Tunneling into and between helical edge states - fermionic approach
D.N. Aristov1, 2 and R.A. Niyazov1, 2

arXiv:1601.07080v2 [cond-mat.str-el] 9 Jul 2016

2

1
NRC “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
Department of Physics, St.Petersburg State University, Ulianovskaya 1, St.Petersburg 198504, Russia
(Dated: May 7, 2018)

We study four-terminal junction of spinless Luttinger liquid wires, which describes either a corner
junction of two helical edges states of topological insulators or the tunneling from the spinful wire
into the helical edge state. We use the fermionic representation and the scattering state formalism,
in order to compute the renormalization group (RG) equations for the linear response conductances.
We establish our approach by considering a junction between two possibly non-equivalent helical
edge states and find an agreement with the earlier analysis of this situation. Tunneling from the
tip of the spinful wire to the edge state is further analyzed which requires some modification of our
formalism. In the latter case we demonstrate i) the existence of both fixed lines and conventional
fixed points of RG equations, and ii) certain proportionality relations holding for conductances
during renormalization. The scaling exponents and phase portraits are obtained in all cases.

I.

INTRODUCTION

The advances in technology stimulate a renewed theoretical interest to the properties of one dimensional (1D)
quantum wires. The practical implementations of such
systems include carbon nanotubes, chains of metal atoms
or weakly side-coupled molecular chains in solids. A new
class of materials is given by two-dimensional topological insulators (TIs), whose edge states are ideal quantum
wires 1,2 . This paper discusses the transport via the junction between these edge states and its renormalization by
interactions.
The transport properties of 1D wires has been theoretically studied since early 1990s in terms of Luttinger liquid and within bosonization formalism3–5 . It was found
that the interaction between fermions renormalizes the
impurity scattering, so that for repulsive interaction the
conductivity of the wire tends to zero in the limit of low
temperatures, even for one impurity. The bosonization
approach considers the interaction in the bulk of the wire
exactly, while the impurity is regarded as perturbation.
In more sophisticated theories the impurity is considered
as certain boundary conditions for the bulk fields, and
the approach is generalized for the case of junctions between several wires6 .
Alternative description using more traditional
fermionic approach and S-matrix formalism was developed in7,8 for case of one impurity, this approach
was later generalized to the case of junctions of several
wires9,10 . It was found that the renormalization of
S-matrix by interaction is described in terms of renormalization group equations, which eventually defined
the scaling exponents of the conductance behavior. The
initial formulation of the fermionic S-matrix approach
assumed that the bulk interaction was considered only
in the lowest order, and this might be regarded as some
disadvantage in comparison with bosonization. In certain cases the lowest order calculation was insufficient,
and the calculation of next-order corrections to S-matrix
was required11,12

It was found, however, that the S-matrix approach
could be essentially improved by summation of simple diagrammatic series in perturbation theory13,14 . The result
of this summation was the modification of the RG equations for S-matrix, so that the scaling exponents found in
limiting fixed points (FPs) of these equations coincided
with those established by bosonization. At the same time
the S-matrix approach has an advantage over bosonizaton in providing full scaling curves for the conductances.
The method was tested for junctions of two leads13 ,
three-lead junctions12,14,15 , also for non-equilibrium16 .
Generalization of this approach for the case of infinite
Luttinger liquid wires was undertaken in17 .
It was recently shown that the case of a junction connecting four quantum wires, which is generally characterized by S-matrix belonging to U (4) group, might be
principally different in the form of RG equation, even in
the lowest order of interaction.18 This difference from the
simpler cases of junctions of two and three wires shows
itself in the discrete Z2 ambiguity in the parametrization
of S-matrix, which is unobservable in terms of conductances of the junction. This “hidden phase” is known to
happen in U (N ) groups at N ≥ 4, and it may indicate
that the bosonization description becomes inadequate already for four wires.
In this paper we continue our studies of junctions of
four quantum wires. First we analyze the renormalization of corner junction between the 1D helical edge states
of topological insulators, earlier discussed in11,19–21 . The
work by Teo and Kane11 allows the direct comparison
with our analysis here, as they provide the second-order
RG analysis for S-matrix, in addition to the bosonization
treatment. It was argued in11 that the time-reversal symmetry determines the specific structure of the S-matrix,
in particular the absence of backscattering in the same
edge state. We extend the analysis of Ref.11 by summing
the perturbation series in bulk interaction and arrive to
non-perturbative RG equations for conductances, whose
FPs and scaling exponents are in exact correspondence
with bosonization results, where available. The above
problem of “hidden” phase in U (4) is absent in case of

