Chiral Y-junction of Luttinger liquid wires at strong coupling: fermionic
representation
D.N. Aristov1, 2, 3 and P. W¨lfle2, 4
o
1

“PNPI” NRC “Kurchatov Institute”, Gatchina 188300, Russia
Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
3
Department of Physics, St.Petersburg State University, Ulianovskaya 1, St.Petersburg 198504, Russia
4
Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
(Dated: April 29, 2018)

arXiv:1304.5167v2 [cond-mat.str-el] 20 Aug 2013

2

We calculate the conductances of a three-terminal junction set-up of spinless Luttinger liquid
wires threaded by a magnetic flux, allowing for different interaction strength g3 = g in the third
wire. We employ the fermionic representation in the scattering state picture, allowing for a direct
calculation of the linear response conductances, without the need of introducing contact resistances
at the connection points to the outer ideal leads. The matrix of conductances is parametrized by
three variables. For these we derive coupled renormalization group (RG) equations, by summing up
infinite classes of contributions in perturbation theory. The resulting general structure of the RG
equations may be employed to describe junctions with an arbitrary number of wires and arbitrary
interaction strength in each wire. The fixed point structure of these equations (for the chiral Yjunction) is analyzed in detail. For repulsive interaction (g, g3 > 0) there is only one stable fixed
point, corresponding to the complete separation of the wires. For attractive interaction (g < 0
and/or g3 < 0) four fixed points are found, the stability of which depends on the interaction
strength. We confirm our previous weak-coupling result of lines of fixed points for special values of
the interaction parameters reaching into the strong coupling domain. We find new fixed points not
discussed before, even at the symmetric line g = g3 , at variance with the results of Oshikawa et al.
The pair tunneling phenomenon conjectured by the latter authors is not found by us.

I.

INTRODUCTION

Electric circuits of one-dimensional quantum wires are
described by laws quite different from those of corresponding classical systems. These laws are non-local, often quantized, and are generally difficult to derive. An
important first step is the exploration of the properties
of junctions of wires. The linear response conductance
of a (symmetric) two-terminal junction in the limit of
zero temperature has been found to be either zero (repulsive) or unity (attractive interaction) in units of the
conductance quantum1,2 . Most of the existing theories
employ the exactly solvable Tomonaga-Luttinger liquid
(TLL) model, in which backward scattering is neglected3 .
Moreover, for simplicity the fermions are assumed to be
spinless, in which case backward scattering only gives
rise to a renormalization of the forward scattering amplitude. Furthermore, most of the early works on the
problem of two TLL-wires connected by a junction employed the method of bosonization, or, for special values
of the interaction parameter, used a mapping on to exactly solvable models4,5 .
An alternative formulation using a purely fermionic
representation in terms of scattering states and a renormalization group (RG) approach for the conductance has
been pioneered by6 . The latter formulation has several
advantages over the bosonic approach. Perhaps the most
important difference is that it allows to describe the physically relevant system of a quantum wire (with interaction between the fermions) adiabatically coupled to reservoirs (with negligible interaction), whereas the bosonic
approach is formulated for the infinite system. The effect

of a finite wire length has only been convincingly derived
for a clean wire7–9 ; in the case of wires connected by
a junction a plausible, but unproven ad hoc procedure
has been proposed10 . As will be discussed below we have
reasons to suspect that the latter procedure is not always
correct. The fermionic scattering state approach is generally applicable, allowing for the study of wires with two
impurities11 , of multi-lead junctions12,13 as well as the inclusion of spin, of backward scattering,6 of out of equilibrium situations,14 point contact in a quantum spin Hall
insulator15 . The original model calculation6 has been
limited to weak coupling. It may be shown, however,
that the method can be extended to work in the strong
coupling regime as well, by summing up an infinite class
of leading contributions in perturbation theory16 . In all
cases considered so far it has been found that the method
provides results in agreement with any available exact results, which builds confidence in its validity. As shown
in16 , in the case of a two-lead junction, the contributions
neglected in the above-mentioned partial summation are
subleading and may be shown to vanish when the fixed
points are approached.
Generally, the transport behavior of one-dimensional
systems at low energy/temperature is dominated by only
a few fixed points (FPs) of the RG flow. The most common and intuitively plausible FPs are those with quantized conduction values, G = 0, 1, in units of the conductance quantum G0 = e2 /h (we consider systems of
spin-less fermions). There may, however, appear additional FPs associated with noninteger conductance values, usually in the case of attractive interaction. A particularly well-studied case is that of a symmetric three-

2
lead junction with broken chiral symmetry, as induced
by a magnetic flux threading the junction. This latter
problem has been studied by the method of bosonization
in10,17 , and by the functional RG method in18 . These
authors have identified a number of new FPs, for attractive interaction. The discovery of these new FPs has
sparked interest in the problem of mapping out the complete fixed point structure of the theory, even though
most of the interesting new physics appears to be in the
physically less accessible domain of strongly attractive
interactions. Only the fully symmetric situation of identical wires (g = g3 , see below) has been considered in10,18 .
As we have shown in19 for weak coupling and will show
below for any coupling strength, important new physics is
missed by restricting the consideration to the symmetric
case.
One result found by10 deserves special attention: it
is the claim that a new fixed point named D (Dirichlet) exists for strongly attractive interaction (Luttinger
parameter K > 3), which is interpreted as describing
multiparticle transport. As a consequence, the conductance values at this FP are outside the domain of values
allowed by a single particle S-matrix description. Put differently, this requires the excitation of particle-hole pairs
at the junction, with say the particle moving into lead 2,
while the hole is going into lead 3. In our view, at least at
zero temperature, and in the linear response regime, such
processes are not occurring, because the phase space for
particle-hole excitations tends to zero in this limit. As
we will show below, we do not find a fixed point with
the characteristics of D, but in contrast we find two new
fixed points, not discussed in Refs. [10,20].
In principle, the Y-junction set-up may be realized experimentally by a one-dimensional tunneling tip contacting a quantum wire. A magnetic flux threading the junction may be created by local magnetic moments at the
junction, although this has not been realized experimentally so far.
In this paper we report results of an RG treatment of
electron transport in the linear response regime through
a junction of three spinless TLL wires threaded by a magnetic flux. We employ a fermionic representation as described in detail in16 . The method has formerly been
used in12,13 in the case of Y-junctions. More recently
this approach has been used to derive the RG equations
for the conductances of a Y-junction connecting three
TLL wires in the absence of magnetic flux, for weak
interaction21 , and in a recent work for any strength of
interaction22 . In21 it was found that even in weak coupling, but beyond lowest order, interesting new structures appear. Even a weak higher order contribution may
change the RG-flow in a dramatic way. In the present
paper we describe a similar effect: an asymmetry of the
three wires of a chiral Y-junction to the effect that in the
tip wire the interaction strength g3 is different from that
in the main wire, g, allows to access certain regions in
interaction parameter space where a whole line of fixed
points rather than a single fixed point is stable. This hap-

pens for attractive interaction only (g, g3 < 0) The line
of stable fixed points is connecting two fixed points at two
special manifolds of interaction values, (a) g = 0; g3 < 0,
−1
or (b) whenever the condition K3 = 2 − (K + K −1 )/2,
1 < K < 2 is met. Along these fixed point lines the
FP values of the conductances are continuously varying. Results on this new aspect of transport through Yjunctions within weak coupling have been reported in19 .
In the present work we extend our theory into the strong
coupling regime. We derive a general expression for the
scale-dependent contribution to the conductances, valid
for a junction with any number of leads and any values
of interaction gj in lead j . This result allows to derive
RG-equations for the set of independent conductances
in the given case. We go on to evaluate these general
expressions for the three-lead junction of the Y -type (interaction g in the main wire and g3 in the tip wire) in
the presence of magnetic flux. Then we determine the
six physically allowed FPs, and discuss their stability as
a function of position in coupling constant space.
In order to compare our results with the recent findings of20 for a fully asymmetric junction, we extend our
analysis by allowing for full asymmetry of the S-matrix
at the junction, while keeping equal interaction in the
main wires. This minimal modification enables us to elucidate the differences between our conclusions and those
in20 . We show that the set of RG equations for the most
general asymmetric chiral case can be easily derived in
our formalism, and allows us to discuss the robustness
of our results obtained for the partially asymmetric case
(g1 = g2 ) with respect to further asymmetry. A full analysis of the case g1 = g2 is, however, beyond the scope of
the present study.
Our findings for the 1-2-symmetric Y-junction are summarized in Fig. 1, where the regions of stability of the
four FPs N , A, χ± and M are displayed in the space
of interaction constants g and g3 . Fixed point N corresponds to the complete separation of all three wires. The
separation of the third wire from the perfectly conducting “main wire” 1-2 is described by FP A. The chiral
FPs χ+ , χ− correspond to maximum chirality. They describe a kind of Hall effect situation (depending on the
orientation of the magnetic flux), where the in-current
from wire j flows into wire j + 1 (χ+ ) or j − 1 (χ− ).
Fixed point M , finally, corresponds to the situation of
maximum transparency. The stability of the M -point in
the region to the right, g > 0, (checkerboard pattern) is
lost for infinitesimally small deviations from perfect 1-2
symmetry of the junction, in favor of a new A-point (A1
or A2 , meaning separation of wire 1 or 2 from the remaining “main wire”). The region of stability of the M -point
on the left, g < 0, remains (see the detailed discussion in
Section VI and Fig. 6). It is worth noting that in certain
regions two FPs (e.g., A and M , or χ± and M ) are stable
and attract trajectories within their respective basins of
attraction.

