ) = G.(Va , Vb , V0 ), with
Va = (V1 − V2 ), Ia = (I1 − I2 )/2,
Vb = (V1 + V2 − 2V3 )/2 Ib = (I1 + I2 − 2I3 )/3, (16)
V0 = (V1 + V2 + V3 )/3, I0 = (I1 + I2 + I3 )/3,
In this notation, we seek the retarded response of the
˜
current Ia (x) = 1 evF ( ψ † λ3 ψ(x) − ψ † λ3 ψ(−x) ), to
2
0
e
the source term , e.g., HV,a = 2 −∞ dxVa ψ † λ3 ψ(x). We
can write these combinations symbolically in the static
limit as
1
HV = 2 Va λ3 +

1
√ Vb λ8
3

1
˜
Ia = 2 (λ3 − λ3 ),

I0 =

1
√ (λ0
6

+

Ib =

1
√ V0 λ0 ,
6
1
˜
√ (λ8 − λ8 ),
3

(17)

˜
− λ0 ) → 0,

As expected, I0 = 0; in addition we may choose the zero
of the electric potentials such that V0 = 0. It can be
shown19 that the conductance in the d.c. limit is proportional to the trace of the product of current and the
source vertices. It follows then that the line and the row,
corresponding to I0 and V0 are identically zero, and we
omit them for clarity below. The remaining four components are non-zero and we obtain the conductance matrix

1
2

1
˜
1 − 2 T r(λ3 λ3 ) ,
1
˜
− 2√3 T r(λ8 λ3 ),

1.

1
˜
− 2√3 T r(λ3 λ8 )
2
3

˜
1 − 1 T r(λ8 λ8 )
2



(18)



Conductance of Y-junction

For the particular choice of S given in (14) above we
have
ˆ
G=
=

1
|t2 | + 2 |t2 |, 0
1
2
0,
2|t2 |
2
1
2 [1

− cos θ cos ψ],
0
0,
sin2 θ

(19)

We see that the new third variable γ does not appear
in any of the components of the conductance. In addition, we show below that γ does not take part in the
renormalization process. Hence, in line with the above
argument, the general symmetric case is completely determined by two independent parameters, which can be
ultimately chosen as the two non-zero components of the
conductance matrix, Gaa ≡ Ga , Gbb ≡ Gb , Eq. (19). We
will present two coupled RG equations for the two conductances Ga , Gb in Sec. IV B below.
It follows from the parametrization (19) of the conductances that the physically accessible area in the twodimensional space of conductances is not simply given by
the unit square, but is defined by
0 ≤ Gb ≤ 1 − 4 Ga −

1 2
2

(20)

We will see below that the boundary of the physical
regime in Ga , Gb -space defined by (20) plays a special
role in that all fixed points of the problem are located
on the boundary. We will use this fact to our advantage
when we consider the simplified structure of the RG-flow
along the boundary curve.
In the most general case, the matrix of conductances
has four independent components, in accordance with the
analysis of the number of relevant parameters characterizing the general S-matrix given in26 .
It is worth to note the following property of the boundary (20). It was proven in27 that the boundary of the
region of allowed conductances, defined by Eq. (15), corresponds to the matrices S which can be made entirely
real by certain “rephasing”, i.e. multiplication of rows
and columns of S by phase factors. It was also proven
that one can recover S from G up to this “rephasing”.27
In our particular case (14) the “rephasing”, leading to
the real-valued last line and last column, is achieved by
diag[ie−iγ , ie−iγ , 1].S.diag[i, i, eiγ ]. It is then clear, that
the boundary corresponds either to ψ = 0, π, or θ = 0, π,
which is indeed the case, as shown below (see also26 ).

6
2.

Conductance in the simple tunneling case

Let us now consider the special case when the S-matrix
is characterized by only a single parameter, θ = θ4 = θ6 .
This corresponds to the simplest model of tunneling from
the tip into the wire. Then we have
˜
λ3 = cos θλ3 −
˜
λ8 = −

√

3
2

1
√
2

sin θ(λ4 − λ6 )

sin2 θλ1 −

3
8

sin 2θ(λ4 + λ6 )

(21)

1 + 3 cos 2θ
+
λ8
4
and the reduced conductance tensor follows as
ˆ
G=

sin2
0,

θ
2

0
sin2 θ

(22)

We note that the conductances satisfy the relation
Gb = 1 − 4(Ga − 1 )2 , implying that the simple tun2
neling case traces the boundary of the physically allowed
region in Ga , Gb -space, see (20). As we will see below this
case is more than a specialization to a very simple physical model. It actually already contains the information
on the fixed point structure of the RG flow of the general
model. Its advantage is that the fixed points and even
the conductances may be calculated analytically. The
stability of the fixed points cannot be decided in this restricted model, as runaway flow away from the boundary
may occur (see Sec. IV B below).

in perturbation theory. To illustrate our approach, we
˜
˜
consider first the special case of the S-matrix, (θ, φ, ψ) =
(θ, 0, 0) in Eq. (10). Our consideration follows closely our
analysis of the simpler case of one impurity in a Luttinger
liquid wire.19 In comparison with this previous case, we
now have a few channels of interaction in the Hamiltonian
(8). For the reader’s convenience, we now outline the
basics of our approach (see also Sec. V).
Consider the causal Green’s function for fermions
†
Gij (x, t; y, t ) = −i Tt ψj,in (x, t)ψj,out (−y, t ) . In the
non-interacting system (gj = 0) and given our model assumption of equal Fermi velocities in the wires, we have
†
Gij (x, t; y, t ) = G(t − t, y − x)Sji , according to (3). Here
the scalar quantity G describes the kinetic part of the
Green’s function in the scattering states representation.
For the Wick-rotated (imaginary time) quantity G we
have
−1
G(iω ; z) = −ivF sign(ω)Θ(ωz)e−ωz/vF .

The renormalization of the junction is obtained by considering the d.c. limit of Glm (x < −L, y > L, ω → +0),
and the contribution of interaction terms order-by-order.
∗
Without interaction we have in this limit −iSlm , when
we set vF = 1. The renormalized S-matrix is then de∗
fined by −iSlm |r = limω→+0 Glm (ω; x < −L, y > L) The
first correction to Glm (x, 0; y, t) in the basis of scattered
states is of the form
δGlm (x, t; y, t ) = 2πgk

dz

dτ ψl (x, 0)

†
× ψi (−z, τ )ρi i ψi (−z, τ )
k

3.

Conductance in the fully symmetric case

†
†
∗
ρk
× ψj (z, τ )˜j j ψj (z, τ ) ψj (y, t) Sjm ,

For the fully symmetric case with t1 = t2 = 1 (e3iθ − 1)
3
we have from Eq. (19)
ˆ
G=

2
3

sin2
0,

3θ
2

8
9

0
sin2

3θ
2

(23)

which shows that the maximum transparency of the fully
symmetric junction is reached at θ = π/3 and corresponds to Ga = 2/3, Gb = 8/9. These values are smaller
than the maximum (unity) values of the individual conductances, which according to (20) are attained at the
points Ga = 1/2, Gb = 1 and Ga = 1, Gb = 0 in the
Ga − Gb - plane.
IV.

RG EQUATION: UNIVERSAL
CONTRIBUTIONS
A.

Simple tunneling case
1.

