Junctions of multiple quantum wires with different Luttinger parameters
Chang-Yu Hou,1, 2 Armin Rahmani,3 Adrian E. Feiguin,4 and Claudio Chamon5
1

arXiv:1205.2125v2 [cond-mat.str-el] 22 Aug 2012

Department of Physics and Astronomy, University of California at Riverside, Riverside, CA 92521
2
Department of Physics, California Institute of Technology, Pasadena, CA 91125
3
Theoretical Division,T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
4
Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA
5
Physics Department, Boston University, Boston, MA 02215, USA
(Dated: May 15, 2018)
Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points
of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical
properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We
present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two
chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed
method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the
analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase
diagram, and compute all fixed-point conductances as a function of the three Luttinger parameters.
PACS numbers: 73.63.Nm, 71.10.Pm, 73.23.b

I.

INTRODUCTION

Conducting quantum wires, at low energies, generically
form a Tomonaga-Luttinger liquid (TLL), characterized by a
Luttinger parameter g, which encodes the effects of electronelectron interactions.1–4 Due to the prominent role of interactions in low dimensions, the nature of these one-dimensional
electronic systems is dramatically different from their higherdimensional counterparts described by Landau’s Fermi-liquid
theory.5 The TLL state of matter in one-dimensional quantum
wires has been realized in numerous experiments over the last
few years.6–13
Transport properties of such quantum wires are of considerable interest: From a fundamental point of view, a large
number of interesting phenomena have been predicted and observed. For instance, at low temperature and low bias voltage,
a TLL with repulsive interactions (g < 1) is totally disconnected in the presence of an impurity, while one with attractive
interactions (g > 1) conducts as in the absence of the impurity.14–19 From a practical viewpoint, junctions of TLL wires
serve as important building blocks of quantum circuits,20,21
and are thus of technological significance. Junctions of three
quantum wires, known as Y junctions, also have highly nontrivial transport properties.21,22 Due to their rich transport behavior, junctions of quantum wires and their networks have
thus attracted much attention. 23–37
Most of the previous works on the transport properties of
junctions of TLL wires focus on wires with the same Luttinger parameter. However, experimentally, there is no reason
for all the TLLs emanating from a junction to be identical.
Moreover, a single TLL can have inhomogeneities; for example, a contact between an interacting TLL and a Fermi-liquid
lead, a key ingredient of most transport measurements, is often studied as an inhomogeneous TLL wire smoothly interpolating between interacting (TLL) and noninteracting (Fermiliquid) regions or as a two-wire junction with the Luttinger parameter abruptly changing at the junction.38–56 A junction of
three quantum wires with different Luttinger parameters has

FIG. 1: Junctions of two and three TLLs with different Luttinger
parameters.

been studied in the weak coupling regime.21,57–59 The experimental importance of junctions of TLL wires with generally
unequal Luttinger parameters motivates an in-depth study of
their properties, which is the main objective of the present paper.
Here, we focus on junctions of two and three nonchiral
Luttinger liquids schematically depicted in Fig. 1. For two
wires, it is known that the transport properties of the junction are fully controlled by one effective Luttinger parameter
ge = 2/(g−1 + g−1 ) as found in Ref. 39. In the context of
1
2
fractional Hall edge states60 , similar results have been found
for tunneling between two chiral-TLL edge states.43 . For two
nonchiral wires, one can reach the same conclusion through
an almost identical argument. In this work, however, we obtain this result within the framework of boundary conformal
field theory (BCFT), using the delayed evaluation of boundary conditions (DEBC) method,61–64 which, as we will see,
has the advantage that it can be readily generalized to junctions of more quantum wires.
Such generalization to a junction of three nonidentical
quantum wires is a key result of this paper. We find the stability regions of the previously identified (in Ref. 27 for three
equal Luttinger parameters) fixed points of such a Y junction
in the (g1 , g2 , g3 ) space, and compute their corresponding conductances as a function of these three Luttinger parameters.
Moreover, we obtain new asymmetric fixed points, which are
only stable for nonidentical TLLs, thereby providing a more

2
complete classification of the conformally invariant BCs for
three TLLs. Such asymmetric fixed points have been identified using perturbative renormalization-group analysis in the
weak coupling regime.21,57,58
Another important result of this paper is a direct numerical
verification, through DMRG computations,65 of the analytical
predictions for the conductance of a junction of two nonidentical wires. Using a recently developed method,66,67 which allows us to extract the conductance from a ground-state calculation in a finite system, we compute, in a microscopic lattice
model, the conductance of a junction of two chains of interacting spinless electrons with different interaction strengths. Our
numerical results show excellent agreement with the DEBC
predictions.
The outline of this paper is as follows. In Sec. II, we set up
the notation and present the model in the bosonization framework. In Sec. III, we present the DEBC analysis of a junction of two spinless wires and show that the scaling behavior of such junctions is governed by a single effective Luttinger parameter ge = 2/(g−1 + g−1 ). For wires with spin1
2
1/2 electrons, we obtains two effective Luttinger parameters
gc,s = 2/(1/gc,s + 1/gc,s), corresponding to charge and spin
e
2
1
sectors. Section IV contains the numerical DMRG calculations of the conductance of a junction of two spinless chains
with different interaction strengths. In Sec. V, we summarize
our results on the Y junction with the detailed DEBC analysis
presented in Appendix A. In Sec. VI, we present the analysis
of the conductance renormalization when the wires are contacted to Fermi-liquid leads. Finally, we conclude in Sec. VII.

fields, can be written in terms of the above bosons through:
√ L,R
√
ηi
ηi
ψiL,R = √ ei 2φi = √ ei(ϕi ±θi )/ 2 ,
2π
2π

where ηi are anticommuting Klein factors, which ensure the
correct fermionic statistics. The Klein factors play no role
in our analysis and are hence neglected throughout the paper.
It is convenient to define complex variables z = τ + ix and
z = τ − ix such that the left- and right-moving bosons, corre¯
sponding to current flowing toward and away from the junction, are, respectively, functions of z and z only. The chiral
¯
current operators can then be written as
i
JiR = √ ∂z θi ;
¯
2π

−i
JiL = √ ∂z θi ,
2π

(2.5)

where ∂z = (∂τ −i∂ x )/2 and ∂z = (∂τ +i∂ x )/2. The total current
¯
is proportional to the difference between the right and the left
currents: Ji = vi (JiR − JiL ).
To analyze the junctions of quantum wires with unequal
Luttinger parameters, it is convenient to introduce rescaled
bosonic fields
√
˜
θi ≡ θi / gi ;

ϕi ≡
˜

√
gi ϕi ,

(2.6)

which effectively have a noninteracting action [i.e., g = 1 in
Eq. (2.1)]. Note that these rescaled fields satisfy the original commutation relations. Similarly, we define the following
rescaled left- and right-moving bosonic fields:
˜i
˜
ϕi = φiL + φR ;
˜

II. GENERAL SETUP

(2.4)

˜
˜i
˜
θi = φiL − φR ,

(2.7)

in terms of which the left- and right-moving fermions become
In this section, we present the model in the bosonization
framework and set up the notation. In the low-energy limit, the
wires are described by TLLs with the Euclidean action:14,15


∞
2

vi g i


(∂ x ϕi )2 + (∂τ ϕi ) 