2
helical edge states due to the symmetry of S-matrix, reducible to the symplectic form.
Next we analyze the generalization of the above setup
by considering different interaction strength in the helical
edge states, whereas the form of the S-matrix remains the
same due to symmetry. The phase portrait in this case is
characterized by two phases; the new phase as compared
to the previous case appears for interaction strength of
different signs, when some FPs disappear or change their
character. We show that if the X-junction is characterized by the “bare” S-matrix close to one of FPs of saddle
point type, then the temperature evolution reveals the
non-monotonic behavior of conductances, similarly to the
case of Y-junction considered in12 .
We further notice that the case of electron tunneling
from a spinful Luttinger liquid wire tip to helical edge
state can be described by the same S-matrix, but different matrix of bulk interactions. This physically important setup was not considered previously and it is different from the tunneling from fully spin-polarized tip to
the helical edge state22 , the latter situation described in
terms of three-lead Y-junction. The different form of the
interaction matrix leads to slightly modified derivation
of RG equation, but otherwise our approach remains the
same. As a result, we obtain the phase portrait describing the tunneling from the spinful wire to the helical edge
state. We find only one fixed point, corresponding to the
absence of tunneling. In addition, we demonstrate one or
two fixed lines of conductances, depending on the interaction. The RG fixed lines in the plane of conductances
was found earlier for chiral Y-junctions at certain values
of the interaction strength.23 The scaling exponents are
in exact agreement with those expected from bosonization arguments.
The rest of the paper is organized as follows. We describe our method in the Section II. The setup of our
model, the particular form of S-matrix and the relevant
set of conductances are introduced, and the renormalization equations are briefly discussed here. In Section III
we discuss tunneling between helical edge states, first in
a simpler case of symmetric junction, and then introducing asymmetry in interaction strength. In Section IV we
discuss tunneling from a spinful wire to the edge state.
This case requires some modification of our method and
we sketch the corresponding derivation. We present the
concluding remarks in Section V.

II.

THE MODEL AND RENORMALIZATION
OF CONDUCTANCES
A.

The model

We consider the two-channel Tomonaga-Luttinger liquid (TLL) model with a local scatterer of arbitrary
strength in the middle of the wire. In particular, such
model incorporates a model of spinful electrons, scattering on a local potential and a model of corner junction

TABLE I: Correspondence between the helical edge states
with projections of spin and the numbering of our channels.

in
out

1
↑, left
↓, left

2
↓, right
↑, right

3
↑, right
↓, right

4
↓, left
↑, left

between the helical edge states of topological insulators.
As in our previous papers, we assume that the shortrange interaction between the fermions is of the forward
scattering type and takes place in wires of finite length
L, contacted by reservoirs. The adiabatic transition from
wire to reservoir 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 is
expressed below by the window function Θ(x) = 1, if
l < x < L, and zero otherwise. The regions x > L are
thus regarded as reservoirs or leads labeled j = 1, 2, 3, 4.
For the linearized spectrum near the Fermi energy we
may write the TLL Hamiltonian in the representation
of incoming and outgoing waves in channel j (fermion
operators ψj,in , ψj,out ) as
∞

dx[H 0 + H int Θ( < x < L)] ,

H=
0

H 0 = vF Ψ† i Ψin − vF Ψ† i Ψout ,
out
in

(1)

4

H int = 2πvF

gjk ρj ρk .
j,k=1

Here Ψin = (ψ1,in , ψ2,in , ψ3,in , ψ4,in ) denotes a vector operator of incoming fermions and the corresponding
vector of outgoing fermions is expressed through the Smatrix as Ψout (x) = S · Ψin (x) at x → 0. We put the
Fermi velocity vF = 1 below. 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 4×4 unitary
S-matrix characterizes the scattering at the junction and
belongs to U (4) group. Depending on the physics of the
problem, the form of the S-matrix and the interaction
matrix, gjk , varies.
Specifically, we consider below the corner junction between the helical edge states in symmetric and asymmetric setup with respect to interaction. We also consider
the tunneling of electrons from the spinful wire tip into
the helical edge state. A general geometry of X-junction
is shown in Fig. 1a. It turns out, that all the above cases
with spinful fermions can be described by the Fig. 1b,
and the difference between these cases is encoded in the
form of gjk . For further convenience, we explicitly give
the correspondence between the channel and the spin index in Table I.

3
of the form Ii =

FIG. 1: (a) Schematically shown X-junction, (b) X-junction
of the helical edge states, with spin projections explicitly indicate, see also Table I.

B.

