Unrestricted electron bunching at the helical edge
Pavel D. Kurilovich, Vladislav D. Kurilovich, Igor S. Burmistrov, Yuval Gefen, Moshe Goldstein
UUnrestricted electron bunching at the helical edge
Pavel D. Kurilovich, Vladislav D. Kurilovich, Igor S. Burmistrov, Yuval Gefen, and Moshe Goldstein Department of Physics, Yale University, New Haven, CT 06520, USA L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
A quantum magnetic impurity of spin S at the edge of a two-dimensional time reversal invarianttopological insulator may give rise to backscattering. We study here the shot noise associated withthe backscattering current for arbitrary S . Our full analytical solution reveals that for S > theFano factor may be arbitrarily large, reflecting bunching of large batches of electrons. By contrast,we rigorously prove that for S = the Fano factor is bounded between 1 and 2, generalizing earlierstudies. Introduction. — Zero-frequency current noise in a con-ductor can reveal information about correlations in elec-tronic transport which cannot be extracted from the av-erage current [1, 2]. Obtaining information about thecorrelations requires going beyond linear response (wherethermal noise is fully determined by linear conductancethrough the fluctuation-dissipation theorem), and study-ing shot noise at voltage larger than the temperature.The ratio between the shot noise and the average currenttimes the electron charge is referred to as
Fano factor . Itis useful for characterizing the unit of effective elementarycharge in correlated electron systems, e.g., quasiparticlecharges in fractional quantum Hall edges [3, 4]. Entan-glement with an external degree of freedom may modifythe effective Fano factor [5, 6].The experimental discovery of 2D topological insula-tors [7] triggered intensive experimental and theoreticalresearch [8, 9]. Electron transport along the helical edgewas theoretically predicted to be protected from elasticbackscattering by time-reversal symmetry. However, thisideal picture was impugned by transport experiments inHgTe/CdTe [7, 10–14] and InAs/GaSb [15–22] quan-tum wells, Bi bilayers [23], and WTe monolayers [24–26]. In order to explain this data, several physical mech-anisms of backscattering were proposed and studied the-oretically [27–48].In contrast to the average current, shot noise at the he-lical edge has attracted much less experimental and the-oretical attention so far [40, 49–53]. The shot noise dueto backscattering of helical edge electrons via anisotropicexchange (which has to break the conservation of the to-tal z -projection of the angular momentum to affect thedc current [29]) with a local spin S = magnetic mo-ment has been calculated in Ref. [52]. The authors ofRef. [52] studied the so-called backscattering Fano factor , F bs , which is the ratio between the zero-frequency noiseof the backscattering current, S bs , and the absolute valueof the average backscattering current, | I bs | , in the limitof large voltage bias V . It was found that F bs is boundedbetween 1 and 2, with the extreme values correspond-ing to independent backscattering of single electrons andbunched backscattering of pairs of electrons, respectively. The considerations of Ref. [52] were limited to the caseof almost isotropic exchange interaction. This assump-tion is natural for the model of charge puddles which actas effective spin- magnetic moments [32, 35]. However,the spin of a magnetic impurity (MI) can be larger than , e.g., S = for a Mn ion in a HgTe/CdTe quan-tum well. Moreover, in the case of a “genuine” MI theexchange interaction is strongly anisotropic [41, 46].In this Letter we study the backscattering shot noiseat the edge of a 2D topological insulator mediated by thepresence of a single quantum MI. We assume that the im-purity is of an arbitrary spin S and the exchange inter-action matrix is of a general form. We find the backscat-tering Fano factor analytically, cf. Eq. (9). Strikingly, forany S > it is not bounded from above, cf. Eqs. (11) and(12); F bs > S > can bunch helical electrons together. Here, in asignificant parameter range, for each value of the impu-rity spin projection S z electrons are backscattered with arate ∝ S z , while S z itself changes slowly. This results ina modulation of the backscattering events into long cor-related pulses (Fig. 1a). For S = this effect is absent(Fig. 1b), and we find a concise exact expression for F bs proving rigorously that 1 (cid:54) F bs (cid:54)
2, cf. Eq. (10). Ourresults elucidate an important facet of the dichotomy be-tween topological properties and electronic correlationsin one-dimensional edges [54], accounting for mechanismsthat break topological protection against backscattering.
