Unrestricted electron bunching at the helical edge

TL;DR

This paper investigates electron bunching phenomena at the edges of topological insulators caused by magnetic impurities, revealing that larger impurity spins lead to arbitrarily large electron bunching, while spin-1/2 impurities have bounded bunching.

Contribution

The study provides an exact analytical solution for shot noise due to magnetic impurities of arbitrary spin at topological insulator edges, extending understanding of electron bunching behavior.

Findings

For $S>1/2$, Fano factor can be arbitrarily large.

For $S=1/2$, Fano factor is bounded between 1 and 2.

Analytical solution for shot noise with arbitrary impurity spin.

Abstract

A quantum magnetic impurity of spin $S$ at the edge of a two-dimensional time reversal invariant topological insulator may give rise to backscattering. We study here the shot noise associated with the backscattering current for arbitrary $S$. Our full analytical solution reveals that for $S>\frac{1}{2}$ the Fano factor may be arbitrarily large, reflecting bunching of large batches of electrons. By contrast, we rigorously prove that for $S=\frac{1}{2}$ the Fano factor is bounded between $1$ and $2$, generalizing earlier studies.

Figures

Equations

$$H=H_{\mathrm{e}}+H_{\mathrm{e}-\mathrm{i}},H_{\mathrm{e}}=iv\int dy{\Psi }^{†}(y){\sigma }_z{\partial }_y\Psi (y),$$

$$H_{\mathrm{e}-\mathrm{i}}=\frac{1}{\nu }{\mathcal{J}}_{ij}{S}_i{s}_j(y_0),s_j(y)=\frac{1}{2}{\Psi }^{†}(y){\sigma }_j\Psi (y).$$

$$\frac{d{\rho }_S^{(\lambda )}}{dt}=-i[H_{\mathrm{e}-\mathrm{i}}^{\mathrm{mf}},{\rho }_S^{(\lambda )}]+{\eta }_{jk}^{(\lambda )}{S}_j{\rho }_S^{(\lambda )}{S}_k-\frac{{\eta }_{jk}^{(0)}}{2}\{{\rho }_S^{(\lambda )},S_k{S}_j\}.$$

$${\Pi }_V^{(\lambda )}=\left(\begin{array}{ccc}{f}_{\lambda }^{+}\left(V/T\right)& -i{f}_{\lambda }^{-}\left(V/T\right)& 0\\ i{f}_{\lambda }^{-}\left(V/T\right)& f_{\lambda }^{+}\left(V/T\right)& 0\\ 0& 0& 1\end{array}\right)$$

$$\frac{d}{dt}⟨m|{\rho }_S^{(\lambda )}|m⟩=\sum _{m^{\prime }=-S}^S{\mathcal{L}}_{m{m}^{\prime }}^{(\lambda )}⟨m^{\prime }|{\rho }_S^{(\lambda )}|m^{\prime }⟩.$$

$${\rho }_{S,\mathrm{st}}^{(0)}\sim {\left(\frac{1+p}{1-p}\right)}^{S_z},p=\frac{2{\mathcal{J}}_{xx}{\mathcal{J}}_{yy}}{{\mathcal{J}}_{xx}^2+{\mathcal{J}}_{yy}^2+{\mathcal{J}}_{yx}^2},$$

$$g={\mathcal{J}}_{xx}^2+{\mathcal{J}}_{yy}^2+{\mathcal{J}}_{yx}^2+{\mathcal{J}}_{zx}^2+{\mathcal{J}}_{zy}^2,q=1-\frac{{\mathcal{J}}_{zx}^2+{\mathcal{J}}_{zy}^2}{g}.$$

$$I_{\mathrm{bs}}=-\frac{\pi gV}{4}⟨R(S_z)⟩,$$

$$F_{\mathrm{bs}}=1+\frac{4}{q(1-p)}\sum _{n=1}^{2S}\frac{⟨P_n[R(S_z)-⟨R(S_z)⟩]{⟩}^2}{n(2S+1-n)⟨R(S_z)⟩{\mu }_n}.$$