Reduced conductances

In the linear response regime our system is characterized by the matrix of conductances defined by Ii = Cij Vj ,
with the current Ii flowing in channel i and the voltage Vj applied to the channel j. The current conservation,
Ii = 0, and the absence of response to the
equal change in voltages lead to the Kirchhof’s rules,
i Cij =
j Cij = 0. It means that we can choose
more convenient linear combinations of Ii , Vj reducing
the number of independent components in Cij . Using the
1
Kubo formula, one has in the d.c. limit Cij = 2 (δij −Yij ),
with Yij = |Sij |2 ; one can also write Yij = Tr(ρi ρj ).15
The appropriate representation for the reduced conductance matrix may be constructed by using generators
of U (4) Cartan subalgebra, which are three traceless diagonal matrices and one unit matrix. We define
√
µ1 = 1/ 2diag(1, −1, −1, 1),
√
µ2 = 1/ 2 diag(1, 1, −1, −1),
√
µ3 = 1/ 2diag(1, −1, 1, −1),
√
µ4 = 1/ 2 diag(1, 1, 1, 1).

(2)

with the property Tr(µj µk ) = 2δjk , j = √ . . . , 4. The
1,
densities may be expressed now as ρj = 1/ 2 k Rjk µk ,
where the 4×4 matrix R is given by


1
1 −1

R= 
2 −1
1

1
1
−1
−1

1
−1
1
−1


1

1

1
1

(3)

and has the properties R−1 = RT , det R = 1. The
outgoing amplitudes are expressed in similar form with
µj replaced by µj = S + µj S. It also means24 that we
work now with the combinations of currents and voltages

new
I1
V1new
new
I2
V2new
new
I3
V3new

k

new
Rik Ik , Vi =

k

new
Rik Vk , or

= (I1 − I2 − I3 + I4 )/2 ,
= (V1 − V2 − V3 + V4 )/2 ,
= (I1 + I2 − I3 − I4 )/2 ,
= (V1 + V2 − V3 − V4 )/2 ,
= (I1 − I2 + I3 − I4 )/2 ,
= (V1 − V2 + V3 − V4 )/2 ,

new
I4 =

j

Ij /2 ,

V4new =

(4)

j

Vj /2 .

Our notation is slightly different from Ref. 11, where the
new
Kirchhof’s law, I4 = 0, was explicitly used. The meannew
ing of the currents Ij , linked to the Fig. 1, is as follows:
new
new
I1
is the charge current moving to the right, I1
new
charge current moving down, and I3
is the spin current. The resulting reduced conductance matrix in our
1
new basis is determined by G = RT C R = 2 (1 − YR )
1
R
with Yij = 2 Tr(µi µj ) and has a general structure
G=

3×3 0
0 0

(5)

For our choice of S-matrix below in Eq. (8), G attains
even simpler diagonal form.

C.

S-matrix for helical edge states

It was noted in11 that in addition to unitarity, S † S = 1,
the time reversal symmetry leads to the additional condition on the S-matrix. Namely, under the time reversal, T , one has T Ψin = EΨout and T Ψout = −EΨin
with E = diag[1, −1, 1, −1]. It results in the relation
S = −ES T E, i.e. the matrix ES is antisymmetric. Additionally assumed constraints on the form of S refer to
the symmetry with respect to interchange of the wires:
1 ↔ 2 and 3 ↔ 4, which should not change the observable conductivities. In terms of the matrix Yij = |Sij |2
below it reads as Y = XY X with


0 1 0 0


 1 0 0 0
X=
(6)

 0 0 0 1
0 0 1 0
These constraints define the S- matrix in the form


0
t
f
r


0 −r∗ f ∗ 
 t
S=
,
−f −r∗ 0 t∗ 
r −f ∗ t∗ 0

(7)

with complex- valued t, f, r and |t|2 + |f |2 + |r|2 = 1. Our
analysis below is invariant with respect to “rephasing”
operations, S → U 1 SU 2 with U 1,2 arbitrary matrices of

4
the form Ukl = δkl eiαk . Using such rephasing we can
represent the S-matrix for our purposes as


0
t
f
r


0
reiu −f eiu 
 t
(8)
S=
,
−f reiu
0
−teiu 
r f eiu −teiu
0
now with real-valued t, f , r subject to the condition t2 +
f 2 + r2 = 1 and arbitrary u. We notice here that the
operation XSX corresponds to a change u → π − u in
(8), up to some rephasing.
We also note that it is always possible to make a partial
rephasing of S in (7) so that f is real-valued, in this case
S becomes symplectic matrix from Sp(2) group. This
latter property of S-matrix for the helical edge states is
principally different from the previously studied case of
X-junction between usual wires,18 where the mentioned
symplectic property is achieved in a very special case,
α1 = −α2 , in notations of Ref. 18.
We may parametrize (8) by two angles as
t = cos β ,

r = cos γ sin β

f = sin γ sin β ,

(9)

which leads to the matrix of conductances in the form:
1
1
G = 2 (1 − YR ) = 2 diag[1 − a, 1 − b, 2 + a + b]
≡ diag[GR , GD , GS ]