3
is expressed by the window function Θ(x; −L, −l) = 1,
if −L < x < −l, and zero otherwise. The regions
x < −L are thus regarded as reservoirs or leads labeled j = 1, 2, 3. We denote the interaction constants
g1 = g2 = g from now on, and put the Fermi velocity
vF = 1. The various incoming and outgoing channels are
illustrated in Fig. 1 elsewhere.19,22 The interaction term
of the Hamiltonian is expressed in terms of density operators ρj,in = Ψ+ ρj Ψ = ρj , and ρj,out = Ψ+ ρj Ψ = ρj ,
where ρj = S + · ρj · S and the density matrices are given
+
by (ρj )αβ = δαβ δαj and (ρj )αβ = Sαj Sjβ . The S-matrix
characterizes the scattering at the junction and (up to
irrelevant phase factors) has the structure (see19 )

1.0

0.5

g3 0.0
0.5

1.0
1.0

0.5

0.0

g

0.5

1.0



FIG. 1: (Color online) The RG phase portrait, showing regions with different stable fixed points. Diagonal red stripes
correspond to stable N point, vertical green stripes - to A
point, horizontal blue stripes - to χ± points, gray shaded and
checkerboard regions indicate stable M point.

II.

PERTURBATION THEORY FOR
CONDUCTANCES
A.

The model

We consider a model of interacting spinless fermions
describing three quantum wires connected at a single
junction by tunneling amplitudes in the presence of a
magnetic flux piercing the junction. In the continuum
limit, linearizing the spectrum at the Fermi energy and
including forward scattering interaction of strength gj
in wire j, we may write the TLL Hamiltonian in the
representation of incoming and outgoing waves in lead j
(fermion operators ψj,in , ψj,out ) as
0

H =
H0 =
H

int

0
int
dx[Hj + Hj ] ,
−∞
v F Ψ† i
in

Ψin − vF Ψ† i Ψout ,
out

(1)

= 2π{g[ρ1 ρ1 + ρ2 ρ2 ] + g3 ρ3 ρ3 }Θ(x; −L, −l) .

Here Ψin = (ψ1,in , ψ2,in , ψ3,in ) denotes a vector operator of incoming fermions and the corresponding vector
of outgoing fermions is expressed through the S-matrix
as Ψout (x) = S · Ψin (−x) . In the chiral representation
we are using positions on the negative (positive) semiaxis corresponding to incoming (outgoing) waves. We
consider quantum wires of finite length L, contacted by
reservoirs. The transition from wire to reservoir is assumed to be adiabatic (i.e. produces no additional potential scattering). The junction is assumed to have microscopic extension l of the order of the Fermi wave length.
Inside the junction interaction effects are neglected. This


r1 t12 t13
S =  t21 r1 t23 
t31 t32 r2

(2)

A convenient representation of 3×3-matrices is in
terms of Gell-Mann matrices λj , j = 0, 1, . . . , 8, the
generators of SU (3) (see22 ). Notice that the interaction operator only involves λ3 and λ8 (besides the unit
2/3 δαβ ). We note Tr(λj ) = 0,
operator (λ0 )αβ =
Tr(λj λk ) = 2δjk , j = 1, . . . , 8, and [λ3 , λ8 ] = 0. We introduce a compact notation by defining a three-component
vector λ = (λ3 , λ8 , λ0 ), in terms of which the densities
may be expressed as ρj = 1/2 µ Rjµ λµ , where the
3 × 3 matrix R is defined as


R=

1
√
2
1
− √2

0

1
√
6
1
√
6

−

2
3

1
√
3
1
√
3
1
√
3





(3)

and has the properties R−1 = RT , det R = 1. The
outgoing amplitudes are expressed in the analogous form
with λj replaced by λj = S + · λj · S. With the aid of the
λj the S-matrix may be parametrized by eight angular
variables (see23 and Appendix A). For the case under
consideration only three of these, θ, ψ, ξ, are relevant:
S = eiλ2 ξ/2 eiλ3 (π−ψ) eiλ5 θ eiλ2 (π−ξ)/2 . The corresponding
elements of the S-matrix are given by
1
(cos θ + eiψ ) sin ξ , r2 = cos θ ,
2
ξ
ξ
= t32 = i cos sin θ , t23 = t31 = i sin sin θ , (4)
2
2
2 ξ
2 ξ
iψ
= cos θ cos
− e sin
, t21 = t12 |ξ→π−ξ .
2
2

r1 =
t13
t12

As will be shown below, the S-matrix and therefore the
angular variables θ, ψ, ξ, will be renormalized by the interaction.

4
B.

Linear response conductances

b

In the linear response regime, the conductances
a

−L

Cjk =
−∞

dy (ρj (−L) − ρj (L))ρk (y)

c

ω→0

relate the current Ij in lead j (flowing towards the junction) to the electrical potential Vk in lead k, Ij = Cjk Vk .
Here the combination (ρj (−L) − ρj (L)) means that the
current is measured at x = −L, and the integration over
y for the incoming densities is over the length of the ideal
leads, to which the electric potential Vk is applied. Denoting the frequency of the applied external field by ω,
we consider the d.c. limit ωL → 0, when the value of
L does not play a role.16 The quantum averaging . . .
involves taking the trace over the matrix products. The
conductance matrix C has only three independent √
ele1
ments, Gaa = 2 (1−a), Gbb = 2 (1−b) , and Gab = c/ 3 ,
3
relating the currents Ia = 1 (I1 −I2 ), Ib = 1 (I1 +I2 −2I3 )
2
3
1
to the bias voltages Va = (V1 −V2 ), Vb = 2 (V1 +V2 −2V3 ).
In compact notation we may define a 3 × 3 matrix of conductances
  1
Gaa Gab 0
2 (1 − a)
c
√
G =  −Gab Gbb 0  = 
3
0
0 0
0


c
− √3 0
2

3 (1 − b) 0
0
0
(5)
The connection between matrices C and G comes from
the observation, that the nonzero elements of the matrix
CR = RT CR are essentially the reduced √
conductances:
R
R
R
R
C11 = 2Gaa , C22 = 3 Gbb , C12 = −C21 = 3Gab , where
2
the numerical factors arise due to the physically motivated asymmetric definitions of the currents and voltages.
It may be shown that the parametrization of the conductance tensor in terms of a, b, c follows quite naturally, by
observing that the initial conductances following from
the Kubo formula are given by Cjk = δjk − Tr(ρjr ρk ) =
r
δjk −|Sjk |2 where the superscript r denotes that the quantity is fully renormalized by interactions22 . By expressR
ing Yjk = Tr(ρjr ρk ) in terms of Yµν = 1 Tr(λr λν ) as
µ
2
R
T
Y = (R · Y · R ), we find by comparison with the conductance matrix, CR = 1 − YR , that YR has nonzero elements given by the conductance parameters introduced
above:



a c 0
YR = −c b 0
0 0 1

N

!+
A

!-

FIG. 2: (Color online) Allowed values of conductances, a, b, c,
lie inside the depicted body, B. The location of the RG fixed
points N , A, χ± , M is also shown. The line of fixed points
connecting FPs A and χ+ is indicated in yellow.

1
to the Euler angle variables: a = − 2 (cos2 θ + 1) cos2 ξ +
√
1
cos θ cos ψ sin2 ξ, b = 2 (3 cos2 θ − 1), c = − 23 sin2 θ cos ξ.
We find therefore that a, b, c are confined within the re√
√
gion a ∈ [−1, 1], b ∈ [−1/2, 1], c ∈ [− 3/2, 3/2]. The
physically allowed points in a-b-c-space, satisfying the
conditions | cos θ| < 1, | cos ψ| < 1 and | cos ξ| < 1, lie
inside a body labelled B as shown in Fig. 2. We may
trace the above restriction on the allowed values of the
conductance back to our consideration of energy conserving scattering. In the linear response regime and at zero
temperature only lead electrons at the Fermi level will
be transported through the interacting wire, and particlehole excitations are excluded. The restriction of the set of
conductances to the domain of allowed values has important consequences for the stability of certain fixed points
of the RG flow, as will be discussed below. We mention
now and will show later that all fixed points are located
on the surface of the body B in Fig. 2.
For the isotropic case discussed below we have a = b.
One possible parametrization of the S-matrix for this case
is given by Eq. (B1) The corresponding subspace of allowed conductance values is the twodimensional domain
∆ enclosed by a deltoid curve, see Eq. (B8) and Fig. 3.

III.