S =
dx 
dτ

4π
v2
0
i
i


(2.1)
∞
2

vi


(∂ x θi )2 + (∂τ θi )  ,



=
dx 
dτ

4πgi
v2
0
i
i
where gi and vi are, respectively, the Luttinger parameter and
the plasmon velocity of wire i. Different wires can have different electron-electron interactions and consequently different Luttinger parameters gi . The boson fields, ϕi and θi , have
the following equal-time commutation relation,
[ϕi (x), θ j (x′ )] = iπδi j sgn(x′ − x),

(2.2)

so the conjugate momenta of θi fields are given by Πθi =
(∂ x ϕi )/2π. Let us also define left- and right-moving boson
fields as
ϕi = φiL + φR ;
i

θi = φiL − φR .
i

(2.3)

The fermions ψi (x) = eikF x ψR (x) + e−ikF x ψiL (x), with ψiL (x)
i
and ψR (x) the linearized left- and right-moving fermionic
i

i
ϕi
˜
√ ˜
ψiL,R = exp √ ( √ ± gi θi ) .
2 gi

(2.8)

In the absence of a junction (boundary), the correlation functions of the rescaled fields are given by:
δi j
ln [(z − w)(¯ − w)] ,
z ¯
2
δi j
˜ ¯ ˜
θi (z, z)θ j (w, w) = −
¯
ln [(z − w)(¯ − w)] ,
z ¯
2
ϕi (z, z)θ j (w, w) =0.
˜
¯ ˜
¯

ϕi (z, z)ϕ j (w, w) = −
˜
¯ ˜
¯

(2.9)

Imaginary time ordering is implied for all correlation functions here and throughout the paper. Different bosonic fields
above are of course uncorrelated. However, as we will show
below, the presence of a junction mixes these fields and effectively reduces the independent bosonic degrees of freedom by
half.
At the end of wires, x = 0, fermions can hop between different wires. this process is described by a single particle hopping Hamiltonian
HB = −

ti j eiαi j ψ† (0)ψ j (0) + H.c. ,
i
i, j

(2.10)

3
where ti j and αi j are the strength and the phase of the hopping amplitude between wires i and j. Without loss of generality22,27 (at least for junctions of two or three wires), we
only consider the symmetric case ti j = t in this paper. The
phases encode the distribution of magnetic fluxes at the junction, which for a junction of three or more wires can play a
crucial role in the properties of some RG fixed points.22,27

III. JUNCTION OF TWO WIRES: DEBC ANALYSIS

In this section, we analyze the stability of RG fixed points,
and compute their corresponding conductances for a junction
of two TLL quantum wires with unequal Luttinger parameters. By using the DEBC method of Ref. 27, we show that
the properties of such junctions only depend on an effective
Luttinger parameter
g−1 = (g−1 + g−1 )/2.
e
2
1

(3.1)

Briefly, the junction of two wires is totally decoupled for
ge < 1, and has a conductance ge (e2 /h) for ge > 1. This
result is consistent with what was found in Ref. 43, where the
tunneling between fractional quantum Hall edge states with
different filling fractions was discussed. The DEBC method
used in this paper, however, has the advantage that it can be
straightforwardly generalized to junctions of more than two
wires. (see Sec.V and Appendix A)

B.

Junctions of two quantum wires

The first step of the DEBC method is to write an ansatz for
the conformally invariant BCs describing the RG fixed points.
The next step is to list all the boundary operators, which can
possibly become relevant and make the fixed point unstable,
and compute their scaling dimensions with such ansatz for every point in the parameter space, g1 and g2 . If for a given
ansatz, none of these boundary operators have a scaling dimension smaller than one (in some region of the parameter
space known as the stability region), we have found a stable
RG fixed point. In case of a junction, a natural ansatz can be
expressed in terms of a rotation matrix R that relates outgoing
to incoming bosonic fields:
˜
˜
φR = RφL ,

˜
˜
˜ L,R
where φL,R ≡ (φ1 , . . . , φiL,R )T are i-component vector fields.
The most important boundary operators, in case of a junction of two wires, correspond to the following processes: tunneling of chiral fermions between the two wires, and backscattering within the individual wires. It is useful to introduce
a compact notation for boundary operators describing the
single-particle tunneling processes,
†

T ba ≡ ψb ψa | x=0 ,
i
j
ji

For one-dimensional quantum impurity problems, one often invokes the conformal symmetry of the bulk system (Luttinger liquids in our case) and assumes that the effect of an
impurity (junction), at low energies, is imposing a conformally invariant boundary condition (BC), which describes
the renormalization-group (RG) fixed point. This methodology of relating the BC and RG fixed points of the system
is called boundary conformal field theory (BCFT), and has
proved greatly successful in the study of quantum impurity
problems.63,64
A useful technique within the framework of BCFT is the
DEBC method, which hugely simplifies the evaluation of the
scaling dimensions, ∆OB , of boundary operators, OB , with a
given BC.27 The scaling dimension, in turn, determines the
leading scaling behavior of a given operator under the RG
flow, and thus govern the stability of the RG fixed points: In
general, an RG fixed point (boundary condition) is stable if
all boundary operators are either equivalent to identity or irrelevant ∆OB > 1. Moreover, the conductance associated with
the given fixed point can be readily computed from the BC.
For a complete description of the DEBC method, we refer the
reader to Refs. 27,29. Here, we simply apply this method to a
junction of two quantum wires with unequal Luttinger parameters, commenting only on some key ingredients.

(3.3)

where a, b = R, L. We can then list these six fundamental
boundary operators in terms of the rescaled boson fields as
follows:
RL
T 21(12) ∼e

A. DEBC method

(3.2)

RL
T 11(22) ∼e
LL(RR)
∼e
T 21

i
±√ (
2

ϕ
˜
√1
g1

√ √
˜
i 2 g1 θ1(2)
ϕ
˜
i
√ ( √1
g1
2

ϕ
˜

2
− √g )
2

e

i
√

2

√ ˜ √ ˜
( g1 θ1 + g2 θ2 )

,
(3.4)

,
√

√

ϕ2
˜
i
˜
˜
− √g ) ± √ ( g1 θ1 − g2 θ2 )
2

e

2

.

The boundary operators corresponding to multiparticle processes are not forbidden and can be generated as higher-order
perturbation processes even they are not presence in the bare
Hamiltonian. In general, they can be constructed from these
fundamental boundary operators and have larger scaling dimensions and are less relevant than the single-particle processes.
All the above boundary operators have the generic form
˜
˜
OB ∼ eia·ϕ+ib·θ , where a, b are vectors that contain the pref˜
actors of the ϕi and θi fields. By eliminating the redundant
˜
degrees of freedom with Eq. (3.2), and using Eq. (2.9), the
scaling dimension of the generic OB above can then be written in terms of R as
∆RB =
O

1 T
|R (a − b) + (a + b)|2 ,
4

(3.5)

where the superscript T represents matrix transpose.
To find all the R matrices that correspond to stable fixed
points, it is convenient to express ϕi fields in terms of the fol˜
lowing ± fields:
√
√
√
√
g2 ϕ1 − g1 ϕ2
˜
˜
g1 ϕ1 + g2 ϕ2
˜
˜
; ϕ− =
.
(3.6)
ϕ+ =
√
√
g1 + g2
g1 + g2

4
Corresponding θ± are defined in a similar manner. The six
fundamental boundary operators can then be written as
RL
T 21(12) ∼e

ϕ−
±i √ge

RL
T 11(22) ∼e±i

LL(RR)
T 21

∼e

i

√

ϕ
√−
ge

,

ge θ−

e

(3.7)

,

√
±i ge θ−

C. Conductance for each fixed point

Here, we will compute the conductance associated with NBC and D-BC. The conductance tensor is defined through the
following current-voltage relation:

.

