Finite frequency backscattering current noise at a helical edge
FFinite frequency backscattering current noise at a helical edge
B. V. Pashinsky,
1, 2, 3
M. Goldstein, and I. S. Burmistrov
3, 5 Skolkovo Institute of Science and Technology, 143026 Moscow, Russia Moscow Institute for Physics and Technology, 141700 Moscow, Russia Russia L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432, Chernogolovka, Russia Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel Laboratory for Condensed Matter Physics, National Research University Higher School of Economics, 101000 Moscow, Russia (Dated: July 17, 2020)Magnetic impurities with sufficient anisotropy could account for the observed strong deviation ofthe edge conductance of 2D topological insulators from the anticipated quantized value. In this workwe consider such a helical edge coupled to dilute impurities with an arbitrary spin S and a generalform of the exchange matrix. We calculate the backscattering current noise at finite frequencies asa function of the temperature and applied voltage bias. We find that in addition to the Lorentzianresonance at zero frequency, the backscattering current noise features Fano-type resonances at non-zero frequencies. The widths of the resonances are controlled by the spectrum of correspondingKorringa rates. At a fixed frequency the backscattering current noise has non-monotonic behaviouras a function of the bias voltage. I. INTRODUCTION
The hallmark of two-dimensional (2D) topological in-sulators is helical edge states [1, 2]. They exist due tospin-momentum locking caused by the presence of strongspin-orbit coupling [3, 4]. The helical edge states havebeen detected experimentally in HgTe/CdTe quantumwells [5–9]. The time-reversal symmetry protects the he-lical edge states from elastic backscattering. As a con-sequence, one expects ballistic transport along the heli-cal edge with the quantized conductance of G = e /h .However, this idealized picture of edge transport wasquestioned by experiments in a number of 2D topo-logical insulators: HgTe/CdTe quantum wells [5, 10–14], InAs/GaSb quantum wells [15–22], WTe monolay-ers [23–25], and Bi bilayers [26]. In order to account forthe experimental data, several physical mechanisms ofbackscattering were proposed and studied theoretically,including the effects of electron-electron interaction [27–36], charge puddles acting as an effective spin-1 / S = [54] andspin S > / S = the the zero-frequencybackscattering shot noise Fano factor is bounded fromabove whereas for spins S > / S (see Fig. 1). Similar toRef. [58] we consider the most general exchange inter-action between magnetic impurity and the helical edgestates. Under the assumption of weak exchange interac-tion we derive analytic expression for the current noise asa function of frequency, ω , voltage, V , and temperature, T . We find that the frequency dependence of the currentnoise has a resonant structure (see Fig. 3). While theresonance at zero frequency has a Lorentzian form, theresonances at non-zero frequencies are of Fano type. Theresonance structure of the current noise is similar to thebehavior of the dynamical spin susceptibility under theconditions of electronic paramagnetic resonance. In ourcase voltage plays a role of magnetic field that lifts degen- FIG. 1. Sketch of the Setup: A helical edge of a 2D topologicalinsulator contaminated by dilute spin- S magnetic impurities.(see text) a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l eracy of the impurity spectrum, whereas finite frequencyallows transitions between the split energy levels. Thebroadening of the resonances is determined by the cor-responding inverse Korringa times. For S > / S = 1 / II. FORMALISMA. Model