(6)

From now on we will drop the superscript r , with the understanding that all quantities are renormalized. Therefore, we may use YR to represent the conductances in
the renormalization group analysis below.
By substituting the S-matrix in the form (2), (4) into
the definition of YR , we find the relation of the a, b, c

M

RENORMALIZATION GROUP EQUATIONS

The renormalization of the conductances by the interaction is determined by first calculating the correction terms in each order of perturbation theory. We are
in particular interested in the scale-dependent contributions proportional to Λ = ln(L/l), where L and l are two
lengths, characterizing the interaction region in the wires
(see above). In lowest order in the interaction the scale
dependent contribution to the conductances is given by19

5
1.0

matrix equation (see appendix C)

c

0.5

gladder = 2(Q − Y)−1 .

0.0

We observe that the effective interaction is found to depend on the conductance components Cjk = δjk − Yjk
in a highly nonlinear way. The matrix Q characterizes
the interaction strength and depends on the Luttinger
parameters Kj = [(1 − gj )/(1 + gj )]1/2 as

0.5

qj = (1 + Kj )/(1 − Kj ) .

Qjk = qj δjk ,
1.0
1.0

0.5

0.0

a

0.5

1.0

FIG. 3: (Color online) Allowed values of conductance a, c
for the isotropic case. The S-matrices (B1) characterized by
parameters Γ, φ lead to two quantities a and c, as described
by (B9). We took 500 pairs Γ, φ at random, these are blue
points on the plot. As discussed in Appendix B, the domain
∆ is bounded by a deltoid curve.

Cjk = Cjk

g=0

+

1
2

Tr Wjk Wlm gml Λ ,

(7)

l,m

where Cjk |g=0 = δjk − Yjk , the Wjk = [ρj , ρk ] are a set of
nine 3×3 matrices (products of W ’s are matrix products),
gml = gm δml is the matrix of interaction constants and
the trace operation Tr is defined with respect to the 3 × 3
matrix space of W ’s. As argued in16 the leading terms in
each order of perturbation theory are certain diagrams of
the ladder type, which may be summed up analytically.
The result is a renormalized interaction matrix gladder
replacing the bare interaction matrix g in Eq.(7). The
components of gladder are obtained from the following

Q1 =

3qq3 − q − 2q3
.
2q + q3 − 3

The RG-equations describe the flow of the conductances

(9)

(Notice that |qj | > 1 for any interaction strength.) In order to remove the redundancy in the conductance matrix
C we now multiply with RT from the left and R from
the right to get the components of YR in the form
R
R
Yjk = Yjk

g=0

−

1
2

ladder,R
R
R
Tr Wjk Wlm gml
Λ.

(10)

l,m

Differentiating these results with respect to Λ (and
then putting Λ = 0) we find the RG equations up to
infinite order in the interaction
d R
1
Y =−
dΛ jk
2

ladder,R
R
R
Tr Wjk Wlm gml
.

(11)

l,m

R
Here the Wjk = {RT · W · R}jk are a set of nine 3 ×
3 matrices and the trace operation Tr is defined with
ladder,R
respect to that matrix space, whereas gml
= {RT ·
ladder
R
R −1
R
g
· R}ml = 2{(Q − Y ) }ml and Q , YR are
scalars with respect to this space. The nine matrices
R
Wjk are best evaluated with the aid of computer algebra,
inserting the S-matrix in terms of the quantities a, b, c as
given above (see also Appendix D). As a result one finds
the following set of RG-equations

da
= D−1 {a2 (3b − 1 − 2Q1 ) + a(3c2 + q(1 − b)) + (Q1 − b)(1 + b) + c2 (2 + q)} ,
dΛ
db
= D−1 {q(1 − b)(1 + 2b) + a(b − 1)(1 + 3b − Q1 ) + c2 (2 + 3b + Q1 )} ,
dΛ
dc
= −cD−1 {1 + q + Q1 − 3c2 + 2b(q − 1) + a(2Q1 − 3b − 2)} .
dΛ

where D = (a − q)(b − Q1 ) + c2 , and

(8)

(12)

upon increasing Λ until a stable fixed point is reached.
The fixed points are found by putting the right hand
sides (the β-functions) simultaneously equal to zero. The
approach towards any given fixed point is characterized
by power laws. The fixed point pattern and the power

6
law exponents depend on the interaction strength. We
note that the β-functions (the right hand sides of Eqs.
(12) ) are highly nonlinear functions of a, b, c and of q, q3 ,
giving rise to a rich manifold of fixed points and RG flow
patterns.

IV.

da
1−K
1
= 6
a+
,
dΛ
2+K
3
dc
(1 − K)(2 − K)
c.
= −3
dΛ
(2 + K)2

FIXED POINTS OF RG FLOW
A.

Chiral isotropic case

In the isotropic case, meaning (i) equal interaction
strength in all wires, q = q3 , and (ii) isotropic tunneling amplitudes, we have Q1 = q and a = b. The RG
equations read
da
dΛ
dc
dΛ

−1
= βa = Dis {(q − a)(1 − a)(1 + 3a) + c2 (2 + q + 3a)} ,

−1
= βc = −cDis {1 + 2q + 4a(q − 1) − 3(a2 + c2 )} .
(13)
where Dis = (a − q)2 + c2 .
By putting the βfunctions equal to zero we find four fixed points, labeled
N, M, χ± , C ± , at positions

aN = 1,
cN = 0,
aM = −1/3, cM = 0,

aχ± = −1/2,
aC ± =

√
cχ± = ± 3/2,

3 − K2
,
3(K − 1)2

cC ± = ±

(14)

2K K(K − 2)
.
3(K − 1)2

The location of these points in the c−a-plane is shown in
Fig. 4. We observe that all the FPs lie on the boundary
curve of the domain of allowed conductances. We discard
the unphysical solution a = q, c = 0, because |a| < 1 <
|q|.
Next we discuss the stability of these FPs. We start
with the N -point, located at a cusp of the boundary curve
∆, see Fig. 2. The cusp is infinitely sharp, leaving only
a single direction of approach from the inside of ∆ : a =
1 − x , c = 0 . Linearizing the RG equation for a in the
small quantity x we find
dx
= −2x(K −1 − 1) .
dΛ
It follows that the N -point is stable for repulsive interaction, when K −1 − 1 > 0 .
We now turn to the χ± -FPs, which are again located
at a cusp of Curve ∆, Fig. 3. Therefore, again there is
only one allowed direction for the trajectory towards or
√
away from χ± : (a, c) = 1 (−1, ± 3)(1 − x) , where x is
2
found to obey
dx
= −2x
dΛ

4K
−1
3 + K2

.

We thus see that the χ± -FPs are stable in the interaction
regime 1 < K < 3 .
The M -point is located at a flat section of curve A and
we therefore have a two-dimensional manifold of possible
trajectories leading to it. Expanding the RG-equations
in linear order around the M -point we find

It follows that the M -point is stable for K > 2 . We note
in passing that the M -point cannot be discussed within
the Abelian bosonization approach, as the criterion a2 +
M
c2 = 1 is not satisfied.23
M
Whereas the N, M, χ± -FPs have been described before, we find in addition a new pair of FPs C ± here. The
location of these FPs is not geometrical, but depends
on the interaction. We first observe that the C ± -points
are physically allowed (i.e. are located within the open
domain ∆) only for sufficiently strong attractive interaction, K > 2 (they are always residing at the boundary
curve ∆). At K = 2 they merge with the M -point. As
K increases the C ± -points start to move away from the
M -point in opposite direction until they end at the χ± points when K = 3 . Upon further increase of K the
C ± -points move beyond the χ± -points along the bound1
ary of ∆ , until they end at a = − 3 , c = ± 2 , in the limit
3
K → ∞ . The stability analysis tells that the C ± -points
are never stable.
As a result, we conclude, that for K > 3 the only stable fixed point is the M point, and not the unphysical D
point a = −1, c = 0 suggested in10 . In Sec. 6.3 of that
paper the authors say: “While the χ± fixed points are
stable against a small change in the flux, we do not know
whether the M fixed point has such a stability. The simplest assumption is that it does not. In this case the RG
flows might go to the χ± fixed points for any non-zero
flux φ, starting from arbitrarily small V . Alternatively,
it is possible that the M fixed point is stable against
adding a small flux. In that case, there would have to be
additional unstable fixed points defining the boundaries
between the basins of attraction of the M , χ+ and χ−
fixed points. So again, an ’economy’ principle suggests
the simple picture with only three stable fixed points”,
for 1 < K < 3. Our analysis shows, however, that this
conjecture is not fully correct, as the new unstable fixed
points C ± appear for K > 2 , allowing the point M to
become stable. Examples of flow trajectories for interaction strengths K = 0.7; 1.5; 2.07; 2.7; 4; 15 are shown in
Fig. 4.

B.

