Non-trivial Interplay of Strong Disorder and Interactions in Quantum Spin Hall Insulators Doped with Dilute Magnetic Impurities Jun-Hui Zheng1, 2, ∗ and Miguel A. Cazalilla2, 3, 4, † arXiv:1609.06227v3 [cond-mat.mes-hall] 16 Jul 2018 1 Institut f¨r Theoretische Physik, Goethe-Universit¨t, 60438 Frankfurt/Main, Germany. u a 2 Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan. 3 National Center for Theoretical Sciences (NCTS), Hsinchu 30013, Taiwan. 4 Donostia International Physics Center (DIPC), Manuel de Lardizabal, 4. 20018, San Sebastian, Spain. We investigate nonperturbatively the effect of a magnetic dopant impurity on the edge transport of a quantum spin Hall (QSH) insulator. We show that for a strongly coupled magnetic dopant located near the edge of a system, a pair of transmission anti-resonances appear. When the chemical potential is on resonance, interaction effects broaden the anti-resonance width with decreasing temperature, thus suppressing transport for both repulsive and moderately attractive interactions. Consequences for the recently observed QSH insulating phase of the 1-T of WTe2 are briefly discussed. I. INTRODUCTION Two-dimensional (2D) topological materials like quantum spin Hall insulators (QSHIs) have become a fascinating research topic, with many potential applications [1– 3]. Theoretically, QSHIs are predicted to possess gapless one-dimensional (1D) edge states [3, 4]. Disorder potentials that are invariant under time-reversal symmetry (TRS) cannot cause Anderson localization, which is otherwise ubiquitous in 1D systems. Indeed, it has been shown [3–6] that for scalar and spin-orbit (SO) disorder potentials, even in the presence of weak electron-electron interactions, the 1D edge channels of QSHIs exhibit perfect transmission, whose hallmark is a quantized conductance at low temperatures [7]. On the other hand, strong interactions can break TRS [4, 5] and lead to complex edge reconstructions [8, 9], which jeopardize the perfect conductance quantization. Experimentally, the QSH effect arising from gapless edge channels has been observed in HgTe/CdTe and InAs/GaSb/AlSb semiconductor quantum wells (QWs) [2], graphene submitted to a strong, tilted magnetic field [10], Bi (111) bilayers [11, 12] and, more recently, in the 1-T phase of the transition metal dichalcogenide WTe2 [13–16]. However, in HgTe/CdTe and InAs/GaSb/AlSb samples, long edge channels (∼ 1 µm) in the topological phase exhibit relatively short meanfree paths, and the conductance deviates from quantization [2, 17–19]. For the monolayer WTe2 , the conductance of the devices with longer edges does not exhibit the expected quantized value [14, 16]. Moreover, the interpretation of the observations in InAs/GaSb QWs [18, 19] has also been questioned after the discovery of rather similar edge conduction features in the trivial phase [17], Deviations from perfect conductance quantization at ∗ Electronic † Electronic address: jzheng@th.physik.uni-frankfurt.de address: miguel.cazalilla@gmail.com low temperatures arise from backscattering (BS) in the edge channels. Several BS mechanisms have been discussed using effective 1D models [3, 20–24]. The latter often involve electron-electron scattering in combination with scalar, spin-orbit coupling and magnetic disorder [6, 21–29]. Indeed, magnetic impurities break TRS above the Kondo temperature, and therefore they cause BS [5, 28–31]. Nevertheless, the connection between the effective 1D models of disorder and the 2D aspects of the physics of QSHIs has not yet been fully investigated to the best of our knowledge. With the exception of a few numerical studies in the non-interacting limit [32, 33], there appears to be no systematic investigation about the validity of these 1D models. Indeed, little is known about whether they actually apply in the strong coupling limit where coupling strength to the impurity becomes comparable or larger than the band gap of the QSHI. The latter is an experimentally relevant regime given the small band gaps exhibited by many of the experimentally realized QSHIs. Below, we shall show that the problem of a magnetic dopant impurity problem can be mapped, in the strong coupling limit, to a generalized 1D Fano model [34] describing two resonant levels coupled to an interacting 1D channel. Using a renormalization group analysis, we show that the transmission coefficient is suppressed at low temperatures for repulsive interactions. Interestingly, when the chemical potential of the edge electrons resonates with one of the in-gap states, we find that the transmission is also suppressed for weak to moderately attractive interactions. The rest of this article is organized as follows: Section III describes the solution of the scattering problem for a toy model of a single magnetic impurity in the neighborhood of a non-interacting QSH edge channel. In section IV, we construct an effective 1D model to describe this system, which allows us to treat the effect of weak to moderate interactions. In this section, we also discuss the effects not included in our toy model, such as the Rashba coupling in the band-structure and the non-planar alignment of the magnetic moment. Finally, in section V we offer the conclusions of this work and provide an outlook 2 for future research directions. The Appendix contains the most technical details of the calculations. Henceforth, we work in units where = 1. II. MODEL In this work, we consider the effect of a magnetic dopant impurity in a QSHI taking into account the electron-electron interactions along the edge. We shall assume a large spin-S magnetic impurity at temperatures T well above the Kondo temperature TK (TK is exponentially suppressed for large S [35]). This allows us to treat the magnetic moment of the dopant classically. For the sake of simplicity, we first solve a model in which the moment lies on the plane perpendicular to the spin-quantization axis of a QSHI, which is described by the Kane-Mele model [7]. The