$$F_{\mathrm{bs}}(S=1/2)=(1-q{p}^4)/(1-q{p}^2).$$

$$F_{\mathrm{bs}}(p=0)=1+(2S-1)(2S+3)(2-3q{)}^2/(45q).$$

$$F_{\rm bs}(p\to 1)=1+\frac{(1-p)\bigl{(}(2-3q)S+q-1\bigr{)}^{2}}{q(1-q)S^{3}}.$$

$$F_{\mathrm{bs}}(q,p\to 1)=\frac{2(1-p)+(1-q)S}{1-p+(1-q)S}.$$

$$F_{\mathrm{bs}}=1+\frac{1}{2S}\frac{(1-p^2)(2-3q{)}^2}{q{p}^3(1-q)}+O(1/S^2),$$

Full text

Introduction. — Zero-frequency current noise in a conductor can reveal information about correlations in electronic transport which cannot be extracted from the average current BB ; LS . Obtaining information about the correlations requires going beyond linear response (where thermal noise is fully determined by linear conductance through the fluctuation-dissipation theorem), and studying shot noise at voltage larger than the temperature. The ratio between the shot noise and the average current times the electron charge is referred to as Fano factor. It is useful for characterizing the unit of effective elementary charge in correlated electron systems, e.g., quasiparticle charges in fractional quantum Hall edges Glattli ; Reznikov . Entanglement with an external degree of freedom may modify the effective Fano factor Birchall ; Barbarino .

The experimental discovery of 2D topological insulators Konig2007 triggered intensive experimental and theoretical research Qi-Zhang ; Hasan-Kane . Electron transport along the helical edge was theoretically predicted to be protected from elastic backscattering by time-reversal symmetry. However, this ideal picture was impugned by transport experiments in HgTe/CdTe  Konig2007 ; Nowack ; Grabecki ; Gusev2013 ; Gusev2014 ; Kvon2015 and InAs/GaSb  Knez2011 ; Suzuki2013 ; RRDu1 ; RRDu2 ; Suzuki2015 ; Mueller2015 ; RRDu3 ; Mueller2017 quantum wells, Bi bilayers Sabater2013 , and WTe2 monolayers Cobden17 ; Jia2017 ; Herrero2018 . In order to explain this data, several physical mechanisms of backscattering were proposed and studied theoretically Xu2006 ; Maciejko2009 ; Tanaka2011 ; Schmidt2012 ; Oreg2012 ; Gornyi2014 ; Goldstein2013 ; Cheianov2013 ; Altshuler2013 ; Goldstein2014 ; Yudson2015 ; Schmidt2015 ; Glazman2016 ; Nagaev2016 ; Kimme2016 ; Meir2017 ; Mani1 ; Hsu1 ; Hsu2 ; Kurilovichi2017 ; Kurilovichi2018 ; Yudson2018 .

In contrast to the average current, shot noise at the helical edge has attracted much less experimental and theoretical attention so far Rosenow2013 ; Khrapai1 ; Khrapai2 ; Nagaev2016 ; VG2017 ; Nagaev2018 . The shot noise due to backscattering of helical edge electrons via anisotropic exchange (which has to break the conservation of the total $z$-projection of the angular momentum to affect the dc current Tanaka2011 ) with a local spin $S=\frac{1}{2}$ magnetic moment has been calculated in Ref. VG2017 . The authors of Ref. VG2017 studied the so-called backscattering Fano factor, $F_{\rm bs}$, which is the ratio between the zero-frequency noise of the backscattering current, $\mathcal{S}_{\rm bs}$, and the absolute value of the average backscattering current, $|I_{\rm bs}|$, in the limit of large voltage bias $V$. It was found that $F_{\rm bs}$ is bounded between $1$ and $2$, with the extreme values corresponding to independent backscattering of single electrons and bunched backscattering of pairs of electrons, respectively. The considerations of Ref. VG2017 were limited to the case of almost isotropic exchange interaction. This assumption is natural for the model of charge puddles which act as effective spin-$\frac{1}{2}$ magnetic moments Goldstein2013 ; Goldstein2014 . However, the spin of a magnetic impurity (MI) can be larger than $\frac{1}{2}$, e.g., $S=\frac{5}{2}$ for a $\mathrm{Mn}^{2+}$ ion in a HgTe/CdTe quantum well. Moreover, in the case of a “genuine” MI the exchange interaction is strongly anisotropic Kimme2016 ; Kurilovichi2017 .

