Spin-polarized tunneling into helical edge states: asymmetry and conductances
D.N. Aristov1, 2 and R.A. Niyazov1

arXiv:1612.09594v2 [cond-mat.str-el] 7 Mar 2017

1

NRC “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
2
St.Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia

We consider tunneling from the spin-polarized tip into the Luttinger liquid edge state of quantum
spin Hall system. This problem arose in context of the spin and charge fractionalization of an injected electron. Renormalization of the dc conductances of the system is calculated in the fermionic
approach and scattering states formalism. In lowest order of tunneling amplitude we confirm previous results for the scaling dependence of conductances. Going beyond the lowest order we show that
interaction affects not only the total tunneling rate, but also the asymmetry of the injected current.
The helical edge state forbids the backscattering, which leads to the possibility of two stable fixed
points in renormalization group sense, in contrast to the Y-junction between the usual quantum
wires.
I.

INTRODUCTION

Topological insulators (TI) is a class of materials apparently important for future electronic devices.1,2 Being
an insulator in the bulk, the TI necessarily has conducting states on its edge. The spin-orbit interaction leads
to electrons spin-momentum locking in the edge states
creating so called helical states. Backscattering on the
nonmagnetic impurities in the same channel of such edge
states is absent due to TI time-reversal symmetry. It
allows a creation of devices with a lossless electron transport.
Electrons on the edge of 2D TI are confined to one
spatial dimension (1D). One dimensional electron transport near the Fermi energy is described by well-known
Tomonoga-Luttinger liquid (TLL) model.3 The TLL corresponds to strongly correlated electron matter, and the
1D helical edge states add new peculiarities to it. This
explains large experimental and theoretical interest to helical TLLs, with several suggested ways of probing their
strongly correlated nature. Investigating the helical edge
states was theoretically proposed by means of magnetic
field in4–6 , by specific spatial construction of helical state
in7–9 and by quantum dots in10,11 .
Another natural way of probing – tunneling from the
tip into the edge state – attracted a lot of attention.
There is experimental evidence12 and theoretical explanation13 that the electron injection even into usual TLL
may be characterized by charge fractionalization resulting in the currents imbalance. For helical TLL probing
by the tip was considered theoretically in papers14–19 .
Experimentally it was confirmed that TI edge state is a
quantum spin Hall state20 , helical behavior of the edge
state was observed in21 , the STM experiments were reported in22,23 .
The transport properties of TLL model may be studied in two main approaches. The bosonization approach3
takes into account the electron interaction exactly, the
impurities are regarded as perturbation. The fermionic
approach treats interaction perturbatively and studies
the renormalization of impurities/junctions in S-matrix
formalism of asymptotic electronic states. Transport in

quantum point contact is described by the conductances
(transparencies) of junctions. The problem with one impurity in quantum wire can also be interpreted as the
point contact of two wires. The latter case was considered within the TLL model in bosonization24–26 and
fermionic27,28 approaches. The conclusion of both approaches is that repulsive interaction renormalizes bare
conductance to zero (total reflection case) whereas the attractive interaction leads to perfect transmission in the
low temperature limit.
Further studies of usual TLL in bosonization approach
addressed Y-junction with equal interaction constant29
and arbitrary interaction in three wires30 . The corner
junction of helical TLLs was considered for two TI’s in
31–33
.
In the fermionic approach the renormalization group
(RG) equations for S-matrix characterizing the junction
for arbitrary number of quantum wires was presented in
34,35
. Conductances of Y and X junctions were analyzed
in the first order of interaction.
A partial summation of the perturbation series was
performed in36 in order to recover the exact scaling exponents for conductance for one impurity. The results
were used for the RG analysis of Y-junction 37–39 and
generalized to arbitrary number of wires 40 . The contact of two helical TLL and tunneling from unpolarized
spinful wire into the helical edge state were considered in
41
. The agreement between bosonization and fermionic
approaches has been obtained when the comparison was
available.
In the present letter we focus on the spin-polarized
tunneling into the helical edge state. This problem was
previously studied within bosonization approach15 , with
the setup consisting of a point contact between fully
spin-polarized tip and the helical edge state (see Fig.1).
Generally, spin polarization in the tip differs from the
spin direction of the edge state fermions by some angle, ξ. In the absence of interaction this angle defines
the left-right current asymmetry of the injected current.
The charge fractionalization was demonstrated for the
injection of electrons even with polarization parallel to
the edge states in case of infinite helical TLL without
Fermi liquid leads. Generalization of this result to time-