The non-interacting 1D helical mode coupled to a mag-netic impurity is described by the following Hamiltonian, H = H e + H e − i . (1)Here the first term is the Hamiltonian of edge electrons, H e = iv (cid:90) dy Ψ † ( y ) σ z ∂ y Ψ( y ) , (2)where v denotes the velocity of the edge states, Ψ † andΨ stand for the creation and annihilation operators, and σ x,y,z are the standard Pauli matrices operating in thepseudospin space of edge states. The interaction betweenhelical electrons and a magnetic impurity located at y = y is assumed to be in the form of local exchange, H e − i = 1 ν J ij S i s j ( y ) , s j ( y ) = 12 Ψ † ( y ) σ j Ψ( y ) . (3)Here ν = 1 / (2 πv ) is the density of states per one edgemode and S i stands for the components of impurity spinoperator.The 3 × J ij , i, j = x, y, z , is not diagonal due to the presence of spin-orbitcoupling in the 2D topological insulators. For example,there are four nonzero components, J xx = J yy , J zz , and J xz , for a magnetic impurity in a HgTe/CdTe quantumwell in the case of negligible interface inversion asym-metry [44, 45, 59]. The inversion asymmetry present inHgTe/CdTe quantum wells [60–67] renders all matrix el-ements J ij nonzero. The exchange interaction (3) is ex-pected to be applicable for other 2D topological insula-tors, e.g., InAs/GaSb quantum wells, WTe monolayers,and Bi bilayers. We assume that dimensionless exchangeinteraction is weak, |J ij | (cid:28)
1. This is fully justifiedin physical systems. For example, for Mn ion in aHgTe/CdTe quantum well |J ij | ∼ − [68].In the Hamiltonian (1) we neglect the local anisotropyof the impurity spin, described by D jk S j S k . This can be justified for |D jk | (cid:28) max {J jk T, |J jk V |} [46]. Withneglect of the local anisotropy the exchange matrix J ij can be brought to a lower triangular form by rotation ofthe spin basis for the impurity spin S i . We thus assumehereinafter that J xy = J xz = J yz = 0 and J xx J yy > J jk that is responsible for a logarithmic de-pendence of exchange interaction on max { T, | V |} [45]. Inaddition, we neglect electron-electron interactions alongthe helical edge in the Hamiltonian (1). We discuss theeffect of the interactions on the backscattering currentnoise in Sec. V. B. Backscattering current noise and thegeneralized master equation
The presence of a magnetic impurity causes thebackscattering of helical states. In the presence of abias voltage V along the edge, scattering helical statesoff magnetic impurity produces a backscattering current.The spin-momentum locking allows one to relate thestatistics of the total pseudospin projection of the he-lical states, Σ z = (cid:82) dy s z ( y ), with the statistics of thenumber of electrons backscattered off the impurity spinduring a large time interval t ,∆ N ( t ) = Σ z ( t ) − Σ z , Σ z ( t ) = e iHt Σ z e − iHt . (4)Thus cumulant generated function for ∆ N ( t ) can be writ-ten as G ( λ, t ) = ln Tr (cid:104) e iλ Σ z ( t ) e − iλ Σ z (0) ρ (0) (cid:105) . (5)Here ρ (0) denotes the initial density matrix of the to-tal system evolving in accordance with the Hamiltonian(1). It is convenient to express the cumulant generatingfunction as (see Ref. [69] for a review), G ( λ, t ) = ln Tr ρ ( λ ) ( t ) . (6)Here the generalized density matrix of the system in thepresence of counting field λ is given as ρ ( λ ) ( t ) = e − iH ( λ ) t ρ (0) e iH ( − λ ) t ,H ( λ ) = e iλ Σ z / He − iλ Σ z / . (7)Tracing out the degrees of freedom of helical edge stateswe can reduce the problem of computation of the cumu-lant generating function to the evaluation of the reducedgeneralized density matrix of the magnetic impurity, G ( λ, t ) = ln Tr S ρ ( λ ) S ( t ) . (8)Using the smallness of the exchange interaction, one canderive the generalized Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation, which governs the time evo-lution of ρ ( λ ) S ( t ) [58], dρ ( λ ) S dt = i (cid:104) ρ ( λ ) S , H mf e − i (cid:105) + η ( λ ) jk S j ρ ( λ ) S S k − η (0) jk (cid:110) ρ ( λ ) S , S k S j (cid:111) . (9)Here H mf e − i = J zz (cid:104) s z (cid:105) S z /ν stands for the mean-fieldpart of H e − i where (cid:104) s z (cid:105) = νV / × η ( λ ) jk = πT ( J Π ( λ ) V J T ) jk (10)controls the non-unitary evolution of the reduced gener-alized density matrix. Here we introducedΠ ( λ ) V = f + λ ( V /T ) − if − λ ( V /T ) 0 if − λ ( V /T ) f + λ ( V /T ) 00 0 1 (11)and f ± λ ( x ) = x (cid:0) e − iλ e x ± e iλ (cid:1) / ( e x − I bs = lim t →∞ ddt dG ( λ, t ) d ( iλ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ =0 . (12)The backscattering current noise at a frequency ω can bealso determined with the help of the cumulant generatingfunction [70, 71], S bs ( ω ) = ω ∞ (cid:90) dt sin( ωt ) ddt d G ( λ, t ) d ( iλ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ =0 . (13)We note that the master