In this Letter we study the backscattering shot noise at the edge of a 2D topological insulator mediated by the presence of a single quantum MI. We assume that the impurity is of an arbitrary spin $S$ and the exchange interaction matrix is of a general form. We find the backscattering Fano factor analytically, cf. Eq. (9). Strikingly, for any $S>\frac{1}{2}$ it is not bounded from above, cf. Eqs. (11) and (12); $F_{\rm bs}>2$ over a wide parameter range, see Fig. 2. This implies that a dynamical magnetic moment with $S>\frac{1}{2}$ can bunch helical electrons together. Here, in a significant parameter range, for each value of the impurity spin projection $S_{z}$ electrons are backscattered with a rate $\propto S_{z}^{2}$, while $S_{z}$ itself changes slowly. This results in a modulation of the backscattering events into long correlated pulses (Fig. 1a). For $S=\frac{1}{2}$ this effect is absent (Fig. 1b), and we find a concise exact expression for $F_{\mathrm{bs}}$ proving rigorously that $1\leqslant F_{\mathrm{bs}}\leqslant 2$, cf. Eq. (10). Our results elucidate an important facet of the dichotomy between topological properties and electronic correlations in one-dimensional edges Thomale2019 , accounting for mechanisms that break topological protection against backscattering.

Model. — Helical edge electrons coupled to a MI are described by the following Hamiltonian (we use units with $\hbar=k_{B}=-e=1$):

[TABLE]

where $\Psi^{\dagger}$ ($\Psi$) is the creation (annihilation) operator of the edge electrons with velocity $v$. The Pauli matrices $\sigma_{x,y,z}$ act in the spin basis of the edge states. The exchange electron-impurity interaction is assumed to be local:

[TABLE]

Here $\nu=1/\left(2\pi v\right)$ is the density of states per one edge mode, operator $S_{i}$ denotes the $i$-th component of the impurity spin, and the couplings $\mathcal{J}_{ij}$ are dimensionless and real. We stress that in general the exchange interaction (2) is strongly anisotropic and violates the conservation of the total $z$-projection of the angular momentum of the system Otten2013 ; Kimme2016 ; Kurilovichi2017 . The latter violation is required to generate persistent backscattering of helical electrons. We assume that the coupling constants are small, $|\mathcal{J}_{ij}|\ll 1$, and we neglect the local anisotropy $H_{\mathrm{anis}}=\mathcal{D}_{kp}S_{k}S_{p}$ of the MI spin which is justified at $|\mathcal{D}_{kp}|\ll\max\{\mathcal{J}_{ij}^{2}T,|\mathcal{J}_{ij}|V\}$ Kurilovichi2018 . In the absence of the local anisotropy we can rotate the spin basis for $S_{i}$ bringing the exchange matrix $\mathcal{J}_{ij}$ to a lower triangular form. We thus assume hereinafter that $\mathcal{J}_{xy}=\mathcal{J}_{xz}=\mathcal{J}_{yz}=0$. In addition we ensure that $\mathcal{J}_{xx}\mathcal{J}_{yy}>0$ with a proper rotation.

