Pis’ma v ZhETF, vol. 109, iss. 9, pp. 639 - 640
© 2019
May 10
Comment on “Noise in the helical edge channel anisotropically coupled
to a local spin”
(Pis’ma v ZhETF 108, 700 (2018))
I. S. Burmistrov+1), P. D. Kurilovich∗, V. D. Kurilovich∗
+L. D. Landau Institute for Theoretical Physics Russian Academy of Sciences, 119334 Moscow, Russia
∗Department of Physics, Yale University, New Haven, CT 06520, USA
Submitted 12 March 2019
Resubmitted 12 March 2019
Accepted 21
March 2019
DOI: 10.1134/S0370274X19090145
In [1] the current noise in the helical edge channel
ing to Eq. (7) of [1] the backscattering current is given
anisotropically coupled to a local spin 1/2 has been com-
by (G0 = e2/h)
puted. In addition to the noise, a result for the backscat-
tering current Ibs was reported. The latter formula (see
INRSbs = -G0TJ2a/(2v2).
(2)
Eq. (7) of [1]) does not coincide with the expression for
Ibs derived in our recent work (see Eq. (22) of [2]) for
a general form of the exchange interaction matrix. Be-
This result should be contrasted with our result [2]:
low we shall argue that, in general, the result of [1] for
the backscattering current is erroneous. Equation (7) of
V
2J2aJ20
[1] gives the correct answer for the diagonal exchange
Ibs = -G0
(3)
2v2 2J2a + J2
z
matrix only. The incorrect result of [1] is a consequence
of the assumption (which was also done in [3]) that the
In addition to a very different dependence of the
density matrix of the impurity spin, ρS , is diagonal in
backscattering current on the elements of the exchange
the eigenbasis of Sz (see Eq. (2) of [1]). As we demon-
matrix, Eq. (2) predicts saturation of the backscattering
strated in [2], a careful analysis of the problem invali-
current at V ≫ T whereas Eq. (3) does not. This satura-
dates this assumption.
tion occurs due to the full polarization of the magnetic
In order to set notations, we define the Hamiltonian
impurity along z-axis by the applied voltage V ≫ T.
describing the exchange interaction between the helical
However, such a polarization is a consequence of an er-
edge states and a magnetic impurity as Hint = JjkSj sk,
roneous assumption that ρS is diagonal in the eigenbasis
where S (s) denotes the operator of the impurity spin
of Sz. In fact, there are no physical reasons for the full
(the spin density of helical electrons) and Jjk is a 3 × 3
polarization (along z-axis) to occur: the magnetic im-
exchange matrix. In [1] the following form of the ex-
purity remains partially polarized in a direction tilted
change matrix was considered
with respect to z-axis for arbitrary large voltage (see
discussion around Eq. (26) in [2]).
2(J0 + J2)
0
2Ja
To be more specific, the polarization along z-axis
J =
0
2(J0 - J2)
0
.
(1)
predicted by [1] follows from a claim that the dephas-
2J1
0
Jz
ing of the impurity spin is mainly induced by the term
JzSzsz in Hint. However, the term 2JaSxsz enters Hint
We note that in our paper [2] we used dimensionless ex-
on the equal grounds and thus has to be taken into con-
change matrix Jjk = νJjk. Here ν = 1/(2πv) stands for
sideration to properly account for the dephasing. In par-
the density of states per edge mode and v denotes the
ticular, if Jz = 0 the magnetic impurity gets polarized
velocity of the helical states.
along x-axis for V ≫ T. In this regime, the backscat-
To illustrate our point we first consider the case
tering is induced by the term 2J0(Sxsx + Sysy) in the
J2 = J1 = 0 and the regime V ≫ T. Then, accord-
Hamiltonian and is insensitive to the precise value of
Ja. This is consistent with our Eq.(3) and not consis-
1)e-mail: burmi@itp.ac.ru.
tent with Eq. (2).
Письма в ЖЭТФ том 109 вып. 9 - 10
2019
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