Chiral anisotropic case

The above RG equations, Eqs. (12), describe the flow
of a, b, c towards the stable fixed points (FPs) inside the

7

!+

!+

K=0.7

K=1.5

0.0

0.5

N
M

c

c

0.5

0.5

!0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.4

0.0

0.2

0.8

1.0

K=2.7

"+
0.5

N
M

c

"+

N

0.0

M

"0.5

"-

!0.4

0.2

0.0

0.2

0.4

0.6

0.8

!-

1.0

0.4

0.2

0.0

0.2

a

0.4

0.6

0.8

1.0

a

!+

!+

"+

K=4

K=15

"+

0.5

0.5

N
M

c

c

0.6

!+

0.5

0.0

0.4

a

K=2.07

0.0

0.2

a

!+

c

M

0.5

!-

0.5

N

0.0

0.0

0.5

M

0.5

!- "-

N

!-

0.4

0.2

0.0

0.2

a

0.4

0.6

0.8

1.0

"0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

a

FIG. 4: (Color online) RG flows for the fully symmetric chiral case, for different values of interaction. The four starting points
are the same in all plots. The non-universal positions of the chiral C ± points are shown by open circles.

8
body B of allowed conductance values (see Fig. 2), depending on the initial conditions. The fixed points are
defined by the zeros of the β-functions on the r.h.s. of
Eqs. (12). Some of the fixed points are outside the allowed region B in a − b − c-space and will be discarded as
unphysical. There are altogether six physical FPs (see table I), which may be classified into two groups, universal
(independent of the interaction) and non-universal FPs.
The three universal FPs are labeled N, A, χ± . These
FPs may be thought to be of geometrical nature, as the
corresponding S-matrix elements are 1 or 0 . The FP N
describes the total separation of the wires, meaning zero
conductance in all components. By contrast, A describes
the separation of wire 3 from the perfectly conducting
wires 1–2. Finally, χ± are the FPs generated by chirality
(left- or right-handed).
The non-universal FPs are labeled M, Q, C ± . They
only exist or are physical in a limited region of interaction constant space, mainly for attractive interaction. As
shown in table I the conductances at these FPs depend
in a complicated way on the interaction constants.
The locations of the M and Q points have been given
in22 . We reproduce the results here in the present notation. Both points can simultaneously exist only for
attractive interaction, g, g3 < 0. Notice that for equal
interactions g = g3 we have q = q3 = Q1 , τ = |1 + q| (see
below) and we return to (14).
The FPs M and Q merge along a line in interaction
constant space, defined by τ = 0, where
τ=

1 + q 2 + 2Q1 .

(15)

The end point of this line is at K = 2, K3 = 4/3, and
was named “tricritical” in Ref. [22].
The location of the FPs C ± is best represented in terms
of the angular variables θ, ψ, ξ as
1
cos ψ = 1 ,
cos θ = − (2 + Q1 ) ,
3√
3
cos ξ = ±
[3 − (2q − Q1 )(2 + Q1 )]1/2 .
5 + Q1

(16)

The C ± points lie on the surface of the body of physically
allowed conductances, since cos ψ = 1 corresponds to the
outmost points of our parametrization. The FPs C ± are
only in the physical domain if the interaction constants
satisfy the requirement | cos θ| ≤ 1 , implying −5 ≤ Q1 <
1, and 0 ≤ cos2 ξ ≤ 1 leading to the bounds
0 ≤ 3 − (2q − Q1 )(Q1 + 2) ≤ 1 (5 + Q1 )2
3

The interval shrinks to zero when Q1 → −5, and the
above inequalities demand q → −3, which in turn gives
q3 → −7. Remarkably, these values again correspond
to K = 2 and K3 = 4/3, dubbed the “tricritical” point
above.
From the Table I one may also verify that the points
C ± merge with χ± at
Q1 = −2

⇔

−1
K −1 + 2K3 = 1

(17)

implying, in the symmetric situation, the condition K =
K3 = 3 .
V.

STABILITY OF FIXED POINTS

The above set of equations (12) allows to perform a
rather straightforward analysis of the stability of FPs.
Assuming that we have a fixed point (a0 , b0 , c0 ), we may
expand the RG equations in terms of the vector of small
deviations x = (a − a0 , b − b0 , c − c0 ) = λ(a1 , b1 , c1 ), to
linear order in λ
1:
d
−M x=0
dΛ

(18)

The stability of the fixed point is defined by the direction
of the RG flow, which is determined by the eigenvalues
µi (i = 1, 2, 3) of the matrix M. All µi are positive for
the fully unstable FP, all µi are negative for stable FPs
and in the the saddle point case (some µi negative, some
positive) the FP is again unstable.
Two remarks are in order here. First, the matrix M is
in general not symmetric, so that its eigenvectors are not
orthogonal. Second, in the case of asymmetric interaction
(q = Q1 ) the expressions for M are not simple for the
non-universal FPs, and should be analyzed numerically.
Simplifications occur at q = q3 = Q1 , as was already
demonstrated for the M point in the non-chiral case in22 .
We mention that not all directions of the vector
(a1 , b1 , c1 ) are permitted. This is because all FPs lie
on the surface of the body of physically allowed conductances. Certain displacements x would take the point
(a, b, c) outside of the body. If the FP lies on a smooth
part of the surface of the body, the requirement for x to
be inside the body is not very restricting, because it only
selects half of all possible directions. But in our case the
situation is more complicated by the fact, that some FPs
lie on ridges of the body. This situation happens with
the universal FPs and is discussed below.
In the following we present a detailed stability analysis
for the different fixed points. The results are collected in
Table II and in Fig. 5, showing the phase boundaries (i.e.
the boundaries, separating different domains of stability)
in the g − g3 -plane.
A corresponding figure for the non-chiral case (c =
0) has been presented as Fig. 4 in our previous
publication22 . By comparing the figures we observe that
phases I–VI and 1,3’,5 are characterized by FPs with
c = 0 and were already discussed in the non-chiral case.
The new phases are labeled 2–4, 6, involving either the
stable chiral FPs χ± or the new unstable FPs, C ± .
It should be emphasized, that the scaling exponents
obtained by our method for the “universal” FPs N , A,
χ± are in exact agreement with the results, found by
boundary conformal field theory20 . Our results for the
non-chiral FPs N , A were reported earlier22 . The particular expressions for the χ± points below, Eq. (24), co-

9
TABLE I: The positions of the fixed points of the RG equations, (12). The quantity τ is defined in Eq. (15).
FP
N
A
χ±
M
Q
C±

1
6

a
1
−1
−1
2
1
(|q| − τ ) sign(q)
3
1
(|q| + τ ) sign(q)
3
2q(Q1 + 2) − Q2 + 1
1

b
1
1
−1
2
1
((|q| − τ )2 − 3)
6
1
((|q| + τ )2 − 3)
6
1
(Q1 (Q1 + 4) + 1)
6

TABLE II: Summary of the RG phase portrait, depicted in
Fig. 5. The existence and stability of the RG fixed points is
shown below. FPs N , A, χ± always exist, and the stability is
indicated by an “s” symbol. The stable and unstable FPs M ,
Q, C ± are denoted by “s” and “u”, the “-” symbol is used if
such a point does not exist.

I
II
III
IV
V
VI
1
2
3
4
5
6
3’

N
u
u
s
s
s
u
u
u
u
u
u
u
u

χ±
u
u
u
u
u
u
u
u
s
s
u
u
u

A
s
s
u
u
u
u
s
s
u
u
s
s
u

M
u
u
u
s
u
s
s
s
s

Q
u
u
-

C±
u
u
u
-

A.

N point

In this case the matrix M is given by

MN



4(1 − Q1 )
Q1 − q
0


= fN 
0
4 − 3q − Q1
0
,
0
0
3(2 − q − Q1 )

fN = [(q − 1)(Q1 − 1)]−1 .

(19)

Formally, there are three (left) eigenvectors (1, −1/3, 0),
(0, 1, 0), (0, 0, 1), corresponding to eigenvalues 4/(1 − q),
1/(1−q)+3/(1−Q1 ), 3/(1−q)+3/(1−Q1 ), respectively.

3
2

1
(Q1
6

− 1)

0
0
3 − (2q − Q1 )(Q1 + 2)

However, as can be seen in Fig. 3, any displacement
with c1 = 0 is not allowed, as it would end outside the allowed domain. Therefore, when discussing the RG flows
starting inside “the body”, we should discard any displacements along c. We thus end with two exponents
governing the RG flow towards N
αN,1 = 4/(1 − q) = −2(K −1 − 1) ,
1
3
−1
αN,2 =
+
= −(K −1 + K3 − 2) ,
1 − q 1 − Q1

(20)

which may be interpreted as the weak link exponents in
the main wire and between the main wire and the tip,
respectively (eq. (46) in22 ). These values are negative
for K, K3 < 1, so that the N point is stable in the case
of repulsive interaction in all wires.
B.

A point

The matrix M for the asymmetric A point is given by

MA
incide with the exponents deduced from Table I of Ref.
[20] for the scaling dimensions of the leading irrelevant
boundary operators, ∆χ . This becomes clear after the
identification of our αχ with their 2 − 2∆χ , and after
putting their g1 = g2 ; g3 equal to our K; K3 .

c
0
0
√



4(Q1 − 1)
Q1 + q
0


= fA 
0
Q1 − 4 − 3q
0
,
0
0
Q1 − 4 − 3q

fA = [(q + 1)(Q1 − 1)]−1 .