equation approach for compu-tation of the current noise is limited to not too high fre-quencies, | ω | (cid:28) max {| V | , T } due to the breakdown ofthe Markovian approximation at short time scales (see,e.g., Ref. [72]).The reduced generalized density matrix ρ ( λ ) S ( t ) and,consequently, the cumulant generating function, dependon the choice of the initial reduced density matrix ρ ( λ ) S (0).Since we are interested in the statistics of the backscat-tering current in the steady state of the system, we choosethe ρ ( λ ) S (0) to be equal to the steady state density matrix ρ st S , for which the right hand side (r.h.s.) of the GKSLEq. (9) equals zero at λ = 0. III. THE BACKSCATTERING CURRENTNOISE FOR AN ARBITRARY IMPURITY SPINA. General expression
We start analysis of the average backscattering currentand noise, cf. Eqs. (12) and (13), from a derivation ofgeneral expressions valid for an arbitrary value S of theimpurity spin. As usual, it is convenient to think of the(2 S + 1) × (2 S + 1) reduced density matrix ρ ( λ ) S as a(super)vector | ρ ( λ ) S (cid:105) of length (2 S + 1) . Then the GKSLequation can be rewritten as follows, ddt | ρ ( λ ) S (cid:105) = L ( λ ) | ρ ( λ ) S (cid:105) , (14) where the (2 S + 1) × (2 S + 1) matrix L ( λ ) denotes the(super)operator corresponding to the r.h.s of Eq. (9).For further analysis, it is convenient to introduce the left(super)vector (cid:104) ˜0 | whose inner product with an arbitrary(super)vector | ρ ( λ ) S (cid:105) gives the trace, Tr ρ ( λ ) S ≡ (cid:104) ˜0 | ρ ( λ ) S (cid:105) .In order to find the backscattering current noise weneed to find | ρ ( λ ) S ( t ) (cid:105) to second order in λ . This canbe done by means of perturbation theory in λ . Let usexpand the matrix L ( λ ) in powers of λ , L ( λ ) = L + λ L + λ L + . . . . (15)The (super)vector | (cid:105) corresponding to the steady statedensity matrix ρ st S is the right eigenvector for the matrix L with zero eigenvalue, L | (cid:105) = 0 . (16)We note that (cid:104) ˜0 | is the left eigenvector of L with zeroeigenvalue, (cid:104) ˜0 | L = 0 . (17)Left and right zero eigenvectors of L satisfy normaliza-tion condition (cid:104) ˜0 | (cid:105) = 1.Solving Eq. (14) perturbatively, we find to the secondorder in λ , | ρ ( λ ) S ( t ) (cid:105) (cid:39) | (cid:105) + t (cid:90) dt e − L t (cid:32) λ L + λ L + λ t (cid:90) dt L e L ( t − t ) L (cid:33) | (cid:105) . (18)Hence, using the relation G ( λ, t ) = ln (cid:104) ˜0 | ρ ( λ ) S ( t ) (cid:105) , we ob-tain the series in λ expansion of the cumulant generatingfunction, G ( λ, t ) = λ (cid:104) ˜0 | (cid:34) t (cid:0) L + λ L (cid:1) + λ t (cid:90) dτ ( t − τ ) L × (cid:16) e L τ − | (cid:105)(cid:104) ˜0 | (cid:17) L (cid:35) | (cid:105) + O ( λ ) . (19)Next, with the help of Eqs. (12) and (13), we find theaverage backscattering current I bs = − i (cid:104) ˜0 | L | (cid:105) , (20)and the backscattering current noise S bs ( ω ) =2 (cid:104) ˜0 | L G L − L | (cid:105)− ω (cid:104) ˜0 | L G (cid:0) L + ω (cid:1) − L | (cid:105) . (21)We note that the first line in Eq. (21) describes the zerofrequency noise, while the second line is the frequencydependent contribution. The matrix G is the pseudo in-verse of L . One needs to work with the pseudo inversematrix since L has zero eigenvalue. The pseudo inverse G satisfies the following relation G L = − | (cid:105)(cid:104) ˜0 | . (22)With the help of this relation, Eq. (21) can be rewrittenas S bs ( ω ) = 2 (cid:104) ˜0 | (cid:104) L L (cid:0) L + ω (cid:1) − L − L (cid:105) | (cid:105) . (23)It is worthwhile to mention that the zero frequence noiseis obtained from Eq. (23) as S ( ω → L with non-zero eigenvalues l α , α = 1 , . . . , S ( S + 1), L | α (cid:105) = l α | α (cid:105) , (cid:104) ˜ α | L = (cid:104) ˜ α | l α . (24)These eigenvectors are assumed to be mutually orthogo-nal, (cid:104) ˜0 | α (cid:105) = (cid:104) ˜ α | (cid:105) = 0 and (cid:104) ˜ α | β (cid:105) = δ αβ . An eigenvalue l α can be either pure real or belong to a complex conju-gate pair. The real parts of all eigenvalues are negative, l (cid:48) α = Re l α <