Lowest order

The renormalization of the S-matrix due to interaction
can be understood by considering the simplest diagrams

where summation over repeated indices is implied, and
the matrices ρk and ρk are given in (4). The two possible
˜
ways of contraction of the fermion operators lead to the
expressions
∗
2πgk G(τ, −z − x)ρlj G(t = 0, 2z)˜j j G(t − τ, y − z)Sjm ,
ρk
k

and
∗
2πgk G(τ, z − x)˜li G(t = 0, −2z)ρi j G(t − τ, y + z)Sjm .
ρk
k

Multiplying these expressions by eiωt and integrating
over τ, t, we note that the dependence on x and y disappears in the limit ω → 0. The renormalized value of
∗
−iSlm is hence of the form
∗
∗
−iSlm |r = −iSlm − 2πgk

dz ρlj G(t = 0, 2z)˜j j
ρk
k

∗
+ ρli G(t = 0, −2z)ρi j Sjm ,
˜k
k

or, symbolically,
∗
Sr .S = 1 − 2πigk

dz G(0, 2z)ρk ρk + G(0, −2z)˜k ρk .
˜
ρ

7
The above correction may be interpreted as the
self-energy Fock diagrams ΣF , since the renormalized
propagator may be represented generally in terms of
the self-energy as Σ as Gr (−L, L) = G0 (−L, L) +
G0 (−L, y)Σ(y, z)G0 (z, L), where integration over y, z is
implied. Using G(0, 2z) = −i/4πz, we may write this
correction (first at g3 = 0) as

Let us now discuss how our method can be extended
to the situation of the Y-junction. First we discuss the
ladder summation, then we present the results of a computer symbolic calculation of the three-loop terms.

2.

Ladder summation

L

dz
([ρ+ , ρ+ ] + [ρ− , ρ− ])
˜
˜
a0 2z
g i(λ4 + λ6 )
√
sin θ (1 + cos θ)
=− Λ
4
2

ΣF = −g

(24)

with Λ = ln(L/a0 ). We see that the matrix Green’s function receives an off-diagonal static correction, signalling
the necessity to redefine the rotation angle θ in the representation (10) of the S-matrix. It is remarkable that
the above correction may be interpreted as a change of
the angle θ, δθ , as it is directed along the initial vector,
i.e., along λ4 + λ6 . This means that in the case of simple
tunneling we are allowed to consider only a single renormalization equation for θ, as opposed to a set of three
RG equations for θ, ψ and γ.
In the presence of the interaction in the third wire, contributing a term g3 ρ3 ρ3 in the Hamiltonian, the changes
˜
in the above expression for Σ in first order of the interaction are minimal. In addition to the above combination
g ([ρ+ , ρ+ ] + [ρ− , ρ− ]) we should add g3 [ρ3 , ρ3 ], but since
˜
˜
˜
˜
[ρ+ , ρ+ ] ∼ [ρ3 , ρ3 ] ∼ [λ8 , λ8 ] we obtain
˜
˜
√
Σ = i(λ4 + λ6 )δθ/ 2,
(25)
δθ = − 1 Λ (g sin θ + (g + 2g3 ) sin θ cos θ)
4
This equation (considering it as a precursor to the RG
equation) is equivalent to Eq.(8) in21 (see also6 ).
In our previous work,19 we showed that the renormalization of the impurity in the Luttinger liquid can be
analyzed within the fermionic formalism. The change of
θ induced by the interaction and calculated in first order
perturbation theory in (25) may now be used to obtain
the renormalization group equation for θ in lowest order
dθ
1
(26)
= − (g sin θ + (g + 2g3 ) sin θ cos θ)
dΛ
4
In our earlier work we showed that higher order terms
in the interaction may be summed in a systematic way,
to access the strong coupling domain. In particular, we
showed that the one-loop contributions to the RG equations for the S-matrix form a ladder series, which can be
resummed by solving an integral equation of the WienerHopf type. The result of this summation reproduces the
known results obtained with the bosonization method for
the weak and strong impurity case. It is found to be universal in the sense that it does not depend on the choice
of regularization of the logarithmic divergences in the
theory. We also showed that two-loop RG contributions
are absent and the three-loop RG corrections are not universal but are not very sizeable in the whole range for a
realistic choice of model parameters.

Our previous solution of the ladder equation19
amounted to a the dressing of the interaction in the presence of the impurity. It thus led to a replacement g → g
in an equation analogous to (26) with
g=

2g
1+

1 − g 2 − gY

(27)

where Y = T r(ρ− ρ− ) ; notice the different sign in front of
˜
gY here, which is a result of the different parametrization
of the S-matrix in our previous work.
˜
In view of the symmetry 1 ↔ 2, we have T r(λ3 .λ8 ) =
˜ 3 .λ8 ) = 0. This means that the symmetric denT r(λ
sity combinations remain orthogonal after the scattering
T r(ρ− ρ+ ) = T r(˜− ρ+ ) = T r(ρ− ρ+ ) = T r(˜− ρ+ ) = 0.
ρ ˜
˜
ρ
This property allows us to perform the ladder summation
separately in each of the channels, as described below.
First we consider the simpler case of g3 = 0, the absence of interaction in the third wire. From Eq. (21) we
1
have T r(ρ− ρ− ) = cos θ and T r(ρ+ ρ+ ) = 4 (3 + cos 2θ).
˜
˜
The ladder summation of the one-loop RG contributions in each channel is performed along the previous
guidelines19 and the resulting RG equation is written as
dθ
g
≡ β(θ) = − sin θ
dΛ
4

2
1 + d − g cos θ
2 cos θ
+
1 + d − (g/4)(3 + cos 2θ)

(28)

with d = 1 − g 2 .
Let us now discuss the simultaneous presence of g, g3 =
0. The result of the summation in the ρ− channel is the
same, whereas the channels ρ+ and ρ3 begin to mix in
higher orders of the interaction. We omit the details
of the derivation below and only provide intuitive arguments for the result obtained.
The dressing of the interaction occurs in the individual
wires, characterized by two density components ρ+ , ρ3 ,
which are orthogonal before the scattering, T r(ρ+ ρ3 ) =
0. If these density components were orthogonal to those
after the scattering, (T r(ρ+ ρ3 ) = T r(ρ+ ρ+ ) = 0, etc.)
˜
˜
then the dressed interaction would simply be given by
(27) with Y = 0. We define renormalized interaction constants in the form of two ”charges” q, q3 by g → 2g/(1 +
−1
2
1 − g 2 ) = 2q −1 , g3 → 2g3 /(1 + 1 − g3 ) = 2q3 and
ˆ = diag[q, q3 ] here.
introduce the diagonal matrix Q
According to Eq. (4) the two density components
ρ+ , ρ3 are connected to a vector of scattering eigenmodes,

8
√
φ = (λ0 , λ8 )/ 2, by
1 ˆ λ0
=√ U
,
λ8
2
√
1
2, √
1
ˆ
U=√
1, − 2
3

ρ+
ρ3

1
dx
1
4x
+
= (1 − x2 )
dΛ
2
Q + 1 − 2x2
q−x
3
(x − x3 )(x − x4 )
,
= (x2 − 1) 2
2
(x − q0 )(x − q)