more general case when the magnetic moment is pointing in an arbitrary direction and the QSHI is described by more realistic extensions of the Kane-Model model will be discussed in Sect. IV C. Once the scattering problem with the dopant impurity is solved, we obtain an effective 1D model by fitting the scattering data. The effective model allows us to introduce the electron-electron interactions and treat them non-perturbatively. With the above assumptions, the impurity potential is written as follows: Ä ä Vimp = λimp c†0 ↑ ci0 ↓ + h.c. = λimp c†0 sx ci0 , (1) i i with c† = (c† , c† ). As we will further elaborate below, i i↑ i↓ for λimp ∆, where 2∆ is the band gap, two bound states appear within the gap when the impurity is located deep inside the bulk of the QSHI. As the position of impurity is shifted from the bulk to the edge, the bound states hybridize with the edge states inducing a pair of anti-resonances in the transmission coefficient. Thus, we show that the two-dimensionality arising from the QSHI physics leads to a much richer interplay between interactions and (magnetic) disorder than the one encountered in simple models of structureless impurities in 1D interacting electron systems [36–43]. These results provide the foundation for future studies based on more realistic models of the microscopic origins of the absence of quantization in the QSH effect at low temperatures. Notice that the model considered here is also drastically different from models based on charge puddles resulting from doping fluctuations [23]. Indeed, the situation envisaged in this work is more relevant to isolated strongly coupled magnetic moments that are well localized on the lattice scale, as it is the case of vacancies in 2D materials [44] or isolated magnetic dopant impurities in general QSHIs. On the other hand, puddles are described [23] as extended quantum dots containing many levels and many electrons, which resonate with the QSH edge states. Furthermore, unlike the study reported below, the authors of Ref. [23] neglected Luttinger liquid effects in their treatment of the edge, which may be a good approximation for the HgTe quantum wells due to the large value of the dielectric constant. In the puddle model, backscattering is induced by the edge electrons dwelling in the quantum dots and undergoing inelastic scattering with other electrons in puddle [23]. Thus, in the absence of interactions, the puddle model will not lead to backscattering, whereas the model considered below backscattering is present even in the absence of interactions. III. SOLUTION OF SCATTERING PROBLEM A. Solution of the clean Kane-Mele ribbon In order to describe the QSHI, we consider the KaneMele (KM) model [7] (cf. Fig. 1), c† cj − iλSO i H0 = −t i,j νij c† sz cj i (2) i,j where λSO describes the intrinsic SO coupling [7] as an imaginary next nearest neighbor hopping and νij = ±1 depends on the electron hoping path; sz is the electron spin projection on the axis perpendicular to the 2D plane. For the sake of simplicity, we first neglect Rashba SO coupling. This approximation does not qualitatively modify our results, as we discuss in section V. In the absence of interactions, the impurity problem is described by the Hamiltonian: H = H0 + Vimp . (3) In order to solve this problem, we first obtain an analytical solution of the clean KM model, Eq. (2), for a zigzag ribbon of width L (cf. Fig. 1). The transmission coefficient of the edge state for the system with an impurity (1) will be evaluated by solving the Lippmann-Schwinger equation in section III B. In the ribbon geometry, the Bloch wavevector parallel to the edge, kx , is a good quantum number. However, ky = −i∂y must be treated as an operator. The wave functions along the y-axis obey open boundary conditions [45]. The Hamiltonian (2) in the Bloch basis can be obtained by using the Fourier transform, ci∈A = k c √kA eik·(Ri +rg ) , ci∈B = Nt k ckB √ eik·Ri . (4) Nt Here Ri∈A(B) is the position of A(B) sublattice sites and Nt is total number of unit cells. Because of the bi-partite structure of the honeycomb lattice, the Fourier transform of H0 is not unique and depends on the relative phase k·rg . This gauge freedom must be fixed by the boundary conditions (BCs). The appropriate choice for the zigzag edge is √ rg = −(a/2 3)ey , (5) 3 B. Effect of the magnetic impurity In order to investigate the effect of the impurity on the electronic transport, we next solve the LippmannSchwinger equation (LSE): |Ψ = |Φ + G0 ( ) Vimp |Ψ , FIG. 1: (Color online) Sketch of (a) the zigzag edge with a single impurity at the edge and (b) the “brick wall” lattice to which it maps. so that the N th row of the A sublattice are effectively shifted (See Eq.(4)) to overlap with the N th row of the B sublattice (See Fig. 1). This maps the honeycomb lattice onto the so-called “brick wall” lattice and thus the BCs become Φ = (ΦB , ΦA )T = 0 for y = ±L/2. s ˆ H0 (α, β)Φs (kx , y) = Φs (kx , y), ˆ dx = −t(2 cos α + cos β), s ˆ dy = −t sin β, s dz s (7) √ 3a 2 ∂y (8) respectively (α = kx a/2). The Pauli matrices σ i (i = x, y, z) is in the pseudo-spin space corresponding to the sublattice (B, A) components. Furthermore, since sz is a good quantum number, s = ±1. Below, we look for solutions that are combinations of plane waves eiky y . We are not interested in finite size effects and therefore take L → ∞. In this limit, the coupling between the two edges vanishes and we obtain the dispersion for the edge states (see Appendix): 6sλSO t sin(kx a) = ±» , 2 t2 + [4λSO sin(kx a/2)] Φ(sσ, r) = s, σ, r |Φ , Ψ(sσ, r) = s, σ, r |Ψ , (11) (12) where σ = (+, −) corresponds to the (B, A) sublattice. Thus, the asymptotic behavior of the wave function becomes |Ψ → (1 + ζt )|Φ for x → +∞, ˜ |Ψ → |Φ + ζr |Φ for x → −∞, ˜ where |Φ = ˆ = sλSO (2 sin 2α − 4 sin α cos β), s (kx ) where G0 ( ) = ( + i0+ − H0 )−1 is the Green’s function for Eq. (2). We assume the magnetic impurity to be located on the B sublattice at the bottom