Here, we have dropped all e terms as they are effectively an
identity at the boundary: Charge conservation requires i JiR −
JiL = 0, which, using Eqs. (2.5) and (2.6), gives ∂τ θ+ = 0 (i.e.,
Dirichlet BC on θ+ ), and makes eiθ+ an effective identity.27 The
simplified boundary operators in Eq. (3.7) then only depend
the effective Luttinger liquid parameter ge .
In terms of the left- and right-moving ± fields defined in a
similar manner to Eq. (2.7), the Dirichlet BC on θ+ gives
L
φR = φ+
+

.
x=0

(3.8)

L,R
Now we only need to specify the BC relating the φ− fields
as:
L
φR = R− φ−
−

x=0

.

(3.10)

j

where current is defined as positive when flowing toward junction. In the linear-response regime, the conductance above can
be evaluated via the Kubo formula:27
Gi j = lim

ω→0+

−

e2 1
ωL

∞

L

dτeiωτ

−∞

dx Ji (y, τ)J j (x, 0) . (3.11)
0

The current correlation function can be rewritten as a sum of
chiral-current correlation functions as

(3.9)

Because there is a single pair of RL fields, only the NeumannN,D
BC (N-BC) and the Dirichlet-BC (D-BC), R− = ±1, are
allowed. By using Eq. (3.5) on boundary operators listed in
Eq. (3.7), we obtain the scaling dimension of each operator
with the N-BC and D-BC:
N
D
OB
∆OB (N-BC) ∆OB
RL
RL
T 21 , T 12
1/ge
RL
RL
T 11 , T 22
0
LL
RR
T 21 , T 21
1/ge

Gi j V j ,

Ii =

iθ+

(D-BC)
0
ge
ge

Here, the scaling dimension 0 indicates that OB is equivalent
to identity operator 1 for the given boundary condition.
1
From the table above, we conclude that the N-BC is stable when ge < 1 and D-BC is stable when ge > 1. As
shown below, the N-BC corresponds to a fixed point where
two wires are disconnected and the D-BC corresponds to a
fixed point where two wires are maximally connected with
conductance G D = ge (e2 /h). Let us recall the well-known results of the Kane and Fisher problem, namely, a single impurity in a spinless TLL wire, which is equivalent to a junction
of two wires with the same Luttinger parameter, g.14 There,
two RG fixed points are identified, totally disconnected fixed
point for g < 1 and maximally connected fixed point for g > 1
with G D = g(e2 /h). As ge → g when g1 = g2 = g, the fixed
points, which we identified, generalize Kane and Fisher’s results to the case of two quantum wires with different Luttinger
parameters.
A similar DEBC analysis can also be applied to junction of
two quantum wires with spin-1/2 electrons.29 In that case, we
obtain two effective Luttinger parameters gc,s = 2/(1/gc,s +
e
1
1/gc,s), corresponding to charge and spin sectors, which gov2
ern the scaling behavior of boundary operators near the fixed
points.68 As a consequence, the stable fixed points of such system are connected to those of equal Luttinger parameters but
with gc,s → gc,s .15,16,18
e

Ji (y, τ)J j (x, 0) =
JiR (y, τ)J R (x, 0) + JiL (y, τ)J L (x, 0)
j
j
− JiR (y, τ)J L (x, 0) − JiL (y, τ)J R (x, 0) , (3.12)
j
j
where the left and right currents are defined in Eq. (2.5). By
using Eq. (2.6), we can write the chiral currents in terms of
rescaled boson fields as follows:
√
√
gi
gi
√ ˜
JiR = + i √ ∂z θi = −i √ ∂z φR ≡ gi JiR ,
¯˜
¯ ˜i
2π
2π
√
√
gi
gi
√ ˜
˜
˜
JiL = − i √ ∂z θi = −i √ ∂z φiL ≡ gi JiL ,
2π
2π

(3.13)

where we have defined left and right currents associated with
the rescaled fields.
To evaluate the above correlation functions given a BC, it is
convenient to first express the boundary conditions R directly
˜
in the rescale boson field, φiL,R , basis. From Eq. (3.8) and
Eq. (3.9), we can derive the rotation matrix:
N

R =1
1;





D

R =



g1 −g2
g√+g2
1
2 g1 g2
g1 +g2

√
2 g1 g2
g1 +g2
g2 −g1
g1 +g2





,




(3.14)

for the N-BC and D-BC, respectively. From Eq. (3.13), the left
˜
˜
and right currents are constrained by the JiR (0, τ) = Ri j J L (0, τ)
j
BC at the origin, which, upon unfolding the current, implies
˜j
˜
JiR (x, τ) = Ri j J L (−x, τ),

(3.15)

for x > 0. Therefore, all the right-moving currents can be
interpreted as left-moving on the x < 0 domain.
Now, the chiral current correlation functions can be evalu-

5
ated:
δi j
gi
,
2 (¯ − z )2
4π zi ¯ j
δi j
gi
JiL (zi )J L (z j ) = 2
,
j
4π (zi − z j )2
√
gi g j
Ri j
,
JiR (¯i )J L (z j ) = 2
z j
4π (¯i − z j )2
z
√
gi g j
R ji
JiL (zi )J R (¯ j ) = 2
z
.
j
4π (zi − z j )2
¯

JiR (¯i )J R (¯ j ) =
z j z

(3.16)

By inserting these correlation functions into the Kubo formula
Eq. (3.11), and after some algebra, we obtain a concise relation between conductances and boundary conditions27
Gi j =

e2 √
gi g j (δi j − Ri j ).
h

(3.17)

With the N-BC and D-BC represented by the rotation matrices
of Eq. (3.14), we immediately conclude that
G N = 0;

G D = ge

e2
h

1 −1
.
−1 1

(3.18)

As expected, the N-BC corresponds to a fixed point with decoupled wires and D-BC corresponds to a fixed point with
conductance ge (e2 /h).
Here, we shall emphasize that the correlation functions
listed in Eq. (3.16) include only the universal part for a given
boundary condition. There are also nonuniversal contributions
to the correlation functions, which, in general, decay faster
and become irrelevant at long distance. However, when the
universal part vanishes, which is the case for correlations between different wires with the N-BC, the higher-order contributions dominate and could lead to nonlinear conductance. In
the next section, we will perform the DMRG calculations and
confirm that Eq. (3.16) indeed represents the universal part of
chiral-current correlation functions.
IV.

JUNCTION OF TWO WIRES: DMRG CALCULATIONS

In this section, we perform numerical computations of the
conductance of a junction of two Luttinger liquids with different Luttinger parameters g1 and g2 . A microscopic lattice
model with the Luttinger-liquid physics, which is suitable for
numerical calculations, is the one-dimensional tight-binding
model of interacting spinless electrons,
Hi =
m

1
1
c† ci,m+1 + H.c. + Vi (ni,m − )(ni,m+1 − ), (4.1)
i,m
2
2

for wire i, where the hopping amplitude is set to unity. A
junction of two wires can be described by
H = H1 + H2 − tc† c2,0 − tc† c1,0 .
2,0
1,0

(4.2)

The parameters gi and vi of the Luttinger-liquid Hamiltonian (2.1) are related to the interaction strength V through the

FIG. 2: a) The junction of two semi-infinite Luttinger-liquid wires
with interaction strength V1 and V2 . Hopping amplitude is set to
unity in the bulk of the the two wires, and is equal to t at the junction.
b) The corresponding finite system used for DMRG calculations according to the method of Refs.66,67 .