(29)
,

,
(32)

with
ˆ
with U 2 = 1 and T r(φi φj ) = δij . Notice that the
modes λ0 and λ8 remain orthogonal both before and
after the scattering, which is reflected in the matrix
1
˜
ˆ
Y = T r(φi φj ) = diag[1, 4 (1 + 3 cos 2θ)]. The result
of the ladder summation is represented by the matrix
ˆ ˆ ˆ ˆ
(Q − U .Y .U )−1 .
The renormalization of the S-matrix in the first order
occurs due to the non-vanishing commutators of densities [ρ+ , ρ+ ], [ρ3 , ρ3 ]. Going to higher orders in the in˜
˜
teraction, we have to take into account also the mixed
commutators, [ρ+ , ρ3 ], [ρ3 , ρ+ ]. Noticing that the only
˜
˜
˜
non-commuting components here are λ8 and λ8 , and using Eq. (29), we find that the effect of renormalizing the
interaction amounts to the replacement

ˆ ˆ ˆ ˆ
(g + 2g3 ) sin 2θ → 6 U .Q.U − Y

−1

sin 2θ ,

(30)

22

which leads to the explicit result for the β-function in the
ladder approximation:
1
sin θ
2 sin 2θ
+
2 Q − cos 2θ q − cos θ
4qq3 − 2q − 3q3 + 1
,
Q=
2q + q3 − 3
1+K
1 + K3
q=
, q3 =
,
1−K
1 − K3
1−g
1 − g3
K=
, K3 =
,
1+g
1 + g3

βL (θ) = −

,

(31)

where we also provide the alternative definition of q, q3
through the Luttinger parameters K, K3 .
The equation (31) is a central result of this paper, and
we analyze it in some detail in the next subsection. It
should be noted that in the limit g3 → 0 and for small g
this equation reduces to the Eq. (1) of Ref. [22], where the
fixed points of the point contact model were discussed.
We confirm Eq. (31) by calculating the perturbative corrections to the S-matrix up to the third order of the
interaction in Sec. V.

3.

Explicit solution of the RG equation

Let us analyze the RG equation (31). Introducing the
variable x = cos θ, which determines the conductivity
components via (22), we rewrite it in the form

1
(Q + 1) ,
2
1
3q0 + q 2 ] ,
= [q
3

q0 =
x3,4

(33)

Here Q is defined in (31). The zeros of the r.h.s. , the
four fixed points (FPs) of the RG equation, are given by
x1,2 = ±1, and x3,4 . For later reference we label the
fixed points x1 , x2 , x3 , x4 as N, A, M, Q. Fixed point N
corresponds to Ga = Gb = 0, i.e the three wires are separated. At fixed point A we have Ga = 1, Gb = 0, i.e. the
third wire is disconnected whereas the main wire is perfectly conducting. The fixed points M, Q are located on
the boundary of the physically accessible region, unless
they are outside the region defined by (20).
For repulsive interaction, 0 < K, K3 < 1, the fixed
point x4 > 1 is outside the physical domain, |x| ≤ 1.
One can easily verify that the two fixed points N, A at
x1,2 = ±1 are stable, i.e. dβ/dx|x1,2 < 0. We will see
below that A turns unstable in the general symmetric
case. Fixed point M at x3 ∈ (−1, 0) is unstable. At
weak coupling, we write K = 1 − g and K3 = 1 − g3 with
0 < g2
g, g3
1, and get x3 −g/(g + 2g3 ), cf.22 .
The equation (32) may be integrated to give the implicit solution for x as a function of the length L in the
form
F [x(L)] = (1 − x)γ1 (1 + x)γ2 |x − x3 |γ3 |x − x4 |γ4
(34)
= F [x(L0 )] L/L0 ,
where
γ1 =

1 (1 − q0 )(1 − q)
3 (1 − x3 )(1 − x4 )

−1
= − K −1 + K3 − 2

γ2 =

−1

,

1 (1 − q0 )(1 + q)
3 (1 + x3 )(1 + x4 )

=−

−1
1
2 (K

−1
+ K − 2) + K3 − 1

−1

(35)
,

2 q − x3 q0 − x2
3
,
3 x4 − x3 1 − x2
3
2 q − x4 q 0 − x2
4
γ4 = −
,
3 x4 − x3 1 − x2
4

γ3 =

We see that the fixed point N at x = 1, corresponding
to three fully detached wires, is characterized by the exponent γ1 , which is determined by the sum of two bound−1
ary exponents, K −1 and K3 . The second fixed point A
at x = −1, corresponding to the ideal wire 1 and 2 and

9
the detached wire 3, relates to the exponent γ2 , which
is governed by the boundary exponent of the third wire,
−1
K3 , and the bulk anomalous dimension of the fermion
operator (K −1 + K)/2, the latter quantity defining the
local density of states in tunneling experiments.
In the important special case of arbitrary but equal
interaction strength, K3 = K we have Q = (4q − 1)/3,
q0 = (2q + 1)/3 = x4 and x3 = −1/3. The expressions
for γ3,4 simplify and we have
−1
γ3 = 6

1−K
,
(2 + K)2

= 2 (K −1 − 1) −
3
−1
γ4 = −6

2 4−K
3 (2+K)2

K −1 + K − 2 ,

(36)

3−K
,
K(3 + K)

For completeness, we also consider the case of attractive interaction K, K3 > 1. First of all, the role of the
fixed points is reversed, so that the fixed points N, A at
x = ±1 are unstable and the third fixed point M at x3
becomes stable. Further, we observe that while for repulsive interaction x4 > 1 always lies outside the physical
region of x, this is different in the attractive case. For
simplicity, let us first consider the case of equal interaction in the wires, K = K3 . We have x3 = −1/3 and the
3+K
value of x4 = (2q + 1)/3 = 3(1−K) reaches the physical
range of x first and coincides with x = −1 at K = 3.
−1
−1
At this point the inverse exponents γ2 = γ4 = 0. A
subsequent increase of K takes −1 < x4 < −1/3 inside
the physical domain. One can verify that the fixed point
Q at x4 in this latter case is unstable, whereas the fixed
point A at the edge x = −1 becomes stable.
Another interesting possibility for attractive interaction is merging of the points x3 and x4 . It happens at
3q0 = −q 2 in (33), or at
K3 = 2K

1 + K3
,
(1 + K)2 (2K − 1)

and the position of FP in this case is
x3 = x4 = 1 (1 + K)/(1 − K) .
3
The last equations show, that in order to have |x3,4 | < 1
one should let K ≥ 2, K3 ≥ 4/3. We return to these
questions below in Sec. VI, when discussing the “phase
diagram” of our model.

B.