(21)

We have three (left) eigenvectors (1, 1/3, 0), (0, 1, 0),
(0, 0, 1), corresponding to eigenvalues 4/(1 + q), 1/(1 +
q) + 3/(1 − Q1 ), 1/(1 + q) + 3/(1 − Q1 ), respectively.
The last eigenvalue is doubly degenerate, and therefore
we again have only two exponents
αA,1 = 4/(1 + q) = 2(1 − K) ,
(22)
1
3
−1
αA,2 =
+
= −( 1 (K −1 + K) + K3 − 2) ,
2
1 + q 1 − Q1

which may be interpreted as i) the weak impurity exponent in the main wire and ii) the tunneling exponent
in the main wire and the boundary exponent in the tip,
respectively (eq. (47) in22 ). At the “tricritical” point,
K = 2, K3 = 4/3, the second exponent vanishes. This
is a manifestation of the fact, that the tricritical point
is the endpoint of the line of stable fixed points in the
interaction parameter space, see Fig. 5.

10
“the body” and should be discarded, similarly to the situation at the N point.
The other two physical exponents are

1.0

II

III

0.5

g3

0.0

IV
αχ,2

0.5

V
VI

1.0
1.0

0.5

0.0

0.5

g

αχ,1 = αχ,2 = 2 − 8K/(3 + K 2 ) .
When the C ± and χ± merge, which happens at Q1 =
−2, the second exponent αχ,2 in (24) vanishes.
For K3 = K the form of the eigenvectors is not transparent, and it might be useful to restore the parametrization in terms of Euler angles. The coordinates of the χ±
points are then given by θ = π/2, ξ = 0, π, for arbitrary
ψ. For χ− , we put ψ = 0 and expand θ = π/2 + θ ,
ξ = π + ξ , with θ , ξ
1. Then, linearizing the resulting RG equations, one finds a decoupled set of differential
equations for θ , ξ

0.2

1

g3

3
0.6

0.8

2

5

6

4

dθ
2(3q − Q1 + 4)
=θ
,
dΛ
(2q + 1)(2Q1 + 1) + 3
dξ
4(Q1 + 2)
=ξ
,
dΛ
(2q + 1)(2Q1 + 1) + 3

3’

1.0
1.0

0.8

0.6

g

0.4

0.2

0.0

D.

χ± point



√
4Q1 + 3q + 19 2Q1 + q + 3 − 3(2q + 1)
2
2
√


= fχ  3(Q1 + 1 )
Q1 + 6q + 19 − 3(2Q1 + 1) ,
2
2
√
√
3(2Q1 − 1 )
3(2q − 1 )
9
2
2

fχ = 2[(2q + 1)(2Q1 + 1) + 3]−1 .

√
The three left eigenvectors are (−3, 4Q1 −6q+5 , 2 3),
2Q1 +1
√
√ √
3(Q1 −1)
+3q+1
( Q1 +3q+2 , 2Q11+3q+2 , − 3), ( 3, 3, −2), corresponding
Q
to the eigenvalues 2fχ (3q−Q1 +4), 4fχ (Q1 +2), 3fχ (Q1 +
q + 4), respectively. The last eigenvector points out of

M , Q and C ± points

As seen in Table I, the position of these FPs is not universal, i.e. depends on the interaction. The linearization
of the corresponding set of RG equations around these
points leads to very cumbersome expressions for the matrix M, which we do not show here. Instead we list the
three scaling exponents for the conductances.
For M fixed point we have (for g < 0):

We consider only one of these chiral points, χ− , for
which we obtain

Mχ

(25)

in correspondence with the above exponents for the conductance components.

FIG. 5: (Color online) The RG phase portrait, showing regions with different set and character of the FPs. The region
of attractive interactions is shown in more detail in the second panel. The numbers in the plot refer to its description
in Table II. The three dots correspond to the special points
K = K3 = 2, K = K3 = 3, and K = 2, K3 = 4/3, discussed
in the text.

C.

(24)

For K3 = K we have both exponents equal :

1.0

0.0

0.4

4(3q − Q1 + 4)
,
(2q + 1)(2Q1 + 1) + 3
8KK3
=2−
,
2K + K3 + K 2 K3
8(Q1 + 2)
=
,
(2q + 1)(2Q1 + 1) + 3
4K(K + K3 )
=2−
.
2K + K3 + K 2 K3

αχ,1 =

I

(23)

αM,1 = −

3τ ((q + τ )2 − 9)
,
2(q + τ )(2q − τ )2

3 (q + τ )2 + 3
,
(26)
(q + τ )(2q − τ )
3(q + τ + 3)(q 2 + 2q − τ 2 + 2τ − 3)
= −
.
2(q + τ )(2q − τ )2

αM,2 = −
αM,3

with
the
corresponding
eigenvectors
q((q+τ )2 −9)
( 3+(5q−3τ )(q+τ ) , −1, 0), (q + τ, −1, 0), (0, 0, 1) and τ
defined in (15).

11
line of fixed points emerges. Examples of trajectories in
that case, demonstrating the existence of a line of fixed
points, have been given in our previous work for the weak
coupling regime19 .

For Q point we obtain (again, for g < 0)
αQ,1 =

3τ (q − τ )2 − 9
,
2(q − τ )(2q + τ )2

3 (q − τ )2 + 3
,
(27)
(q − τ )(2q + τ )
3(q − τ + 3)(q 2 + 2q − τ 2 − 2τ − 3)
= −
,
2(q − τ )(2q + τ )2

αQ,2 = −
αQ,3

q((q−τ )2 −9)
( 3+(5q+3τ )(q−τ ) , −1, 0),

and the eigenvectors are
τ, −1, 0), (0, 0, 1), correspondingly.
For the C ± point we get

12
12
−
− 6,
−3q + Q1 + 2 Q1 − 1
3 (3q − Q1 + 4) (Q1 + 5)
=
2 (Q1 − 1) (−3q + Q1 + 2)

VI.

(q −

αC,1 =
αC,2(3)

×

Q1 + 1 1
±
Q1 + 5 3

1+

(28)

8 (Q1 + 2) (3q − Q1 − 2)
(Q1 − 1) (3q − Q1 + 4)

while the expressions for eigenvectors are too complicated
to be presented here. One can check that at Q1 = −2 the
exponent αC,2 vanishes, it corresponds to C ± merging
with χ± , as discussed after Eq. (24).
E.

Lines of fixed points

In our previous work19 on transport through a chiral
Y -junction at weak coupling we found that at special
lines in the g − g3 -plane lines of fixed points may appear.
We conjectured that this may happen at any boundary
in the g −g3 -plane separating phases with different stable
FPs, which still exist upon approach to the boundary. A
precondition is that the two different stable fixed points
should exist as separate fixed points at the respective
boundary in the g − g3 -plane, i.e. the FPs should not
have merged or otherwise disappeared upon approach to
the boundary. By inspecting the many phase boundaries
in Fig.4 in that respect we find that only at two phase
boundaries, (1) separating phases 3 and V I, at K = 1;
K3 > 1 (i.e. g = 0, g3 < 0) and (2) separating phases
−1
1 and 3, at 1 (K + K −1 ) + K3 − 2 = 0, 1 < K < 2 a
2

da
dΛ
db
dΛ
dc
dΛ
d¯
c
dΛ

,

FULLY ANISOTROPIC Y-JUNCTION

In the previous sections we presented a rather detailed
analysis of the 1 − 2 - symmetric Y-junction in the presence of magnetic flux. We now consider the effects of the
absence of symmetry. As in previous cases, one might expect that some of the stable fixed points found above will
become unstable if the additional freedom of the RG-flow
in the absence of the symmetry restriction is admitted.
One may distinguish two ways in which the symmetry
with respect to wires 1-2 may be broken. The first is
provided by different properties of the wires 1, 2, in particular different interaction strengths g1 = g2 . A second symmetry characterizes the junction, i.e. the way
in which the third wire is attached. For instance, if the
third wire is non-perpendicular to the main wire, one
may expect some asymmetry in hopping to wire 1 and to
wire 2. The resulting asymmetry in the S-matrix and the
conductance matrix may grow in the process of renormalization. Such a scenario is implied in the work20 . In order
to compare our results with those in20 , we now discuss a
possible anisotropy in the conductance tensor.
In the following we confine ourselves for simplicity to
the asymmetry of the second type, keeping in mind that
the completely general case is fully covered by the general form of the RG-equations. The relevant formulas are
listed in the Appendices A, D. The S-matrix is now characterized by four Euler angles, and instead of Eq. (6) we
have four conductance parameters


a c 0


YR =  c b 0
¯
0 0 1

(29)

with c = −c. In the asymmetric non-chiral situation
¯
we have c = c, see23 . Using the general RG equation
¯
(11) and formulas in Appendix D, we obtain after some
calculation:

−1
= D1 {a2 (3b − 1 − 2Q1 ) + a(−3c¯ + q(1 − b)) + (Q1 − b)(1 + b) + c2 − c¯q + c2 } ,
c
c
¯
−1
= D1 {q(1 − b)(1 + 2b) + a(b − 1)(1 + 3b − Q1 ) − c¯(2 + 3b + Q1 )} ,
c
−1
= D1 {c (a(3b − 2Q1 + 2) − q(2b + 1)) + c −2b − 3c2 + Q1 + 1 } ,
¯
−1
c
= D1 {¯ (a(3b − 2Q1 + 2) − q(2b + 1)) + c −2b − 3¯2 + Q1 + 1 } .
c