Bethe ansatz (see, e.g., Refs. 66,69). We use the method of
Refs. 66,67 to compute the conductance. This method allows
us to extract the conductance from a ground-state static calculation in a finite system as explained below. The semi-infinite
junction of Hamiltonian (4.2) is depicted in Fig. 2(a), while
the corresponding finite system used in the numerics is shown
in Fig. 2(b).
We extract the conductance Gi j from the following asymptotic (large x) relationship in the finite system shown in
Fig. 2(b):
j
i
JR (x)JL (x) ≃

h
π
Gi j 4 ℓ sin x
ℓ
e2

−2

,

(4.3)

j

i
where JR (x)JL (x) is the ground-state correlation function of
chiral currents in wires i and j in Fig. 2b. In terms of total
charge N and current J operators, which can be modeled on
the lattice, we generically have67
j
i
JR (x)JL (x) = −

1
1
J i (x)J j (x) −
N i (x)J j (x) . (4.4)
2vi v j
2v j

Note that in the time-reversal symmetric case considered here,
the second term in the expression above vanishes, and we only
need to compute a static current-current correlation function
J i (x)J j (x) . In terms of the lattice creation and annihilation
operators appearing in Hamiltonian (4.1), we have
1
J i (m + ) = i(c† ci,m − c† ci,m+1 ).
i,m
i,m+1
2
All we need to do now is to numerically compute J 1 (x)J 2 (x)
for the above current operator, and divide it by 2v1 v2 to obtain
1
2
JR (x)JL (x) . The numerical calculations are done for system
of 180 sites in each of the two wires. The truncated number
of states in our DMRG computations is 1100.
A. Repulsive effective interaction ge < 1

For ge < 1, we have G12 = 0, which implies that the
1
2
leading term [Eq. (4.3)] in the JR (x)JL (x) correlation func-

6
ℓ
tion vanishes. If, as a function of π sin π x , the computed
ℓ
1
2
JR (x)JL (x) decays faster than a power law with exponent −2,
we have a signature of a vanishing G12 . Our numerical results
indeed confirm this: for any combination of g1 and g2 with

−
12

−α(ge )

ℓ
1
2
ge < 1, we find that JR (x)JL (x) decays as π sin π x
ℓ
with α(ge ) > 2. The exponent α only depends on ge , and is
independent of individual gi and the hopping amplitude t. The
prefactor of the correlation function depends on the hopping
amplitude t since universality is a property of the leading term,
and the coefficient of the subleading term observed here can
depend on microscopic details such as t.
Note that the correlation functions for different combinations of g1 and g2 , which have the same ge , collapse not only
in the large x limit but also close to the microscopic length
scales. This strongly suggests that the single parameter ge
determines all the subleading corrections to the correlation
function. This behavior can be understood by noting that all
the boundary operators in Eq. (3.7) depend on ge (as opposed
˜
to individual g1 and g2 ) after dropping the ϕ+ and θ+ fields
˜
due to current conservation. Hence, the measured correlation
functions should also be determined only by the effective Luttinger parameter ge and the hopping strength t. Interestingly,
the exponent α is indeed close to the scaling dimension of the
leading irrelevant operator (i.e., 2/ge ) but we are not able to
make a definitive statement due to finite-size effects and limited numerical precision. It is worth mentioning that we have
also considered combinations of g1 and g2 where one wire has
attractive interactions (g1 > 1), but this does not affect the behavior of the junction as long as ge < 1.
The results are summarized in Fig. 3. We have considered
two values of ge = 0.871, 0.83 and four combinations of g1
and g2 for each ge as shown below.

V1

V2

g1

g2

ge

0.463

0.463

0.871

0.871

0.871

0

0.9

1

0.771

0.871

0.347

0.576

0.9

0.843

0.871

−0.285

1.145

1.1

0.720

0.871

0.632

0.632

0.830

0.830

0.830

0

1.2

1

0.709

0.830

0.347

0.904

0.9

0.770

0.830

−0.285

1.415

1.1

0.666

0.830

Here, we have considered two different values of hopping amplitude, t = 0.5, 0.7.

−
16

−
20

0

In this case, we expect any tunneling amplitude t 1 at the
junction to heal, and result in a universal conductance ge e2 /h.
In our numerics, we have used two values of tunneling amplitude, t = 0.7, 0.9. Similarly to the repulsive case, we consider
two values of ge = 1.175, 1.258; for each ge , we consider four
combinations of g1 and g2 shown below.

2

3

4

−
10

−
14

−
18
1

2

3

4

1
2
FIG. 3: The correlation function JR (x)JL (x) for different combinations of g1 and g2 corresponding to two values of ge < 1. The data
exhibits very good collapse, and indicates a vanishing conductance.

V1

V2

g1

−0.4625

−0.4625

1.175

1.175

1.175

1

1.423

1.175

0.347

−1.196

0.9

1.690

1.175

−0.637

1.1

1.260

1.175

−0.632

−0.632

1.258

1.258

1.258

1

1.690

1.258

0.347

−1.460

0.9

2.087

1.258

1.1

1.468

1.258

−0.9

0

−0.285

−1.2

0

−0.285

−0.960

g2

ge

Again, we obtain very good agreement with the analytical predictions. The results are summarized in Fig. 4. With fixed ge ,
the data points collapse for different combinations of g1 and
g2 and different values of hopping t. The numerical results approach the analytical prediction for a conductance of ge e2 /h
(solid black line) in the asymptotic limit.
V.

B. Attractive effective interaction ge > 1

1

JUNCTION OF THREE WIRES

A junction of three quantum wires with equal Luttinger parameters has three distinct types of fixed points described by a
rotation matrix ansatz: decoupled fixed point, chiral-χ± fixed
points, and Dirichlet fixed points, which are respectively stable for g < 1, 1 < g < 3 and g > 3. (There is an additional less understood time-reversal-invariant M fixed point
for 1 < g < 3, which we do not consider in this work.)22,27,66,67
Thus, it is reasonable to expect that these fixed points remain

7
ters. Such analysis with equal Luttinger parameters was performed in Ref. 67, but is beyond the scope of the present paper. We discuss the physical properties of each of these stable fixed points in the remainder of this section. The detailed
analysis, based on the DEBC method, can be found in Appendix A. As the conductances of fixed points can be evaluated in the similar way as in Sec. III C, we simply write down
the results in the following discussion.

a. Decoupled fixed point

1
2
FIG. 4: The correlation function JR (x)JL (x) for different combinations of g1 and g2 corresponding to two values of ge > 1. The data
exhibits very good collapse, and indicates a conductance of ge e2 /h
because of the asymptotic agreement with the exact theoretical prediction shown by a sold black line.