General Y junction

Let us now analyze the general case of the S-matrix,
Eq. 10. As shown above in Sec. IV A 1, the RG flow to
first-order in the interaction is determined by two contri˜
˜
butions, which are proportional to [λ3 , λ3 ] and [λ8 , λ8 ].
We write
˜
˜
S −1 .Sr − 1 = Λ(α3 [λ3 , λ3 ] + α8 [λ8 , λ8 ]) ,

(37)

where α3 = −g/4, α8 = −(g + 2g3 )/12 are introduced
for brevity. Comparing the right-hand side of the above
expression with the parametrization (12) of the S-matrix,
we can find the corresponding change of the parameters
˜
˜
θ, φ, ψ in the renormalized scattering matrix Sr .
After some calculation we find the following set of RG
equations
˜
dθ
˜
˜
˜
˜
= α3 (cos ψ sin θ + sin ψ cos θ sin φ)
dΛ
˜
˜
+ 3α8 cos θ sin θ cos2 φ,
˜
cos φ sin ψ
dφ
= α3
− 3α8 sin φ cos φ,
˜
dΛ
sin θ
˜
dψ
˜
˜
˜
˜
= 3α3 (cos ψ sin θ sin φ + sin ψ cos θ) .
dΛ

(38)

At first sight, there are three equations for three parameters. However, one can check that dψ/dΛ =
3 cos2 θd(sin φ tanθ)/dΛ, irrespective of the interaction,
so that only two of the equations are independent.
Indeed, using the equivalent parametrization of the Smatrix in the form Eq. (14), we obtain the RG equations
in the new variables θ, ψ, γ as :
dθ
= sin θ(α3 cos ψ + 3α8 cos θ),
dΛ
dψ
1
= α3 sin ψ 3 cos θ +
,
dΛ
cos θ
sin ψ
dγ
= α3
dΛ
cos θ

(39)

Therefore, we have only two independent RG equations
(39) for the variables θ and ψ, which define the components of the conductance matrix (19). The third component γ is not independent and is determined by θ and ψ
; it does not enter the conductance.
It is also instructive to express the RG equations directly in terms of the components of the conductance
tensor (18). We introduce
1
˜
a = 2 T r(λ3 λ3 ) = cos θ cos ψ,
˜
b = 1 T r(λ8 λ8 ) = 1 (1 + 3 cos 2θ),
2

(40)

4

ˆ
so that the conductance matrix is G = diag( 1 (1 −
2
2
a), 3 (1 − b)). From (39) we find
da
= −2α3 (1 + b − 2a2 ) − 2α8 (1 − b)a,
dΛ
db
= −2(1 − b)(α3 a + α8 (1 + 2b)),
dΛ

(41)

or, directly in terms of conductance,
dGa
3
= α3 8Ga (1 − Ga ) − 2 Gb + 3 α8 Gb (1 − 2Ga ),
2
dΛ
dGb
= 2α3 Gb (1 − 2Ga ) + 6α8 Gb (1 − Gb ),
(42)
dΛ

10
1

Comparing Eq. (39) to the above special case, (ψ =
γ = 0), we observe that the arguments, leading to the
possibility of the ladder summation in Sec. IV A 2, remain valid. Hence we can use the previous result, which
amounts to the substitution
α3 = −

2
= 3 (1 − b) + a2 = 1 − cos2 θ sin2 ψ ,

(43)
0.2

N

A

0
0

(44)

due to Eq. (42) has a form
dG⊥
∝ (1 − G⊥ )
dΛ

M
0.4

with q, q3 defined in (31). The equations (41), (42), (43)
are the main result of this paper.
Notice that the RG equation for the quantity
G⊥ = Gb + (2Ga − 1)2 ,

0.6

Gb

1 1
1
1
=−
,
2q−a
2 q − 1 + 2Ga
1
1
1 1
=−
,
α8 = −
2 Q1 − b
2 Q1 − 1 + 3 Gb
2
3qq3 − q − 2q3
Q1 =
2q + q3 − 3

0.8

(45)

which, together with (20) and second line in (42), shows
that the boundaries for the observable conductances for
free fermions, G⊥ = 1 and Gb = 0, are the RG fixed lines
in the interacting case.
Concerning the character of the fixed points in this
more general situation, there is a qualitative change. In
the tunneling case, and for repulsive interaction, we had
two stable FPs , A, N , corresponding to i) one wire detached (x = −1) and ii) three wires detached (x = 1).
The unstable FP M (x3 ) was located between these two
limiting cases.
Now we have two independent components of the conductance. We find only one truly stable FP : N , the case
of all three wires detached, Ga = Gb = 0. The previous
stable FP A with one wire detached (Ga = 1, Gb = 0)
transmutes into a saddle point, so it is not truly stable.
The third (unstable) FP M is at the boundary of the
region of conductances in the Ga − Gb -plane allowed by
unitarity and is unstable in both directions.
The RG flows for repulsive interactions of strength
g = 0.2, g3 = 0.03 are depicted in Fig. 2. It is interesting to note that the three representative flow trajectories emanating from fixed point M indicate nonmonotonic behavior of Gb (black) or of Ga (red) as a function
of, e.g. temperature. As shown in22 this behavior at
g3 = 0 appears first in second order perturbation theory in the interaction g. It was demonstrated there, that
the intermediate M point at g3 = 0 appears due to the
competition between the bulk zero-bias anomaly and the
scattering off Friedel oscillations induced by the third
wire. This underlines the difference in the origin of the
intermediate point in Ref. [22] and the origin of the intermediate point in this and other studies6,7,16,18,21 with
g3 = 0.

0.2

0.4

0.8

0.6

1

Ga

FIG. 2: (Color online) Ladder approximation for the βfunction and RG flows for g = 0.2 and g3 = 0.03. Three
fixed points are shown by dots and are : (i) stable one at
Ga = Gb = 0, (ii) unstable one at Ga 0.8542, Gb 0.4979,
and (iii) saddle-point type fixed point at Ga = 1, Gb = 0.
The values of conductances, allowed by unitarity, are shown
as a shaded region.

The scaling behavior in the case of attractive interaction is even more interesting (although attractive effective interaction is not easily realized in nature!). The
conductances at M are given by Ga = (1 − x3 )/2 and
Gb = 1 − x2 , where x3 is given in (33). As mentioned
3
above, fixed point N becomes unstable, while M is stable. In the case of not too strong attractive interaction,
1 < K, K3 < 3 , M is the only stable fixed point. Again
the behavior of both, Ga and Gb may be nonmonotonic.
For even stronger attractive interaction the fourth fixed
point Q enters the physically accessible region as a further unstable FP. This leads to a switch of fixed point
A from unstable to stable. In Fig. 3 the flow diagram is
shown for strong attractive interaction, K = K3 = 3.7 .
It is useful to analyze the RG equations (41) around the
N and A fixed points. For the N point which corresponds
to a = b = 1 we write a = 1 − a , b = 1 − b and find in
the leading order of the small deviations a > 0, b > 0:
d
a − 1b
3
dΛ
d
b
dΛ

2 1 − K −1
2−K

−1

1
a − 3b

−

−1
K3

,
(46)

b .

Similarly, around the A with a = −b = −1 we expand
a = −1 + a , b = 1 − b to obtain:
d
1
a − 3b
dΛ
d
b
dΛ

1
2 (1 − K) a − 3 b
−1
1 − K3 −

,

(1 − K)2
2K

(47)
b .