2
TLL model with the interaction between electrons of
short-range forward-scattering type. All three quantum
wires of length L are adiabatically connected to the reservoirs or leads such that additional scattering is absent.
Scattering electron states flowing in and out the junction
are connected by the S matrix.
TLL model Hamiltonian with linearized spectrum near
the Fermi energy and in the scattering states formalism
reads
∞

FIG. 1: Schematically shown scanning tunneling microscopy
of the helical edge state of topological insulator by the spinpolarized tip. ξ is an angle between electron spin in the tip
and spin quantization axis of the helical state.

dx[H 0 + H int Θ( < x < L)] ,

H=
0

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

(1)

3

H int = 2πvF
16

dependent injected pulses was done in . The effect of
metallic leads for the visibility of fractionalization phenomena in transport measurement in helical TLL was
analyzed in 18 . It was argued there that the fractionalization cannot be observed for dc currents, while information about helical TLL physics can be extracted from
studying the transient dynamics.
In our approach below we use the fermionic formalism
of scattering states, which implies the existence of Fermi
liquid leads. It results in the absence of charge fractionalization in dc limit for the parallel polarization of the tip.
For non-parallel polarization we show that the left-right
asymmetry of the dc current, injected into helical TLL,
is not defined entirely by the polarization angle but is
renormalized along with the total tunneling conductance,
with the resulting coupled RG equations. The asymmetry renormalization appears in first order of interaction
strength and first order of tunneling conductance, and
we discuss it in detail. Simultaneous renormalization of
two quantities is tied to the existence of saddle point RG
fixed point (FP) in the phase portrait of the system.42
In the absence of backscattering in the helical TLL, this
non-universal FP separates two truly stable FPs. This in
turn may lead to qualitatively different renormalization
of asymmetry and the total tunneling rate for different
interaction strength.

II.

THEORETICAL FORMALISM
A.

The model

We consider Y-junction of the quantum wires shown
in Fig. 1. Two quantum wires labeled by j = 1, 2 correspond to the TI helical edge state and the wire with
j = 3 is the tunneling tip with fully polarized electrons.
The polarization vector of electrons in the tip is different from the quantization axis in the TI edge state. We
denote this difference by the angle ξ, so that ξ = 0(π) describes the tip polarization parallel (antiparallel) to the
right-moving fermions in the edge state.
The helical state of the main wire is described by the

gj ρj ρj .
j=1

Density operators are defined by 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 . Electron interaction constant in the
edge state is defined by g1 = g2 = g and g3 = 0 in the
tip ; we set vF = 1 below. Here we introduce window
function Θ( < x < L) such that Θ(x) = 1, if < x < L
and zero otherwise. The ultraviolet cutoff at x = is required for our model point-like interaction, H int , and for
finite-range interaction is associated with the screening
length, see Appendix B.
Ingoing and outgoing electronic waves ψj,in , ψj,out
are connected at the junction by the S matrix
Ψout (x) = S · Ψin (x) at x → 0, where Ψin(out) =
(ψ1,in(out) , ψ2,in(out) , ψ3,in(out) ). The S-matrix characterizes the scattering in the junction and belongs to the unitary group U (3). Our setup implies the chiral property,
|S13 | = |S32 |, |S23 | = |S31 |. Due to the time-reversal invariance of TI, the backscattering in the edge state without the tip is forbidden. The same is true in case in which
the polarization of the tip is parallel to the spin direction
of right or left mover, ξ = 0, π.
In general, the chiral S-matrix is characterized by three
parameters39 . The absence of backscattering reduces the
appropriate S matrix to the following two-parametric
form


r1 t12 t13
S = t21 r1 t23 
t31 t32 r2
r1 = sin2
t12
t13

θ
sin ξ,
2

r2 = cos θ,

(2)

ξ
ξ
= − cos θ cos2 − sin2 , t21 = t12 |ξ→π−ξ ,
2
2
ξ
ξ
= t32 = cos sin θ, t23 = t31 = sin sin θ.
2
2

One parameter, θ, is responsible for the tunneling amplitude and θ = 0 corresponds to the fully detached tip, in
this case there is a perfect transmission through the edge
state. Second parameter ξ is the above angle between

3
electron spin in the tip and spin quantization axis of the
helical state (see Appendix A).

B.