stable if we slightly change the Luttinger parameters and make
them unequal. Here, we first determine this region of stability around the g1 = g2 = g3 line. In addition, we identify three asymmetric fixed points, only realized for unequal
Luttinger parameters, in which one of the wires is decoupled
from the junction and the other two wires are fully connected.
These asymmetric fixed points have important consequences
for the stability of N, D, and χ ones: there are regions of the
parameter space near the transition points gi = 1, 3 on the
g1 = g2 = g3 line where small perturbations normal to the
equal-g line would drive the system into one of the asymmetric fixed points.
In Table I, we first summarize the scaling dimensions of
the leading boundary operators as well as their corresponding
conductances as a function of the three Luttinger parameters
for each boundary condition. These scaling dimensions determine the stability of the fixed point: when all them are larger
than one in certain parameter region, the given fixed point is
stable. From Table I, we observe that the three scaling dimensions of the leading irrelevant operators for the asymmetric
fixed points are each the inverse of those of the decoupled,
Dirichlet and chiral fixed points. Hence, in any given point of
the (g1 , g2 , g3 ) parameter space, there exists at least one stable
fixed point. In other words, the decoupled, Dirichlet, chiral
and asymmetric fixed points fully cover the phase diagram of
a Y junction of spinless TLL wires.
In this paper, the DMRG analysis is not applied to the junction of three quantum wires with unequal Luttinger parame-

The decoupled fixed point corresponds to the Neumann BC
for all the bosonic fields ϕ. From Table I, we see that the scal˜
ing dimensions of the leading irrelevant operators are equal to
1/gi, j for gi, j = 2gi g j /(gi + g j ), which is the same as Eq. (3.1)
e
e
for a pair of wires. Hence, the decoupled fixed point (N-BC)
is then stable when the N fixed point is stable for all three possible pairs of wires. One can simply check that these scaling
dimensions reduce to 1/g when the Luttinger parameters are
all equal. In Fig. 5, the stability region of the decoupled fixed
N
point, ∆OB > 1, is shown in red. As expected, the conductance
of the decoupled fixed points is simply
GiNj = 0.

(5.1)

b. Dirichlet fixed point
As discussed in Appendix A, one can construct three independent linear combinations of the bosonic field ϕ such that,
˜
akin to ϕ+ field for junctions of two wires, one of them, known
˜
as the center of mass field, always satisfies the Neumann BC
due to charge conservation. The Dirichlet fixed point corresponds to imposing the D-BC on the other two combinations.
None of single particle processes becomes identity with such
boundary condition. Instead, some of two- or more-particle
processes reduce to identity, which suggests that the Dirichlet fixed point is associated with a certain type of “Andreev”
reflection that enhances the conductance.
From Table I, the scaling dimensions of all leading irreleD
vant operators, ∆OB = gi (g j + gk )/2(g1 + g2 + g3 ) ∀ i j k,
reduce to g/3 when g1 = g2 = g3 = g. Hence, the D-BC
becomes stable at g > 3, consistent with Ref. 27. By requirD
ing ∆OB > 1, we obtain the stability region, shown in green
in Fig. 5, of the Dirichlet fixed point for unequal Luttinger
parameters.
The conductance of the Dirichlet fixed point is given by
GD = 2
jk

g j gk
e2
g j δ jk −
,
h
g1 + g2 + g3

(5.2)

where we have made a cyclic identification g0 ≡ g3 and
g4 ≡ g1 . When all the Luttinger parameters are equal, we
have G D = g(e2 /h)(2δ jk − 2/3), which reproduces the result
jk
in Ref. 27.
c. Chiral-χ± fixed points

8
TABLE I: The scaling dimensions of leading irrelevant boundary operators and the conductance tensor for each stable fixed point of a Y
junction. The detailed analysis for obtaining these scaling dimensions is given in Appendix A with the corresponding operators listed in
Table II. The asymmetric fixed point Ai represents a boundary condition where the wire i, for i = 1, 2, 3, is decoupled from the junction.
Here, the scaling dimensions of all leading irrelevant operators run over the indices for all possible combinations of i j k. We have also
introduced following notations: the cyclic identification g0 ≡ g3 and g4 ≡ g1 ; the two indices antisymmetric tensor ǫ j, j±1 = ±1 and 0 otherwise;
and the index m satisfies m j k. The conductance tensors are given units of e2 /h and are defined through I j = k G jk Vk .
Fixed point

Conductance G jk [e2 /h]

Scaling dimensions ∆

Decoupled (N-BC)

(gi + g j )/2gi g j

Chiral (χ± -BC)
Asymmetric (Ai -BC)

0

gi (g j + gk )/2(g1 + g2 + g3 )

Dirichlet (D-BC)

2[g j δ jk − g j gk /(g1 + g2 + g3 )]

2gi (g j + gk )/(g1 g2 g3 + g1 + g2 + g3 )
(g1 + g2 + g3 + g1 g2 g3 )/2gi (gi+1 + gi−1 )
2gi+1 gi−1 /(gi+1 + gi−1 )
2(g1 + g2 + g3 )/gi (gi+1 + gi−1 )

The chiral-χ± fixed points have a particular transport feature: the realization of χ+ or χ− fixed points, with the incoming current respectively flowing clockwise or counterclockwise into one of the adjacent wires, depends on the direction
of the threaded magnetic field into the ring.27 (This point will
become more apparent in the next section when discussing
Fermi-liquid leads) When g1 = g2 = g3 = g, the scaling dimensions listed in Table I for both χ± fixed points reduce to
4g
3+g , and hence χ± fixed point is stable for 1 < g < 3, which is
consistent with what found in Ref. 27. Again, we obtain the
stability region of the chiral fixed points through ∆χ±B > 1. In
O
Fig. 5, such region is shown in orange.
The conductances for chiral-χ± fixed points are, in turn,
give by
Gχ± = 2
jk

e2 g j (g1 + g2 + g3 )δ jk + g j gk (∓gm ǫ jk − 1)
,
h
g1 g2 g3 + g1 + g2 + g3

where ǫ j, j±1 = ±1 while ǫ jk = 0 for j = k, and gm

(5.3)

g j , gk .

d. Asymmetric fixed points
The asymmetric-Ai fixed points have the property that wire
i is decoupled from the junction. Such fixed points are a new
feature of a system with unequal Luttinger parameters. As
shown in Table I and mentioned earlier, the scaling dimensions of the leading irrelevant operators of Ai fixed points are
inverse to those of the decoupled, Dirichlet and chiral fixed
points. Hence, the stability regions of Ai fixed points are complimentary to those of the other fixed points, and the entire
parameter space is covered by at least one of the stable fixed
point presented in this work. In Fig. 5, the stability regions of
A1,2,3 fixed point are shown in yellow, gray, and blue, respectively. The regions where two asymmetric fixed points overlap
are shown in white.
The conductances of the asymmetric fixed points Ai are give
by
i+1,i−1
G Ai = ge
jk

e2
(−1 + δi j + δik + 2δ jk − 3δi j δik ),
h

(5.4)

m,n
where ge is the effective Luttinger parameter for the pair of
wires m and n. To give an idea of the properties of asymmetric

2

g j (g1 +g2 +g3 )δ jk +g j gk (∓gm ǫ jk −1)
g1 g2 g3 +g1 +g2 +g3

2gi+1 gi−1
(−1
gi+1 +gi−1

+ δi j + δik + 2δ jk − 3δi j δik )

fixed points, we hereby write the conductance tensor explicitly
for A1 ,




2g2 g3 e2  0 0 0 



 0 1 −1  .