Model. — Helical edge electrons coupled to a MI are de-scribed by the following Hamiltonian (we use units with (cid:126) = k B = − e = 1): H = H e + H e − i , H e = iv (cid:90) dy Ψ † ( y ) σ z ∂ y Ψ( y ) , (1)where Ψ † (Ψ) is the creation (annihilation) operator ofthe edge electrons with velocity v . The Pauli matrices σ x,y,z act in the spin basis of the edge states. The ex-change electron-impurity interaction is assumed to be lo-cal: H e − i = 1 ν J ij S i s j ( y ) , s j ( y ) = 12 Ψ † ( y ) σ j Ψ( y ) . (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Here ν = 1 / (2 πv ) is the density of states per one edgemode, operator S i denotes the i -th component of theimpurity spin, and the couplings J ij are dimensionlessand real. We stress that in general the exchange interac-tion (2) is strongly anisotropic and violates the conser-vation of the total z -projection of the angular momen-tum of the system [41, 46, 56]. The latter violation isrequired to generate persistent backscattering of helicalelectrons. We assume that the coupling constants aresmall, |J ij | (cid:28)
1, and we neglect the local anisotropy H anis = D kp S k S p of the MI spin which is justified at |D kp | (cid:28) max {J ij T, |J ij | V } [47]. In the absence of thelocal anisotropy we can rotate the spin basis for S i bring-ing the exchange matrix J ij to a lower triangular form.We thus assume hereinafter that J xy = J xz = J yz = 0.In addition we ensure that J xx J yy > Cumulant generating function. — The average backscat-tering current and its zero frequency noise can be ex-tracted from the statistics of the number of electronsbackscattered off a MI during a large time interval t :∆ N ( t ) = Σ z ( t ) − Σ z , where Σ z ( t ) = e iHt Σ z e − iHt andΣ z = (cid:82) dys z ( y ). The cumulant generating function for∆ N can be written as G ( λ, t ) = ln Tr[ e iλ Σ z ( t ) e − iλ Σ z ρ (0)],where ρ (0) stands for the initial density matrix of thefull system [57]. It is convenient to write G ( λ, t ) =ln Tr ρ ( λ ) ( t ) where ρ ( λ ) ( t ) = e − iH ( λ ) t ρ (0) e iH ( − λ ) t is thegeneralized density matrix of the system at time t and H ( λ ) = e iλ Σ z / He − iλ Σ z / . Tracing out the degrees offreedom of the helical electrons, we obtain G ( λ, t ) =ln Tr S ρ ( λ ) S ( t ), where ρ ( λ ) S ( t ) denotes the reduced gener-alized density matrix of the impurity. Generalized master equation. — In order to find G ( λ, t )we derive a generalized Gorini-Kossakowski-Sudarshan-Lindblad equation, which governs the time evolution of ρ ( λ ) S ( t ) (see Supplemental Material [55]): dρ ( λ ) S dt = − i [ H mfe − i , ρ ( λ ) S ]+ η ( λ ) jk S j ρ ( λ ) S S k − η (0) jk { ρ ( λ ) S , S k S j } . (3)Here H mfe − i = J zz (cid:104) s z (cid:105) S z /ν is the mean-field part of H e − i with the average non-equilibrium spin density (cid:104) s z (cid:105) = νV /
2. Additionally, we have introduced η ( λ ) jk = πT ( J Π ( λ ) V J T ) jk , whereΠ ( λ ) V = f + λ ( V /T ) − if − λ ( V /T ) 0 if − λ ( V /T ) f + λ ( V /T ) 00 0 1 (4)and f ± λ ( x ) = x (cid:0) e − iλ e x ± e iλ (cid:1) / ( e x − V (cid:29) T . The first termon the r.h.s. of Eq. (3) is then much larger than the othertwo terms. Consequently, one may implement the rotat-ing wave approximation to simplify Eq. (3). Within itsframework ρ ( λ ) S is diagonal in the eigenbasis of H mfe − i , i.e., of S z . Denoting the impurity state with S z = m as | m (cid:105) ( m = S, ..., − S ) we obtain a classical master equation forthe occupation numbers ddt (cid:104) m | ρ ( λ ) S | m (cid:105) = S (cid:88) m (cid:48) = − S L ( λ ) mm (cid:48) (cid:104) m (cid:48) | ρ ( λ ) S | m (cid:48) (cid:105) . (5)Here L ( λ ) is a (2 S + 1) × (2 S + 1) tridiagonal ma-trix. The tridiagonal form indicates that S z changesby not more than unity in each elementary scatter-ing process. Nonzero elements of L ( λ ) are given by L ( λ ) m +1 ,m = e − iλ η + [ S ( S + 1) − m ( m + 1)] / L ( λ ) m,m +1 =( η − /η + ) L ( λ ) m +1 ,m , and L ( λ ) mm = − e iλ L ( λ ) m +1 ,m − e iλ L ( λ ) m − ,m +( e − iλ − η (0) zz m , where η ± = η (0) xx + η (0) yy ± i ( η (0) xy − η (0) yx ). Itis worthwhile to note that by Eq. (5), the characteristicfunction of (cid:104) m | ρ ( λ ) S | m (cid:105) obeys the Heun equation [58]. Results. — At λ = 0 Eq. (5) describes the timeevolution of populations of the impurity energy levels.Through this equation we establish that the steady statedensity matrix of the impurity at V (cid:29) T is given by ρ (0) S, st ∼ (cid:18) p − p (cid:19) S z , p = 2 J xx J yy J xx + J yy + J yx , (6)The dimensionless parameter p determines the polariza-tion of the impurity, i.e., for p = 1 only the state S z = S is occupied, whereas for p = 0 all levels are equally popu-lated. Physically, at p = 1, J xx = J yy , J yx = 0, and theimpurity spin can be flipped down only by backscatteringan edge electron carrying spin down. We note, though,that at large voltage the current is carried mainly byspin-up electrons. Thus, a steady state of the impurity isestablished in which S z = S with essentially unit proba-bility. At p < J zx and J zy do not enter intothe expression (6) for p because the corresponding termsin the Hamiltonian do not induce impurity spin flips.To express I bs and S bs in a compact form we introducetwo parameters: g = J xx + J yy + J yx + J zx + J zy , q = 1 − J zx + J zy g . (7)Then we find that η ± = πgV (1 ± p ) q/ η (0) zz = πgV (1 − q ) /
2. Notice that 0 < p, q (cid:54) g (cid:28) I bs = (cid:104) ∆ N (cid:105) /t = − ( i/t ) ∂G ( λ, t ) /∂λ , where the limits t → ∞ and λ → λ , we find I bs = − πgV (cid:104) R ( S z ) (cid:105) , (8)where R ( S z ) = qS ( S + 1) − qpS z + (2 − q ) S z and (cid:104) ... (cid:105) =Tr (cid:0) ... ρ (0) S, st (cid:1) . We note that (cid:104) R ( S z ) (cid:105) >
0, hence I bs isnegative. (a) t , time | d Δ N / d t | (b) t , time | d Δ N / d t | (c) t , time | d Δ N / d t | (d) t , time | d Δ N / d t | FIG. 1. (Color online) Sketches of the backscattering currentas a function of time in different regimes: (a) q (cid:28) S = 1;(b) q (cid:28) S = 1 /
2; (c) p = 1; (d) 1 − p (cid:28)
1. Red andblue peaks correspond to backscattering processes with andwithout the impurity flips, respectively. Transitions betweenimpurity levels are depicted above each spin-flip process.
The backscattering current noise at zero frequencyis given by the second cumulant of ∆ N as S bs = (cid:104)(cid:104) (∆ N ) (cid:105)(cid:105) /t = − t − ∂ G ( λ, t ) /∂λ at t → ∞ and λ → S bs from Eq. (5) we employ secondorder perturbation theory in λ [55]. The noise can bewritten as S bs = F bs | I bs | , where the backscattering Fanofactor reads F bs = 1 + 4 q (1 − p ) S (cid:88) n =1 (cid:104) P n [ R ( S z ) − (cid:104) R ( S z ) (cid:105) ] (cid:105) n (2 S + 1 − n ) (cid:104) R ( S z ) (cid:105) µ n . (9)Here P n = (cid:80) Sm = S − n +1 | m (cid:105)(cid:104) m | is a projector on the sub-space of n impurity states with largest S z projection and µ n = (cid:104) S + 1 − n | ρ (0) S, st | S + 1 − n (cid:105) . Notice that Eq. (9)implies F bs (cid:62)
1. So far, we considered the model of non-interacting edge states. Accounting for electron-electroninteraction results only in the common factor for I bs and S bs that leaves F bs intact [55].The most striking feature of Eq. (9) is the divergenceat q → (cid:54) p < unrestricted from above. Theonly exception is the case of S = 1 /
2, for which Eq. (9)gives F bs ( S = 1 /
2) = (1 − qp ) / (1 − qp ) . (10)This expression indicates that F bs is restricted to therange between 1 and 2 for S = 1 /
2. Eq. (10) extends the p, q → p and q .The divergence of F bs at q → S > / S = 1. The in-equality q (cid:28) |J zx | , |J zy | (cid:29) |J xx | , |J yy | , |J yx | and, therefore, the backscattering predominantly hap-pens without spin flips of the impurity. By Fermi’s goldenrule, the rate of such reflection processes is proportionalto S z , rendering the backscattering current, d ∆ N/dt , very sensitive to the spin state of the impurity. Theprocesses associated with J xx , J yy , and J yx in H e − i are incapable of producing a significant contribution to d ∆ N/dt on their own, but they can transfer the impu-rity from one spin state to another, switching efficientbackscattering on ( S z = ±