edge since the wavefunction of edge states on this edge is mostly localized on the B sublattice (See appendix). In order to extract the transmission and reflection coefficients of the edge electrons, we assume the incident electron has a 0 Bloch wave number kx on the right-moving edge channel, 0 0 i.e. |Φ = kx , s = −1 . Therefore, its energy is − (kx ) 0 and its group velocity is v = ∂kx − (kx )|kx =kx . Let us introduce (6) After identifying the boundary conditions, we proceed to solve the 1D Schr¨dinger equation: o ˆ where we have used the following notation: β = −i s and H0 = i di σ i , with s (10) (9) where the + (−) sign corresponds to the bottom (top) edge at y = −L/2 (y = +L/2) and s = ±1. The bands of edge states cross at kx = π [7] (for a bearded edge a they cross at kx = 0 [46], see appendix). For kx ≈ π , a Eq. (9) agrees with the semi-analytic results of Ref. 47. For the bottom edge states, below we use the notation |kx , s . A plot of the bands [7] for a wide zigzag ribbon and the corresponding wavefunctions can be found in the Appendix. 2π a (13) (14) 0 − kx , s = +1 , and Ψ(++, r0 )Φ∗ (−+, r0 ) , v ˜ Ψ(−+, r0 )Φ∗ (++, r0 ) ζr = −iλimp Lx . v ζt = −iλimp Lx (15) (16) Here Lx is the normalization length of system along the edge and r0 ∝ Ri0 is the impurity position. From the above results, the transmission and the reflection coefficients are obtained from ζr as follows: T ( ) = |1 + ζt | 2 R( ) = |ζr | . 2 (17) (18) The energy dependence of the transmission coefficient is shown in Fig. 2. Note that, when the magnetic impurity is located on the first atomic row (i.e. N = 1), the transmission coefficient is essentially energy independent, which makes it similar to a BS impurity in a purely 1D channel. This behavior arises from weak coupling between the edge and bulk states via the impurity (owing to the small weight of the bulk states on the N = 1 row). This holds true even for relatively large values of λimp . Thus, scattering is dominated by the 1D edge states. However, we believe this behavior is not a robust feature but a peculiarity of present KM model. On the other hand, for the second atomic row and beyond (i.e. 4 IV. 1D EFFECTIVE MODEL A. FIG. 2: Transmission coefficient T ( ) for an impurity on a B sublattice site on (a) the first atomic row (i.e. N = 1), (b) N = 2, (c) N = 3, and (d) N = 4. The spin-orbit coupling is λSO = 0.06 t. N ≥ 2), the weight of the bulk states is larger, and a strong impurity can thus lead to a sizable coupling between bulk and edge states. As a consequence, for large values of λimp , a pair of narrow scattering anti-resonances appears within the energy gap. In the neighborhood of the anti-resonances, the transmission coefficient changes very rapidly with energy and, on resonance, it vanishes for large λimp . In order to understand the emergence of the pair of scattering anti-resonances, we need to consider the poles of the T-matrix, Non-interacting limit After finding a non-perturbative solution to the scattering problem of the edge electrons with a magnetic dopant impurity, in this section we construct a 1D lowenergy effective model that describes a non-interacting edge channel in the presence of magnetic impurity at √ large λimp /∆, where ∆ = 3 3λSO (2∆ is the bulk band gap). The effective model is valid at energies and temperatures smaller than ∆ and therefore only involves the degrees of freedom of the 1D edge and the in-gap states. The Hamiltonian of the effective 1D model describing the coupling between the edge electrons and the ingap states is constrained by the existence of a number of symmetries of H = H0 + Vimp . The KM model in the ribbon geometry described by H0 (cf. Eq. 2), is invariant under TRS (T ), spin rotations about the z-axis −1 (i.e. Uθ = exp(−iθsz /2), Uθ H0 Uθ = H0 ), particlehole symmetry (C), and lattice translations along the edge direction. The impurity potential, Vimp , breaks all those symmetries, but the composite system described by H = H0 + Vimp is invariant under the subgroup span by the combined Uπ T and CT transformations. Therefore, according to the above discussion, the effective model takes the form of a generalized Fano model [34], describing two discrete levels coupled to the continuum of edge states. Furthermore, this model is invariant under Uπ T and CT . Since for |λimp | → ∞ the position of the resonances approaches the center of the band gap at = 0, we shall focus in the neighborhood of kx = π , where linearization of the edge state spectrum, a i.e. ± (kx ) = vF k, is a good approximation. Thus, the effective Hamiltonian can be written as follows: Heff = HB + H+ [u, t+ ψ(0)] + H− [d, t− ψ(0)] , −1 T ( ) = [1 − Vimp G0 ( )] Vimp (19) For a strong impurity potential located within the bulk, of the QSHI, the poles of the T-matrix are obtained from the condition det 1 − λimp GB (r0 , r0 , ) sx = 0 0 (20) where GB is the Green’s function constructed from bulk 0 states. The latter is real for within the energy gap since the density of states vanishes there and it is odd in (due to the particle-hole symmetry of H0 ), therefore vanishing at = 0, i.e. the middle of the gap. Thus, GB (r0 , r0 , ) ∝ 0 for small . Hence, at large λimp , two bound in-gap states appear at ∝ ±t2 /λimp , corresponding to the two eigenvalues of sx . As the impurity location is shifted towards the edge, the bound states hybridize with the continuum of edge states, leading to the anti-resonances in the transmission coefficient. We will generalize this argument below in section IV C when discussing the effect of extensions to the present toy model. (21) dx ψ † sz ∂x ψ + VB a0 ψ † (0)sx ψ(0), (22) HB = ivF H± [f, χ] = ± 0 f †f − 1 2 1/2 + Vc a0 f † χ + h.c. (23) Ä † ä † where ψ † (x) = ψL (x), ψR (x) is the spinor field operator describing the edge states, u† and d† are the creation operators of electrons in the bound states with sx eigenvalue and energy sx = +1, = + 0 and sx = −1, = − 0 , respectively, and t± = (±1, 1); a0 = vF /∆ is a