Cumulant generating function. — The average backscattering current and its zero frequency noise can be extracted from the statistics of the number of electrons backscattered off a MI during a large time interval $t$: $\Delta N(t)=\Sigma_{z}(t)-\Sigma_{z}$, where $\Sigma_{z}(t)=e^{iHt}\Sigma_{z}e^{-iHt}$ and $\Sigma_{z}=\int dys_{z}(y)$. The cumulant generating function for $\Delta N$ can be written as $G(\lambda,t)=\ln\operatorname{Tr}[e^{i\lambda\Sigma_{z}(t)}e^{-i\lambda\Sigma_{z}}\rho(0)]$, where $\rho(0)$ stands for the initial density matrix of the full system Esposito . It is convenient to write $G(\lambda,t)=\ln\operatorname{Tr}\rho^{(\lambda)}(t)$ where $\rho^{(\lambda)}(t)=e^{-iH^{(\lambda)}t}\rho(0)e^{iH^{(-\lambda)}t}$ is the generalized density matrix of the system at time $t$ and $H^{(\lambda)}=e^{i\lambda\Sigma_{z}/2}He^{-i\lambda\Sigma_{z}/2}$. Tracing out the degrees of freedom of the helical electrons, we obtain $G(\lambda,t)=\ln\operatorname{Tr}_{S}\rho_{S}^{(\lambda)}(t)$, where $\rho_{S}^{(\lambda)}(t)$ denotes the reduced generalized density matrix of the impurity.

Generalized master equation. — In order to find $G(\lambda,t)$ we derive a generalized Gorini-Kossakowski-Sudarshan-Lindblad equation, which governs the time evolution of $\rho_{S}^{(\lambda)}(t)$ (see Supplemental Material SM ):

[TABLE]

Here $H_{\rm e-i}^{\rm mf}=\mathcal{J}_{zz}\langle s_{z}\rangle S_{z}/\nu$ is the mean-field part of $H_{\rm e-i}$ with the average non-equilibrium spin density $\langle s_{z}\rangle=\nu V/2$. Additionally, we have introduced $\eta_{jk}^{(\lambda)}=\pi T(\mathcal{J}\Pi_{V}^{(\lambda)}\mathcal{J}^{T})_{jk}$, where

[TABLE]

and $f_{\lambda}^{\pm}(x)=\frac{x}{2}\left(e^{-i\lambda}e^{x}\pm e^{i\lambda}\right)/\left(e^{x}-1\right)$.

Below we focus on the regime $V\gg T$. The first term on the r.h.s. of Eq. (3) is then much larger than the other two terms. Consequently, one may implement the rotating wave approximation to simplify Eq. (3). Within its framework $\rho_{S}^{(\lambda)}$ is diagonal in the eigenbasis of $H_{\rm e-i}^{\rm mf}$, i.e., of $S_{z}$. Denoting the impurity state with $S_{z}=m$ as $|m\rangle$ ($m=S,...,-S$) we obtain a classical master equation for the occupation numbers

[TABLE]

Here $\mathcal{L}^{(\lambda)}$ is a $(2S+1)\times(2S+1)$ tridiagonal matrix. The tridiagonal form indicates that $S_{z}$ changes by not more than unity in each elementary scattering process. Nonzero elements of $\mathcal{L}^{(\lambda)}$ are given by $\mathcal{L}_{m+1,m}^{(\lambda)}=e^{-i\lambda}\eta_{+}[S(S+1)-m(m+1)]/4$, $\mathcal{L}_{m,m+1}^{(\lambda)}=(\eta_{-}/\eta_{+})\mathcal{L}_{m+1,m}^{(\lambda)}$, and $\mathcal{L}_{mm}^{(\lambda)}=-e^{i\lambda}\mathcal{L}_{m+1,m}^{(\lambda)}-e^{i\lambda}\mathcal{L}_{m-1,m}^{(\lambda)}+(e^{-i\lambda}-1)\eta_{zz}^{(0)}m^{2}$, where $\eta_{\pm}=\eta_{xx}^{(0)}+\eta_{yy}^{(0)}\pm i(\eta_{xy}^{(0)}-\eta_{yx}^{(0)})$. It is worthwhile to note that by Eq. (5), the characteristic function of $\langle m|\rho_{S}^{(\lambda)}|m\rangle$ obeys the Heun equation DMLF .