1) and off ( S z = 0). Conse-quently, the backscattering current as a function of timelooks like a sequence of long pulses, each consisting ofa large number (proportional to 1 /q ) of backscatteredelectrons (see Fig. 1a). This peculiar bunching of helicalelectrons results in 1 /q divergence of F bs . For S > d ∆ N/dt : impu-rity rarely jumps between states of different S z changingthe intensity of backscattering ∝ S z . As q →
0, impuritybackscatters increasingly large number of electrons dur-ing its stay in a state with a given S z , which results inthe divergent backscattering Fano factor. We note thatfor S = 1 / S z = 1 /
4. Because ofthat d ∆ N/dt has no pulses (see Fig. 1b) and F bs is notsingular at q → F bs can be expressedas 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 anunpolarized impurity, p = 0, we find F bs ( p = 0) = 1 + (2 S − S + 3)(2 − q ) / (45 q ) . (11)In this regime the Fano factor scales as S at large S .This fact has a simple physical interpretation. Since for p = 0 each of the 2 S + 1 spin states of the impurityis occupied with the same probability, the dynamics ofimpurity flips between states with different S z is diffu-sive. If the MI starts its motion in a state S z , on average ∼ S transitions occur before S z returns to its initialvalue. Therefore, approximately S subsequent spin flipsof the impurity are correlated. These correlations in thedynamics of the impurity spin are mirrored by the cor-relations in the electron backscattering and result in the S scaling of F bs at small p .For an almost fully polarized MI, 1 − p (cid:28) − q , Eq. (9)yields F bs ( p →
1) = 1 + (1 − p ) (cid:0) (2 − q ) S + q − (cid:1) q (1 − q ) S . (12)As follows from Eq. (12) the Fano factor at p = 1 isequal to unity for q <
1. This result is expected sincefor p = 1 the spin of a MI is locked to the state S z = S .Therefore, the only allowed backscattering processes oc-cur due to the J zx and J zy terms in H e − i , which do notrequire spin-flips of the impurity to scatter helical elec-trons. Consequently, the impurity does not keep memoryabout backscattered electrons, which results in a Poisso-nian single-electron reflection process with F bs = 1 (see (a) q p S = / (b) q p S = (c) q p S = / (d) q p S = (e) q p S = / (f) q p S = FIG. 2. (Color online) The backscattering Fano factor as a function of q and p for different values of the spin: (a) S = 1 / S = 1, (c) S = 3 /
2, (d) S = 2, (e) S = 5 /
2, (f) S = 5. Uniform black color corresponds to F bs > . For S = 1 / F bs isbounded between 1 and 2. For S > / F bs diverges in the limit q →
0, except for the line p = 1, at which F bs = 1. Fig. 1c). For 1 − p (cid:28)
1, rare two-particle reflectionsare involved in addition to the single-particle backscat-tering. They are accompanied by short-time excursionsof the impurity spin from the state S z = S to the state S z = S − − p (cid:28) S the deviation of F bs from unity is additionally suppressed by a factor 1 /S inthe considered limit [cf. Eq. (14)].The behavior of F bs at the point q = p = 1 is non-analytical owing to I bs = 0. The value of the Fano factordepends on the direction in the ( q, p ) plane at which thispoint is approached. For a fixed ratio − p − q , Eq. (9) yields: F bs ( q, p →
1) = 2(1 − p ) + (1 − q ) S − p + (1 − q ) S . (13)The overall behavior of F bs ( q, p ) for different valuesof S is shown in Fig. 2. For S = 1 / ≤ F bs ≤ S > / F bs in the vicinity of q = 0. The divergence appearsto be more pronounced as S increases. However, thistrend breaks down for large S . Eq. (9) implies that for S (cid:29) / [ p (1 − q )(2 − q )] the Fano factor behaves as F bs = 1 + 12 S (1 − p )(2 − q ) qp (1 − q ) + O (1 /S ) , (14)i.e., it gradually decreases and approaches unity as S getshigher. Thus, the limit of large spin corresponds to thebackscattering of helical electrons by a classical MI.Along with the line p = 1 the Fano factor equals unityat the isolated point ( q, p ) = (2 / , S (see Eq. (11)). At this point the backscatter-ing rate is independent of the impurity state and, there-fore, the backscattering current statistics has a Poisso-nian single-particle character. The interplay between an S scaling of F bs at p → q, p ) = (2 / ,