short distance cut-off. In the above model, VB describes a renormalized backscattering amplitude for the edge electrons, and Vc the tunneling into and out of the bound states. The reflection coefficient for the effective 1D model reads: 2 R( ) = p 2 iVc p=±1 ( +p 0 )∆ + (1 − p iVB ) 2∆ , (24) which accurately fits the results obtained (numerically) for T ( ) = 1 − R( ) from the non-perturbative solution 5 effective model: Heff = HKF + H− [d, t− ψ(0)] + d† d − 1 2 x × UF ψ † (0)ψ(0) + UB ψ † (0)s ψ(0) , HKF = HB + U FIG. 3: (Color online) (Left) Transmission coefficient for an impurity strength λimp = 40 ∆ (∆ is the band gap). Dots are the transmission coefficient obtained numerically for the Kane-Mele model with a backscatterer at the edge. The red line is the fit to the effective model (cf. Eq. 21). (Right) Effective model parameters as a function of λimp . of the scattering problem. The left panel of Fig. 3 shows the quality of fit of the transmission coefficient as a function of energy for a magnetic dopant impurity located in the second atomic row (i.e. N = 2). The behavior of the fitted parameters Vc , VB , and 0 as functions of the impurity potential strength λimp is shown on the right panel. As expected from the above discussion, 0 decreases as λimp → +∞. Note that Vc , VB ∆, which is consistent with the assumption that the 1D model, Eq. (21) describes only the edge and in-gap states. dx ρR ρL . Interaction effects Finally, we study the effect of electron interactions on the transport properties of the QSHI with a magnetic dopant. Interactions are treated non-perturbatively using the bosonization method [43]. Their characteristic energy scale is ∼ e2 /a0 (where e is the electron charge), which is assumed to be smaller than the band gap, 2∆. In order to apply bosonization to the interacting model, we further project the effective 1D model in Eq. (21) onto the subspace of excitations with in the neighborhood of the Fermi energy, F . In particular, when F is away from ± 0 , the bound states can be integrated out. To leading order, this yields a renormalized backscattering amplitude ï VB VB − Vc2 0− F Vc2 + 0+ ò . (25) F and thus the 1D model reduces to the impurity model in a 1D interacting channel studied by Kane and Fisher [36, 38] (cf. HKF in Eq. 27 below) with an impurity potential whose backscattering amplitude VB = VB . On the other hand, on resonance, i.e. for F + 0 ( F − 0 ), we can integrate out only the non-resonant level at F − 0 ( F + 0 ). Assuming (without loss of generality) that F − 0 yields the following low-energy (27) The interactions between the edge electrons (with amplitude U ) and between the edge electrons and the resonant level (with amplitudes UF and UB ) have been included in the Hamiltonian. We note that integrating out the non-resonant level at = + 0 renormalizes the amplitude of VB − UB /2 in Heff by an amount Vc2 /( F − 0 ) −Vc2 /2 0 . In addition, forward scattering is also generated but it is dropped since it can be eliminated by a unitary transformation [36, 43]. The Hamiltonian Heff in Eqs. (26,27) is akin to a model of a (side-coupled) resonant level in an interacting 1D channel [48, 49]. Thus, we apply an analysis similar to the one carried out by Goldstein and Berkovits in Ref. [48] to Heff . After bosonizing [43] Eq. (26), we perform a unitary transformation in order to eliminate the forward interaction term ∝ UF at the expense of renormalizing the scaling dimension (∆c ) of the operator (Oc ∝ Vc ) describing the tunneling between the 1D edge channel and the resonant level. Thus, † Oc (τ )Oc (0) ∼ B. (26) Vc2 τ 2∆T , (28) where τ is the imaginary time and (see Ref. [48] and appendix) and î 2ó 1 FK ∆T (K, UF ) = 4 K + K −1 1 − Uπv . (29) In this expression   K= 2πvF − U 2πvF + U is the Luttinger parameter and   Å ã2 U v = vF 1 − 2πvF (30) (31) the velocity of the edge plasmons [43]. Hence, tunneling into the resonant level becomes relevant in the renormalization-group (RG) sense for ∆c (K, UF ) < 1. There are two different interaction regimes for which this happens: For repulsive interactions (i.e. K < 1) and for weak to moderate attraction (i.e. K 1). In the former case, both tunneling Vc and the BS (∝ VB , UF ) are renormalized to strong coupling by the charge-density wave fluctuations dominant in the 1D channel with K < 1 [43]. At T = 0, transmission through the edge channel is completely suppressed. [48, 49] Interestingly, on resonance the transmission through the edge channel of the QHSI is also suppressed for moderately attractive interactions i.e. K 1. In this regime, 6 backscattering is na¨ ıvely irrelevant [36] and therefore UB is initially suppressed by the dominant superconducting fluctuations in the edge channel (see below). However, the tunneling amplitude Vc is still a relevant perturbation since ∆c (K, UF ) < 1. Physically, this is because tunneling is a strongly relevant perturbation in 1D, also in the presence of interactions (see e.g. [43], chapter 8). As the tunneling amplitude renormalizes to strong coupling with decreasing energy scale/temperature, the 2nd order RG flow equations (where yB ∝ UB , yt ∝ Vc , δF ∝ UF , etc. are dimensionless couplings,m see appendix D for derivation details): dyB 2 = (1 − K) yB + yt , d ln ξ dyt = 1 − K/4 − (1 − δF )2 K −1 /4 yt d ln ξ + yt (yB + vB ), dδF 2 = 4(1 − δF )yt , d ln ξ dvB = (1 − K)vB . d ln ξ (32) (34) (35) The main results obtained using the toy model introduced above can be easily generalized to account for the Rashba spin-orbit coupling in the band structure, i.e. adding to Eq. (2) a term of the form (dij is the vector joining the two nearest neighbor sites i and j on the honeycomb lattice): HR = iλr × dij )cj (36) i,j and to the case of a more general coupling to the magnetic impurity (n is a unit vector): ¯ Vimp = λimp c†0 (s · n) ci0 . i (37) In absence of Rashba and for n perpendicular to the spin-quantization z-axis, we can implement rotation along sz direction to change the magnetic