Results. — At $\lambda=0$ Eq. (5) describes the time evolution of populations of the impurity energy levels. Through this equation we establish that the steady state density matrix of the impurity at $V\gg T$ is given by

[TABLE]

The dimensionless parameter $p$ determines the polarization of the impurity, i.e., for $p=1$ only the state $S_{z}=S$ is occupied, whereas for $p=0$ all levels are equally populated. Physically, at $p=1$, $\mathcal{J}_{xx}=\mathcal{J}_{yy}$, $\mathcal{J}_{yx}=0$, and the impurity spin can be flipped down only by backscattering an edge electron carrying spin down. We note, though, that at large voltage the current is carried mainly by spin-up electrons. Thus, a steady state of the impurity is established in which $S_{z}=S$ with essentially unit probability. At $p<1$ the impurity spin can be flipped down by electrons with spin up, resulting in depolarization of the impurity. We stress that $\mathcal{J}_{zx}$ and $\mathcal{J}_{zy}$ do not enter into the expression (6) for $p$ because the corresponding terms in the Hamiltonian do not induce impurity spin flips.

To express $I_{\rm bs}$ and $\mathcal{S}_{\rm bs}$ in a compact form we introduce two parameters:

[TABLE]

Then we find that $\eta_{\pm}=\pi gV(1\pm p)q/2$ and $\eta^{(0)}_{zz}=\pi gV(1-q)/2$. Notice that $0<p,q\leqslant 1$ and $g\ll 1$.

The average backscattering current can be found as $I_{\rm bs}=\langle\Delta N\rangle/t=-(i/t)\partial G(\lambda,t)/\partial\lambda$, where the limits $t\to\infty$ and $\lambda\to 0$ are assumed. Solving Eq. (5) within the first order perturbation theory in $\lambda$, we find

[TABLE]

where $R(S_{z})=qS(S+1)-qpS_{z}+(2-3q)S_{z}^{2}$ and \langle...\rangle=\mathrm{Tr}\bigl{(}...\>\rho^{(0)}_{S,\mathrm{st}}\bigr{)}. We note that $\langle R(S_{z})\rangle>0$, hence $I_{\rm bs}$ is negative.

The backscattering current noise at zero frequency is given by the second cumulant of $\Delta N$ as $\mathcal{S}_{\rm bs}=\langle\kern-5.16663pt~{}\langle(\Delta N)^{2}\rangle\kern-5.16663pt~{}\rangle/t=-t^{-1}\partial^{2}G(\lambda,t)/\partial\lambda^{2}$ at $t\to\infty$ and $\lambda\to 0$. In order to compute $\mathcal{S}_{\rm bs}$ from Eq. (5) we employ second order perturbation theory in $\lambda$ SM . The noise can be written as $\mathcal{S}_{\rm bs}=F_{\rm bs}|I_{\rm bs}|$, where the backscattering Fano factor reads

[TABLE]

Here $P_{n}=\sum_{m=S-n+1}^{S}|m\rangle\langle m|$ is a projector on the subspace of $n$ impurity states with largest $S_{z}$ projection and $\mu_{n}=\langle S+1-n|\rho_{S,\mathrm{st}}^{(0)}|S+1-n\rangle$. Notice that Eq. (9) implies $F_{\rm bs}\geqslant 1$. So far, we considered the model of noninteracting edge states. Accounting for electron-electron interaction results only in the common factor for $I_{\rm bs}$ and $\mathcal{S}_{\rm bs}$ that leaves $F_{\rm bs}$ intact SM .

The most striking feature of Eq. (9) is the divergence at $q\to 0$ for $0\leqslant p<1$ (cf. Eq. (12)). It indicates that in general the Fano factor is unrestricted from above. The only exception is the case of $S={1}/{2}$, for which Eq. (9) gives

[TABLE]

This expression indicates that $F_{\mathrm{bs}}$ is restricted to the range between $1$ and $2$ for $S={1}/{2}$. Eq. (10) extends the $p,q\to 1$ result of Ref. VG2017 to arbitrary values of $p$ and $q$.