(10)

where a = 2 sin2 β cos2 γ − 1, b = cos 2β. The region of
the allowed values of conductance in the (a, b) plane is
given by a triangle defined by vertices (−1, −1), (1, −1),
(−1, 1), as shown in Fig. 3 below. This statement is
initially made for non-interacting fermions, but we also
verify below, that the interaction-induced RG flows of the
parameters never drive the system beyond this triangle,
and the conductances are defined in terms of a, b. This
means that we have a relation

two lengths, characterizing the interaction region in the
wires.
The first logarithmic correction for the S-matrix leads
to the renormalization group (RG) equation which was
obtained in general form in9 . Second-order subleading
corrections which might be important in certain cases
were analyzed in the language of S-matrix in12 . It was
shown, however,15 that RG flow may appear in some of
the phases of S-matrix, which are irrelevant for the observable conductances. It is thus reasonable to reformulate the RG procedure entirely in terms of conductances,
which allows one-to-one correspondence for Y-junction.15
This unique correspondence between RG equation for Smatrix and RG equation for matrix of conductances C
is valid for Y-junction and may be explicitly lost for Xjunction, as shown in18 . This ambiguity stems from the
impossibility to recover the phases of unitary S-matrix,
belonging to U (N ) group, from the absolute values of its
elements, for N ≥ 4. In our case here we also have this
ambiguity in the form of appearance of the phase factor
eiu in (8) mixing left and right isoclinic S-matrices. However, in contrast with18 we do not have the ambiguity in
RG flows for the considered model of interactions gij .
In lowest order in the interaction the scale dependent
contribution to the conductances is given by23
Cjk = Cjk

g=0

+

1
2

Tr Wjk Wlm gml Λ ,

(11)

l,m
1
where Cjk |g=0 = 2 (δjk − Yjk ), the Wjk = [ρj , ρk ] are a
set of sixteen 4 × 4 matrices (products of W ’s are matrix
products), gml is the matrix of interaction constants appearing in (1) and the trace operation Tr is defined with
respect to the 4×4 matrix space of W ’s.
If we multiply Cij with RT from the left and R from
the right then we get the components of YR in the form
R
R
Yjk = Yjk

g=0

−

1
2

R
R
R
Tr Wjk Wlm gml Λ .

(12)

l,m

GS = 2 − GR − GD .
In the absence of spin-flip processes, f = 0, one has a =
−b, i.e. the region of allowed conductances is reduced to
a segment in (a, b) plane.
In the presence of interactions, the structure of (8) is
unchanged, but the elements vary. The main effect in
the d.c. limit can be described by the renormalization
of the S-matrix,15 which translates to the renormalized
quantities a, b in Eq. (10).

R
Here Wjk = [µj , µk ] = {RT · W · R}jk are again a set
R
of 4×4 matrices, but now we have Wjk = 0 for j = 4 or
R
k = 4, so only nine matrices Wjk are non-zero. We also
R
T
defined gml = {R · g · R}ml . The nine nonzero matriR
ces Wjk are evaluated with the aid of computer algebra.
Differentiating these results with respect to Λ (and then
putting Λ = 0) we find the RG equations

d R
1
Yjk = −
dΛ
2
D.

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/ ), where L and are

R
R
R
Tr Wjk Wlm gml .

(13)

l,m

In higher orders of perturbation theory in gml we find
subleading contributions of two types.15 One type, appearing first in the third order, provides a three-loop
contribution to RG equations and does not influence the
scaling exponents around the RG fixed points (FP). Second type of contributions is more important, it is given

5
by the ladder sequence of diagrams, and defines the scaling exponents around FPs. Due to peculiarities of 1D
model with linear dispersion, each diagram in this ladder
sequence is formally a one loop contribution, providing
subleading linear-in-Λ corrections. After the summation
of the ladder series14,25 (for diagonal gij = gi δij ) one ob¯
tains the renormalized interaction matrix g replacing the
bare interaction matrix g in Eq.(11). The components
¯
of g are obtained from the following matrix equation
¯
g = 2(Q − Y)−1 .

(14)

The matrix Q characterizes the interaction strength and
depends on the Luttinger parameters Kj = [(1−gj )/(1+
gj )]1/2 as
Qjk = qj δjk ,
III.

qj = (1 + Kj )/(1 − Kj ) .

(15)

TUNNELING BETWEEN EDGE STATES
A.

Symmetric corner junction

For quantum spin Hall insulator the channels correspond to four helical edge states, with possible tunneling
contact between them. The interaction between the different edge states is absent. In other words, the wires 1
and 4, 2 and 3 do not interact, and we have the simple
interaction matrix:
g = g1.