In the equations (46), (47) the combination a − 1 b > 0
3
measures the distance to the parabola b = (3a2 − 1)/2
of the simple tunneling case ψ = 0 in (40). Thus we
confirm that the border lines of the physical sector of

11
1

obtain from Eq. (41)
M

d¯
a
(1 − a)(1 + 3¯)
¯
a
=
+ O(c2 ) ,
dΛ
q−a
¯
1 + 3¯2 − 4¯q
a
a
dc
c + O(c2 ) .
=
2
dΛ
(q − a)
¯

0.8

Gb

0.6

0.4

This set of equations shows that the RG flows form a
“spindle” shape around the line a = b, with two edge
FPs at a = b = 1 (N point) and a = b = −1/3 (M point).
The first equation in (48) is easily integrated with the
result (cf. (34))

Q

0.2

A

N

0
0

0.2

0.4

0.6

0.8

1

Ga

FIG. 3: (Color online) Ladder approximation for the βfunction and RG flows for K = K3 = 3.7. Four fixed points
are shown by dots and are : (i) unstable one at Ga = Gb = 0,
(ii) stable one at Ga = 1, Gb = 0, (iii) stable one at Ga = 2/3,
Gb = 8/9 and (iv) new (appearing at K > 3) unstable fixed
point at Ga
0.9136, Gb
0.3159. The values of conductances, allowed by unitarity, are shown as a shaded region.

conductances, Eq. (20), are the “fixed lines” of our RG
equations.
The meaning of scaling exponents in (46), (47) is as
follows. The quantity a − 1 b is the difference of the
3
Y-junction from the simple tunneling case which may
arise primarily due to impurity scattering located in the
main wire. It is then quite natural that it scales at N
point with the usual double exponent, 2(1 − K −1 ), of
the weak tunneling between two equal Luttinger liquids.
The quantity b stands for the tunneling conductance and
its exponent combines the weak tunneling contributions
from the main and the third wire. At the fixed point A
2
we have the conductance Gb = 3 b defined by the weak
−1
tunneling from the third wire, (1 − K3 ), into a perfect
Luttinger liquid with a bulk exponent (1 − K)2 /2K. The
small value of a − 1 b in this case corresponds to a weak
3
barrier in the main wire, which scales with the exponent
2(1 − K).22,28

C.

(48)

˜a
F [¯(L)] = (1 − a)γ2
¯

1
3

+a
¯

γ3

˜a
= F [¯(L0 )] L/L0

(49)

−1
here γ2 = −2(K −1 − 1) follows from (35) and
−1

(γ3 )

=6

1−K
.
(2 + K)

(50)

Notice that the similar-looking γ3 in (36) defines the scaling exponent for c in (48).

V.

PERTURBATION THEORY AND
NON-UNIVERSAL TERMS

The ladder summation discussed above captures contributions of a certain type (one-loop) to all orders in
the interaction. By its structure it is similar to the RPA
summation scheme, which works perfectly well for the
description of the bulk properties of a Luttinger liquid,
due to the absence of multi-tail fermionic loops in the
linearized dispersion model29 . There remains the question of the importance of higher loop contributions. In
our work on the Luttinger liquid with barrier we were
able to show in perturbation theory that the two-loop
contribution vanishes and the three-loop contribution is
subdominant in the neighborhood of the fixed points. As
a result, the exponents of the power laws of the conductance in length L or temperature T turned out to be
in exact agreement with those obtained by methods of
bosonization, and the thermodynamic Bethe ansatz. In
this section we explore to which extent a similar result is
true in the case of a Y-junction.

Fully symmetric Y junction
A.

In this section we consider the case of full symmetry
between the wires, which means equal strength of interactions, K = K3 . The asymmetry at the Y-junction may
still lead to an asymmetry in the physical properties of
our system. However, it is a remarkable fact that the
symmetric Y-junction, a = b, in other terms Gb = 4 Ga ,
3
remains a fixed line in this case.
We observe that for K = K3 we have q = q3 = Q1 and
1
α3 = 2 (a − q)−1 , α8 = 1 (b − q)−1 in (43). Introducing
2
the symmetrized variables, c = b − a, a = (a + b)/2, we
¯

Simple tunneling case

The symbolic computation of the diagrams up to the
third order can be performed as explained in the previous
work.19 The diagrammatic rules are similar, with the only
difference that we have 3 × 3 matrices for the vertices
now. This means that the kinematic structure for each
diagram is the same both for the Y -junction and for the
barrier in the Luttinger liquid. Therefore we can use the
previously obtained results for the individual diagrams,
while supplying them with different matrix prefactors.

12
We may represent the renormalized scattering matrix,
Sr , in the form
S † .Sr = 1 + gΣ1 + g 2 Σ2 + g 3 Σ3 + . . . ,

(51)

(in the presence of both g and g3 we consider Σj as dependent on the ratio g3 /g). In the next step, we verify
the functional form of the matrix S † .Sr = exp iδθ(λ4 +
√
λ6 )/ 2 and determine the scalar quantity δθ as a function of g, θ, Λ, g3 /g. We have for the renormalized value
θr = θ + δθ(θ) = f (θ, gΛ, g) and inverting this equality
we may write θ = f ∗ (θr , gΛ, g). Demanding now that the
initial ”bare” value θ should not depend on Λ, we may
derive the RG equation.19
Following these steps, we arrive at an expression for
the β-function, which we keep up to third order in g, g3 .
We compare this perturbative result with the Taylor expansion of Eq. (31) and confirm the identity of the two
expressions up to second order in the interaction. In the
third order of g, the direct calculation of the β-function
provides the ladder contribution given by the corresponding term in the Taylor expansion of Eq. (31) and in addition a three-loop contribution to β(θ) beyond the ladder
result. In accordance with Ref. [19], we find that this
three-loop correction is a subleading contribution, stemming from the non-universal parts of the diagrams. As
was explained in the previous work, performing the evaluation of the diagrams we find a few generic integrals,
which contain non-universal numbers b2 , b3 , b4 which for
T = 0 are equal to π 2 /6, 2 ln 2, 2(ln 2 − 1), respectively.31
The extra term in the β-function, which should be added
to (31) is given by
g3
sin3 θ (b2 f2 + b3 f3 + b4 f4 )
128
f2 = 1 + 8z + 2z 2 + 4z 3 + (1 + 2g3 /g)2

β3 (θ) =

f3 = (1 − z)2 − 4g3 /g

third fixed point x3 is affected by the β3 term at g3 = g;
and the critical exponent γ3 is affected in this case, too.

(52)

f4 = 2(1 + z)(1 + z 2 )
z(cos θ) = (1 + 2g3 /g) cos θ
Comparing with the case of the single wire with barrier
we see that the fj are now smooth functions of θ, rather
than constants.
Let us consider the influence of β3 (θ) on the fixed point
values defined by βL = 0 of the reduced RG equation
(31), as analyzed in the previous subsection. Near the
two fixed points N, A, at θ = 0, π, corresponding to
x = ±1, the term (52) is small, proportional to the third
power of sin θ. This means that neither the positions of
these two fixed points, nor their scaling exponents γ1,2 in
(35) are affected by β3 (θ).
The position of the third fixed point, x3 , is defined in
the ladder approximation by (g + 2g3 )x3 + g = 0, up
to higher order corrections. The latter condition reads
z(x3 ) = −1 in (52), and we find accordingly that the fj
at x3 are given by f2 = −4(2 + g3 /g)(1 − g3 /g), f3 =
4(1 − g3 /g), f4 = 0 . It follows that the position of the

B.