The divergence of $F_{\rm bs}$ at $q\to 0$ for impurities with $S>1/2$ can be explained on physical grounds. For the sake of simplicity we first consider $S=1$. The inequality $q\ll 1$ implies $|\mathcal{J}_{zx}|,|\mathcal{J}_{zy}|\gg|\mathcal{J}_{xx}|,|\mathcal{J}_{yy}|,|\mathcal{J}_{yx}|$ and, therefore, the backscattering predominantly happens without spin flips of the impurity. By Fermi’s golden rule, the rate of such reflection processes is proportional to $S_{z}^{2}$, rendering the backscattering current, $d\Delta N/dt$, very sensitive to the spin state of the impurity. The processes associated with $\mathcal{J}_{xx}$, $\mathcal{J}_{yy}$, and $\mathcal{J}_{yx}$ in $H_{\mathrm{e-i}}$ are incapable of producing a significant contribution to $d\Delta N/dt$ on their own, but they can transfer the impurity from one spin state to another, switching efficient backscattering on ($S_{z}=\pm 1$) and off ($S_{z}=0$). Consequently, the backscattering current as a function of time looks like a sequence of long pulses, each consisting of a large number (proportional to $1/q$) of backscattered electrons (see Fig. 1a). This peculiar bunching of helical electrons results in $1/q$ divergence of $F_{\rm bs}$. For $S>1$ the backscattering current looks differently because the reflection intensity remains finite between the pulses. Still, many-electron correlations are present in $d\Delta N/dt$: impurity rarely jumps between states of different $S_{z}$ changing the intensity of backscattering $\propto S_{z}^{2}$. As $q\rightarrow 0$, impurity backscatters increasingly large number of electrons during its stay in a state with a given $S_{z}$, which results in the divergent backscattering Fano factor. We note that for $S=1/2$ for both spin states $S_{z}^{2}=1/4$. Because of that $d\Delta N/dt$ has no pulses (see Fig. 1b) and $F_{\rm bs}$ is not singular at $q\rightarrow 0$.

The exact analytical result (9) for $F_{\mathrm{bs}}$ can be expressed as a rational function of $p$ and $q$ for any given value of $S$. However, even for $S=1$ such an expression is lengthy. We thus focus on relevant limiting cases below. For an unpolarized impurity, $p=0$, we find

[TABLE]

In this regime the Fano factor scales as $S^{2}$ at large $S$. This fact has a simple physical interpretation. Since for $p=0$ each of the $2S+1$ spin states of the impurity is occupied with the same probability, the dynamics of impurity flips between states with different $S_{z}$ is diffusive. If the MI starts its motion in a state $S_{z}$, on average $\sim S^{2}$ transitions occur before $S_{z}$ returns to its initial value. Therefore, approximately $S^{2}$ subsequent spin flips of the impurity are correlated. These correlations in the dynamics of the impurity spin are mirrored by the correlations in the electron backscattering and result in the $S^{2}$ scaling of $F_{\rm bs}$ at small $p$.

For an almost fully polarized MI, $1-p\ll 1-q$, Eq. (9) yields

[TABLE]

As follows from Eq. (12) the Fano factor at $p=1$ is equal to unity for $q<1$. This result is expected since for $p=1$ the spin of a MI is locked to the state $S_{z}=S$. Therefore, the only allowed backscattering processes occur due to the $\mathcal{J}_{zx}$ and $\mathcal{J}_{zy}$ terms in $H_{\mathrm{e-i}}$, which do not require spin-flips of the impurity to scatter helical electrons. Consequently, the impurity does not keep memory about backscattered electrons, which results in a Poissonian single-electron reflection process with $F_{\rm bs}=1$ (see Fig. 1c). For $1-p\ll 1$, rare two-particle reflections are involved in addition to the single-particle backscattering. They are accompanied by short-time excursions of the impurity spin from the state $S_{z}=S$ to the state $S_{z}=S-1$ and lead to the enhancement of the Fano factor above unity (cf. Eq. (12)). In total, for $1-p\ll 1$ the backscattering of electrons represents superposition of the independent single- and two- particle Poisson processes (see Fig. 1d). For large $S$ the deviation of $F_{\rm bs}$ from unity is additionally suppressed by a factor $1/S$ in the considered limit [cf. Eq. (14)].