(16)

The first-order RG equations for S-matrix of the form
(8) are trivial :
dYR
= 0,
dΛ

(17)

which is the result of the diagonal form of the interaction
(16) and the absence of backscattering.9
To advance further we take into account the higher
orders of interaction as explained above, i.e. we replace
¯
g by g (14). In such a way we obtain non-trivial RG
equations:
b+1
a−1
+
b(K − 1) + K + 1 a(K − 1) + K + 1
a+b
+
(a + 1)(K − 1),
(a + b)(K − 1) − 2
db
da
=
.
dΛ
dΛ a↔b
da
=
dΛ

a
b
stability

-1
-1
s

-1
0
u

-1
1
s

-1/3
-1/3
u

0
0
u

0
-1
u

1
-1
s

The RG equations (18) reveal seven fixed points: three
of them are stable for any value of Luttinger parameter, the others are unstable (see Table II). This result
is in agreement with second order calculation in11 and
the correspondence between our notation and Ref.11 is
R = (1 + a)/2, T = (1 + b)/2, F = −(a + b)/2.
Each of these FPs is characterized by generally two
scaling indices in the plane (a, b), depending on the direction of RG flow with respect to the given FP. We found
that three stable FPs (vertices of the triangle) have the
same exponent in both directions, equal to
α1 = 2 − 1/K − K < 0,

(19)

which corresponds to the weak tunneling process between
the TLL wires and agrees with the result in11 .
One unstable FP at the triangle’s center is characterized by rather unusual exponents, equal to 3 + 27/(2 +
K)2 − 18/(2 + K) > 0, in both directions. Three other
FPs residing at the middle of the triangle’s edges show
the scaling exponents 2 + 8/(1 + K)2 − 8/(1 + K) > 0 and
3 − K − 4/(1 + K) < 0 in the direction along and perpendicular to the edge. The latter FPs are of the saddle
point type.
It was shown in12 that the full scaling curves for
conductances in case of Y-junction may reveal nonmonotonic behavior with the scaling parameter Λ. We
note here that such behavior should generally happen if
the RG trajectories pass near the saddle point FP. To
demonstrate this we plot the full scaling curves for the
present case of X-junction in Fig. 2, choosing reasonable
values for the Luttinger parameter and the appropriate
values of bare conductances.

B.

Asymmetric corner junction.

(18)

Expanding these RG equations to the lowest nonvanishing order ∼ g 2 we recover the Eq. (3.40) in Ref.
[11]. The second-order diagrams in Fig. 9 there correspond exactly to the truncation of the series leading to
our Eq. (18). It is worth noting that the phase u is absent
in (18), it is present in Wjk in Eq. (11) but disappears in
the quantity Tr Wjk Wlm .

TABLE II: Fixed points for symmetric corner junction. The
positions of FPs are given by the coordinates (a, b) in the
plane of conductances, the stability of the FPs is also shown.

In this subsection we consider more general setup, allowing different strength of interaction in the upper (1-2)
and lower (3-4) edge states in Fig. 1b. It is interesting to observe that when the edge states are non-equal,
one could expect more complicated expression for the Smatrix instead of Eq. (8). However the above general
arguments leading to Eq. (8) involve only a few discrete
symmetries which are not broken by the non-equivalence
of the upper and lower edge states. This means that
the system remains at a fixed surface in space of conductances, which is protected with respect to asymmetric

6
1.0

TABLE III: The position of non-universal fixed points for
the case of non-equal interaction in the wires.

0.9
0.8

edge
−1

(g1 −g2 )2
(g1 +g2 )2

b

0.6

edge

a

0.7

2)
− (g1 −g2 )2
(g1 +g

2

−a

median
−1 +
3

2
1 (g1 −g2 )
2
2
3 g1 +g1 g2 +g2

−1 − 2a

0.5
0.4

0

2

4

6

8

10

12

second order of interaction and write

14

da
1
2
= − (1 + a)((1 + b)2 g1
dΛ
8
2
+ 2(−1 + 2a2 + 2ab + b2 )g1 g2 + (1 + b)2 g2 ),
db
1
(21)
2
= − (1 + b)((−1 + b2 )g1
dΛ
8
+ 2(1 + 2a(1 + a) + 2ab + b2 )g1 g2

1.0
0.9
0.8
0.7

2
+ (−1 + b2 )g2 ) .