0) results in a bottleneckfeature in the vicinity of the latter in Figs. 2b - 2f. The1 /S term in Eq. (14) cancels at the line q = 2 / F bs ( q = 2 / − ∼ /S for p (cid:54) = 0. Conclusions. — To summarize, we have investigatedthe zero-frequency statistics of the backscattering cur-rent induced by a magnetic impurity of an arbitrary spin S located near the edge of a two-dimensional topolog-ical insulator. We addressed the limit of large voltage | V | (cid:29) max { T, |D kp / J ij |} where it is possible to neglectthe thermal contribution to the noise, as well as the ef-fect of the local anisotropy of the magnetic impurity. Ouranalytical solution for the average backscattering currentand its zero-frequency noise underscores several strikingfeatures: (i) the dependence of the average backscatter-ing current and noise on the elements of the exchangematrix is determined by three parameters ( g, p, q ) onlyinstead of a-priory six different parameters (the numberof non-zero elements of the exchange matrix J ij ); (ii) for S > / q →
0. This implies thatthe backscattered electrons can be bunched together inlong pulses. Observation of electron bunching is a novelchallenge for experimentalists which might shed light onthe nature of backscattering at the helical edge, and onhow emerging strong correlations can undermine topolog-ical protection against backscattering; (iii) for S = 1 / Acknowledgements. — We thank V. Kachorovskii fordiscussions. Hospitality by Tel Aviv University, theWeizmann Institute of Science, the Landau Institute forTheoretical Physics, and the Karlsruhe Institute of Tech-nology is gratefully acknowledged. The work was par-tially supported by the programs of the Russian Ministryof Science and Higher Education, DFG Research GrantSH 81/3-1, the Italia-Israel QUANTRA, the Israel Min-istry of Science and Technology (Contract No. 3-12419),the Israel Science Foundation (Grant No. 227/15), theGerman Israeli Foundation (Grant No. I-1259-303.10),and the US-Israel Binational Science Foundation (GrantNo. 2016224), and a travel grant by the BASIS Founda-tion (Russia). [1] Ya. M. Blanter, M. Buttiker,
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2, 3
Yuval Gefen, and Moshe Goldstein Department of Physics, Yale University, New Haven, CT 06520, USA L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia Condensed-matter Physics Laboratory, National Research University Higher School of Economics, 101000 Moscow, Russia Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
In these notes, we (i) present a derivation of the generalized master equation governing the timeevolution of the cumulant generating function of the number of backscattered electrons (Eq. (3)of the main text) and (ii) obtain analytical expressions for the backscattering current and thebackscattering Fano factor (Eqs. (8) and (9) of the main text, respectively). We also demonstratethat the backscattering Fano factor is insensitive to the presence of electron-electron interaction atthe helical edge.
S.I. DERIVATION OF THE GENERALIZED MASTER EQUATION
In this section we derive a generalized Gorini-Kossakowski-Sudarshan-Lindblad equation that describes the timeevolution of the reduced generalized density matrix [S1] of a magnetic impurity spin, ρ ( λ ) S ( t ), i.e., Eq. (3) of the maintext. Knowledge of ρ ( λ ) S ( t ) is required to calculate the cumulant generating function G ( λ, t ) of the number of electrons∆ N ( t ) backscattered by the magnetic impurity in a time interval t (we use units where ~ = k B = − e = 1): G ( λ, t ) = ln Tr S ρ ( λ ) S ( t ) , ρ ( λ ) S ( t ) = Tr e e − iH ( λ ) t ρ (0) e iH ( − λ ) t . (S1)Here Tr S and Tr e are the traces over the impurity spin states and over the edge electronic degrees of freedom,respectively, H = H e + H e − i is the full Hamiltonian of the system (the definitions of H e and H e − i are provided inEqs. (1) and (2) of the main text), H ( λ ) = e i Σ z λ/ He − i Σ z λ/ , Σ z = Z dys z ( y ) = 12 Z dy Ψ † ( y ) σ z Ψ( y ) , (S2)and ρ (0) ∼ S ⊗ exp ( − ( H e − Σ z V ) /T ) is the initial density matrix of the whole system. In the latter expression1 S is a unity matrix in the impurity spin subspace, T is the temperature, and V is the voltage applied to the helicaledge.To determine the time evolution of ρ ( λ ) S ( t ) we first study the dynamics of the generalized density matrix of thewhole system: ρ ( λ ) ( t ) = exp (cid:0) − iH ( λ ) t (cid:1) ρ (0) exp (cid:0) iH ( − λ ) t (cid:1) . (S3)A Liouville-type equation governing its evolution is as follows dρ ( λ ) ( t ) dt = − i h H e , ρ ( λ ) ( t ) i − ie i Σ z λ/ H e − i e − i Σ z λ/ ρ ( λ ) ( t ) + iρ ( λ ) ( t ) e − i Σ z λ/ H e − i e i Σ z λ/ . (S4)Importantly, at finite voltage the electron impurity interaction H e − i = J ij S i s j ( y ) /ν acquires a mean-field expectationvalue of H mfe − i = J iz S i Tr ( ρ (0) s j ( y )) /ν = J iz S i V /