Reduced conductances

In the linear response regime the conductances are defined by Ii = Cik Vk where Ij and Vj are the current flowing towards in the junction and the electric potential in
jth lead, respectively. In the dc limit one easily obtains
from Kubo formula Cij = δij −Yij , with Yij = |Sij |2 . The
total current conservation and the absence of current for
equal voltages lead to conditions i Cij = j Cij = 0,
which also stem from the unitarity of S-matrix. This
means that we can introduce new current and voltage
combinations to show the condition explicitly.43
We choose new combinations according to relations
new
Ii = k Rki Ik , Vinew = k Rki Vk , where the matrix
1
√
2
1
− √

2

1
√
6
1
√
6


R=

−

0

1 
√
3
1
√ 
3
1
√
3

2
3

(3)

has properties R−1 = RT , det R = 1.
The corresponding matrix of conductances CR =
RT C R for the above form of S, Eq. (2), becomes


1 − a −c 0
1 − b 0 .
CR =  c
(4)
0
0 0
with
a = 1 (1 + cos2 θ) cos2 ξ − cos θ sin2 ξ,
2
b = 1 (1 + 3 cos 2θ),
4
√

c=

3
2

(5)

cos ξ sin2 θ.

It is convenient to analyze the quantity YR = 1 − CR
below. Clearly, only two components of YR are independent. The chiral component of conductance, c, depends
on the asymmetry parameter, cos ξ, which includes the
polarization angle, ξ.
We can also introduce more “physical” combinations of
currents, Iphys = (Ia , Ib ), and voltages, Vphys = (Va , Vb ),
as
1
Ia = 2 (I1 − I2 ) ,

Va = V1 − V2 ,

Ib = 1 (I1 + I2 − 2I3 ) ,
3
Vb =

1
2 (V1

+ V2 − 2V3 ) .

(6)

In terms of these, the matrix of conductances Iphys =
GVphys is given by
G=

Ga Gab
Gba Gb

=

1
2 (1 −
c
√
3

a)

with the property, 0 ≤ Ga,b ≤ 1, see43 .

c
− √3
.
− b)

2
3 (1

(7)

C.

Renormalization group equations

The dc transport through the chiral Y-junction was
studied in fermionic formalism in38,39 . We sketch here
the derivation of the corresponding formulas. The principal first order interaction correction to the conductances
reads38
Cjk = Cjk

g=0

+

1
2

Tr Wjk Wlm gml Λ ,

(8)

l,m

where we defined Cjk |g=0 = δjk − Yjk , a matrix gml =
gl δml with interaction constants gl in the Hamiltonian (1), nine 3 × 3 matrices Wjk = [ρj , ρk ] and the
trace operation Tr is defined with respect to the 3×3
matrix space of W ’s. The scale-dependent term Λ =
ln(min[L, ξ]/ ) with the temperature correlation length
ξ = vF /T .
The above correction is the leading contribution, ∼ gΛ,
to conductance. It was shown43 that the higher-order
corrections obey the scaling hypothesis for the conductance and are generated by a set of the differential RG
equations. In its simplest form, these equations are obtained by differentiating Eq. (8) over Λ (and then putting
Λ = 0) which gives
1
d R
Yjk = −
dΛ
2

R
R
R
Tr Wjk Wlm gml ,

(9)

l,m

where all quantities with superscript R are defined as
AR = RT A R.
The main source of linear-in-Λ subleading corrections
corresponds to a ladder series of diagrams, which can
be summed analytically43 . These corrections are independent of the scheme of regularization in RG procedure
and lead to the scaling exponents for conductances, coinciding with those obtained by bosonization method.
The result of the ladder summation39,40 is the RG
equation similar to above Eq. (9) with the replacement
¯
g → g = 2(Q − Y)−1 .

(10)

The matrix Q depends on the Luttinger parameters Kj
of individual wires and has the form
Qjk = qj δjk ,

qj = (1 + Kj )/(1 − Kj ) ,

Kj = [(1 − gj )/(1 + gj )]1/2 .

(11)

In our case we have g1 = g2 = g and g3 = 0. The latter
equality corresponds to q3 → ∞, whose limit is easily
taken in (10).
Now we are in a position to obtain RG equations of the
conductances. However, the equivalent RG equations in
terms of parametrization (2) have simpler form, and we
choose it below. After some algebra we obtain from (5),

4
the edge state, Va = 0, we have Ib = Gb Vb , Ia = Gab Vb
and obtain from Eqs. (5), (7) :

(9), (10) the following set of equations
dθ
1
F1 (θ) + F2 (θ) cos 2ξ
= − (1 − K) sin θ
,
dΛ
8
F3 (θ)D(θ, ξ)
dξ
1
(1 − cos θ) sin 2ξ
= (1 − K)
,
dΛ
4
D(θ, ξ)
θ
F1 = 2(1 − K) − (3 − K) sin2
2
(12)
− 3(1 − K) sin2 θ,
θ
θ
F2 = sin2 (2 − 3(1 − K) sin2 ),
2
2
2 θ
F3 = 1 − (1 − K) sin ,
2
θ
2 θ
D(θ, ξ) = K + (1 − K) sin
1 − sin2 sin2 ξ .
2
2
These equations are analyzed in the next section. Notice
that the asymmetry parameter p = cos ξ is now determined not only by the initial polarization angle, but also
by the interaction strength and tunneling amplitude.