moment in Eq. (37) to the form Eq. (1), which maps the problem to the toy model studied above. The presence in the system of a uniform Rashba SOC, Eq. 37, violates the conservation of the total sz as well as the particle-hole symmetry of the model. Yet, for weak to moderate Rashba SOC, the topological phase is stable and exhibits robust helical edge states [7]. In the following, we prove that in the limit λimp → ∞, a magnetic (38) where GB (r0 , r0 , ) is the local Green’s function on the 0 B sublattice, which is a 2 × 2 matrix in spin space. However, TRS implies that its off-diagonal elements vanish [50, 51] GB (r0 , r0 , ) = GB (r0 , r0 , ) = 0 and 0,↑↓ 0,↓↑ GB (r0 , r0 , ) = GB (r0 , r0 , ). Hence, GB (r0 , r0 , ) is 0 0,↑↑ 0,↓↓ indeed proportional to the unit matrix, i.e. (33) Rashba SOC and general magnetic moments c† (s i det 1 − λimp (n · s) GB (r0 , r0 , ) = 0, 0 GB (r0 , r0 , ) = 0 show that this runaway flow of yt ∝ Vc drags along both the backscattering amplitude yB ∝ UB and δF ∝ UF . This ultimately leads to an effective suppression of the transmission through the edge channel as the temperature (or the energy scale) is reduced [48, 49]. C. dopant impurity in the bulk still generates in-gap bound states, which can resonate with the edge states when the impurity is located near the boundary of the insulator. For an arbitrary orientation of the magnetic dopant in the bulk of a QSHI, the positions of bound states are determined by the equation (see Eq. (19)): gB ( ) 1, 2 (39) where the function gB ( ) is related to the local density of states (LDOS) on the B sublattice. If we apply a rotation to align the spin quantization axis with the direction of n, i.e. U † (n) (n · s) U (n) = sz , Eq. (38) yields the following conditions for the existence of in-gap bound states: gB ( ) = ±2λ−1 imp (40) The function gB ( ) becomes real for within the band gap because the LDOS vanishes there. In addition, since the LDOS is positive for outside the band gap, KramersKronig relationships imply that gB ( ) must have a zero within the gap, i.e. gB ( ) = z −1 ( − c ), where z −1 is the proportionally constant and c is an energy within the band gap. For the KM model, particle-hole symmetry further requires that c = 0, which corresponds to the middle of the gap. Rashba SOC breaks particle-hole symmetry and, in general, we expect c = 0. Hence for sufficiently large λimp , the in-gap states will be located at the energies: ± 0 = c ± 2z . λimp (41) However, notice that for λimp ∼ ∆ and/or large particlehole asymmetry (i.e. c ∼ ∆), one or both solutions to Eq. (40) may not be real. Indeed, this the case when energy of the in-gap states overlaps with the continuum of states in the conduction or valence bands. However, the above analsys shows that for λimp ∆, two in-gap states will always be present. The existence of the in-gap bound states can be further explicitly demonstrated by numerically computing the LDOS of QSHI in the presence of the magnetic dopant impurity. Fig. 4 shows the results obtained for the KM with a Rashba SOC of λr = 0.06t and n along the x-axis. We have also checked the existence of the in-gap bound state(s) for other choices of λr and n (not shown here). As the position of the magnetic impurity is shifted towards the edge, the in-gap states hybridize with the topological edge states, which results in anti-resonances 7 in edge channel transmission. This phenomenon is still described by the generalized Fano model introduced in section III with different energy values for the energy the in-gap state(s) and the tunneling Vc treated as an energy dependent function. Nevertheless, provided the Fermi level of the 1D edge ( F ) is off resonance, both in-gap states can be integrated out, resulting in a local backscattering potential, which can be treated as a nonmagnetic impurity in an interacting 1D channel [7, 39]. For F on resonance with one of the in-gap state(s), the other non-resonant state can be integrated out, giving rise to the similar model to the one studied at the end of section IV B, Heff (cf. Eq. 27, the possible energy dependence of Vc being irrelevant in the RG sense). A similar argument applies even when the impurity strength is not weak or the particle-hole symmetry strong, so that only one bound state exists. An exception to the phenomena described the effective model of Eq. (27) is found when there is a symmetry that prevents the hybridization between the in-gap bound states and the electronic states at the edge. Although this is not generic, it is indeed the case for a dopant whose magnetic moment n points along the spin-quantization axis of the KM, Eq. (2). Thus, the total sz is conserved and the Hilbert space of the problem splits into two subspaces labeled by different sz without any matrix element connecting them. Thus, conservation of total sz prevents the existence of backscattering [28]. Therefore, although we have based our calculations in a simplified model of the QSHI and the impurity, the phenomena described above does not depend on the specific microscopic details of the model in the large λimp limit. The emergence of transmission anti-resonances and the interaction induced renormalization of the anti-resonance linewidth [48, 49] stems from the coupling between the edge states and the impurity-induced in-gap states. This will generically be present as long as the wave functions of the edge states and the states bound by the magnetic impurity overlap. Similar arguments can be applied to magnetic dopants described by more sophisticated models of of Z2 topological insulators. However, if λimp is decreased continuously, the bound states will merge into the continuum of bulk states (together or one by one, depending on the degree of particle-hole asymmetry) and finally the resonances will disappear. V. SUMMARY AND OUTLOOK In summary, we have investigated the transport properties of a quantum spin-Hall insulator in the presence of a strongly coupled magnetic impurity. By obtaining