2. We thus divide H e − i = H mfe − i + : H e − i :, where the normal orderingof any operator O is defined by : O := O −
Tr ( O ρ (0)). Then by noticing that [ s j ( y ) , Σ z ] = iǫ jzk s k ( y ) we rewriteEq. (S4) in the form dρ ( λ ) ( t ) dt = − i h H e + H mfe − i , ρ ( λ ) ( t ) i − i ν J ( λ ) ij S i : s j ( y ): ρ ( λ ) ( t ) + iρ ( λ ) ( t ) 1 ν J ( − λ ) ij S i : s j ( y ): , (S5)where we introduced J ( λ ) ij = J ik R ( λ ) kj , R ( λ ) = cos λ − sin λ λ cos λ
00 0 1 . (S6)At λ = 0 Eq. (S5) has a form of a regular Liouville equation and describes the time evolution of the usual densitymatrix of the system ρ ( t ) ≡ ρ ( λ =0) ( t ). The only modification arising for λ = 0 is the rotation of the exchange matrixin accordance with (S6). Notice that this rotation is in different directions for the second and the third terms on ther.h.s. of Eq. (S5). This implies that the trace of ρ ( λ ) ( t ) is not conserved at finite λ and thus the cumulant generatingfunction G ( λ, t ) is non-trivial.In order to obtain the equation for the dynamics of the reduced generalized density matrix of the magnetic impurity ρ ( λ ) S ( t ), it is necessary to trace out the electronic degrees of freedom in Eq. (S5). To do that, the standard perturbativetechnique (in |J ij | ≪
1) of the theory of open quantum systems may be employed (for a review, see [S2]). Suchderivation for λ = 0 was conducted recently by the present authors [S3]. Taking into account the modification of theexchange matrix by a finite λ and using the results of [S3], we find dρ ( λ ) S dt = − i [ H mfe − i , ρ ( λ ) S ]+ η ( λ ) jk S j ρ ( λ ) S S k − η (0) jk { ρ ( λ ) S , S k S j } . (S7)In this equation η ( λ ) jk = πT (cid:0) J ( λ ) Π (0) V (cid:0) J ( − λ ) (cid:1) T (cid:1) jk , where T superscript stands for matrix transposition and Π (0) V denotes the Hermitian part of a zero-frequency spin-spin correlation function of the helical edge electrons: (cid:0) Π (0) V (cid:1) pq = 1 ν Z + ∞−∞ dτ Tr (cid:0) : s q ( y , τ ):: s p ( y , ρ (0) (cid:1) , s q ( y , τ ) = e iH e τ s q ( y ) e − iH e τ . (S8)Performing a straightforward calculation we findΠ (0) V = V T coth V T − i V T i V T V T coth V T
00 0 1 . (S9)By introducing Π ( λ ) V = R ( λ ) Π (0) V R ( λ ) we can alternatively rewrite η ( λ ) jk = πT (cid:0) J Π ( λ ) V J T (cid:1) jk and thus obtain Eq. (4) ofthe main text. Notice that in the shot noise limit of V ≫ T the expression for Π (0) V reduces toΠ (0) V ≈ V T − i i . (S10)For future use we note that the right hand side of Eq. (S7) can be thought of as a linear evolution super-operatoracting on the reduced density matrix of the magnetic impurity.Importantly, while Eq. (S9) is specific for the non-interacting Hamiltonian H e given by Eq. (1) of the main text,Eqs. (S7) and (S8) are quite general and, therefore, can be readily extended to more complicated models. In particular,electron-electron interactions at the helical edge can be easily incorporated into the above considerations by using theLuttinger liquid formalism [S4]. Their sole effect is to modify the expression for the spin-spin correlation function. Ifthe Luttinger parameter equals K (we assume K > /
2, i.e., that the interaction is not too strong) one finds [S5, S6]Π (0) V = F V,T V T coth V T − i V T i V T V T coth V T
00 0 0 + 1 K , (S11)where F V,T = (cid:18) πT au (cid:19) K − B (cid:18) K − i V πT , K + i V πT (cid:19) sinh V TV T . (S12)Here B ( x, y ) is the Euler beta function [S7], u is the excitation velocity (which, in general, differs from v in thenon-interacting Hamiltonian), and a is the ultraviolet cutoff length scale. At K = 1 the expression (S9) is triviallyreproduced. We note that at large voltages, V ≫ T , F V,T ≈ ( aV /u ) K − / Γ(2 K ) and, therefore, for K > / (0) V ≈ K ) (cid:18) aVu (cid:19) K − V T − i i , (S13)where Γ( x ) is the Gamma function [S7]. Remarkably, the matrix structure of asymptotic expressions (S10) and (S13)is similar. We will show in Sec. S.II that this is the reason for the independence of F bs on the electron-electroninteraction, as stated in the main text. S.II. ANALYTICAL EXPRESSIONS FOR THE AVERAGE BACKSCATTERING CURRENT AND FORTHE BACKSCATTERING FANO FACTOR