III.

SPIN-POLARIZED TUNNELING
A.

Lowest order calculation

Let us first consider the limit of weak tunneling into
the helical edge state. This means that both tunneling
from the tip and the backscattering in the edge state due
to this tunneling are approximately zero. In terms of the
S-matrix (2) we write |r2 |2
1 and |r1 |2
0. First
condition gives θ 0 or θ π, and the second condition
states that if θ
π then ξ
0 or ξ
π. These three
seemingly unconnected regions form a neighborhood of
the fixed point (FP) A, as is discussed below.
Confining ourselves to the linear-in- θ terms, we get
from Eqs. (12):
2

dθ
(1 − K)
=−
θ,
dΛ
4K

dξ
= 0.
dΛ

(13)

Hence, the parameter θ, which is related to the barrier
transparency, is renormalized to zero with the scaling
law θ = θ0 exp(−νΛ/2), and the tunneling exponent ν =
(1 − K)2 /2K. The asymmetry ratio t↓ /t↑ = tan ξ/2 is
not renormalized.
This result is in exact agreement with the earlier
work15 , where the renormalization of the tunnel amplitudes t↑ and t↓ was studied in the bosonization approach.
Equation (9) of the above paper obtains the currents
flowing to the right IR and to the left IL ends of the
edge state. One has
IL + IR = I0 ,

IL − IR = −KI0 cos ξ .

(14)

In our notation (6) the current I0 equals to Ib , and the
difference of the currents IL − IR is 2Ia . For zero bias in

2Ia = −Ib cos ξ

(15)

This equation coincides with (14) apart from the factor
K, which is absent in dc limit in our setup with the Fermi
leads. The absence of this factor in the similar setup was
also obtained in the work 18 in bosonization approach.
Now we consider the next terms in the expansion of
RG equations in θ. Quadratic-in θ term appears only in
RG equation for asymmetry angle:
dξ
1−K 2
=
θ sin 2ξ.
dΛ
8K

(16)

Solution of the equation shows that the asymmetry angle
is renormalized according to the law
tan ξ = tan ξ0 e−(θ

2

2
−θ0 )/2(1−K)

.

(17)

Since θ is renormalized to zero (see eq. (13)), we find the
whole renormalization of ξ insignificant for small bare
tunneling amplitude θ0 . At the same time the renormalization of ξ in (16) is first order interaction effect for
1, whereas the tunneling
small interaction 1−K ≈ g
K
exponent for θ in Eq. (13) is given by the second order
2
of interaction, (1−K) ≈ g 2 .
K
We see that if we discard the small-in-θ terms in RG
equation for ξ, then we do not obtain renormalization of
asymmetry at all. However the smallness-in-θ may be
compensated by the less power of interaction in the RG
flow of ξ. As a result for small interaction and tunneling
amplitude the renormalization of θ and ξ may be of the
same order as demonstrated in Fig. 2.
We also note here that Eq. (17) can be rewritten in
terms of the asymmetry p = cos ξ = −2Ia /Ib as
dp
1−K 2
=−
θ p(1 − p2 ).
dΛ
4K

(18)

This equation possesses three fixed points (FPs) p =
0, ±1, and if θ2 were non-vanishing at Λ → ∞, then we
would obtain only p = 0 as a stable FP for repulsive interaction, K < 1. In the leading order of interaction, we
can combine first equation in (13) and (18) to represent
the RG trajectory in the form
θ2 − g log

B.

p2
= const.
1 − p2

(19)

M point

The terms of the order θ3 appear only in the equation
for θ :
(1 − K)2
(1 − K)(κ − cos 2ξ) 3
dθ
=−
θ+
θ ,
dΛ
4K
16K

(20)

5
0.5

TABLE I: The position of the universal fixed points.

0.4

θ
0 π π π
ξ arb. 0 π π/2
Ga 1 1 1 0
Gb 0 0 0 0
Gab 0 0 0 0
FP A A A N

0.3
0.2
0.1
0.0
0.0

0.2

0.4

0.6

0.8

π/2
0
3/4
1
1/2
χ−

π/2
π
3/4
1
−1/2
χ+

1.0

(a)
Already at the level of approximate Eqs. (20), (16) one
can expect that for θ ≈ θM the RG flow for the asymmetry cos ξ may exceed the RG flow of barrier transparency
θ. We discuss this point in more detail in the next subsection.

1.10
1.05
1.00
0.95

C.

0.90
0.85
0

(b)

2

4

6

8

10

Now we go beyond the weak tunneling approximation and discuss the general RG Eqs. (12) for K < 1.
In this rather complicated case we have five universal FPs in the (θ, ξ)-plane, one interaction dependent
(i.e. non-universal) fixed point M (ξM = π/2, θM =
1
arccos 3

FIG. 2: The panel (a) shows RG flows in the plane (ξ, θ)
for g = 0.26 (K = 0.77). Relative renormalization of ξ and θ
for the flow with ξ0 = 0.3, θ0 = 0.4 is shown in panel (b).

where κ ≡ (1−K)(2/3+K)+1/K. For small interaction
κ ≈ 1 and the equation (20) becomes
dθ
g2
g(1 − cos 2ξ) 3
=− θ+
θ .
dΛ
4
16

(21)

First, one can see here that the θ3 -term is unimportant
in parallel polarization case, ξ = 0, π; the right-hand side
of Eq. (16) is zero. Thus the lines ξ = 0, π are RG fixed
lines.
Otherwise the combination of (20) with (16) reveals
the existence of non-universal fixed point M at ξM = π/2,
√
θM ≈ 2g. Similar to above Eq. (16), the next-order
term in θ expansion contains smaller power of interaction strength. This situation was discussed for non-chiral
symmetric Y-junction in42 .
Solving RG equations for asymmetry p and θ gives for
the RG trajectory
θ2 p − g log

1+p
= const .
1−p

(22)

Despite the apparently different form of Eqs. (22) and
(19), these equations yield numerically close curves for
small θ < θM . Indeed the existence of the next term in
equation (20) does not seriously affect RG flows in this
limit.

Non-perturbative RG results

q 2 + 6q − 3 − q and the lines of fixed points

( ξ ∈ (0, π) and θ = 0), see Fig. 3 and Tab. I. The points
(ξ = 0, θ = π), (ξ = π, θ = π) and the fixed line θ = 0
correspond to one point A in terms of conductances.
For repulsive electron interaction the fixed line θ = 0
(the point A) is stable and corresponds to effective detachment of the tip from the edge state. It exists simultaneously with the stable point N which corresponds to full
breaking of the junction. The line in (ξ, θ) plane which
separates these two basins of attraction connects the unstable points χ± with the saddle point M (see Fig. 3).
The chiral FPs χ± were discussed in detail in29,39 and
they become stable for attractive interaction.
The whole plane (θ, ξ) is divided into two regions in
Fig. 3, depending on the interaction strength. One region corresponds to a basin of attraction to the point
N , the other to the point A. The situation is similar to
purely tunneling case (PTC) for symmetric non-chiral Y junction42 . It was shown there that for PTC the points A
or N are stable and separated by the unstable M point.
Any deviation from PTC, which means the backscattering in the main wire, leads to an additional RG flow
driving the junction to truly stable N point while A becomes of saddle point type. In our present model we have
the chiral PTC junction, and any additional modification
of S-matrix is forbidden due to topological character of
edge state and the absence of backscattering in the main
wire.
Notice that for vanishing interaction, g, the FPs A and
M merge, and the basin of attraction for A point vanish.
More precisely, A point becomes of saddle-point type in
this limit and is stable only for parallel polarization of

6
0

1.05

1.0

1.0

0.8
0.6

0.8

0.4

0.6

0.2
0.0
0

(a)

0.4

2

4

6

8

10

12

2

4

6

8

10

12

14

1.0
0.8

0.2

0.6

0.0
0.0

0.2

0.4

0.6

0.8

0.4

1.0

0.2

FIG. 3: RG flows (blue arrows) demonstrate the simultaneous existence of the stable fixed line (blue line) at point A
and stable point N, depending on the initial conditions (ξ, θ)
for electron interaction g = 0.72 (K = 0.4). Red line corresponds to the RG flow plotted in terms of the conductances
in Fig. 4b.

the tip, ξ = 0, π. All RG trajectories in this limit obey
the equation
cos ξ tan2 (θ/2) = const.

(23)

Since the position of M point depends on the interaction strength, a qualitatively different behavior of renormalized conductances may occur even for their equal bare
values at different interactions. Two examples of full scaling curves for conductances calculated for sizable interaction strength are shown in Fig. 4. The values of bare conductances are the same, but the panel (a) demonstrates
RG flows going to the point A, whereas the panel (b)
shows the system driven to the point N . This difference
stems from the fact that changing interaction strength
shifts the separating curve between the basins of attraction of the corresponding FPs.
IV.