G A1 =
(5.5)




g2 + g3 h  0 −1 1 

The stability of the different fixed points in the g1,2,3 parameter space is inferred from the scaling dimensions of Table I, and shown in Fig. 5. Since each stable fixed point implies a phase of the junction, we shall also refer to the graph
defining the regions of stability as a phase diagram. It is convenient to illustrate this three-dimensional phase diagram by
some cross sections. As all the scaling dimensions of leading irrelevant operators have a cyclic symmetry on wire indices, the stability regions show a threefold rotation symmetry
around the g1 = g2 = g3 axis. Thus, a natural choice for these
cross sections is given by planes normal to the equal-g axis:
(g1 + g2 + g3 )/3 = g, where the parameter g, the average of the
¯
¯
three Luttinger parameters, labels each cross section. Excluding negative gi , these cross sections are equilateral triangles
shown in Fig. 5.
We first notice that the decoupled fixed point becomes predominant when g < 2/3, in which none of other fixed points
¯
are stable. One can also show that chiral fixed points appear
when g > 1 and the Dirichlet fixed point only appears when
¯
g > 3. From Figs. 5(b), 5(d), and 5(f), we observe that the de¯
coupled, chiral and Dirichlet fixed points are realized around
the equal-g axis for g < 1, 1 < g < 3 and g > 3, respectively.
¯
¯
¯
These results are consistent with a junction of three identical
TLL wires (indicated as black points at the center of triangles),
and show how far the Luttinger parameters can deviate from
equal-g axis before these phases break down. We find that,
in most regions, these phases are stable under a small perturbation away from the equal-g line. This is relevant for the
realization of these phases experimentally, as the TLL wires
attached to a junction are likely nonidentical.
As shown in Figs. 5(c) and 5(e), the asymmetric fixed points
become important around two marginal points g1,2,3 = 1, 3.
As a small deviation of Luttinger parameters from these two
points easily realizes and switches between asymmetric fixed
points, the resultant fixed points are thus highly sensitive to all

9

FIG. 6: The voltages for the incoming and outgoing chiral currents
in the Fermi-liquid and connected TLL wire.

the three Luttinger parameters, and not just the averaged Luttinger parameter g. Therefore, precise control over the TLL
¯
wires become essential around these points. Note that it may
be possible to alter between different asymmetric fixed points,
and form a nano-switch if one can tune the Luttinger parameters. We mention in passing that the Luttinger parameters of
wires can, in principle, be modified by an external gate capacitively coupled to the wire.44
Finally, it is worthwhile to compare our findings with those
of Aristov and W¨ lfle,58,59 where two identical wires are cono
nected to a wire with unequal Luttinger parameter, by setting g1 = g2 = g. In the repulsive and weak attractive interaction regime, g ≈ g3 < 3, in which the N, χ± and Ai
fixed points predominate, our results show excellent agreement with their findings. In the strong attractive interaction
regime, g ≈ g3 > 3, the D fixed point identified in the present
work and in Ref. 27 (by nonperturbative boundary-conformalfield-theory methods), however, was not found by the perturbative renormalization-group approach of Refs. 58,59.

FIG. 5: We plot the stability regions of the different fixed points
in the triangular cross sections, shown in upper panel, of fixed
g = (g1 +g2 +g3 )/3 in Luttinger parameters space, g1 , g2 , and g3 . In all
¯
panels, the decoupled, chiral and Dirichlet fixed points are painted,
respectively, in red, orange, and green, and, the asymmetric fixed
points A1,2,3 are pained in yellow, gray, and blue, respectively. The
white areas represent an overlap of two Asymmetric fixed points and
the equal-g points are indicated by a black dot at the center of triangle. (a) For g < 2/3, only the decoupled fixed point is realized. (b)
¯
The χ± and Dirichlet fixed points are not stable when g < 1. (c) The
¯
point g1,2,3 = 1 is an exactly marginal point surrounded by asymmetric fixed points. (d) The χ± fixed points appear at the center of cross
section for 1 < g < 3, surrounded by asymmetric fixed points. The
¯
stability region for the N-BC are pushed to the corners where two of
the three Luttinger parameters become much less than one. (e) The
point g1,2,3 = 3 is another exactly marginal point, again surrounded
by asymmetric fixed points. The chiral fixed points have extremely
small stability regions (difficult to see in the figure) located between
any two asymmetric fixed points. (f) The Dirichlet fixed point appears at the center of triangle for g > 3. Note that the figure is not a
¯
schematic, and represents the exact domain boundaries for the shown
g.
¯

VI. CONDUCTANCE RENORMALIZATION FOR WIRES
CONTACTED TO FERMI-LIQUID LEADS

The linear conductances of different fixed points were calculated in Secs. III C and V. Here, we discuss the effect of
attaching the wires to Fermi-liquid leads. Remarkably, we
find that the conductance of each fixed point, in the presence
of Fermi-liquid leads, renormalizes to values that are independent of the Luttinger parameters. This generalizes the following interesting effect for the TLL quantum wire with Luttinger
parameter g: When attached to leads, the measured conductance is quantized at e2 /h, which is different from ge2 /h. This
discrepancy has been resolved by Maslov and Stone and by
Safi and Schulz in Refs. 38,39. There, they studied an inhomogeneous Luttinger liquid and concluded that the conductance of a TLL wire will only depend on the Luttinger parameter at the contact. As a Fermi-liquid (metal) contact can
be thought of as a TLL with Luttinger parameter g = 1, the
measured conductance becomes simply e2 /h.
An alternative way to understand this renormalization of
conductance due to a Fermi-liquid contact is to introduce a
contact resistance, 1/Gc = (g − 1)(2g)(h/e2), at both contacts
in series with the theoretical predicted resistance 1/g(h/e2).
This would give the total conductance exactly at e2 /h. Here,
the contact conductance causes a voltage drop when matching

10
with the wire and gives rise to a current/voltage relation
¯
Ii = Gc (Vi − Vi ),
i

(6.1)

where Vi is the potential of the carriers injected into wire i and
¯
Vi is the applied voltage at the contact connected to the wire
i. We can understand this result by thinking in terms of the
right- and left-moving currents inside the wire and the Femiliquid lead as seen in Fig. 6. The incoming electrons from
¯
the Fermi-liquid side are at the voltage Vi of the reservoir.
¯
The outgoing current may be at a different potential Viout at
the contact point, even though the electrons are expected to
relax to the equilibrium voltage as they propagate in the lead.
Similarly, the right- and left-moving currents in the TLL wire
are respectively at voltage Vi and Viin . The junction of the
Fermi-liquid lead and the TLL has an effective conductance
ge e2 /h with
1/ge = (1 + 1/g) /2.
The current is then related to the difference in the voltages of
the two incoming currents as
¯
Ii = ge (Vi − Viin ).
For the TLL wire, we can similarly write
Ii = g(Vi − Viin ).
Combining the above two equations leads to Eq. (6.1). Generally, it is useful to define a contact conductance tensor
Gc = δi j (e2 /h)(2gi)/(gi − 1) for a junction of three wires.
As the measured conductance is based on the applied voltage at contact, one can define a renormalized conductance ten¯
¯ ¯
sor, Gi j , as Ii = j Gi j V j . To connect this conductance with
one we found in the previous subsection, we invoke current
conservation:
Ii =

¯ ¯
Gi j V j .