(30)

12
with D1 = (a − q)(b − Q1 ) − c¯. In the limit c = −c, we
c
¯
return back to (12).
The investigation of the whole phase portrait defined
by the set of equations, (30), is beyond the scope of
the present study. For our purposes it suffices to check
the stability of the already discussed FPs in the slightly
asymmetric situation, which can be done as follows. Our
previous analysis was confined to the surface c + c = 0
¯
in the space of conductance components a, b, c, c. We
¯
now allow for a small asymmetry of the Y-junction,
|c + c|
¯
1 and expand the RG equations (30) to first
order in h = c + c. As a result, we recover the three
¯
previously found equations (12) and the fourth equation
takes the form:
dh
= hD−1 (1−q +Q1 +3c2 −2b(q +1)−a(2Q1 −3b−2))
dΛ
(31)
A negative (positive) coefficient of h on the right-hand
side of the last equation corresponds to stability (instability) of the previously determined FPs with respect to
the asymmetric distortion of the S-matrix.
It is clear, that if the FP does not exist in some region
of interaction parameter space, or is unstable, then the
additional check of the stability provided by (31) is not
required. Only otherwise stable FPs should be subjected
to this additional test.
Performing this analysis, we find that the FPs collected
in Table II are stable against asymmetry, with only one
exception. Namely, the FP M is unstable with respect
to the asymmetry perturbation h = 0 in region VI. In
all other situations the character of the listed FPs is unchanged. Particularly, the exponents for h around the
points N and χ± are given by αN,2 and αχ,2 , Eq. (20)
and (24), respectively. The RG flow from the M point
in the region VI should apparently lead to one of two
asymmetric FPs, different from A and discussed in23 .
These points A1,2 are defined by a = 1/2, b = −1/2,
√
c = c = ± 3/2 and correspond to the cases of either
¯
the first or the second wire detached from the remain¯
ing two wires. We will denote these points by A below.
In addition to these, we also may have the counterparts
of the point Q discussed above, at strong attraction.
The unstable Q point separates the A and M points,
as discussed in22 . Two new asymmetric Q1,2 points appear for strong repulsion and their position is given by
a = 1 2q (Q1 + 2) + Q2 − 1 , b = 1 (Q1 (Q1 + 4) + 1),
1
6
6
1
c = c = ± 6 (Q1 − 1) (2q + Q1 ) (Q1 + 2) + 3.
¯
We notice also that the asymmetric FP A cannot be
affected by a finite value of h without violating unitarity,
meaning that the scaling behavior is determined by only
two exponents, Eq. (22). This situation is completely
similar to what was discussed above, where the chiral
perturbation was found impossible around the N point,
and one of the perturbations impossible around the chiral
χ± points. The corresponding boundary (deltoid) curve
in the asymmetric non-chiral case, c = c, was shown in
¯
Fig. 1 of Ref.23 .

12

10

VI

3’

8

K3

6

6

V
4

2

4

2

3

5

IV
III

0
0

1
I

II
1

2

3

4

5

6

K
FIG. 6: (Color online) The RG phase portrait, showing regions with different character of stable FPs. The numbers in
the plot refer to the description in Table II. The three dots
mark the points K = K3 = 2, K = K3 = 3, and K = 2,
K3 = 4/3, discussed in the text. The three straight red lines
correspond to (2K +K3 )/3 = 2, 3, 4, respectively. The dashed
line indicates the condition K = K3 .

Below we compare our results on the stability of different FPs with the results summarized in Fig. 5 of Ref.20 .
In order to facilitate the comparison, we re-plot our Fig.
5 in terms of the Luttinger parameters, K, K3 and show
the phase portrait in Fig. 6.
We observe full correspondence of our results with
those shown in panels (a), (b), (c), (d) in Fig. 5 of Ref.20
for relatively small interaction strength. However, we
only partly agree with results obtained in20 for strong
attraction, panels (e), (f), as discussed below.
The three straight red lines in Fig. 6 designating the
manifolds (2K + K3 )/3 = 2, 3, 4, correspond to the vertical lines drawn through the top vertex of the triangles
shown in panels (d), (e), (f) in Fig. 5 of Ref.20 , respectively.
We see that the first line, (2K + K3 )/3 = 2, consecutively intersects the regions IV, V, VI, 3, 1, I, II, when
one increases K from K = 0 to K = 2. This sequence
corresponds to the appearance of stable FPs in the fol¯
lowing order: N , A, χ± , A, in full agreement with Fig.
5(d) of Ref.20 .
The situation is more involved at stronger attraction.
The second line, (2K + K3 )/3 = 3, consecutively intersects the regions IV, V, VI, 3, 4, 3’, 5, 1, I, II, in Fig. 6
upon the increase of K from K = 0 to K = 4.5. From
our Table II we see that this sequence corresponds to the
¯
FPs: N , A, χ± , (χ± +M ), M , (M + A), A. It should
¯
¯
be compared to the sequence N , A, χ± , A, A, shown
in Fig. 5(e) in20 . We notice two differences here: one is

13
related to the fact that Hou et al.20 do not include the
M point in their analysis, therefore they do not show
the combination of several stable FPs, e.g. (χ± +M ) in
region 4, cf. Fig. 4 at K = 2.07 above. Another difference concerns the region 3 which appears, e.g. at K = 2,
K3 = 5. Our analysis shows that, given the initial symmetry with respect to wires 1 and 2, the RG does not
flow away from this symmetry, and the stable FP in this
case is M . This is in contradiction to the conclusion of
the authors of20 , who observe that the asymmetric points
¯
A are stable and hence any RG flow should end there. In
fact, one can show from Eqs. (30) that the symmetric
point M is separated in the region 3’ from the asymmetric points A1,2 by the unstable FPs Q1,2 , mentioned
above. Note that the FPs A1,2 and Q1,2 appear only in
the fully asymmetric case and therefore are not shown in
Table II.
The biggest difference occurs at even stronger attraction. The third line, (2K + K3 )/3 = 4, in our Fig. 6 intersects the regions IV, V, VI, 3, 4, 3’, 6, 2, 1, I, II, these
¯
correspond again to the sequence of stable FPs : N , A,
±
±
χ , (χ +M ), M , (M + A), A. This sequence should be
compared to Fig. 5(f) in20 , which provides the list: N ,
¯
¯
A, χ± , A, D, A. We see that the bosonization analysis, apart from systematically ignoring the existence of
the M point, proposes the appearance of the new fixed
point D. The “Dirichlet” FP D violates the unitarity
of the S-matrix, i.e. the conservation of the number of
particles; the authors of10,20 explain that this property
may be attributed to the formation of superconducting
pairs close to the Y-junction. In particular, the D point
is proposed as a stable FP even in the fully symmetric
situation, K = K3 . Our analysis does not indicate the
existence of a FP with the properties of the D point,
which lies outside the physically accessible region ∆ indicated in Fig. 3 at the point a = b = −1, c = 0. The
relative position of the D point with respect to the body
B of allowed conductance values, available in the fluxless
asymmetric case, is shown in Fig. 7
It should be stressed again, that the scaling exponents at the points N , χ, A obtained in20 coincide with
those, calculated within our approach earlier and in the
present paper. Let us look closer at the argument of the
authors20 in favor of the D point. They observe that the
pair hopping operator at the FP A becomes formally relevant for large attraction, which means that the RG flow
generated by such a perturbation should move the system away from the A point. At the same time, they show
that the perturbations around the proposed D points are
irrelevant and the D point is thus stable. They do not actually suggest any RG flow, connecting the FPs A and D,
because these points are proposed by an ansatz,20 rather
than determined as zeros of the beta-function of a set of
RG equations.
We observe further that the bosonization approach
suggests the existence of the stable FP D at strong attraction, whereas the existence of such a point is not
excluded at weaker attraction, when it only becomes un-

A1
A3

N

M
D

A2

FIG. 7: (Color online) The position of the Dirichlet D point
is shown in relation to the body of allowed conductances in
the asymmetric non-chiral case. The fixed points N , Ai , M
are also shown.

stable. Therefore it should be meaningful to discuss values of conductance, which exceed the maximum allowed
value for one-particle processes, for arbitrary interaction,
in particular also for weak interaction. One may then
hypothetically find examples of contributions from pair
hopping, which may increase the conductance over its
maximum value, even though the subsequent renormalization (i.e. higher order corrections) reduces this value
to usual one.
From a formal perspective, contributions to the Kubo
formula of conductance involving fermion pair processes
are certain vertex corrections, first discussed in the
context of superconducting fluctuation contributions.
These are the Maki-Thompson and Aslamazov-Larkin
corrections24 . We showed, however, in our previous
work22 that vertex corrections vanish at zero temperature in the limit ωL → 0. It is known that the criterion ωL
1 is crucial for obtaining the correct value
of the conductance of the clean Luttinger liquid with
leads7–9 . This criterion however does not appear in the
analysis of10,20 , where the presence of leads at x > L
is modeled at the end of the calculation by introducing
“contact” resistances, which may be positive or negative
depending on the interaction. After this modeling the
authors10,20 conclude that the D point exists also in the
presence of leads and is characterized by the above values
a = b = −1, in violation of the unitarity of the singleparticle S-matrix. The proposed picture raises a natural
question about the role of the length of the interacting
region L. In our fermionic analysis small L correspond
to the absence of any renormalization. By contrast, the

14
result in10,20 reduces to the statement, that a sufficiently
long finite region around the Y-junction should lead at
strong attraction between fermions to the peculiar behavior that a fermion coming from one Fermi-liquid lead
is scattered to another lead, and takes with it a partner
fermion from a third lead. It is apparently suggested that
the finite region close to the junction acts as a pump. If
valid, such a picture should have a precursor at moderate strength of interaction. However, we quoted above
the absence of such contributions in the direct perturbative calculations up to the third order in the interaction.
We conclude that the most plausible resolution of the
contradiction is that the ad hoc procedure of modeling
the non-interacting outer leads used by10,20 is not correct. There is no reason why the “contact resistances”
should be the same for the ideal (clean) TLL-wire and
wires connected by a junction, as assumed in10,20 .