CONCLUSIONS

We analyzed the simultaneous renormalization of
asymmetry and total tunneling conductance from fully
polarized tip into the helical edge state. In particular we
showed that the RG flow for these two quantities is sensitive to the appearance of non-universal FP with asymme2
try cos ξM = 0 and tunneling conductance Gb θM 2g
for small interaction, g.
Our method assumes the existence of Fermi leads,
which are needed for consistent definition of the S-matrix

0.0
0

(b)

14

FIG. 4: Full scaling curves for conductances are shown in
panel (a) for electron interaction g = 0.47 (K = 0.6). The
RG flow corresponds to going to the point A, i.e. effective
detachment of the tip. The panel (b) shows the RG flows
for stronger electron interaction g = 0.72 (K = 0.4). The
conductances are renormalized to the point N , i.e. effective
full breaking of the junction. Bare initial conductances on
both panels are the same and the qualitative change in the
RG flows is caused by the change in the interaction strength.

describing the Y-junction in terms of asymptotically free
states. Experimentally it can be realized by the extended
leads over the helical edge state. The helical edge state,
on the other hand, is a topological object which cannot
have disconnected ends. The dc current injected from
the tunneling tip into the edge state may exit at another
drain Y-junction. This implies the ring geometry with
potentially important interference effects. The above
picture of the renormalization of the single Y-junction
should be valid when the temperature correlation length
ξ = vF /T is smaller than the circumference of the edge
state multiplied by |r1 |2 . The importance of interference
effects is then reduced44 but the corresponding analysis
is beyond the scope of this study.

Acknowledgments

We are grateful to P. W¨lfle, D.B. Gutman and V.Yu.
o
Kachorovskii for useful discussions. This work was partly
supported by RFBR grant No 15-52-06009. The work
of D.A. was supported by Russian Science Foundation
Grant (Project No. 14-22-00281).

7
Appendix A: Rotation of polarization axes

Our setup depicted in Fig. 1 is characterized by the
angle ξ between spin quantization axis in helical edge
states and polarization direction in the tip. This junction is described by the S-matrix (2). We show that
the two-parametric S matrix can be obtained from oneparametric one by rotation of the edge states basis.
The Kramers spinor of scattered states is given by rearrangement of out- states, ψ1,out ↔ ψ2,out , by the matrix


0 1 0


(A1)
F = 1 0 0 .
0 0 1
The rotation of spin quantization axes is described by
 1

1
e 2 i(α+γ) cos β e− 2 i(α−γ) sin β 0
2
2
 1

1
U = −e 2 i(α−γ) sin β e− 2 i(α+γ) cos β 0 ,
(A2)
2
2
0
0
1
where the upper left 2×2 matrix is eiγσ3 /2 eiβσ2 /2 eiασ3 /2
with α, β, γ are Euler angles and σ2,3 Pauli matrices.
For injected spin polarization parallel to the helical edge
state, ξ = 0, the form of S matrix is simplified :


0 − cos θ sin θ


S↑ = −1
(A3)
0
0 .
0
sin θ cos θ

to cases of parallel (β = 0) or antiparallel (β = π) orientation of the quantization axes in individual edges. The
choice between these FPs is defined by the value of relative orientation without interaction, β < π/2 or β > π/2,
respectively.

Appendix B: Ultraviolet cutoff

The logarithmic corrections to conductance come from
consideration of the interaction term, g dx dy f (x −
y)ρ(x)˜(y), with the range function f (x − y) of
ρ
the screened Coulomb interaction having the property
dx f (x) = 1. In case of Luttinger model the interaction is point-like and f (x − y) = δ(x − y), which leads
to the Hamiltonian (1). However this Luttinger form of
interaction leads to ultraviolet divergence which is eventually cut off by another model assumption, formulated
in the first line of Eq. (1) in the main text, namely, the
absence of interaction in the vicinity of the junction on
the scale . We show now that the latter model assumption is not necessary if one assumes the finite range of
the interaction, r0 .
To see it, we recall that the logarithmic type of integral
(in the linear response) corresponds to the loop integra∞
L
tion of the Green’s function δI = 0 dω 0 dx dy f (x −
45
y) sin ω(y + x), see Eq. (19) in Ref. . Evaluating this
integral we have
∞

δI

1
2

0
∞

If ξ = 0(π) then we have to rotate the quantization basis
of the helical edge states:
0

S = U S↑ F U † F.