a non-pertubative solution of the scattering problem, we have derived a 1D effective low-energy Hamiltonian describing the system. In the strong coupling limit, the impurity induces in-gap bound state, which in proximity to the edge state broaden into transmission anti-resonances. When the chemical potential of the edge electrons is not resonant with any of the in-gap FIG. 4: Local density of state at the position of a magnetic dopant impurity located in the bulk of a QSHI insulator described by the Kane-Mele model (see Eqs.2 and 36) with a strength of the bulk Rashba spin-orbit coupling (SOC) λr = 0.06t. The impurity magnetic moment points along the x-axis (see Eq. 1). Notice that the positions of the sharp peaks indicating the existence of impurity-induced in-gap states is not symmetrical with respect to the center of the band gap. This is a consequence of the particle-hole symmetry breaking caused by the Rashba (SOC). states induced by the magnetic impurity, the system can be effectively mapped to the problem of a nonmagnetic impurity in a Tomonaga-Luttinger liquid [36–38] with a renormalized backscattering strength at sufficiently low energy/temperatures (the latter energy scale being set by the separation between the Fermi level and the nearest resonant state). For strong attractive interactions in the channel, this suppression is absent and the 1D channel becomes increasingly transparent at low T . On the other hand, when the Fermi energy is on resonance, repulsive and weak to moderately attractive interactions lead to temperature-dependent broadening of the transmission anti-resonance, which effectively suppresses the conductance of the edge channel as the temperature T is decreased. For many of the current physical realizations of QSHIs [2, 14, 16], the regime in which λimp ∆ is not at all unrealistic as the size of the band gap is typically rather small [2, 3, 14–16], and its size can be tuned close to the topological transition. In addition, in two-dimensional materials, localized moments can appear e.g. from dangling bonds at vacancies [44], rather than from magnetic dopants alone. Based on the analysis provided here, we believe that the presence of such localized magnetic defects in proximity to the edge of the recently observed can induce significant backscattering in the newly observed QSHI in the 1-T phase of WTe2 . The mechanism described here provides additional backscattering sources to accounts for the experimentally observed [14, 16] deviations from conductance quantization at low temperatures. Indeed, if the chemical potential of the edge electrons happens to be at (or near) resonance with in-gap states induced by a magnetic dopant, tunneling in/out of the in-gap states will suppress conductance through the edge channel more ef- 8 fectively than ordinary backscattering (for comparable strength of the bare backscattering yB , vB and tunneling yt dimensionless couplings, cf. Eqs. 32 to 35). This is because tunneling in/out of the (nearly resonant) in-gap state is a more relevant perturbation than backscattering, as manifested by its smaller scaling dimension (i.e. typically ∆(K, UF ) < K, cf. Eq. 29), for both repulsive and moderately attractive interactions. A more detailed analysis relevant to this system will be reported in a future publication. Furthermore, in the future we also plan to study extensions to the model studied here beyond the dilute impurity regime (i.e. the multi-impurity case). Another interesting direction is to treat the spin degrees of the magnetic impurity quantum mechanically. This is especially important to describe spin- 1 impurities below 2 the Kondo temperature. Finally, another interesting research direction, relevant to the study of Majorana bound states, is to the study of the competition of the type of magnetic disorder considered here and the proximity to a nearby s-wave superconductor [52]. We thank L. Glazman, T. Giamarchi, F. Guinea, Y.-H. Ho, C.-L. Huang, and S.-Q. Shen, X.-P. Zhang for useful discussions. M.A.C. gratefully acknowledges support by the Ministry of Science and Technology (Taiwan) under Contract No. 102- 2112-M-007-024-MY5, and Taiwan’s National Center of Theoretical Sciences (NCTS). pseudo-spin space corresponding to the sublattice (B, A) components of the single-particle spin wave function and ˆ dx = −t(2 cos α + cos β), s y ˆ d = −t sin β, s dz s ˆ = λv + sλSO (2 sin 2α − 4 sin α cos β), (A2) (A3) (A4) √ 3a ˆ with α = kx a/2, and β = −i 2 ∂y . We set a = 1 for ˆ simplicity. In addition, we treat β as an operator and β as its eigenvalues. Substituting Φs (kx , y) = χs eκy to Eq. (7), we get the following secular equation: Xf 2 + Y f + Z = 0, where the variables √ 3κ f ≡ cosh , 2 X = −(4λSO sin (A5) (A6) kx 2 ) , 2 (A7) kx kx (λv + 2sλSO sin kx ) − 4t2 cos , 2 2 (A8) kx 2 2 Z = 2 − t2 − 4t2 (cos ) − (λv + 2sλSO sin kx ) . 2 (A9) Y = 8sλSO sin Appendix A: Spectrum and wavefunctions Hence, we obtain the following two roots: Here we provide the analytical approach to solve for the spectrum and the wavefunctions of both bulk and edge states for a generalized Kane-Mele (KM) model [7], c† cj i ˆ H0 = −t i,j νij c† sz cj i − iλSO ξi c† ci . i + λv i,j i (A1) where a staggered potential with ξi = +1 for Ri ∈ B and ξi = −1 for Ri ∈ A sublattice has been included for generality. As mentioned in the main text, there is a gauge degree of freedom for the Fourier transformation of c† (or ci ) due to the bi-particle structure of the lattice. ˆi ˆ The gauge freedom allows to effectively shift the lattice yielding different geometries for the edge. Besides of zigzag edge of interest in the main text, it is also interesting to consider the beard edge in parallel. The correspond to two different gauge choices: √ 1) Zigzag edge: rg = −(a/2 3)ey and 2) Beard edge: √ rg = (a/ 3)ey . In our convention, σ z = (+, −) denotes the sublattice pseudo spin components corresponding to the (B, A) sublattices. a. Spectrum of edge states For the case with