The behavior of $F_{\rm bs}$ at the point $q=p=1$ is non-analytical owing to $I_{\rm bs}=0$. The value of the Fano factor depends on the direction in the $(q,p)$ plane at which this point is approached. For a fixed ratio $\frac{1-p}{1-q}$, Eq. (9) yields:

[TABLE]

The overall behavior of $F_{\rm bs}(q,p)$ for different values of $S$ is shown in Fig. 2. For $S=1/2$ (Fig. 2a) the backscattering Fano factor is bounded by $1\leq F_{\mathrm{bs}}\leq 2$. For $S>1/2$ (Figs. 2b - 2f) there is a divergence in $F_{\mathrm{bs}}$ in the vicinity of $q=0$. The divergence appears to be more pronounced as $S$ increases. However, this trend breaks down for large $S$. Eq. (9) implies that for $S\gg 1/[p(1-q)(2-3q)]$ the Fano factor behaves as

[TABLE]

i.e., it gradually decreases and approaches unity as $S$ gets higher. Thus, the limit of large spin corresponds to the backscattering of helical electrons by a classical MI.

Along with the line $p=1$ the Fano factor equals unity at the isolated point $(q,p)=(2/3,0)$. This is universal for all $S$ (see Eq. (11)). At this point the backscattering rate is independent of the impurity state and, therefore, the backscattering current statistics has a Poissonian single-particle character. The interplay between an $~{}S^{2}$ scaling of $F_{\mathrm{bs}}$ at $p\rightarrow 0$ and the presence of a degenerate point $(q,p)=(2/3,0)$ results in a bottleneck feature in the vicinity of the latter in Figs. 2b - 2f. The $1/S$ term in Eq. (14) cancels at the line $q=2/3$ and $F_{\mathrm{bs}}(q=2/3)-1\sim 1/S^{3}$ for $p\neq 0$.

Conclusions. — To summarize, we have investigated the zero-frequency statistics of the backscattering current induced by a magnetic impurity of an arbitrary spin $S$ located near the edge of a two-dimensional topological insulator. We addressed the limit of large voltage $|V|\gg\max\{T,|\mathcal{D}_{kp}/\mathcal{J}_{ij}|\}$ where it is possible to neglect the thermal contribution to the noise, as well as the effect of the local anisotropy of the magnetic impurity. Our analytical solution for the average backscattering current and its zero-frequency noise underscores several striking features: (i) the dependence of the average backscattering current and noise on the elements of the exchange matrix is determined by three parameters ($g,p,q$) only instead of a-priory six different parameters (the number of non-zero elements of the exchange matrix $\mathcal{J}_{ij}$); (ii) for $S>1/2$ the backscattering Fano factor can be arbitrary large, diverging in the limit $q\to 0$. This implies that the backscattered electrons can be bunched together in long pulses. Observation of electron bunching is a novel challenge for experimentalists which might shed light on the nature of backscattering at the helical edge, and on how emerging strong correlations can undermine topological protection against backscattering; (iii) for $S=1/2$ the backscattering Fano factor is limited to the range between $1$ and $2$; (iv) the backscattering Fano factor is independent of the electron-electron interaction.

Acknowledgements. — We thank V. Kachorovskii for discussions. Hospitality by Tel Aviv University, the Weizmann Institute of Science, the Landau Institute for Theoretical Physics, and the Karlsruhe Institute of Technology is gratefully acknowledged. The work was partially supported by the programs of the Russian Ministry of Science and Higher Education, DFG Research Grant SH 81/3-1, the Italia-Israel QUANTRA, the Israel Ministry of Science and Technology (Contract No. 3-12419), the Israel Science Foundation (Grant No. 227/15), the German Israeli Foundation (Grant No. I-1259-303.10), and the US-Israel Binational Science Foundation (Grant No. 2016224), and a travel grant by the BASIS Foundation (Russia).