0. Using the system of the eigenvectors,Eq. (23) can be written as follows, S bs ( ω ) = 2 S ( S +1) (cid:88) α =1 l α (cid:104) ˜0 | L | α (cid:105)(cid:104) ˜ α | L | (cid:105) l α + ω − (cid:104) ˜0 | L | (cid:105) . (25)In the sum over α in the above equation the eigenvalueswith zero imaginary part, l (cid:48)(cid:48) α = Im l α = 0, contribute tothe Lorentzian resonance at ω = 0. If an eigenvalue hasnon-zero imaginary part, it contributes to the resonancesat ω = ± l (cid:48)(cid:48) α . The form of these resonances are of the Fanotype and can be described as, [ c (cid:48) α l (cid:48) α + c (cid:48)(cid:48) α ( l (cid:48)(cid:48) α ± ω )] / [ l (cid:48) α +( l (cid:48)(cid:48) α ± ω ) ]. Here we define c (cid:48) α + ic (cid:48)(cid:48) α = (cid:104) ˜0 | L | α (cid:105)(cid:104) ˜ α | L | (cid:105) .The width of the resonance is proportional to | l (cid:48) α | .As follows from the GKSL equation (9), the real partof l α is of the order of J max {| V | , T } , whereas the imag-inary part is of the order of JV . Here J is an absolutevalue of a typical value of the exchange matrix elements J jk . Therefore, at a small bias voltage, | V | (cid:28) JT , thebackscattering noise has a single Lorentzian resonanceat zero frequency with a width of the order of J T . Atlarger voltage, JT (cid:28) | V | (cid:28) T , additional side resonancesin S bs become resolved since their widths are still of theorder of J T . In the case of large voltage, | V | (cid:29) T , thereare a number of resonances at frequencies of the order of J | V | . The widths of these resonances are proportionalto J | V | . Below we shall discuss this evolution of thebackscattering noise with increasing voltage in more de-tail. B. The backscattering current noise for | V | (cid:28) JT As is well-known, the fluctuation dissipation theoremrelates the current noise at zero voltage with the linear conductance. Below we shall demonstrate that this rela-tion holds for the backscattering current and noise.We start from expansion of the matrix Π ( λ ) V , Eq. (11),at V = 0 in series to the second order in λ . Then fromEq. (23) we find for the current noise at V = 0, S bs ( ω ) = πT ( Jπ J T ) jk Tr[ S j ρ st S S k ]+ πT (cid:88) σ = ± ( Jπ J T ) jk Tr[ S j δρ σ S k ] . (26)Here we introduce two matrices π = − , π = . (27)Then δρ σ satisfies the following equation, πT ( JJ T ) jk [ S j δρ σ S k − { δρ σ , S k S j } /
2] + iωσδρ σ = πT ( Jπ J T ) jk S j ρ st S S k . (28)Using the explicit form of matrices π , , we obtain S bs ( ω ) = πT S ( S + 1)6 ( Jπ J T ) jj + iπT X j (cid:88) σ = ± Tr[ S j δρ σ ] , (29)where X j = 2 (cid:15) jkl J kx J ly . Solving Eq. (28), we findTr[ S j δρ σ ] = S ( S + 1)3 i (Γ σ ( ω )) − jk X k , (30)where we introduce the matrix(Γ σ ) jk ( ω ) = ( J J T ) jk + 2 iσωπT δ jk − δ jk ( J J T ) ll . (31)Now we perform the singular value decomposition ofthe exchange matrix J = R < Λ R > . Here R <,> are theSO(3) matrices and Λ = diag { λ , λ , λ } . Combining theresults above, we finally obtain the following expressionfor the backscattering current at V = 0 S bs ( ω ) = πT S ( S + 1)3 (cid:0) R − > Φ( ω ) R > (cid:1) zz , (32)where the matrix Φ( ω ) is given asΦ( ω ) =diag (cid:40)(cid:0) λ + λ (cid:1) (cid:0) λ − λ (cid:1) + 4 ω / ( πT ) ( λ + λ ) + 4 ω / ( πT ) , (cid:0) λ + λ (cid:1) (cid:0) λ − λ (cid:1) + 4 ω / ( πT ) ( λ + λ ) + 4 ω / ( πT ) , (cid:0) λ + λ (cid:1) (cid:0) λ − λ (cid:1) + 4 ω / ( πT ) ( λ + λ ) + 4 ω / ( πT ) (cid:41) . (33)It is instructive to compare the result (32) with theresult for the average backscattering current at V → I bs = ∆ GV, ∆ G = − πS ( S + 1)6 (cid:0) R − > Φ(0) R > (cid:1) zz . (34) - ���� - ����� � ����� ���� � ωπ �������� � �� � � � �� FIG. 2. The dependence of the normalized absolute value ofbackscattering admittance, V S bs / (2 T | I bs | ) at V (cid:28) JT , onthe dimensionless frequency 2 ω/ ( πT ). The red solid curve isplotted for J xx = J zx = J yy / J zz . The green dashedcurve corresponds to J xx = J zx = J yy / J yx = 5 J zy = J zz . The blue dotted curve is plotted for J xx / J zx = J yy = J zz . For all curves J zz is equal to 0 . Then we find the Nyquist-type relation for the backscat-tering current noise at zero frequency and voltage, S bs ( ω = 0) = 2 T | ∆ G | . (35)We mention that the Nyquist-type relation (35) impliesthe absence of correlations between the total current I and the backscattering current I bs in the equilibrium.Indeed, the incoming current