In the present section we use Eq. (S7) to derive analytical expressions for the average backscattering current and forthe backscattering Fano factor (Eqs. (8) and (9) of the main text, respectively). To this end we express the averagebackscattering current I bs and zero-frequency noise S bs in terms of the cumulant generating function G ( λ, t ): I bs = h ∆ N i t = − it ∂∂λ G ( λ, t ) , S bs = hh (∆ N ) ii t = − t ∂ ∂λ G ( λ, t ) , t → ∞ , λ → . (S14)We recall that G ( λ, t ) = ln Tr S ρ ( λ ) S ( t ). Then G ( λ, t → ∞ ) = const + tǫ ( λ ), where ǫ ( λ ) is the eigenvalue of theevolution super-operator featured in the master equation (S7) with the largest real part. At λ = 0 this eigenvaluevanishes, ǫ ( λ = 0) = 0. Indeed, Eq. (S7) evaluated at λ = 0 describes the usual dissipative dynamics of the magneticimpurity spin, which has a unique steady state [S3, S8]. At finite but small λ the eigenvalue ǫ ( λ ) is pushed awayfrom zero, ǫ ( λ ) = ǫ (1)0 λ
1! + ǫ (2)0 λ
2! + O ( λ ) , (S15)implying that I bs = − iǫ (1)0 and S bs = − ǫ (2)0 . In general, finding the coefficients ǫ (1 , for the full evolution super-operator is a formidable task. However, when the regime |J ij | ≪ V ≫ T is considered, the unitary dynamics of themagnetic impurity is much faster than its relaxation dynamics. This allows us to use the rotating wave approximation(RWA) [S2], and calculate ǫ (1)0 and ǫ (2)0 considering the evolution of only the diagonal components of the densitymatrix (in the eigenbasis of H mfe − i ). Within the RWA we obtain from Eq. (S7) ddt h m | ρ ( λ ) S ( t ) | m i = S X m = − S L ( λ ) mm ′ h m ′ | ρ ( λ ) S ( t ) | m ′ i . (S16)Here | m i with m = S, S − , ..., − S are the eigenstates of H mfe − i . Since the exchange matrix J ij is assumed to belower triangular (see the discussion of the model in the main text), the mean-field interaction is directed along z axis, H mfe − i = J zz S z V /
2. Therefore, states | m i are eigenstates of S z : S z | m i = m | m i . The matrix L ( λ ) mm ′ has the followingnon-zero elements ( L ( λ ) m ± ,m = e − iλ η (0) ± ( S ( S + 1) − m ( m ± / , L ( λ ) m,m = (cid:0) e − iλ − (cid:1) η (0) zz m − e iλ (cid:0) L ( λ ) m − ,m + L ( λ ) m +1 ,m (cid:1) . (S17)To find the coefficients ǫ (1)0 and ǫ (2)0 we calculate the shift of zero eigenvalue of a non-Hermitian matrix L (0) mediatedby the perturbation ∆ L ( λ ) = L ( λ ) − L (0) up to the second order in λ . A. First order in λ : average backscattering current We start by calculating the coefficient ǫ (1)0 using the first order non-degenerate perturbation theory in λ . First wefind the right and left eigenvectors of L (0) with the eigenvalue ǫ ( λ = 0) = 0. The right eigenvector corresponds tothe stationary state solution for the density matrix at λ = 0. Direct check shows that L (0) | ii = 0 for | ii = 1 Z (cid:0) ϑ S , ϑ S − , ..., ϑ − S (cid:1) T , ϑ = 1 + p − p , Z = S X m = − S ϑ m . (S18)The left eigenvector hh ˜0 | , obeying hh ˜0 |L (0) = 0, is given by hh ˜0 | = (1 , , ...,
1) (S19)as dictated by the conservation of the trace of the density matrix at λ = 0. In formulae above we introduced doublebra and ket notations to distinguish eigenvectors of L ( λ ) from usual quantum-mechanical states. The vectors (S18)and (S19) are normalized so that hh ˜0 | ii = 1.As a next step we write down explicitly the perturbation matrix:∆ L ( λ ) mm ′ = ( e − iλ − h η zz m δ mm ′ + (1 − δ mm ′ ) L (0) mm ′ i = ( e − iλ − L (0) mm ′ + ( e − iλ − δ mm ′ h η zz m − L (0) mm i . (S20)Because hh ˜0 |L (0) = 0 and L (0) | ii = 0, the part of ∆ L ( λ ) mm ′ that is proportional to L (0) mm ′ does not generate a shift ofthe eigenvalue ǫ in the first two orders of perturbation theory in λ . After some straightforward algebra we find thatthe part of the perturbation