General Y-junction

We start as described above in Sec. V, but have to
slightly modify our approach afterwards. It turns out
that in the case of a general Y-junction the intermediate
expressions produced in computer calculation show enormous complexity in the third order. The formulation in
terms of the S-matrix followed in the above analysis becomes impractical.
However, we may find the logarithmic corrections to
the conductance directly, in the spirit of our treatment
of the impurity in the Luttinger liquid.19 Indeed, ultimately we need to analyze only the partial contributions
to the two conductances, Eq. (40), rather than the full
corrections to the S-matrix, i.e. 3 × 3 matrix quantities.
Thus we may keep only these two partial contributions
from each diagram, which drastically simplifies the calculation.
Another simplification occurring in the present analysis of the d.c. limit is the absence of vertex corrections in the diagrams for the conductances.19 As a result, it suffices to consider only the self-energy parts of
the Green’s functions, corresponding to the above Eq.
(51). The absence of vertex corrections is easily proven
in case of T = 0 which we consider in this paper. The
analysis at T = 0 which was undertaken for the impurity
in the Luttinger liquid,19 is more involved in case of the
Y-junction and shall be given elsewhere.
Below we explain a few points of this analysis in
more detail. Each diagram contributing, e.g., to the
component Ga = (1 − a)/2 in (40), stems from the
A
generic expression a = 1 T r(λ3 GR
ω=+0 λ3 Gω=+0 ). Here
2
we introduced the retarded (advanced) Green’s functions GR(A) = iϑ(±t) [ψ(x, t), ψ † (y, 0)] with x → −∞,
y → ∞. When Fourier-transformed and taken at zero
energy, the Green’s functions are stripped from the coordinate dependence and become simply proportional to
the S-matrix, GR = −iS † , GA = iS, (we set vF = 1).
The expansion (51) leads to the renormalized quantity
ar = a0 + ga1 + g 2 a2 + g 3 a3 + . . .

(53)

1
˜
where a0 = 2 T r(λ3 λ3 ) and

a1 =

1
2Tr

˜
λ3 (Σ1 λ3 + λ3 Σ† ) ,
1

a2 =

1
2Tr

˜
λ3 (Σ2 λ3 + Σ1 λ3 Σ† + λ3 Σ† ) ,
1
2

a3 =

1
2Tr

˜
λ3 (Σ3 λ3 + Σ2 λ3 Σ† + Σ1 λ3 Σ† + λ3 Σ† ) ,
1
2
3

(54)

and similarly for br , where the components Σj of the self
energy have been defined in (51). The advantage of directly calculating corrections to ar , br is most apparent

13
when dealing with the long expressions for Σ3 . The partial contribution to a3 has the form
1
2Tr

˜
˜
λ3 (Σ3 λ3 + λ3 Σ† ) = 1 T r λ3 Σ3 λ3 + h.c.
3
2

˜
= Re(T r(λ3 λ3 Σ3 )).
The expressions ar , br obtained in this way show both,
scale-dependent terms in the form of the logarithm of
Λ = L/a and in addition scale-independent terms, as is
explained at length elsewhere.19 The result of this calculation provides expressions for the renormalized values of
conductance, ar , br , in terms of the bare ones, a, b. In the
next step of the Callan-Symanzik scheme we are considering here, we invert these expansions, and express the
bare values via the renormalized ones, keeping terms up
to g 3 in the corresponding series.
Then we require that the bare values be independent
of the scale in consideration, i.e. the logarithm Λ,
da
∂a
∂a ∂ar
∂a ∂br
=
+
+
=0
dΛ
∂Λ ∂ar ∂Λ
∂br ∂Λ

(55)

and similarly for db/dΛ. Solving this system of linear
equations for the quantities dar /dΛ, dbr /dΛ, we find:
d
dΛ

ar
br

=−

∂a/∂ar , ∂a/∂br
∂b/∂ar , ∂b/∂br

−1

d
dΛ

a
b

(56)

This calculation is best done by means of computer algebra, since the intermediate expressions are quite complicated.
We keep the terms of the order of g 3 in the final expressions. A criterion for the correctness of the calculation is
the absence of any Λ-dependence on the right-hand side
of Eq. (56) up to this order.
When we compare the final expressions, found in this
direct calculation, with the first terms of the Taylor expansion of the corresponding expressions (41), (43), we
find complete agreement of the universal (regularization
independent) parts of the β-functions to third order. In
addition to these universal contributions of the “oneloop” ladder summation (43), we also find non-ladder
contributions β3 , which first appear in third order and
explicitly contain the above mentioned regularizationdependent coefficients bj . These contributions are rather
complicated and are listed in Appendix.
Comparing these three-loop contributions with the
simpler case of one impurity in the Luttinger liquid, we
should make several remarks. First of all, one can explicitly check that these contributions satisfy Eq. (45) so
that the curve G⊥ = 1 remains the fixed line. Second,
for small g, g3 these terms do not lead to the appearance
of extra FPs, but the situation with strong interaction
|g|, |g3 | ∼ 1 is, strictly speaking, unclear. However, we
make a plausible conjecture, that all FPs of full RG beta
function (which contains three loop, four loop etc. contributions, beyond the re-summed one-loop terms (42),

(43) ) lie on the borderline of allowed conductances for
the non-interacting case.
The next remark concerns the universality of the phase
diagram, proposed below in Sec.VI on the basis of the
expression (42), (43). What is the evidence that no new
FPs appear at strong interaction, and the structure of the
phase diagram is independent of regularization ? The answer to this question goes along several lines. From the
actual form of three-loop contributions we see that regularization does not change the position of the interactiondependent FP M only in fully symmetric case. However,
in the latter case the three loop terms are apparently
unimportant even in the strongly interacting regime, as
is suggested by comparison of our results with those in18 .
The bosonization approach by Oshikawa et al. showed
changes in the character of A fixed point at strong attraction, K = K3 = 3, and we see the appearance of FP
Q at this value. The scaling dimensions of leading perturbations, found in Sec. 10.1 and 10.4.2 of18 , agree with
our one loop formulas (46), (47) for the whole range of
interaction strength.
In summary, we again find that the non-universal
terms in the β-functions do not influence the behavior
at the fixed points N, A, which means that the ladder
summation is sufficient in the case of repulsive interactions. As for the fixed point M , of importance for attractive interaction, we find that the non-universal terms
are relevant and may change the power law exponents in
principle. We address this question in some detail in the
next subsection.

C.

Fully symmetric Y junction

In the important case of equal interactions, g3 = g,
the position of the third fixed point (M ) in Eq. (32) is
cos θ = x3 = −1/3 and remains unaffected by β3 in (52).
Expanding β3 around x3 we have
√
2 3
β3 =
g (x + 1/3)(4b2 − b3 + b4 ) + . . . ,
(57)
18
so that the scaling exponent along the limiting parabola,
γ3 in Eq. (36), depends on the non-universal coefficients
bj , which were found to depend on the regularization of
the theory. The situation is however more delicate, because the scaling exponent γ3 in (50) in the direction perpendicular to parabola remains unchanged by the three
loop contributions. To see that we expand the expressions for β3 listed in Appendix B. In terms of variables
a, c we have the additional three loop contributions to
¯
Eq. (48)
d¯
a
2b2 + b4
= − g3
(1 − a)2 (1 + 3¯)2 + O(g 3 c2 ) ,
¯
a
dΛ
16
dc
a(1 − a)
¯
¯
(58)
=g 3 c
(2b2 (5 + 7¯) + b3 (¯ − 1)
a
a
dΛ
8
+4b4 (1 + 2¯)) + O(g 3 c2 ) .
a

14
The second equation here at the M fixed point a = −1/3
¯
corresponds to (57). The first equation (58) together
with (48) show that near the fixed points the three loop
contributions do not change the scaling exponent γ3 of
Eq. (50).

VI.