Gi j V j =
j

(6.2)

j

Together with Eq. (6.1), one can show that27
¯
G = (1 + GG−1 )−1G,
c

(6.3)

¯
G−1 = G−1 + G−1 ,
c

(6.4)

or equivalently

which has a simple interpretation of resistances connected in
series.
Besides the decoupled fixed point that has obvious vanish¯
ing conductance G N = 0, we shall now apply Eq. (6.3) and
obtain the measured conductances of the other fixed points.
For Dirichlet fixed point, we have the renormalized conductance
2
¯ j 2 e (3δi j − 1).
GiD =
3 h

(6.5)

As the largest conductance one can obtain from single-particle
unitary scattering is GU = −(4/9)(e2/h) for j
k,20 the
jk

enhanced conductance above demonstrates the role of multiparticle scattering processes.
Upon attaching the wire to external Fermi-liquid leads, the
measured conductances of chiral fixed points become
2
¯ χ± 1 e (3δ jk − 1) ∓ ǫ jk
G jk =
2 h

(6.6)

(i.e., the currents flow only from lead 1 to 2, 2 to 3, and 3
to 1 for the χ+ fixed point and in reversed order for χ− fixed
point). Finally, the measured conductance of asymmetric-Ai
fixed points reads
e2
¯
GiAji = (−1 + δi j + δik + 2δ jk − 3δi j δik ),
h

(6.7)

which simply indicates a decoupled wire i with the rest of two
wires fully conducting.
¯
¯
We note that all renormalized conductance G D , Gχ± , and
¯ Ai , are the same as the unrenormalized conductance G with
G
all gi = 1. This result highlights that the dc conductance
of a junction of TLL wires depends only on the asymptotic
value of the Luttinger parameters of the wires, and in the case
when Fermi liquid leads are attached, this asymptotic value
is gi = 1. Thus, when in contact to Fermi liquid leads, the
conductance tensor of the junction will take on the universal
values listed above for the different fixed points. Notice, however, that which fixed point is selected still depends on the gi ’s
of the interacting wire segments.

VII. CONCLUSION

In summary, we applied the DEBC method to a two-wire
and a Y junction of TLL wires with generally unequal Luttinger parameters. For two spinless wires, we successfully reproduced the prediction that all properties of the junction are
determined by a single effective Luttinger parameter ge . We
verified this prediction by direct numerical calculations on the
lattice with the method of Refs. 66,67: We observed numerically that as long as ge < 1, even if one of the wires has attractive interactions gi > 1, any impurity leads to a vanishing
linear conductance. Moreover, we found that the nonuniversal corrections to the correlations across the junction, which
come from perturbations with irrelevant boundary operators
to the decoupled fixed point, are independent of the individual Luttinger parameters and only depend on ge and the local
microscopic structure of the junction. For ge > 1, we explicitly found a universal conductance of ge e2 /h regardless of the
individual Luttinger parameters and the microscopic details.
For a Y junction of nonidentical TTLs, we found that the
N-BC, χ-BC and the D-BC are stable within regions of the
(g1 , g2 , g3 ) parameter space, which we explicitly determined.
By identifying three more asymmetric fixed points, corresponding to only one decoupled wire, and determining the region of stability of each fixed point, we determined the full
phase diagram of the Y junction. We also obtained explicit
formulas for the conductance of all fixed points. The findings
of this work have direct experimental relevance. In particular,

11
our results shed light on the issue of connecting interacting
TLLs to Fermi-liquid leads for the measurement of transport
properties. Our work also provides an important theoretical
extension of the well-known results on transport through junctions of two and three identical TLL wires.
As an outlook, we finally discuss how to generalize the current method to a junction of N > 3 wires. The key to such
generalization is to identify the possible fixed points via the
rotation matrix R. These rotation matrices are constrained
by charge conservation and can, in general, be expressed as
S O(N − 1) matrices after eliminating the total-charge (centerof-mass) mode (cf., Appendix A). Then, one has to identify
the corresponding N-BC, D-BC, chiral-like BC, and asymmetric BC and analyze their stability. One useful trick for
identifying the possible stable fixed points is to utilize the fact
that a stable BC would make certain boundary operators effectively equal to identity, c.f. the discussion in Appendix Ac.
Of course, with the number of wires increasing, the number of
possible fixed points also increases and the analysis becomes
more complicated.

acknowledgements

We thank I. Affleck and M. Oshikawa for enlightening discussions, and for collaboration on closely related work. We
are also grateful to P. W¨ lfle for helpful discussions regarding
o
his results.58,59 This work was supported in part by the DOE
Grant No. DE-FG02-06ER46316 (A.R., C.C., C.-Y.H.), the
U.S. Department of Energy through LANL/LDRD program
(A.R.), the NSF Grant No. DMR-0955707 (A.E.F.), and the
DARPA-QuEST program (C.-Y.H.).

Appendix A: Junction of three wires with unequal Luttinger
parameters: DEBC analysis

In this Appendix, we present the details of the DEBC analysis of the stability of the following fixed points for a junction of three quantum wires: decoupled, chiral-χ± , Dirichlet, and Asymmetric-Ai fixed points. The system is described
by the action (2.1) with i = 1, 2, 3 and the hopping Hamiltonian (2.10) with αi j = γ/3, where γ is the magnetic flux
through the ring at the junction.
To simplify the notation, we drop the overhead tilde (˜)
symbol for the rescaled fields throughout this appendix. (All
the bosonic fields are rescaled.) To employ the DEBC method,
it is convenient to choose a proper basis for the rescaled boson fields. In the first step, we identify the following centerof-mass field, which always satisfies the N-BC due to charge
conservation:
1
√
√
√
Φ0 = √
( g1 ϕ1 + g2 ϕ2 + g3 ϕ3 ).
g1 + g2 + g3

(a) ± cycle
√

g1 g g3
i √g +g2 +g (ˆ×K3 )·Θ
z

RL
T 21(12) ∼ e±iK3 ·Φ e

1 2 3
√
g g g

i √ 1 2 3 (ˆ×K1 )·Θ
z
e±iK1 ·Φ e g1 +g2 +g3

RL
T 32(23)

∼

RL
T 13(31)

∼ e±iK2 ·Φ e g1 +g2 +g3
(b) Backscattering

√

g g g
i √ 1 2 3 (ˆ×K2 )·Θ
z

√

2 g1 g g3
−i √g +g 2+g (ˆ×K1 )·Θ
z
1 2 3

RL
T 11 ∼ e

√
2 g g g

1
3 z
−i √g +g 2+g (ˆ×K2 )·Θ
1 2 3

RL
T 22 ∼ e

√
2 g1 g2 g3

−i √

(ˆ×K )·Θ
z

3
RL
T 33 ∼ e g1 +g2 +g3
(c) LL-RR processes

±i

LL(RR)
T 21
∼ eiK3 ·Φ e

√

√
g3 (g1 −g2 )Θ2
2g g Θ
√ 1 2 1+ √
g1 +g2
2(g1 +g2 )(g1 +g2 +g3 )

√

√
g3 (g1 +2g2 )Θ2
2(g1 +g2 )(g1 +g2 +g3 )
√
√
g g Θ
g3 (2g1 +g2 )Θ2
∓i √ 1 2 1 + √
iK2 ·Φ
2(g1 +g2 )
2(g1 +g2 )(g1 +g2 +g3 )
g g Θ
∓i √ 1 2 1 − √

LL(RR)
T 32
∼ eiK1 ·Φ e
LL(RR)
T 13
∼e

2(g1 +g2 )

e

then becomes a constant and can be simply neglected. We
then define another two orthonormal boson fields:
√
√
g2 ϕ1 − g1 ϕ2
,
Φ1 =
√
g1 + g2
(A3)
√
√
[ g1 g3 ϕ1 + g2 g3 ϕ2 − (g1 + g2 )ϕ3 ]
Φ2 =
,
√
√
g1 + g2 + g3 g1 + g2
as well as their dual fields, Θ1,2 . Note that the choice of basis above is arbitrary but, as will become apparent, is a convenient one. We will organize the fields above into a vector
Φ = (Φ1 , Φ2 )T and its dual vector Θ = (Θ1 , Θ2 )T .
It is useful to define the following vectors:
√
√
g1
g1 + g2 + g3
K1 =(−
,
),
2g2 (g1 + g2 )
2g3 (g1 + g2 )
√
√
g2
g1 + g2 + g3
K2 =(−
,−
),
(A4)
2g1 (g1 + g2 )
2g3 (g1 + g2 )
√
g1 + g2
K3 =(
, 0),
2g1 g1
which will further simplify the notation. Notice that these
three Ki vectors add up to 0 for any gi .
By using the notation in Eq. (3.3) and neglecting Θ0 (due
to the N-BC on Φ0 ), the boundary operators for single-particle
processes are categorized in four classes, and listed in Table II.
As in the two-wire case, the higher-order processes can be
constructed from these single-particle boundary operators.