zigzag edge, after the Fourier transs ˆ formation, we obtain H0 (α, β) = i di σ i , we have used s the notation where Pauli matrices σ i (i = x, y, z) is in the f1,2 = (−Y ± Y 2 − 4XZ)/2X, (A10) Thus, there are four roots for κ, corresponding to ±κ1,2 2 with κ1,2 = √3 cosh−1 f1,2 . For the edge states, we have that Re κ1,2 = 0. Thus, we use the convention that 2 Re κ1,2 > 0 for the function κ1,2 = √3 cosh−1 f1,2 . Note that only two linearly independent wavefunctions satisfy the open boundary conditions corresponding the zigzag edge, namely Φs (kx , ±L/2) = 0 for each value of . They are 1 2 gc (kx , y) − gc (kx , y), (A11) 1 gs (kx , y) (A12) − 2 gs (kx , y), where [45] cosh(κi y) , cosh(κi L/2) sinh(κi y) i gs (kx , y) = . sinh(κi L/2) i gc (kx , y) = (A13) (A14) The eigenfunctions can be expressed as the linear combination of the above wavefunctions. By introducing a 2 × 2 matrix of coefficients L = [lij ], the eigenfunctions can be written as follows: ï 1 ò 2 gc (kx , y) − gc (kx , y) Φs (kx , y) = L 1 . (A15) 2 gs (kx , y) − gs (kx , y) 9 i Substituting this function into Eq. (7), and using that gc,s are linearly independent, we obtain the following conditions relating the column vectors of the matrix L: L2 = tanh(κ1 L/2)M1 L1 , L2 = tanh(κ2 L/2)M2 L1 , 1 L1 = M1 L2 , tanh(κ1 L/2) 1 M2 L2 , L1 = tanh(κ2 L/2) and D0 = t2 (2 cos α + cos β1 )(2 cos α + cos β2 ) + (λv + 2sλSO sin 2α − 4sλSO sin α cos β1 )(λv + 2sλSO sin 2α − 4sλSO sin α cos β2 ) − 2 . For L → ∞, T = 1 because Re κ1,2 > 0. Thus, it becomes (A16) (A17) 2 2 2 Dx + Dy + Dz = 0, (A18) which gives the dispersion (A19) ± s where L1 = (l11 , l21 )T , =± t(λv + 6sλSO sin 2α) . t2 + (4λSO sin α)2 (A32) (A20) T L2 = (l12 , l22 ) , Mi = σ y {(−2t cos α − t cos βi ) σ x + (λv + 2sλSO sin 2α −4sλSO sin α cos βi ) σ z − } /(t sin βi ), √ 3 κi , βi = −i 2 (A21) Note that Eq.(A24) gives an additional constraint for the spectra. From Mt L2 = 0 and T = 1, we obtain (A22) (Dx σx + Dy σy + Dz σz )L2 = 0. (A23) Combining it with L2 = M1 M2 L2 (Eq.(A24)), we obtain the following the constraint: respectively. Note that, in the above derivation, we have i ˆ i used the fact cos βgc,s (kx , y) = cos βi gc,s (kx , y), i i ˆ sin βgc (kx , y) = sin βi tanh(κi L/2)gs (kx , y) and sin i i ˆ i sin βgs (kx , y) = tanh(κβL/2) gc (kx , y). i The combinations of equations in the same column in Eq.(A16) give the secular equation (A5), which relates κi and spectrum . The other two independent equations are obtained by combining diagonal terms in Eq.(A16), which yields: L2 = T M1 M2 L2 = 1 M2 M1 L2 , T (A24) where T = tanh(κ1 L/2) . tanh(κ2 L/2) (A25) Expressing κi as functions of , this equation is exactly the constraint for spectrum . In the following, we will solve this equation. Eq.(A24) implies that Mt L2 = 0, (A26) 1 where Mt ≡ T M1 M2 − T M2 M1 . To have a nontrivial solution for L2 , the condition det Mt = 0 is required, which gives (T + (A31) 1 2 2 2 2 2 2 ) = 4D0 /(D0 − Dx − Dy − Dz ), T (A27) (A33) − D0 = t2 sin β1 sin β2 . (A34) The second constraint is Re κ1,2 > 0. These constraints restrict a region kx ∈ [Λ− , Λ+ ], where edge states exist. s s We show the resulted spectra in Fig. 5. b. Wavefunctions for a semi-infinite system To investigate the wavefunctions at one of edges only, it is helpful to shift the coordinate origin so that the QSHI occupies the upper half plane (0 ≤ y ≤ L with L → ∞). For a semi-infinite system, the wavefunction satisfying the boundary condition Φskx (0) = Φskx (L → ∞) = 0 has a much simpler form: Φskx (y) = Cs (kx )χs (kx )(e−κs,1 y − e−κs,2 y ). (A35) Thus, what is left is to determine the 2 × 1 matrix χs (kx ) and the normalization factor Cs (kx ). For each kx , we have obtain the spectra s and the wave number 2 κs,i = √3 cosh−1 fs,i with i = 1, 2 in the last section. Substituting Eq. (A35) into the Schr¨dinger Eq.(7), we o obtain where Dx = t(cos β1 − cos β2 ) , Dy = it(λv + 6sλSO sin 2α)(cos β1 − cos β2 ), Dz = 4sλSO sin α(cos β1 − cos β2 ) , (A28) (A29) (A30) χs (kx ) = s s − [H0 ]12 / {[H0 ]11 − 1 s} ≡ χs,1 . 1 (A36) Explicitly, 10 FIG. 5: Band structure of a √ zigzag ribbon described by Eq. (A1). Left panel: √ SO = 0.06t and λv = 0; Central panel: λ λSO = 0.06t and λv = 0.1t < 3 3λSO ; Right panel: λSO = 0.06t and λv = 0.4t > 3 3λSO . χs,1 √ x 2t cos k2 + t exp( 3κs,1 /2) √ = λv + 2sλSO [sin kx − 2 sin(kx /2) cosh( 3κs,1 /2)] − Recall that the above wavefunctions only make sense when evaluated on the discrete set of points of the honeycomb lattice: ψskx (n) = Cs (kx )χs (kx )(e−κs,1 ny0 − e−κs,2 ny0 ), (A38) √ where y0 = 3a/2. The normalization factor is 0 Cs (kx ) = [Υ (2 Re κs,1 ) + Υ (2 Re κs,2 ) −Υ κ∗ + κs,2 − Υ κs,1 + κ∗ s,1 s,2 −1/2 (A40) For the bulk states with periodic boundary conditions, crystal momentum k = (kx , ky ) is treated as good quantum number in both the x-direction and y-direction. √ 3k Thus, upon setting κ = iky in Eq. (A5) (with β = 2 y ), we obtain the (bulk) dispersion: » t2 + 4t2 cos α cos β + 4t2 cos2 α + [λv + 2sλSO (sin 2α − 2 sin α cos β)]2 , where s, η = ±1. However, for open boundary conditions and in the limit L → ∞, the spectrum of bulk state is not modified from the above form because the boundary effects become negligible in the thermodynamic limit. On other hand, wavefunctions are modified and become different from Bloch waves because of the scattering with the boundary. Thus, from the secular equation (A5), for each κ1 = iky (ky is √ 3k Wavefunctions for Bulk States (A39) where Esη (k) = Esη (kx , ky ) = η and Υ(k) ≡ 1/[1 − exp(−ky0 )]. Upon denoting Φskx ,σ (r) as the σ components of Φskx (r), we find |Φskx ,+ (r)|2 |Φskx ,− (r)|2 for the case λv = 0 and λSO t, which suggesting the bottom edge states ‘prefer’ B-sublattice. c. 2 0 Cs (kx ) = (1 + |χs,1 | )−1/2 Cs (kx ) (A37) s real) and thus f1 ≡ cos 2 y , we can find another root, Y f2 = − X − f1 . In total four different roots for κ ex2 ist, i.e. ±κ1,2 with κ1,2 = √3 cosh−1 f1,2 , corresponding to a same energy . Note that f1 and