0.6

0.5

0.4

0

2

4

6

8

10

12

14

FIG. 2:
Full scaling curves for conductances GR =
new
new
I1 /V1new and GD = I2 /V2new , defined in Eq. (4), and
the Luttinger parameter K = 0.4. Slight difference in the initial conductances results in the opposite qualitative behavior.
The panel (a) and (b) show RG flows tending to the FP of the
perfect transmission in the vertical and horizontal direction,
respectively, in terms of Fig. 1. The non-monotonic behavior
stems from the RG flow passing near the FP of saddle point
type at a = b = 0 in Fig. 3.

perturbations of the S-matrix. In other words, only consistent changes in the transmission coefficients within the
wires 12 and 34 are allowed, so that |S12 | = |S34 | = t.
This may be contrasted with interaction-induced asymmetric perturbations in case of Y-junction, which can
drive the system from the symmetrical point, as discussed
in Sec. VI of Ref. 14.
We take the matrix of the dimensionless interaction
constants in the form :

g = diag[g1 , g1 , g2 , g2 ] ,

(20)

and use the previous S-matrix (8). The result of the
ladder summation, Eqs. (13), (14) leads to rather complicated RG equations. In order to illustrate the qualitative
picture, we first expand the coupled RG equations to the

We observe that these equations formally have seven FPs,
but these FPs not always lie in the physical region. Three
FPs with −1 < b < 1 are now non-universal and their
position depends on the ratio of the interactions g1 /g2
as shown in and Table III. We also show the tendency
of these intermediate FPs to change their position in
Fig. 3a. The two previously stable FPs at a = ±1 ,
b = −1 keep their position but their stability now also
depends on g1 /g2 . As can be seen from Table III, if
g1 /g2 > 0 and in particular g1 = g2 then all three nonuniversal points lie either within or on the border of the
triangle of physically allowed conductances. If g1 /g2 < 0
then non-universal points disappear in the physical region and this is accompanied by the change in the character of some of the remaining FPs.
This qualitative picture obtained in the second order
of perturbation is confirmed by the analysis of the full
form of the RG equations. Summarizing here, we have
two different regions in the plane of Luttinger parameters K1 , K2 (see Fig. 4). The blue region is characterized
by the existence of seven FPs, and three of these points
are stable, whereas other three FPs are non-universal in
their position. This region remains in qualitative correspondence with the case of equal interactions discussed in
the previous subsection. The new region marked white is
characterized by the existence of four FPs, one of them is
stable, two (previously stable) FPs are unstable, and the
fourth point remains of the saddle point character. The
only stable point in this white region corresponds to fully
disconnected upper and lower wires, which are perfectly
transmitting within themselves. On the lines dividing
white and blue regions in Fig. 4 three non-universal FPs
coincide with three lowest FPs in Fig. 3, i.e. a = −1, 0, 1,
b = −1. The largest value of b for non-universal FPs is
achieved when the interactions are equal, g1 = g2 .
The yellow FP in Fig. 3, a = −b = −1, is always stable
and corresponds to disconnected lower and upper wires;
it is CC-point in notation of the work11 . The right red
FP in Fig. 3, a = −b = 1, corresponds to the perfect

7
3.0

2.5

2.0

1.5

1.0

0.5

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

FIG. 4: Phase portrait for conductances in a setup with different values of interaction in the edges states, characterized
by Luttinger parameters, K1,2 . Depending on the sign of
(1 − K1 )(1 − K2 ) one has either one or three stable fixed
points, as described in text.

and − (K1 −1)(K2 −1)(2K2 K1 +K1 +K2 )
(K1 +1)(K2 +1)(K1 +K2 )
respectively.

IV.

FIG. 3: RG flows (red lines, the direction of flow shown by
arrows) and fixed points are shown for different values of interaction in the edges states. The panel (a) corresponds to
Luttinger parameters K1 = 0.3, K2 = 0.5 and panel (b) is
for K1 = 1.2, K2 = 0.5. The disappearance of three nonuniversal unstable FPs is visible in panel (b), which is accompanied by the change of the character of the lower red
FPs.

transmission in the vertical direction of Fig. 1 ; it is II
-point in in notation of11 . The left red FP in Fig. 3,
a = b = −1, corresponds to spin current conductance
GS = 0 and was called a perfect spin flip transmission
point in11 .
Summarizing here, we list the scaling exponents of the
conductances near the corresponding FPs of Eq. (21).
Near the points (a, b) = (±1, −1) we have the same
exponent equal to −2(K2 − 1)(K1 − 1)/(K1 + K2 );
near (a, b) = (−1, 1) we have the exponent
1
− 2 (K1 − 1)2 /K1 + (K2 − 1)2 /K2 . Near the unstable
(universal) FP, (a, b) = (0, −1), we find two different exponents, 2(K1 − 1)(K2 − 1)/((K1 + 1)(K2 + 1))

along a and b,

TUNNELING TO HELICAL EDGE STATE
FROM SPINFUL WIRE

As was argued above, the form of S-matrix (8) does
not imply the equivalence between the upper and lower
wires. In particular, we may view the lower wire 3 − 4
as the tip of the spinful wire, whereas the channels 1 and
2 remain associated with the helical edge states. In the
absence of the tunneling between this tip and the edge
state we have f = r = 0, t = 1 which in the usual sense
corresponds to a perfect reflection of the electrons at the
end of the spinful tip.
The main difference of this setup is a new form of interaction matrix:


g1 , 0,
0,
0


0 
 0, g1 , 0,
(22)
g=
,
 0, 0, g2 /2, g2 /2
0, 0, g2 /2, g2 /2
which describes the charge-charge interaction in the spinful wire. The non-diagonal form of g requires certain
adjustment of our analysis. The first-order RG equation
(13) is unchanged and the result of summation of higher
order terms (14) should be revised. The appropriate way
of doing it can be found in the idea of the spin-charge
separation in the spinful wire tip and the details of our

8
derivation of (14) in14,25 . We notice that dressing of interaction gj → Kj happens in the bulk of the wire away
from the contact, and does not involve S-matrix. The
whole procedure of such dressing for non-diagonal g can
be then performed by first diagonalizing the interaction
by appropriate orthogonal transformation, thus passing
to a description in terms of charge and spin density in
the spinful wire. Second step involves the dressing of
the bulk interaction effects, by summing the ladder diagrams, it is encoded in matrix quantity C in14,25 . The
third step consists of the inverse orthogonal transformation and properly consideration of the contact, described
by the above matrix Y. Introducing the orthogonal ma√
trix V = (E + X)/ 2 = V T = V −1 , with E, X defined
in Eq.(6) and before it, this sequence of steps is shown
schematically as
¯
g → V gV → 2Q−1
−1
¯
¯
¯
¯
→ 2Q (1 + YQ−1 + YQ−1 YQ−1 + . . .)

(23)

¯
¯
→ 2V Q−1 V (1 − YV Q−1 V )−1
−1 −1 −1
¯
now with Q−1 = diag[q1 , q1 , q2 , 0] and qj defined in
(15). Overall, instead of (14) we write

¯
¯
g = 2(V QV − Y)−1 .

(24)

¯
with the singularity in Q is resolved according to the last
line in (23).
Using the reduced conductances, Eq. (10) we find the
RG equations in the form:
da
= (a + 1)F (b) ,
dΛ
db
= (b − 1)F (b) ,
(25)
dΛ
2
2(b + 1) b(q1 + q2 − 2) + q1 − q1 + q2 − 1
.
F (b) =
(b + 2q1 + 1)(b(q1 + q2 − 2) − 2q1 q2 + q1 + q2 )
It follows then that the set (25) is in fact reduced to
second equation, and
d 1 − GR
d a+1
=
= 0,
dΛ b − 1
dΛ GD

(26)

It means that the RG flows in the plane (a, b) lie on
straight lines passing through the point (−1, 1) and the
ratio, a+1 , defined for bare quantities, see Fig. 6 beb−1
low. In terms of conductances (10) we see that the ratio
(1 − GR )/GD is unchanged under renormalization, i.e.
the “up-to-down” conductance remains proportional to
the value of “left-to-right” conductance complementary
to conductance quantum.
The dependence of F (b) on the interaction is not transparent and we show a few first terms in powers of gj . The
expansion of (25) begins with the second order g1 , and
with the first order in g2 . Keeping these lowest order

3.0

2.5

2.0

1.5

1.0

0.5

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

FIG. 5: RG phase portrait for tunneling between the spinful
wire and helical edge state. The brown region at K2 < 1 is
the stability region for the FP corresponding to the absence
of tunneling. In the blue region the situation is characterized
by one stable FP and a stable fixed line, as shown in Fig. 6b.
In the white region only the fixed line b = −1 is stable.

terms, we have
da
1
1 2
= − (a + 1)(b + 1) g2 + 2 g1 (b + 1)
dΛ
4
1
db
2
= (1 − b2 ) g2 + 1 g1 (b + 1) .
2
dΛ
4

(27)

These equations are rather unusual, because of the existence of one FP and one or two fixed lines. The fixed
point is defined by a = −1, b = 1, corresponds to the
absence of tunneling to the edge state and is stable ev2
erywhere except for the region g2 < −g1 . The analysis
of the full expression shows that this FP becomes un2
stable at K2 > K1 /(3K1 − K1 − 1), this is depicted by
white region in Fig. 5. The poles of the latter expression
determine a finite range of interaction K1 in the edge
state when the stability of the FP may be lost, namely
√
√
1
5 < K1 < 1
5+3 .
2 3−
2
In addition, we have two fixed lines, one at b = −1,
which is unstable for g2 > 0 (i.e. at K2 < 1, shown
as brown region in Fig. 5) and is stable otherwise. The
second fixed line at b = b0 is determined by the condition
2
F (b0 ) = 0 which gives b0 = −(q1 −q1 +q2 −1)/(q1 +q2 −2),
revealing its non-universal character. It is unstable and
is indicated as the black line in Fig. 6b. The domain of
existence of this fixed line correspond to a blue region in
Fig. 5.
We find the scaling exponent around the FP (a, b) =
(−1, 1) given by
αF P =