DISCUSSION AND CONCLUSIONS

In this paper we presented a theory of charge transport
through a junction of three quantum wires, modelled by
Luttinger liquids. We focused on the case of a Y-junction,
a set-up symmetric with respect to interchanging wires
1 and 2. We allow for different interaction strengths g
and g3 in wires 1, 2 and wire 3, respectively. Our method
employs a purely fermionic representation, which has the
advantage that the connection to ideal (noninteracting)
leads is naturally incorporated. The transition from the
noninteracting leads to the interacting wire is assumed
to be adiabatic. We find that at zero temperature the
scattering process is completely described by elastic scattering (no excitation of real particle-hole pairs). Virtual
excitations of multi particle-hole pairs are all-important;
these processes are described in terms of the renormalized single particle S-matrix. In terms of diagrams for the
conductances this amounts to the absence of any vertex
corrections (at T = 0).
We extended a theory previously applied to a two-wire
junction to the Y-junction problem. That theory employs perturbation theory with respect to the interaction
in fermionic language (using, however, the concepts of
current algebra to systematize the bookkeeping) to derive the renormalization group β-function for the conductance. As shown by us in Ref.19 an RPA type ladder
summation of an infinite class of terms of perturbation
theory may be performed to generate all of the known results on the scaling behavior of the conductance, power
law exponents, crossover behavior, and more, for any interaction strength and any scattering characteristic of the
barrier.
In the general time-invariant case the tensor of conductances features two independent components Ga , Gb .
These components are confined to an area bounded by a
curve B in the fundamental domain 0 < Ga , Gb < 1. We
derive the coupled set of RG-equations for conductances
in the ladder approximation. It is interesting to note that
the fixed points of these equations are all located on the
boundary curve B. In fact a simplified tunneling model
leads to conductances located on the boundary curve B,
and allows for an analytical determination of the fixed
points and the conductances.
We probe the validity of the ladder summation by evaluating all contributions up to and including third order (several thousands diagrams). We classify the contributions into universal (ladder summation) and nonuniversal, with respect to the regularization (finite length
L or finite temperature T ). For repulsive interaction we
find that the non-universal contributions to the RG-β-

1.0

NAM

0.5

g3

NAM

NA

0.0

NA

0.5

NAM

NAMQ

NAM

1.0
1.0

0.5

0.0

g

0.5

1.0

FIG. 4: (Color online) The number and the type of physically
available fixed points is shown for various strength of interactions g and g3 , as defined by the ladder summation. The
positions of N and A points are independent of interaction,
the positions of M and Q points are discussed in text. The
stable, unstable and saddle-point FPs are denoted by large,
small and underlined capital letters, respectively. See text for
additional details.

functions are subleading in the scaling regime, indicating
that the ladder summation is fundamentally correct in
the vicinity of the stable fixed point. For attractive interaction we find that the non-universal contribution in
fact changes the location of the fixed points and the values of the exponents.
We find a rich scenario of fixed points. In total there
are four fixed points N, A, M, Q, but not all of them are in
the physically accessible regime. Figure 4 shows the distribution of fixed points in the coupling constant g − g3 plane. In each regime the stable, unstable and saddlepoint (unstable) fixed points are indicated by large, small
and underlined capital letters. For instance, in the case
of repulsive interactions FP N describes the totally separated wires; it is the stable fixed point . FP A becomes
a stable FP for strong attractive interaction. It stands
for wire 3 separated from the ideal wire 1 − 2. For any
attractive interaction with g = g3 the FP M is stable. It
corresponds to finite conductances in all ways, the value
depending on the interaction strengths g, g3 . We conjecture that M corresponds to the ”mystery point” discussed in18 for the totally symmetric junction threaded
by magnetic flux. The FP Q, finally, is always unstable,
of the saddle point character, in the limited region where
it enters the physical domain.
Most neighboring regions in Fig. 4 differ by one FP
M , which appears either at the A point or at the N
point, with the corresponding change in the character of
this latter point. The situation at the interface between

15
NA and NAMQ is different, as both “floating” FPs M
and Q appear at one point on the parabola (see Fig. 3),
away from A or N . We recall, that for equal interaction
strength, g = g3 , the Q point appears first at K = K3 =
3.
It might be also interesting to note the existence of
the “tricritical” point between the phases NAM, NA and
NAMQ, which happens at K = 2, K3 = 4/3, (i.e., at
g = −3/5, g3 = −7/25) see Sec. IV A 3. This tricritical
point corresponds to the situation when both M and Q
points merge with the A point.
Comparing our findings with previous studies, we observe that there is a correspondence between the scaling
−1
exponents γ3 in our Eq. (36) and (1 − K3 ) for b in (47),
and those numerically obtained by fRG method in17 , reHF
spectively γ2 (3) in Eq. (56) and γ1 (3) in Eq. (48) there.
We confirm that the exponents obtained by our
method around N point, Eqs. (46), and A point, (47), at
K3 = K, coincide with those obtained by bosonization
in18 , Eq. (10.23), and Eqs. (10.106), (10.107), respectively. At the same time, the exponent (1 − K −1 ) around
the N point both in our work and Ref. [18] differs from
the exponent 1−(3K)−1 reported in Eq. (2.11) of Ref. [3]
; we note that in the latter case the RG flow exists even
in the absence of interaction, K = 1. It was argued in3,18
that the bosonic theory of the Y-junction has the duality property K → 3/K, which corresponds to a change
from the case of totally separated wires (N point) to the
case of maximally open Y-junction (D point). At the
latter the conductance exceeds the value allowed by the
unitarity of the single-particle S-matrix, as argued there
possibly due to the formation of Cooper pairs at strong
attraction. The scaling dimension of the leading perturbation around this hypothetic D point was thus found as
K/3, in Eq. (10.30) of18 , and as K/9 in Eq. (3.3) of3 , the
latter value evidently arising due to the above additional
factor 1/3 around the N point. As a result, bosonization
studies predict a qualitative change in the scaling behavior of the system at K > 3,18 or at K > 9.3 Our analysis
also shows a qualitative change at K = K3 > 3, which
corresponds to the appearance of extra FPs, but not of
the exotic D type. We stress again, that the RG flows
in our study always end at the surface of the (generally
four-dimensional) body describing the conductance matrix in the absence of interaction. Particularly, the D
point (which is a = b = −1 or Ga = 1, Gb = 4/3 in our
notation, see26 ) is not a fixed point of Eqs. (41) and the
RG trajectories do not end there, even if we start from
outside this body. Our above analysis of the three loop
RG contributions confirms this picture. We have looked
for contributions violating the unitarity condition as proposed in18 , but did not find any. In our formalism such
contributions would be generated by vertex corrections.
However, at T = 0, all vertex corrections vanish in the
d.c. limit.
In summary, we have derived a renormalization group
theory description for the two independent conductances
characterizing a Y-junction of a Luttinger liquid wire

(1,2) with interaction constant K and a tunneling tip
Luttinger liquid wire with interaction constant K3 . We
summed up infinite classes of contributions in perturbation theory (ladder approximation) to obtain the RG
β-functions. Additional contributions appearing in third
order were employed to decide whether the result of the
ladder approximation in the neighborhood of the stable
fixed points remained unchanged. This was found to be
the case for repulsive interaction (where arguments can
be made that all higher order non-ladder terms should
also be negligible). In the case of attractive interaction
non-ladder contributions might change the critical behavior in certain cases. The existence of further fixed points,
not captured by the ladder approximation, cannot be excluded, although it is not very likely. Nonetheless, we
find it remarkable that our method allows to determine a
rich scenario of fixed points and RG-flows, including the
crossover behavior. The corresponding conductances as
a function of the scaling parameter are readily accessible.