VII.

CONCLUSION

In this paper we presented results for the effect of
fermion-fermion interaction on the transport properties
of three quantum wires joined at a Y-junction. We
employed the Luttinger liquid (LL) model of spinless
fermions to calculate the S-matrix and the conductances
of this system for the case of equal interaction constants g
in the main wire and different interaction g3 in the third
wire. Using a purely fermionic representation we are
able to directly describe the physical situation of wires of
length L adiabatically attached to reservoirs. We argue
that in this case (at least at zero temperature) the scattering is purely elastic (excitation of particle-hole pairs is
excluded) and therefore described by the single-particle
S-matrix (the S-matrix is strongly renormalized by interaction effects). We calculate the renormalization of
the S-matrix from a set of three renormalization group
equations for three conductances. The RG-equations are
derived by calculating the leading scale dependent contributions to all orders in the interaction. These contributions are obtained by analytically solving a set of integral equations (ladder summation). In our earlier work
we have presented arguments suggesting that the selected
contributions are the dominant ones, meaning that additional contributions do not change the scaling behavior near the fixed points16 . Additional support for the
correctness of our method is coming from the fact that
our results agree with known results in all cases where
these results are well founded. The relative simplicity of
our method allows, however, to consider more complex
problems not addressed before, e.g. in the present case
anisotropic Y-junctions at strong coupling. As demonstrated in our earlier work, the addition of even small
non-symmetric perturbances to a highly symmetric system may change the RG-flow pattern in a qualitative way.
For example, a second order contribution to the tunneling
from a tip into a quantum wire is found to destabilize the
fixed point corresponding to the tip separated from the

ideally conducting wire21 . Similarly, a small asymmetry
in the interaction constants of tip and wire of a chiral Yjunction leads to the appearance of lines of fixed points19 .
The fully symmetric chiral Y-junction has been studied in
great detail by10 . These authors found the emergence of
a pair of chiral FPs, χ± , which we confirm in our present
analysis, extending the considerations to anisotropic interactions g = g3 . They also conjectured a new and quite
unusual fixed point termed D, which in their interpretation involved ”Andreev scattering”, i.e. the tunneling of
fermion pairs. Such processes necessarily involve particlehole excitations, which we exclude from the beginning
(these processes have vanishing phase space in the limit
of zero excitation energy). The existence of the new FP
was argued in10 to be necessary to complete the RG flow
pattern. We agree that new fixed points are necessary
at large attractive interaction. However, we find in the
isotropic case a set of additional unstable chiral FPs, C ± ,
in the domain of attractive interaction, K > 2, serving
a similar purpose. The emergence of C ± allows to stabilize the FP M . In the anisotropic case a further unstable
FP, Q, is appearing. In a follow up to10 the properties
of a fully asymmetric Y-junction have been considered
recently20 . Our detailed comparison with the results obtained by the latter authors reveals agreement in many
aspects of this multifaceted problem. At strong attraction, however, our results differ from theirs in two respects: we do not find their fixed point D, which violates
the unitarity of the single particle S-matrix, but instead
find additional fixed points not discovered by them. Finally we like to point out that our formulation is rather
general, allowing to derive RG-equations for junctions
connecting any number of leads in any asymmetric way
(see Eq.(11)). The challenge in that case is to find a
suitable parametrization of the S-matrix. Work in this
direction for the four-lead junction is in progress.

Acknowledgments

We are grateful to I.V. Gornyi, A.W.W. Ludwig, D.G.
Polyakov, and A. Rahmani for useful discussions. The
work of D.A. was supported by the German-Israeli Foundation (GIF), the Dynasty foundation, and a BMBF
grant. The work of D.A. and P.W. was supported by the
DFG-Center for Functional Nanostructures at KIT. One
of us (PW) acknowledges the support by an ICAM Senior
Scientist Fellowship during his stay as visiting professor
at the University of Wisconsin, Madison.

Appendix A: S-matrix parametrization and
re-phasing

The traceless Gell-Mann matrices, λj , with j = 1, . . . 8
are the generators of the SU(3) group and are listed
elsewhere.23 Together with the unit matrix λ0 , they form
a basis for representing the S-matrix of a three-lead

15
junction. We note the property Tr[λj λk ] = 2δjk , with
j, k = 0, . . . 8.
Up to an overall phase, an arbitrary matrix S, belonging to representations of the SU(3) group, may be defined
by its Euler angles parametrization as follows23 :
√

¯
S = U eiλ5 θ U eiλ8 α4 (

3/2)

,

U = eiλ3 α2 /2 eiλ2 ξ/2 eiλ3 ψ/2 ,
¯
¯
¯
U = eiλ3 φ/2 eiλ2 ξ/2 eiλ3 α3 /2 .

(A1)

While this expression for S contains eight Euler angles,
for our purposes it is convenient to introduce an additional degree of freedom by adding a ninth angle and
defining
S1 = e

√
iλ8 α1 ( 3/2)

S.

(A2)

One may now use the redundancy in the latter representation to derive a more convenient representation of
S. Using symbolic computer calculations, one can verify
that S1 is invariant under the following change
α1 → α1 − β,

α4 → α4 + β,
¯
¯
φ → φ − 3β.

ψ → ψ + 3β,

It is natural to parametrize the S-matrix in a minimal way by the expression S = exp [iθM ], where θ =
2Γ in the limit Γ
1. It turns out, however, that
the expressions for conductances are simplified using a
slightly different parametrization. Instead of the pair
(θ cos(φ/3), θ sin(φ/3)) in Eq. (B2) it is better to use the
pair (θ1 , θ2 ) with
√
θ cos(φ/3) = θ1 /3, θ sin(φ/3) = θ2 / 3 .
After this convention, we arrive at the S-matrix in the
form
S = exp

Appendix B: Correspondence with other authors

S = e−2ik 1 − eik ΓM

−1

1 − e−ik ΓM ,

(B1)

where k is the wave-vector, and Γ is the amplitude in the
tight-binding Hamiltonian,describing the hopping from
one wire to another. The matrix M is given by M =
eiφ/3 B † + e−iφ/3 B, and B =

0, 1, 0
0, 0, 1
1, 0, 0

. Here φ measures

chirality, i.e. is a normalized flux through the Y-junction.
For small Γ we have (up to an overall phase) S
1 + 2i sin k ΓM . In the tight-binding, particle-hole symmetric case, considered below, we have k
kF = π/2
and S 1 + 2iΓM .
We can express M in terms of Gell-Mann matrices λi
as
φ
φ
M = cos (λ1 + λ4 + λ6 ) + sin (λ2 − λ5 + λ7 ) . (B2)
3
3

i
√ θ2 (λ2
3

− λ5 + λ7 )

(B3)

with
r = 1 (eiθ1 + 2 cos θ2 ),
3
√
t± = 1 (eiθ1 − cos θ2 ± 3 sin θ2 ) .
3

(B4)

The connection with parametrization (B1) is established
by taking θ1,2
1:
√
(B5)
θ1 = 3Γ cos(φ/3), θ2 = 3Γ sin(φ/3),
here Γ denotes the hopping amplitude, and φ = πn is
standing for the flux, piercing the junction, with n the
number of flux quanta. We have in special cases
|θ1 | = 3Γ, θ2 = 0,
|θ1 | = |θ2 | = 3Γ/2,

n = 0 mod 3
n = ±1 mod 3

(B6)

The components of conductance, a = b, c, Eq. (6), are
given by
3
a = 2 |r|2 −
√

Let us first establish the correspondence with the notation of Oshikawa et al.10 in their Appendix A. They
obtain the S-matrix in the form

+ λ4 + λ6 ) +

= e−iθ1 /3 (r + t+ B + t− B † )


r, t+ , t−


= e−iθ1 /3 t− , r, t+ 
t+ t− r

(A3)