1
2

−K1 −

1
K1

−

1
K2

+3 ,

(28)

9
and the scaling exponent near the fixed line at b = −1 is
αF L = −2

2
K1 (K2 − 1)
.
(K1 + 1)(K1 + K2 )

(29)

The border between the blue and white region in Fig.
5 corresponds to a condition b0 = 1 which translates to
αF P = 0. Similarly, the border between the brown and
blue region in Fig. 5 is given by the condition b0 = −1
and αF L = 0, i.e. K2 = 1.
Two examples of RG flows are shown in Fig. 6, where
the upper panel corresponds to the brown region in Fig.
5 and the lower panel, Fig. 6b - to the blue region in
Fig. 5. If we fix K1 and increase K2 , then at smallest K2
we see the qualitative picture of RG flows shown in Fig.
6a. Then at K2 = 1 the second fixed line appears in the
triangle of physical conductances in (a, b) plane. Upon
further increase of K2 this line moves upwards where the
RG flows follow the pattern of Fig. 6b. Finally, the nonuniversal fixed line crosses the FP with a = −1, b = 1,
and disappears; this corresponds to the white region in
Fig. 5, where only the line b = −1 is stable.
We compare our findings (28) with the bosonization
approach, particularly with Ref.11 , where the the scaling
exponent (19) of the conductance in the weak backscattering limit was found. The exponent in this case can be
regarded as twice the scaling dimension of the tunneling
operator at the edge state, (K − 1)2 /2K. In the considered setup one TI is substituted by the semi-infinite spinful wire. The contact between a semi-infinite quantum
wire and a 3D metal was analyzed in26 , with the con−1
−1
ductance scaling exponent found as 1 (Kρ + Kσ ) − 1.
2
For spin isotropic interaction in our case we should put
Kσ = 1 and Kρ = K2 . Combining both cases we obtain
the overall scaling exponent in the form (K1 + 1/K1 )/2 +
(1 + 1/K2 )/2 − 2 which expression exactly corresponds
to Eq. (28).
Summarizing here, we obtained the phase portrait for
the case of the tunneling from the spinful wire tip to the
helical edge state, we find the scaling exponents which
are in agreement with bosonization approach when available. We note that the appearance of the RG fixed lines
in the plane of available conductances is rather unusual
phenomenon, discussed previously in the case of chiral
Y-junction for certain relations between the interaction
parameters.23
V.

CONCLUSIONS

In this paper we consider four-terminal junctions, involving helical edge states of topological insulators. The
time reversal symmetry arguments for such junctions define the general structure of single-particle S-matrix, describing the tunneling processes between the edges. In
presence of interactions the conductances characterizing
the junction are renormalized, which effect is described
in the proposed formalism by a set of non-perturbative
RG equations. In order to validate our approach we first

FIG. 6: Two examples of RG flows are shown. Typical behavior in the brown region of Fig. 5 is shown in panel (a)
for K1 = 1.55, K2 = 0.51. The situation in blue region of
Fig. 5 is qualitatively reproduced in panel (b) at K1 = 1.38,
K2 = 1.03.

analyze the setup with two helical edge states and show
that our results agree with the bosonization studies, when
available. The fixed point structure, scaling exponents
and the overall phase portrait of this setup is obtained.
We further observe that the same symmetry arguments
can be applied to the S-matrix for the tunneling from the
tip of spinful wire to the helical edge state. This important physical setup was not previously considered. The
difference of this setup from the contact between two helical edge states is in the form of interaction, which requires
certain modification of our formalism. The phase portrait
is now more involved and, depending on the interaction,
we find the possibility of RG fixed points and fixed lines
of conductances. We show that the calculated scaling
exponents coincide with those expected from bosoniza-

10
tion. We predict that in the discussed setup the scaling
is defined by one equation, and certain proportionality relations between three different conductances are obeyed
during renormalization. This prediction may hopefully
be checked in future experiments.
The advantage of the employed S-matrix approach is
the possibility to obtain both the full scaling curves for
the conductances and exact scaling exponents at the fixed
points. We demonstrate that if the RG flow drives the
system near by unstable fixed points then non-monotonic

1

2

3

4

5
6

7

8

9
10

11

12

13

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Acknowledgments

We are grateful to P. W¨lfle for numerous useful discuso
sions. This work was supported by the Russian Scientific
Foundation grant (project 14-22-00281) and by RFBR
grant No 15-52-06009.

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