I in = I − I bs from reser-voirs splits into the transmitted I and backscattered I bs currents. Under equilibrium conditions, the incomingcurrent carries thermal noise, S in = 2 T G , correspond-ing to the ideal transport channel. The noise of the to-tal current should obey the fluctuation-dissipation theo-rem, i.e., is given by S I = 2 T ( G + ∆ G ). Hence, usingEq. (35), we find that the cross correlation of the totalcurrent I and the backscattering current I bs vanishes, S cross ∝ (cid:104)(cid:104) I bs I (cid:105)(cid:105) = 0.Extending the arguments above to the case of the finitefrequency but still zero bias voltage, we obtain that theright hand side of Eq. (32) determines the absolute valueof the backscattering admittance at the helical edge. Thedependence of the backscattering admittance on the fre-quency is depicted in Fig. 2. Irs absolute value growswith increasing the frequency. At | ω | ∼ J T the admit-tance crosses over into a constant in agreement with Eq.(32). C. The backscattering current noise for | V | (cid:29) T Now we consider the limit of large voltage | V | (cid:29) T .In this regime the backscatteing current noise determinesthe full current noise S I [54, 56]). At large voltage | V | (cid:29) T the matrix (11) simplifies toΠ ( λ ) V −→ V T e − iλ sgn V − i i . (36)Therefore, both the backscattering current noise andthe backscattering current are proportional to the volt-age and are independent of temperature. The ratio S bs ( ω ) / | I bs | is a function of the dimensionless parame-ter ω/V . At zero frequency the ratio coincides with thebackscattering Fano factor F bs = S bs (0) / | I bs | . This Fanofactor is bounded from below by unity, F bs (cid:62) S = 1 / F bs (cid:54) S > / F bs is unboundedfrom above due to bunching of electrons backscatteredoff a magnetic impurity [58].The behavior of the backscattering current noise as afunction of frequency at voltage bias | V | (cid:29) T is shown inFig. 3 for several magnitudes of the spin of the magneticimpurity. For S = 1 / S bs ( ω ) has a maximum at ω = 0and minima at ω = ± J zz V /
2. At larger frequencies thebackscattering current noise tends to a constant. In thecase S = 1 the resonances at the frequencies ± J zz V / ± J zz V . The frequency dependence of the backscat-tering current noise for S = 3 / S = 2 resembles theone for S = 1. Additional resonances which are possiblein the case S = 3 / S = 2 are not visible since theiramplitude is of higher order in the small anisotropic partof the exchange interaction matrix J .At larger frequencies, J | V | (cid:28) | ω | (cid:28) | V | , S bs is fullydetermined by the backscattering current. Indeed, theform of the matrix Π ( λ ) V at | V | (cid:29) T , cf. Eq. (36), impliesthe the following relation, L = − i sgn V L /
2. Hence, inthe limit of large frequencies, J | V | (cid:28) | ω | (cid:28) | V | , we findthat the backscattering current noise coincides with thebackscattering current, S bs ( ω ) / | I bs | →
1. This impliesthat backscattering is completely uncorrelated. We notethat a similar result has been derived in Ref. [56] forthe case of S = 1 /
2. It is worthwhile to mention thatdepending on the parameters of the exchange matrix J jk ,the ratio S bs ( ω ) / | I bs | can be even smaller than unity atintermediate frequencies, as illustrated in Fig. 3. D. The Korringa relaxation rates
The imaginary parts of eigenvalues l α determine theposition of the Fano-type resonances. These resonancesare due to the transitions between levels of the mean fieldHamiltonian, H mf e − i = J zz V S z /
2. At voltage | V | (cid:29) T theresonances are well separated since their width is of theorder J V . The width of the resonance can be estimatedmore accurately as follows. Let us introduce the eigenba-sis of S z , S z | m (cid:105) = m | m (cid:105) with m = − S, . . . , S . Neglectingthe terms of the second order in J in GKSL equation, wefind that the set of eigenvalues { l α } can be approximated (a) � = � / � � �� = ���� � �� = � �� = � �� = ���� � �� = � �� = ������ / � = ��� / � = ��� / � = ��� / � = �� - ��� - ��� � ��� ��� � ωπ ������������������������������������� � �� ( ω ) π � (b) � = � � �� = ���� � �� = � �� = � �� = ���� � �� = � �� = ������ / � = ��� / � = ��� / � = ��� / � = �� - ��� - ��� � ��� ��� � ωπ ���������������� � �� ( ω ) π � (c) � = � / � � �� = ���� � �� = � �� = � �� = ���� � �� = � �� = ������ / � = ��� / � = ��� / � = ��� / � = �� - ��� - ��� � ��� ��� � ωπ �������������� � �� ( ω ) π � (c) � = � � �� = ���� � �� = � �� = � �� = ���� � �� = � �� = ������ / � = ��� / � = ��� / � = ��� / � = �� - ��� - ��� � ��� ��� � ωπ ���������������������� � �� ( ω ) π � FIG. 3. The dependence of the backscattering current noise, 4 S bs / ( πT ), on the dimensionless frequency 2 ω/ ( πT ) at differentvalues of V /T (cid:29) S . The black lines are guides for an eye, marking the positions of the resonances. as − i J zz V ( m − m (cid:48) ) /
2, where m, m (cid:48) = − S, . . . , S . Theomitted terms can then be taken in account by the firstorder perturbation theory. Denoting − Re l α as 1 /τ m,m (cid:48) ,we obtain τ − m,m (cid:48) = πV g (cid:26) q (cid:20) S ( S + 1) − m + m (cid:48) (cid:21) + (1 − q ) ( m − m (cid:48) ) − qp ( m + m (cid:48) ) (cid:27) , (37)where g = ( J T J ) xx + ( J T J ) yy = J xx + J yy + J yx + J zx + J zy , and q = J xx + J yy + J yx J xx + J yy + J yx + J zx + J zy ,p = 2 |J xx J yy |J xx + J yy + J yx . (38)The width of the resonance at ω = 0 is determined by theset of Korringa rates 1 /τ m,m . It is worthwhile to mentionthe relation, τ m,m ∝ /q , that gives rise to bunching ofthe backscattering electrons and to unlimited backscat-tering Fano factor in the limit q → q , all resonances have widths which are of thesame order. IV. THE BACKSCATTERING CURRENTNOISE FOR S = 1 / In the case of spin S = 1 / | ρ ( λ ) S (cid:105) = 12 (cid:110) Tr ρ ( λ ) S , Tr[ ρ ( λ ) S S ] (cid:111) . (39)In this representation the matrices L , , can be writtenexplicitly. In particular, we find L = πT (cid:18) V X /T − Γ , (cid:19) (40)where 3 × jk = 1 πT (cid:18) δ jk η ll − η jk + η kj V J iz ε ijk (cid:19) . (41)We note that at V = 0 the matrix Γ coincides with thematrix Γ σ ( ω = 0), cf. Eq. (31). The right and left zeroeigenvectors of L can be written explicitly, | (cid:105) = (cid:18) V Γ − X /T (cid:19) , (cid:104) ˜0 | = { , , , } . (42)Next, the matrix L is given by L = − iπV (cid:18) g − coth (cid:0) V T (cid:1) X T (cid:0) V T (cid:1) X Q (cid:19) . (43)Here we introduce the symmetric matrix Q jk = J jx J kx + J jy J ky − gδ jk / . (44)The matrix L can be cast in the following form, L = − πV (cid:18) g coth (cid:0) V T (cid:1) − X T X (cid:0) V T (cid:1) Q (cid:19) . (45)Finally, using Eq. (23), we obtain the expression for thebackscattering current noise for spin S = 1 / S bs ( ω ) = πV T coth (cid:18) V T (cid:19) X T (cid:32) + (cid:18) ωπT (cid:19) Γ − (cid:33) − × (cid:34) V T (cid:18) g − V T coth (cid:18) V T (cid:19) X T Γ − X (cid:19) Γ − − Γ − (cid:18) coth (cid:18) V T (cid:19) + VT Q Γ − (cid:19)(cid:35) X + πV (cid:18) g coth (cid:18) V T (cid:19) − V T X T Γ − X (cid:19) . (46)We mention that this result for the backscattering cur-rent noise generalizes the result found in Ref. [56] tothe case of arbitrary ratio between J zz and the othercomponents of the exchange matrix (see the discussionin Refs.[73, 74]).For a sake of completeness, we present here also theresult for the average backscattering current for spin S =1 / I bs = πV (cid:18) V T coth (cid:18) V T (cid:19) X T Γ − X − g (cid:19) . (47)As one can easily check from Eqs. (46) and (47), thebackscattering current noise S bs ( ω ) = | I bs | for J | V | (cid:28) ω (cid:28) | V | and S bs ( ω = 0) = 2 T | I bs /V | for V (cid:28) JT .As follows from Eq. (46), the positions and widths ofthe resonances in S bs ( ω ) are determined by the eigenval-ues Ω − , , of the matrix πT Γ /
2. In the case | V | (cid:29) T ,they can be found asΩ = iτ / , / + iτ − / , − / ≡ iτ K , Ω ± = ± J zz V + iτ / , − / . (48)To illustrate the origin of the resonances we con-sider the lower triangular exchange matrix with |J xx −J yy | , |J zx | (cid:28) |J xx | , |J yy | , |J zz | , and J yx = J zy = 0. � / ��� / ��� � � � � � � � � ������������ � �� ( ω � � ) � �� ( ω � � ) FIG. 4. The dependence of the backscattering cur-rent noise normalized by its value at zero bias voltage, S bs ( ω, V ) / S bs ( ω, V / (2 T ),for S = 1 / , , / , and 2. The exchange interaction isthe same as in Fig. 3, i.e., J xx = J zx = J zz = 0 . J zy = J yx = 0 . J yy = 0 .