¯
for arbitrary β. This means that we may set, e.g., φ = 0,
¯
by choosing β = φ/3. Doing this, we arrive at an equivalent representation of the S-matrix in terms of eight parameters. This representation contains four re-phasing
angles α1 , . . . α4 . These do not appear in the expressions
for the conductances. In the main text we therefore discarded the re-phasing angles, which leads to a simpler
parametrization of the S-matrix. In sections II-IV, we
¯
set ξ = π − ξ and work with the three angles θ, ψ, ξ.
In sections V, VI, Appendix D we also refer to the most
¯
general case with four parameters, θ, ψ, ξ, ξ.

i
3 θ1 (λ1

c=

3
2 (|t+ |

1
2
2

1
= 3 (cos 2θ2 + 2 cos θ1 cos θ2 )

− |t− |2 ) =

2
3

sin θ2 (cos θ1 − cos θ2 )

(B7)

We note that a and c reach their extremal values at
cos θ1 = ±1, and these values describe the boundary of a
domain of the form
1
a = 3 (cos 2θ2 + 2 cos θ2 ),

c=

2
3

sin θ2 (1 − cos θ2 ), (B8)

This curve is called deltoid, and is depicted in Fig. 3.
The Abelian bosonization approach may be applied to
the problem when the condition a2 + c2 = 1 is satisfied23 .
One observes that there are only three such points in Fig.
3, the corners of the deltoid.
The simple relation (B5) holds only for small Γ. For finite values of Γ we calculate the conductance components
directly from (B1) and obtain
12 Γ2 (1 + Γ2 )
(1 + 3Γ2 )2 + 4Γ6 cos2 φ
√
8 3 Γ3 sin φ
c=−
(1 + 3Γ2 )2 + 4Γ6 cos2 φ

a=1−

(B9)

16
These expressions correspond to Eq. (91) in25 after a
change Γ → t. The quantities (a, c) lie within a deltoid
curve for arbitrary Γ ∈ (0, ∞), φ ∈ (0, 2π), as shown in
Fig. 3. The proper deltoid curve is given by Eqs. (B9)
at φ = π/2 and Γ ∈ (0, ∞)
It was argued in10 that the free-fermion description
of the S-matrix is not fully applicable in the interacting
case. Instead, it was proposed to use the bosonization
approach and to find an analog of small Γ in the bosonic
Hamiltonian. When abandoning the fermionic formulation and working directly with currents (densities), one
does not find restrictions on the values of a, c, except
for the condition a2 + c2 ≤ 1, mentioned above. It was
hence implied in10 that such restrictions, imposed by the
unitarity of S-matrix, are relaxed in the interacting case.
For completeness, we quote here the lowest-order RG
equations in terms of θ1,2
d
dΛ θ1
d
dΛ θ2

= −g sin θ1 cos θ2 ,

= − 1 g sin θ2 (cos θ1 + 2 cos θ2 ) .
3

(B10)

The resulting RG flows are shown qualitatively in the two
first panels in Fig. 4, obtained for moderate interaction
strength, K = 0.7, 1.5.
The point of maximally open Y-junction, or M -point,
is given by a = −1/3, c = 0 and is obtained in two cases.
√
First, in the absence of the flux, φ = 0, and Γ = 1/ 2,
and second in the case of maximum flux φ = ±π/2, and
infinitely strong hopping, Γ → ∞. It is worth to compare this observation with Fig. 12 of Ref.10 , where the
case, φ = ±π/2, Γ → ∞, was associated not with the M
point, but rather with a different FP labeled D. In our
analysis we showed that the M point is the only stable
FP for K > 3, as indicated in Fig. 4 above. Taken from
this perspective, our results for K > 3 do not contradict
√
the results obtained in10 , since both φ = 0, Γ = 1/ 2
and φ = ±π/2, Γ → ∞ correspond to the same FP, the
M point. The important distinction is, however, that
our analysis describes the renormalization of the conductances fully in terms of the one-particle S-matrix, where
the D point would be in the unphysical regime. By contrast, we find the new FPs C ± , not discussed in10 .
Appendix C: RG equation for conductances

In this section we sketch the derivation of the formula
(8). This derivation closely follows our paper26 , dealing
with a junction of two non-equal Luttinger liquid wires.
The formalism presented in that paper can be naturally
generalized to an arbitrary number of wires, connected
by a single junction.
In the lowest order of perturbation theory the renormalization of the S-matrix and the conductance matrix
is proportional to the strength of interaction, Eq. (7).
In higher orders of interaction the interaction constant,
entering the RG equation for the conductance, is an effective quantity, described by a function which is non-local
and includes retardation effects.

−y
L

−y

x

+

x

x
−y

=

x

−z1

z2

=

−y
C

−y

−y

−x

z

x

+

x

FIG. 8: Feynman diagrams depicting the integral equations
for the renormalized interaction, Eqs.(C2) and (C1). A simple wavy line stands for the initial interaction matrix g, the
double wavy line defines the quantity L, leading to the effective interaction strength in the RG equations for the conductances.

As shown in16 the leading terms in perturbation theory
form a ladder series, which may be summed up analytically to give a function L. The latter obeys the integral
equation:
L(x, y; ω)
L2 (x, y; ω)
×

= 2πgδ(x − y)

1
0

L

− 2π

gYΠ(x + z, ω), gΠ(x − z, ω)
gΠ(z − x, ω),
0

dz
l

L(z, y; ω)
L2 (z, y; ω)

,

with the fermionic loop Π(x, ωn ) = (2π)−1 (δ(x) −
|ωn |θ(xωn )e−xωn ).
It can be shown, that in the case with different strength
of interaction in different semi-wires, it is more convenient to solve this equation in two steps. This is graphically depicted in Fig. 8, whereas the details are presented
elsewhere.26
At the first step, the auxiliary quantity C is introduced
as L, taken at Y = 0. It satisfies the equation
L
1
C(x, y; ω) = 2πgδ(x − y) − 2 ωgg

× e

−ω(x+z)

+e

−ω|x−z|

dz
l

(C1)

C(z, y; ω) .

2
with gj = gj /d2 and dj = 1 − gj . The above condition
j
Y = 0 is not intuitively evident, and it describes the
absence of correlations between the densities of incoming
and outgoing electrons. In this special case, the effects
of scattering are absent and the effects of the interaction
amount to the usual dressing of the incoming density by
the outgoing one. Strictly speaking, the condition Y = 0

17
cannot be realized for any S-matrix, as it violates the
charge conservation, and should be viewed as vaguely
corresponding to the situation “in the bulk”, far away
from the junction.
Solving Eq. (C1), we then obtain the renormalization
of the plasmon velocity and certain prefactors of the interaction strength in the individual wires, which eventually become Luttinger coefficients.
At the second step, we use the expression for C thus
obtained in the equation for L, which now includes the
knowledge about the scattering Y :

×

ω
2π

L

dz1 dz2
l
C(x, z1 ; ω)Ye−ω(z1 +z2 ) L(z2 , y; ω) ,

L(x, y; ω) = C(x, y; ω) +

(C2)

The general equation (11) contains a quantity with four
R
R
indices, Fjklm = −Tr Wjk Wlm . This quantity is entirely defined by the parameters of S-matrix, as opposed
ladder,R
to the interaction, entering the definition of gml
. It
is clear from the properties of Gell-Mann matrices, that
Fjklm = 0 if any of its indices equals 3; hence there are
only 24 = 16 non-zero components of Fjklm . An analysis similar to the one presented in the main text for the
1 − 2 symmetric junction shows that Fjklm can be expressed through the conductance components, a, b, c, c
¯
in a compact way. In view of definition (29) it is convenient to refer to pairwise combinations of indices as
{11} = a, {12} = c, {21} = c, {22} = b, so that F1121
¯
is now denoted as Fa¯ ; Fcb stands for F1222 etc. In this
c
notation we have

The solution of the above integral equation (C2) is relatively simple. Substituting the result into the expression
for the conductances one observes that retardation and
non-locality effects, contained in L, disappear in the limit
ωL → 0. We then arrive at the RG equation containing
the expression for the effective interaction in the form
gladder = 2(Q − Y)−1 . This is Eq. (8) quoted in the
main text of the paper.

Faa = 1 + b − 2a2 ,

Fcc = 1 − b − 2c2 , Fcc = 1 − b − 2¯2 ,
c
¯¯
Fab = Fba = Fc¯ = Fcc = a(1 − b) − c¯ ,
c
c
¯
Fac = Fca = c − 2ac , Fa¯ = Fca = c − 2a¯ ,
¯
c
c
¯
Fbc = Fcb = −(2b + 1)c ,
Fb¯ = Fcb = −(2b + 1)¯ ,
c
c
¯

Appendix D: Fully asymmetric Y-junction

In the most general case the S matrix, (A1), is defined by eight parameters, but only four parameters, θ,
¯
ψ, ξ, ξ define the conductance. The parameters of the
conductance matrix (29) are given by
a=
b=
c=

1

2
3

4

5

6

7

8
9
10

11

12

cos2 θ+1
¯
¯
cos ξ cos ξ − cos θ cos ψ sin ξ sin ξ ,
2
2
1
2 (3 cos θ − 1) ,
√
√
¯
¯
− 23 cos ξ sin2 θ , c = − 23 cos ξ sin2 θ .

(D1)

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26

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