Acknowledgments

We thank D.A. Bagrets, A.P. Dmitriev, V.Yu. Kachorovskii, A.W.W. Ludwig, I. Safi, A.G. Yashenkin for
useful discussions. We are grateful to I.V. Gornyi and
D.G. Polyakov for careful reading of the manuscript and
valuable suggestions. The work of D.A. was supported
by the German-Israeli Foundation, the DFG-Center for
Functional Nanostructures, and the Dynasty foundation,
RFBR grant 09-02-00229. The work of P.W. was supported by the DFG-Center for Functional Nanostructures.

Appendix A: Generators of SU(3) group

For reader’s convenience, we list here the matrices λj
used in the main part of the paper. The traceless GellMann matrices, λj , with j = 1, . . . 8 are the generators
of the SU(3) group, discussed, e.g., in30 .




0 −i 0
0 1 0
λ1 = 1 0 0 , λ2 =  i 0 0 ,
0 0 0
0 0 0




1 0 0
0 0 1
λ3 = 0 −1 0 , λ4 = 0 0 0 ,
0 0 0
1 0 0




(A1)
0 0 −i
0 0 0
λ5 = 0 0 0  , λ6 = 0 0 1 ,
i 0 0
0 1 0




0 0 0
1 1 0 0 
0 1 0
λ7 = 0 0 −i , λ8 = √
3 0 0 −2
0 i 0
Together with the unit matrix λ0 , they have the property T r[λj λk ] = 2δjk , with j, k = 0, . . . 8. After this

16
normalization, the structure of algebra is determined
by the structure constants fijk according to [λj , λk ] =
2i l fjkl λl . In the familiar case of SU(2) algebra (which
is a subalgebra of SU(3), spanned by λ1,2,3 ), one has
fjkl = jkl , with totally antisymmetric tensor; it leads to
simple mnemonic rules. For the present SU(3) case such
simple rules are absent, and in most cases we resorted to
symbolic computer calculations, which are readily done,
e.g., in Mathematica.

β3θ = g 3 sin θ sin2 ψ cos ψ
+

g3
sin3 θ(f2 b2 + f3 b3 + f4 b4 ),
128

β3ψ = g 3 sin3 ψ cos θ
+

The perturbative calculation of the corrections to the
conductances, as described above, eventually leads to
RG β-functions containing one-loop and three-loop contributions. The one-loop contributions are resummed
into expressions (43), and the three-loop contributions
to da/dΛ, db/dΛ are found as functions of a, b, Eq. (40).

(B5)

g
sin ψ sin2 θ(f2 b2 + f3 b3 + f4 b4 )
128

with the functions
f2 = 4κ cos θ(2 cos2 ψ + κ2 cos2 θ)
+ cos ψ(8 cos 2ψ − 7 + κ2 (2 + cos 2θ))
f3 = cos ψ(3 − 2κ(1 + cos θ cos ψ) + κ2 cos2 θ)
2

(B1)

β3b = g 3 (F2 b2 + F3 b3 + F4 b4 )

2b2 + b4
4

(B4)

3

Appendix B: β3 for the general Y-junction

β3a = g 3 (F2 b2 + F3 b3 + F4 b4 ),

2b2 + b4
16

2

(B6)

2

f4 = 2κ cos θ(cos ψ + κ cos θ
+ κ cos θ cos ψ) + 2 cos 3ψ
and

where
1
−432a4 − 36a3 (b − 1)κ + a2 −24b2 κ2
F2 =
864
+b 39κ2 + 405 − 15κ2 + 459 − 4aκ 2b3 κ2
−3b2 κ2 − 9b + κ2 + 9 + 3 4b3 κ2 − b2 5κ2

−3b

2

2

κ + 3 + κ + 9 − 27(b + 1)

2

1 − 3 cos2 θ
(3 − 2κ + κ2 cos2 θ
cos θ
− 2κ cos θ cos ψ)

f3 =

+27) + b κ2 − 81 − 36 ,
1−b 2
a − b 6(a + 1)κ − (2b + 1)κ2 − 9 ,
F3 =
288
1
F4 =
−108a4 − 18a3 (b − 1)κ − 3a2 b2 κ2
432
−2b κ2 + 18 + κ2 − 36 − aκ 2b3 κ2
2

3 cos2 θ − 1 2
(κ cos 2θ − 1)
cos θ
+ 8κ cos ψ + 24 cos θ cos 2ψ

f2 =

(B7)

1 − 3 cos2 θ
+ 12 cos θ cos 2ψ
cos θ
+ 2κ(1 + 3 cos2 θ) cos ψ

f4 = 2

.
(B2)

1
(b − 1) 216a3 + 72a2 (b − 1)κ
864
+3a 4b2 κ2 + b κ2 − 45 − 5κ2 − 27

F2 = −

+4(b − 1)(2b + 1)2 κ3 ,
1
F3 =
a(b − 1)2 6(a + 1)κ − (2b + 1)κ2 − 9 ,
288
1−b
F4 =
54a3 + 9a2 (b − 1)κ + 3a 2b2 κ2
432
−b κ2 + 9 − κ2 − 9 + (b − 1)(2b + 1)2 κ3 ,
(B3)
and κ = 1 + 2g3 /g. These expressions are used for the
analysis of the fixed point M in Sec. V C. In order to
compare to the result of the simple tunneling case, Eq.
(52), and to our previous work, it is more convenient to
go now from the ”conductances” a, b to the angular quantities θ, ψ and discuss three-loop contributions to dθ/dΛ
and dψ/dΛ, which we denote by β3θ and β3ψ , respectively.

The previously considered case of impurity in the Luttinger liquid19 corresponds to setting θ = 0 in the above
equations. From the form of (B5) and Eq. (39) one verifies that the scaling exponents at the RG fixed points N
(θ = 0, ψ = 0) and A (θ = π, ψ = 0) are not modified by
the presence of β3θ , β3ψ , as the latter functions contain
higher powers of θ, ψ in those points.
Appendix C: Fixed point M in asymmetric case

In this section we consider the position of the FP M
in case when all three bulk interaction terms gj in Eq.
(2) are different. We restrict ourself by the first order in
gj , and use the precursor to the RG equation (37) in its
general form
3

S −1 .Sr − 1 = − 1 Λ
2

gj [˜j , ρj ] .
ρ

(C1)

j=1

We parametrize26 the S-matrix in T -symmetric case by
S = eiλ2 ξ/2 eiλ3 (π−ψ)/2 eiλ5 θ eiλ2 ξ/2 .

(C2)

17
The conductance matrix in (18) is then given by
Gaa =

1
2

sin2 ξ (1 − cos θ cos ψ) +

Gab = Gba = cos ξ sin2 θ ,

1
4

cos2 ξ sin2 θ ,

Gbb = sin2 θ .

2

3

4

5
6
7

8

9

10

11

12

13
14

15

16

−1
−1
−1
−1
cos θ = −g3 /(g1 + g2 + g3 ) ,

(C3)

so that the Eq. (19) is restored at ξ = π/2.
Simple calculations show that r.h.s. of (C1) is zero at
N fixed point, θ = 0, ψ = 0, and at three A points : i)
θ = π, ψ = 0, ξ = π/2, considered above, ii) θ = π/2,

1

ξ = 0, iii) θ = π/2, ξ = π. The position of M point is
defined by conditions ψ = 0 and

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