02. The frequency is equalto 2 ω/ ( πT ) = 0 . Simplifying the general expression (46), we find S bs ( ω ) = 2 πV (cid:40)
14 ( δ J ) + ( δ J ) ωτ K ) + J zx δ J|J xx | + |J yy | (cid:88) s = ±
11 + 4( ω − sJ zz V / τ K (cid:41) , (49)where δ J = |J xx | − |J yy | . As one can see from Eq. (49),the appearance of side resonances at ω = ±J zz V / J xx and J yy should be different and, in addition, J zx should benonzero. Depending on the sign of δ J the current noiseat ω = ±J zz V /
S > / S bs with V becomes even more involved asshown in Fig. 4. V. DISCUSSION AND CONCLUSIONA. The effect of the electron-electron repulsion
In order to take into account the electron-electron re-pulsion we must use the Luttinger liquid formalism [75].The sole effect of the electron-electron interaction at thehelical edge is the modification of the expression for thespin-spin correlation function [76, 77]. Therefore, thekernel Π ( λ ) V , cf. Eq. (11), will be transformed toΠ ( λ ) V = F V,T f + λ ( V /T ) − if − λ ( V /T ) 0 if − λ ( V /T ) f + λ ( V /T ) 00 0 K − F − V,T . (50)Here the Luttinger liquid parameter K is assumed to bewithin the range 1 / < K (cid:54)
1, corresponding to moder-ate repulsion ( K = 1 corresponds to the noninteractingcase). The function F V,T is defined as follows F V,T = (cid:18) πT au (cid:19) K − TV sinh (cid:18) V T (cid:19) × B (cid:18) K − i V πT , K + i V πT (cid:19) . (51)Here B ( x, y ) stands for the Euler beta function, u is therenormalized velocity of helical edge states, and a denotesthe length scale which corresponds to the ultraviolet cut-off.In the case of large bias voltage, | V | (cid:29) T , the matrixΠ ( λ ) V simplifies to (cf. Eq. (36)),Π ( λ ) V = V K ) T (cid:18) aVu (cid:19) K − e − iλ sgn V − i i . (52)In the case of lower triangular exchange matrix J jk , thisform of Π ( λ ) V implies the following replacements in thefinal result (23) for the backscattering current noise at | V | (cid:29) T , J jk → J jk (cid:112) Γ(2 K ) (cid:18) aVu (cid:19) K − , j = x, y, z, k = x, y, J zz → J zz . (53)With this simple prescription for the inclusion ofelectron-electron repulsion in hands, we are able to makethe following predictions. At | V | (cid:29) T the positionsof the resonances in S bs ( ω ) are essentially insensitiveto the presence of interaction. However, the repulsiveelectron-electron interaction broadens the resonances. Inthe regime of large frequencies, J | V | (cid:28) | ω | (cid:28) | V | , therelation S bs ( ω ) / | I bs | → | V | (cid:28) JT , one can use the re-sults for the backscattering current noise, cf. Eq. (32),and for the average backscattering current, cf. Eq. (34),provided the following substitutions are performed: J jk → J jk Γ( K ) (cid:112) Γ(2 K ) (cid:18) πT au (cid:19) K − , j = x, y, z, k = x, y, J zz → J zz / √ K. (54)In particular, this implies that the Nyquist-type rela-tion (35) still holds in the presence of repulsive electron-electron interaction. B. Dilute magnetic impurities
Experimentally, the helical edge can be contaminatedby many magnetic impurities. Let us assume that theyare situated in a δ -layer in a HgTe/CdTe quantum wellheterostructure. Then in the limit of dilute magneticimpurities the average backscattering current and noiseare given as sum of independent contributions from eachmagnetic impurity. Since the exchange interaction J jk depends exponentially on the distance from the edge [45],significant backscattering is caused by magnetic impuri-ties situated close to the helical edge only. As the conse-quence, the approximation that all magnetic impuritiesrelevant for backscattering have the same matrix J jk ,is well justified. Thus the average backscattering cur-rent and noise are proportional to the number of mag-netic impurities situated at the helical edge. Dispersionin parameters of the exchange interaction for magneticimpurities at the edge is small and, thus, will only giverise to weak broadening of the resonances. In the caseof magnetic impurities scattered randomly in the direc-tion of quantum well growth, the exchange interaction J jk for each magnetic impurity situated at the helicaledge is also random. Therefore, the average backscatter-ing current and noise for this case can be obtained fromaveraging of the results for a single impurity over J jk with a proper distribution. Inevitably, this would lead toa significant broadening of the resonances in the currentnoise.Due to the indirect exchange interaction, the spinsof magnetic impurities can become correlated, which,in turn, can affect the backscattering events. In thepresence of electron-electron interaction the indirect ex-change interaction depends on a distance R between im-purities as a power law, ( a/R ) K − [78]. Since we as-sumed that K > /
2, the indirect exchange interac-tion decays with increase of R . On the other hand, for K < / R (cid:29) a [36]. Therefore, in the considered case, K > /
2, the condition n imp a (cid:28)
1, determines the dilutelimit for magnetic impurities. Here n imp stands for theone-dimensional concentration of magnetic impurities atthe helical edge. C. Relation to experiments