that affects zero eigenvalue in the first two orders of the perturbation theory is given bythe following diagonal operator:∆ e L ( λ ) = ( e − iλ − πgV R ( S z ) , R ( S z ) = (2 − q ) S z − pqS z + qS ( S + 1) . (S21)Then we obtain ǫ (1)0 = hh ˜0 | ∂ λ ∆ e L ( λ ) | ii (cid:12)(cid:12)(cid:12) λ =0 = − i πgV h R ( S z ) i , h R ( S z ) i = S X m = − S ϑ m Z R ( m ) . (S22)Equation (S22) results in Eq. (8) of the main text, which determines the average backscattering current. B. Second order in λ : zero-frequency noise and backscattering Fano factor The second order correction to the zero eigenvalue of L (0) consists of two contributions, ǫ (2)0 = ǫ (2 , + ǫ (2 , . Thefirst contribution, ǫ (2 , , is obtained by expanding the perturbation matrix ∆ e L ( λ ) to the second order in λ and thenapplying first order perturbation theory. The resulting expression is ǫ (2 , = hh ˜0 | ∂ λ ∆ e L ( λ ) | ii (cid:12)(cid:12)(cid:12) λ =0 = − πgV h R ( S z ) i . (S23)Finding ǫ (2 , requires a more cumbersome procedure of second order perturbation theory. The first step is to findorthogonal complements to the vectors hh ˜0 | and | ii . We consider the following sets of vectors: | n ii = 1 Z ( ϑ S , ϑ S − , ..., ϑ S − n +1 | {z } n components , , , ..., T − S X m = S − n +1 ϑ m Z | ii , n = 1 , ..., S, (S24) hh ˜ r | = (1 , , , ..., , | {z } r components , , , ..., − S X m = S − r +1 ϑ m Z hh ˜0 | , r = 1 , ..., S. (S25)Direct check shows that sets | n ii and hh ˜ r | are orthogonal to hh ˜0 | and | ii , respectively, i.e., hh ˜0 | n ii = 0 and hh ˜ r | ii = 0for n, r = 1 , ..., S . The vector sets introduced here are particularly appealing because they diagonalize L (0) : hh ˜ r |L (0) | n ii = − δ rn πgV q (1 − p ) S ( S + 1) − ( S + 1 − n )( S − n )4 ϑ S +1 − n Z , r, n = 1 , ..., S. (S26)Note, however, that hh ˜ r | n ii 6 = 0. Overall, we find ǫ (2 , = − S X n =1 hh ˜0 | ∂ λ ∆ e L ( λ ) | n ii (cid:12)(cid:12)(cid:12) λ =0 hh ˜ n | ∂ λ ∆ e L ( λ ) | ii (cid:12)(cid:12)(cid:12) λ =0 hh ˜ n |L (0) | n ii . (S27)After straightforward algebraic manipulations we obtain ǫ (2 , = − πgVq (1 − p ) S X n =1 hh ˜0 | R ( S z ) | n iihh ˜ n | R ( S z ) | ii n (2 S + 1 − n ) Z ϑ S +1 − n . (S28)The explicit structure of vectors hh ˜ r | and | n ii allows us to rewrite this expression in terms of the averages of R ( S z ).To this end we note that hh ˜0 | R ( S z ) | n ii = hh ˜ n | R ( S z ) | ii = h P n [ R ( S z ) − h R ( S z ) i ] i , P n = S X m = S − n +1 | m ih m | . (S29)Therefore, the expression for ǫ (2 , simplifies to ǫ (2 , = − πgVq (1 − p ) S X n =1 h P n [ R ( S z ) − h R ( S z ) i ] i n (2 S + 1 − n ) Z ϑ S +1 − n . (S30)Combining ǫ (2 , and ǫ (2 , and introducing µ n = ϑ S +1 − n / Z , we obtain the expression for the zero-frequency noise: S bs = πgV h R ( S z ) i q (1 − p ) S X n =1 h P n [ R ( S z ) − h R ( S z ) i ] i n (2 S + 1 − n ) h R ( S z ) i µ n ! . (S31)Eqs. (S22) and (S31) lead to Eq. (9) in the main text for the backscattering Fano factor F bs . We stress that, despitethe (1 − p ) factor in the denominator of Eq. (S31), the result is actually finite for p →
1, since then the impuritybecomes fully polarized and h P n [ R ( S z ) − h R ( S z ) i ] i → ( λ ) V is insensitive to the presence of interactions in the shot noise limit V ≫ T (compare Eqs. (S10) and (S13)):nonzero interaction just results in a voltage-dependent multiplicative factor. This feature allows us to modify theexpressions for I bs and S bs to account for K = 1: I bs = πgV h R ( S z ) i K ) (cid:18) aVu (cid:19) K − , S bs = πgV h R ( S z ) i K ) (cid:18) aVu (cid:19) K − q (1 − p ) S X n =1 h P n [ R ( S z ) − h R ( S z ) i ] i n (2 S + 1 − n ) h R ( S z ) i µ n ! . (S32)Strikingly, the ratio between S bs and I bs – that is, the backscattering Fano factor F bs – does not depend on K at all,and thus is insensitive to the electron-electron interaction at the helical edge. [S1] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics inquantum systems , Rev. Mod. Phys. , 1665 (2009).[S2] ´A. Rivas and S. F. Huelga, Open Quantum Systems , Springer, New York, (2012).[S3] V. D. Kurilovich, P. D. Kurilovich, and I. S. Burmistrov, M. Goldstein
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