(A4)

The dependence of rotated S-matrix on angles α and γ
is removed by additional rephasing. The matrix (A4)
coincides with (2) after the substitution β → −ξ.
In our paper 41 the tunneling from the unpolarized
tip into the helical edge states was discussed. This X
junction was modeled by 4×4 S matrix since there are two
spin channels in unpolarized case. It was shown that a
certain ratio of the conductances is not renormalized, see
equation (26) in41 . Applying the above arguments to the
situation with X-junction, we confirm that the discussed
ratio corresponds to the angle between the quantization
axes for helical edge states and the tip (the quantity −2γ
in41 is the above Euler angle β). This observation clarifies
the scale invariance of the ratio, as the choice for the
spin quantization basis can be made arbitrarily without
changing physical consequences.
The case of the corner X-junction between two helical
edge states41 can also be discussed in terms of quantization axes rotation. Three stable FPs were obtained for
repulsive interaction. One of them corresponds to the detachment of helical edge states independent of directions
of the spin quantization axes. Two other FPs correspond

∞

dω
dω
2ω

2L

dr f (r)
−∞
∞

dR sin ωR
|r|

(B1)

dr f (r)(cos ωr − cos 2ωL)
−∞

It is clear that the finite range of interaction, r0 , becomes
−1
important in the limit of large ω
r0 , cutting off the
logarithmic divergence. Therefore r0 plays the same role
as the model scale and for our purposes we can take
the larger of them as the ultraviolet scale of the theory.

Appendix C: Fixed point A

Renormalization group (RG) analysis of the conductances (7) was performed in terms of parameters θ and
ξ because the RG equations are more transparent in this
form. However, one may ask why the same point A described as θ = 0 is stable and at the same time unstable
when parametrized by θ = 0, ξ = 0(π). This situation
is clarified by considering the surface of conductances in
Fig. 5. It is seen that domains of different character (stable and unstable) near the point A are separated by three
ridges. Two of them are the lines ξ = 0, θ ∈ (0, π) and
ξ = π, θ ∈ (0, π) depicted by the red lines on the conductances surface, the third ridge connects A and N points.
The domain 1 corresponds to θ ≈ 0 and the domain 2 (3)
corresponds to θ ≈ π but ξ ≈ 0(π). All these ridges are

8

2 3
1

FIG. 5: 2D fixed surface of physical conductances (7) in 3D conductances space is shown. Fixed points are depicted by black
dots. Fixed lines corresponding to ξ = 0, θ ∈ (0, π) and ξ = π, θ ∈ (0, π) are shown in red. The panel (b) shows that three
domains near the point A are separated by ridges, which cannot be crossed by any RG flow. This explains why the point A is
stable for repulsive electron interaction in the domain 1 ( θ ≈ 0 ), and at the same time unstable in domains 2 and 3 (θ ≈ π,
ξ ≈ 0(π)).

RG fixed lines, and any RG flow starting on these lines
continues along them. It also means that any RG flow
cannot pass across these ridges.
For completeness we provide here the scaling exponents at the FPs A and N . The scaling exponent for
the point A depends on the direction of approach to this
FP as explained above. In the domain 1 in Fig. 5 i.e. at
θ 0 we have only one scaling exponent for θ, equal to

1

2

3

4

5
6

7

8

9

10

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
(2010).
X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057
(2011).
T. Giamarchi, Quantum Physics in One Dimension, vol.
121 of The international series of monographs on physics
(Clarendon; Oxford University Press, 2004), illustrated
edition.
B. Braunecker, C. Bena, and P. Simon, Phys. Rev. B 85,
021017 (2012).
M. Kharitonov, Phys. Rev. B 86, 165121 (2012).
A. Soori, S. Das, and S. Rao, Phys. Rev. B 86, 125312
(2012).
G. Dolcetto, F. Cavaliere, D. Ferraro, and M. Sassetti,
Phys. Rev. B 87, 085425 (2013).
A. Calzona, M. Carrega, G. Dolcetto, and M. Sassetti,
Phys. Rev. B 92, 195414 (2015).
A. Calzona, M. Acciai, M. Carrega, F. Cavaliere, and
M. Sassetti, Phys. Rev. B 94, 035404 (2016).
S.-P. Chao, S. A. Silotri, and C.-H. Chung, Phys. Rev. B

(1 − K)2 /4K. In the domains 2 and 3 in Fig. 5, i.e. at
the points θ = π, ξ = 0(π) we have additional scaling
exponent, the RG flows away from A point along ξ with
a weak impurity exponent (1 − K). Near the stable N
1
point the exponents along θ and ξ are 2 (K −1 − 1) and
−1
(K − 1).