(A1)
Using the ansatz (3.2), we then write a rotation matrix:

The dual field to Φ0 , i.e.,
1
√
√
√
Θ0 = √
( g1 θ1 + g2 θ2 + g3 θ3 )
g1 + g2 + g3

TABLE II: The boundary operators corresponding to single-particle
processes at the Y junction.

(A2)

Rξ =

cos ξ sin ξ
,
− sin ξ cos ξ

(A5)

12
where ξ is a rotation angle. The rotation matrix above relates
φL = (Φ + Θ)/2 to φR = (Φ − Θ)/2. In terms of Rξ , the
scaling dimension of boundary operators is given by Eq. (3.5).
We now proceed to the stability analysis of the four Y-junction
fixed points.
a. Decoupled fixed point
The decoupled fixed point corresponds to the N-BC for Φ
field, and makes the Θ field a pure number. Therefore, all
backscattering processes are effectively identity. The rotation
N
matrix is simply equal to Rξ=0 = 1 From Eq. (3.5), the scal1.
ing dimension of an arbitrary operator with the N-BC becomes
N
∆OB = |a|2 . The explicit scaling dimensions for the operators
in Table II are listed below.

where ξ± are the rotation angles of χ± -BC, respectively. The
rotation matrices are obtained by plugging in the respective
rotation angles into Eq. (A5). Here, we do not show them
explicitly.
Let us first focus on the χ+ fixed points. The scaling dimensions of the single particle processes (excluding + cycle)
are:
∆χ+B
O
2g1 (g2 +g3 )
g1 g2 g3 +g1 +g2 +g3
2g2 (g1 +g3 )
g1 g2 g3 +g1 +g2 +g3
2g3 (g1 +g2 )
g1 g2 g3 +g1 +g2 +g3

OB

RL
RL
LL
RR
T 11 , T 23 , T 13 , T 21
RL
RL
LL
RR
T 22 , T 31 , T 21 , T 32
RL
RL
LL
RR
T 33 , T 12 , T 31 , T 13

N
Note that condition ∆OB > 1 determines the stability region of
the N-BC, shown in red in Fig. 5.

Notice that these scaling dimensions are cyclic in three indices
and hence all operators are important for determining the stability of the χ+ fixed point.
As for the χ− fixed point, one can show that all the leadingorder operators have exactly the same scaling dimensions
listed in the table above. Thus, both χ± fixed points share
exactly the same stability. In Fig. 5, the stability region of
chiral-χ± fixed point is shown in orange.

b. Dirichlet fixed point

d. Asymmetric fixed points

The Dirichlet fixed point corresponds the D-BC on the
Φ field (i.e., Φ is effectively a constant at boundary). The
D
rotation matrix of D-BC simply reads Rξ=π = −1 From
1.
Eq. (3.5), the scaling dimension of an arbitrary operator with
N
D-BC becomes ∆OB = |b|2 . Unlike the N-BC, none of the
single-particle processes becomes identity with D-BC. However, some of the two-or-more-particle processes do become
RL RL †
RL RL †
identity under D-BC, for instance: T 21 T 12 , T 32 T 23 , and
RL RL †
T 13 T 31 . This indicates that the Dirichlet fixed point is associated with the Andreev reflection.
Hereby, we only list the scaling dimensions for the ± cycle
as they are the leading irrelevant operators.

Although the use of rotation matrices Rξ is useful for other
fixed points, it is most convenient to identify rotation matrices
˜
directly in the rescaled boson field, φiL,R , basis. Because the
decoupled wire effectively has the N-BC for itself and the connected wires should follow mutual D-BC, by using Eq. (3.14),
the rotation matrix of asymmetric fixed point, A1 , has the form

OB

RL
RL
LL
RR
T 21 , T 12 , T 21 , T 21
RL
RL
LL
RR
T 32 , T 23 , T 32 , T 32
RL
RL
LL
RR
T 13 , T 31 , T 13 , T 13

N
∆OB (N-BC)

|K3 |2 = (g1 + g2 )/(2g1g2 )
|K1 |2 = (g2 + g3 )/(2g2g3 )
|K2 |2 = (g3 + g1 )/(2g3g1 )

D
∆OB (D-BC)

OB

RL
RL
T 21 , T 12
RL
RL
T 32 , T 23
RL
RL
T 13 , T 31

g3 (g1 + g2 )/2(g1 + g2 + g3 )
g1 (g2 + g3 )/2(g1 + g2 + g3 )
g2 (g3 + g1 )/2(g1 + g2 + g3 )

Now, the Dirichlet fixed point is stable only when all these
D
scaling dimensions ∆OB > 1. In Fig. 5, the stability region of
Dirichlet fixed point (D-BC) is painted in green.
c. Chiral-χ± fixed points
The chiral-χ± fixed points are defined as follows: the BC
corresponding to the χ+ (χ− ) fixed point would effectively
make all operators in + (−) cycle equal to identity for all Luttinger parameters. With this in mind, we could derive the following relationship for the rotation angles of the corresponding rotation matrices:
tan ξ± = ±

g1 + g2 + g3
,
g1 g2 g3

(A6)

RA1


1





=0




0

0
g2 −g3
g√+g3
2
2 g2 g3
g2 +g3

0

√
2 g2 g3
g2 +g3
g3 −g2
g2 +g3







,






(A7)

while those of A2,3 can be constructed by permuting the indices in the corresponding matrix elements.
By using this rotation matrix (A7), it is straightforward to
show that the following single-particle tunneling processes are
RL
RL
RL
equal to identity: T 32 , T 23 , and T 11 . In addition, the scaling
dimensions of the leading relevant/irrelevant operators read as
follows.
OB

RL
RL
LL
RR
RL
RL
LL
RR
T 21 , T 12 , T 21 , T 21 , T 13 , T 31 , T 13 , T 13
RL
RL
LL
RR
T 22 , T 33 , T 32 , T 32
†

RL RL
RL RL
T 21 T 12 , T 13 T 31

†

A
∆O1B (A1 -BC)

g1 +g2 +g3 +g1 g2 g3
2g1 (g2 +g3 )
2g2 g3
g2 +g3
1
1
2( g1 + g2 +g3 )

Here, we notice that some leading order operators are two particle processes. To obtain the scaling behaviors of operators
near the A2,3 fixed points, one can simply permute the indices
of the Luttinger parameters with the corresponding operators.
In Fig. 5, the stability regions of A1,2,3 fixed points are shown
in yellow, gray, and blue, respectively.

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