As shown in Fig. 4, we predict a non-monotonic de-pendence of the backscattering current noise on voltageat fixed frequency. We note that similar non-monotonicdependences of the current noise (but at ω = 0) havebeen observed for a 2D hole system in the regime ofhopping conductivity [79], and for tunneling via inter-acting pairs of localized states in a 2D electron system[80, 81]. In the latter case the non-monotonicity in thecurrent noise has been attributed to correlations betweentunneling events via different localized states. We stressthat in our case the non-monotonicity of S bs is due to amechanism similar to electronic paramagnetic resonance.The bias voltage plays the role of an effective magneticfield that lifts the spin degeneracy for the magnetic im-purity. Transport at finite frequency allows to inducetransitions between split energy levels of the magneticimpurity. Then backscattering off the magnetic impurityleads to the Fano-type resonances in the current noise atnon-zero frequencies which are similar to an asymmetricelectronic paramagnetic resonance [82, 83]. Also S bs ( ω )can be contrasted with the resonances of the Lorentzianform measured in the frequency dependent shot noise intransport via single electron transition [84]. It is worth-while to mention that recently the Fano-type resonancesin the frequency current noise have been predicted in adouble dot Aharonov-Bohm interferometer [85]. Theirorigin was an interplay between Coulomb blockade andRabi interference in the presence of non-zero Aharonov-Bohm flux.Recently, the zero-frequency current noise measuredby scanning tunneling microscope situated at the top ofa magnetic adatom has been demonstrated to be a tool toresolve its energy structure [86–88]. Our results suggestthat the frequency resolved backscattering current noisecan serve as a sensitive probe to measure various phys-ical characteristics of a dynamical impurity spin. Mea-surement of the resonant frequencies in the dependenceof S bs on ω allows one to estimate a value of the dimen-sional exchange interaction J zz . Measurement of widthsof the resonances enables to estimate a magnitude of theother components of the exchange matrix J jk . Observa-tion of more that one non-zero resonant frequency signalsthat the impurity spin is larger than 1 /
2. However, wemention that we are not aware of measurements of thefrequency dependence of the current noise at the helicaledge in 2D topological insulators to date.Let us estimate the resonant frequency in the case of2D topological insulator in a HgTe/CdTe quantum wellwith Mn ions. For a typical bias current, I ≈ V ∼ hI/e ≈ . · − V. We note that this voltagecorresponds to the temperature of the order of 0 . f ∼ eJV /h ≈ J ≈ − . The widths of theresonances can be estimated, roughly, as ∆ f ∼ Jf ≈ ω (cid:28) v/L . Here L denotes the length of the helicaledge. Estimating velocity of the helical edge as v ≈ m/s [2], we find v/L ≈
100 GHz for L = 1 µ m. In the consideration above we neglected the localanisotropy of the impurity spin Hamiltonian, D jk S j S k ,that can be present in a real system. We remind thatthis can be justified for |D jk | (cid:28) max { J T, | JV |} . Thelocal anisotropy results in splitting of energy levels for theimpurity spin. In turn, this affects the average backscat-tering current [46]. We expect that the modulation of theaverage backscattering current due to local anisotropycan lead to non-monotonic dependence of the shot noiseon a bias voltage already at zero frequency. D. Conclusion
To summarize, we have studied the helical edge cou-pled to the dilute dynamical magnetic impurities. Weconsidered the case of an arbitrary spin S and a generalform of the exchange interaction allowed by the symme-tries. Under the assumption of weak exchange interac-tion, we derived analytic expressions for the backscatter-ing current noise at finite frequency and studied its de-pendence on the temperature and applied bias voltage.Our main finding is that in addition to the Lorentzianresonance at zero frequency the backscattering currentnoise has additional Fano-type resonances. Such resonantstructure of the current noise as a function of frequencytransforms into a non-monotonic behaviour of S bs as afunction of the bias voltage. We proposed the backscat-tering current noise measured at finite frequency as asensitive tool to access a fine structure of the impurityspin Hamiltonian. ACKNOWLEDGMENTS
The authors are grateful to V. Khrapai for very use-ful comments. The authors are indebted to Y. Gefen,P. Kurilovich and V. Kurilovich for collaboration at theinitial stage of this project. The research was partiallysupported by the Russian Ministry of Science and HigherEducation, the Alexander von Humboldt Foundation,the Israel Ministry of Science and Technology (ContractNo. 3-12419), the Israel Science Foundation (Grant No.227-15) and the US-Israel Binational Science Foundation(Grant No. 2016224). Hospitality by Tel Aviv Univer-sity, the Weizmann Institute of Science, Landau Institutefor Theoretical Physics, and National Research Univer-sity Higher School of Economics is gratefully acknowl-edged. M.G. acknowledges a travel grant by the BASISFoundation (Russia). [1] M. Z. Hasan and C. L. Kane, “Colloquium: Topologicalinsulators,” Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, “Topological insulators andsuperconductors,” Rev. Mod. Phys. , 1057 (2011). [3] C. L. 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