11

12

13

14

15
16
17

18

19

88, 085109 (2013).
G. Dolcetto, N. T. Ziani, M. Biggio, F. Cavaliere, and
M. Sassetti, physica status solidi (RRL) - Rapid Research
Letters 7, 1059 (2013).
H. Steinberg, G. Barak, A. Yacoby, L. N. Pfeiffer, K. W.
West, B. I. Halperin, and K. L. Hur, Nat Phys 4, 116
(2007).
K. L. Hur, B. I. Halperin, and A. Yacoby, Annals of Physics
323, 3037 (2008).
S. Pugnetti, F. Dolcini, D. Bercioux, and H. Grabert, Phys.
Rev. B 79, 035121 (2009).
S. Das and S. Rao, Phys. Rev. Lett. 106, 236403 (2011).
I. Garate and K. Le Hur, Phys. Rev. B 85, 195465 (2012).
U. Khanna, S. Pradhan, and S. Rao, Phys. Rev. B 87,
245411 (2013).
A. Calzona, M. Carrega, G. Dolcetto, and M. Sassetti,
Physica E: Low-dimensional Systems and Nanostructures
74, 630 (2015), ISSN 1386-9477.
R. A. Santos, D. B. Gutman, and S. T. Carr, Phys. Rev.
B 93, 235436 (2016).

9
20

21

22

23

24

25
26

27

28

29

30

31

S. H. Kim, K.-H. Jin, J. Park, J. S. Kim, S.-H. Jhi, T.H. Kim, and H. W. Yeom, Physical Review B 89, 155436
(2014).
T. Li, P. Wang, H. Fu, L. Du, K. A. Schreiber, X. Mu,
X. Liu, G. Sullivan, G. A. Cs´thy, X. Lin, et al., Phys.
a
Rev. Lett. 115, 136804 (2015).
C. Pauly, B. Rasche, K. Koepernik, M. Liebmann,
M. Pratzer, M. Richter, J. Kellner, M. Eschbach, B. Kaufmann, L. Plucinski, et al., Nat Phys 11, 338 (2015), ISSN
1745-2473.
R. Wu, J.-Z. Ma, S.-M. Nie, L.-X. Zhao, X. Huang, J.-X.
Yin, B.-B. Fu, P. Richard, G.-F. Chen, Z. Fang, et al.,
Phys. Rev. X 6, 021017 (2016).
T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325
(1988).
A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 (1993).
C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233
(1992).
K. A. Matveev, D. Yue, and L. I. Glazman, Phys. Rev.
Lett. 71, 3351 (1993).
D. Yue, L. I. Glazman, and K. A. Matveev, Phys. Rev. B
49, 1966 (1994).
M. Oshikawa, C. Chamon, and I. Affleck, Journal of Statistical Mechanics: Theory and Experiment 2006, P02008
(2006).
C.-Y. Hou, A. Rahmani, A. E. Feiguin, and C. Chamon,
Phys. Rev. B 86, 075451 (2012).
C.-Y. Hou, E.-A. Kim, and C. Chamon, Phys. Rev. Lett.
102, 076602 (2009).

32

33

34
35

36

37
38

39

40

41

42

43

44

45

J. C. Y. Teo and C. L. Kane, Phys. Rev. B 79, 235321
(2009).
A. Str¨m and H. Johannesson, Phys. Rev. Lett. 102,
o
096806 (2009).
S. Lal, S. Rao, and D. Sen, Phys. Rev. B 66, 165327 (2002).
S. Das, S. Rao, and D. Sen, Phys. Rev. B 70, 085318
(2004).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 80, 045109
o
(2009).
D. N. Aristov, Phys. Rev. B 83, 115446 (2011).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 86, 035137
o
(2012).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 88, 075131
o
(2013).
D. N. Aristov and P. W¨lfle, Lithuanian Journal of Physics
o
52, 89 (2012).
D. N. Aristov and R. A. Niyazov, Phys. Rev. B 94, 035429
(2016).
D. N. Aristov, A. P. Dmitriev, I. V. Gornyi, V. Y. Kachorovskii, D. G. Polyakov, and P. W¨lfle, Phys. Rev. Lett.
o
105, 266404 (2010).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 84, 155426
o
(2011).
A. P. Dmitriev, I. V. Gornyi, V. Y. Kachorovskii, and D. G.
Polyakov, Phys. Rev. Lett. 105, 036402 (2010).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 90, 245414
o
(2014).