thus f2 are real. Thus there are two different cases: 1) |f2 | > 1, the plane wave decays at the edge; and 2) |f2 | ≤ 1, different modes interference with each other: (A41) 2 Case 1: For |f2 | > 1, we have κ2 = √3 cosh−1 f2 with Re κ2 > 0. Thus the full solutions of the secular equation (A5) for κ are ±iky and ±κ2 . The mode ∼ eκ2 y diverges for y → ∞, so it will not emerge and there are only there modes left: e±κ1 y and e−κ2 y . After using the boundary condition Φsη,k (y = 0) = 0, only two linear independent wavefunctions are left. The general wavefunction has the following form: Csη (k) exp(iky y) − exp(−κ2 y) L , exp(−iky y) − exp(−κ2 y) Ny (A42) where L = [lij ]2×2 is a 2×2 matrix, and Csη (k) is the normalization constant. Obviously, such a kind of wavefunction is a combination of extended state and local state, which decays at the edge. Φsη,k (y) = 11 Now we need to calculate out the matrix L. Substituting Eq.(A42) into Schr¨dinger equation (7), and using o the fact that exp(±iky y) and exp(−κ2 y) are linear independent, we obtain the following results: l1 , 1 L2 = c2 L1 + L2 = ∗ l1 , 1 l2 , 1 L1 = c1 Csη (k ). Note that these two eigenstates are not orthogonal. In the following, we construct an orthogonal and symmetric basis by means of |+ = |1 + ϑ |2 , |− = |2 + ϑ |1 . (A52) (A43) Using the orthogonality condition together with 1 |1 = 2 |2 = 1, we obtain (A44) 2 |ϑ| = 1, Re ϑ = −Re 1 |2 , (A53) 2 where 1 |2 = Csη (k)Csη (k )[−c∗ (|l1 |» +1)−c2 (|l2 |2 +1)]. 2 where 2 L1 = (l11 , l21 )T , (A45) L2 = (l12 , l22 )T , s [H0 (α, βi )]12 li = − s [H0 (α, βi )]11 − Esη (k) We use the convention that Im ϑ = 1 − (Re ϑ) ≥ 0, and finally, we obtain the orthonormalized wavefunctions (A46) √ (A47) Φsηk (n) = √ and c1 , c2 are constants, β1 = 23 ky , β2 = i 23 κ2 . Solving l2 −l∗ these equations, we find c1 = l1 −l1 and c2 = −l2 +l1 . ∗ ∗ l1 −l1 1 The next step is to calculate the normalization coefficient Csη (k). For large L limit, exp(−κ2 y) does not influence normalization. Using the orthogonality of exp(±iky y), we obtain 1 » Csη (k) = » . 2 2 2 |c1 | + |c2 | |l1 | + 1 (A48) k,sη Φsη,k (n) = |k, s, η k, s, η| + + i0+ − Esη (k) kx ,s |kx , s kx , s| , + i0+ − s (kx ) (B1) where |k, s, η = Φsηk . In the real space, ˆ Gs 0,σσ (r, r , ) = r,s, σ| G0 ( ) |r ,s, σ , (B2) where σ = ±1 represents the different components of σ z . Thus, using r,s, σ |k, s, η = Φsηk,σ (r) and r,s, σ |kx , s = Φskx ,σ (r), where Φsηk,σ (r) and Φskx ,σ (r) are the σ components of Φsηk (r) and Φskx (r) respectively, we have Gs 0,σσ (r, r , ) = k,η 1 exp(iky y) − exp(−iky y) |1 = Csη (k)L , exp(−iky y) − exp(−iky y) Ny (A50) where L is same as the one in Eq. (A42) except for the replacement of κ2 with iky and thus the normalization becomes: 1 . (|c1 |2 + |c2 |2 )(|l1 |2 + 1) + (|l2 |2 + 1) (A51) The second eigenstate |2 can be obtained by the replace∗ ments: ky → ky (which implies that l2 → l1 ). We denote the corresponding parameters as L1 , L2 , c1 , c2 and (A55) So far, we have obtained the eigenvalues and the complete set of eigenfunctions for the model of Eq. A1. Hence, the Green’s function can be expressed in terms of them: ˆ G0 ( ) = Csη (k) = (A54) Appendix B: Green’s function for the Kane-Mele model in a semi-infinite system As a result, in real space, we have Csη (k) exp(iky ny0 ) − exp(−κ2 ny0 ) L , exp(−iky ny0 ) − exp(−κ2 ny0 ) Ny (A49) 2 Case 2: For |f2 | ≤ 1, we have κ2 = √3 cosh−1 f2 = iky with ky ≥ 0. The full solutions of the secular equation (A5) for κ are ±iky and ±iky . The boundary conditions Φsη,k (y = 0) = 0 require these four running waves inference with each other, and thus there are only three linear independent wavefunctions. Following the method used in previous case, we can construct the eigenfunctions by combining the three wavefunctions. However, we shall proceed in a different way here. Similar to the previous case, there is one eigenfunction, 1 |+ , 2 + 2Red [ϑ 1 |2 ] 1 |− . 2 + 2Re [ϑ 2 |1 ] Φsηk (n) = Φsηk,σ (r)Φ∗ sηk,σ (r ) + i0+ − Esη (k) + kx Φskx ,σ (r)Φ∗ x ,σ (r ) sk . + i0+ − s (kx ) (B3) Appendix C: Spectrum of the beard edge For comparison purposes, we also study the edge spectrum for√ beard edge. Using a gauge choice where the rg = (a/ 3)ey , and following the same steps as for the zigzag case, we obtain the spectrum for edge state: ± s =± t[2sλSO sin α + cos α(λv + 2sλSO sin 2α)] . (C1) (t cos α)2 + (2λSO sin α)2 12 FIG. 6: Band structure of a bearded-edge ribbon described by Eq. A1. Left panel: λSO = 0.06t and λv = 0; Central panel: √ √ λSO = 0.06t and λv = 0.1t < 3 3λSO ; Right panel: λSO = 0.06t and λv = 0.4t > 3 3λSO . In this case, the constraint becomes − D0 = 4t2 cos2 α sin β1 sin β2 , (C2) where 2 2 D0 = t w1 w2 + u1 u2 − , wi = 1 + 2 cos α cos βi , ui = λv + 2sλSO sin 2α − 4sλSO sin α cos βi . (C3) The resulting band structure is shown in Fig. 6. Note that the edge states intersect at kx = 0 [46]. ∂x φ (x) + ζπδ (x), the forward scattering term ∝ UF can be eliminated from Heff (cf. Eq. (27)), and the resulting Hamitonian, Heff = S † Heff S reads: vB Heff = H∗ + UR UL e2iφ0 + UL UR e−2iφ0 ξ Å ã 2yB 1 + d† d − UR UL e2iφ0 + UL UR e−2iφ0 ξ 2 ä yt î † Ä d UR e−i(φ0 −λθ0 ) − UL ei(φ0 +λθ0 ) + ξ Ä ä ó + UR ei(φ0 −λθ0 ) − UL e−i(φ0 +λθ0 ) d , (D5) where Appendix D: Renormalization group analysis H∗ = Next, in order to deal with the effects of interactions in a nonperturbative way, we shall rely upon the bosonization technique. The resulting model is analyzed along the lines of the analysis reported is Ref. 48. In bosonization the electron field operator for the right (R) and left moving (L) edge electron can be expressed in terms of a set of bosonic fields θ(x) and φ(x) as follows: UR(L) −i[±φ(x)−θ(x)] ψR(L) (x) = √ e , 2πvξ (D1) where ξ is a short-distance cutoff, v is the plasmon velocity (cf. Eq. 31), UR and UL are the so-called Klein factors satisfying {Ur , Ur } = 2δr,r , which allows to satisfy the anti-commutation relations between the two fermion chiralities R and L. The bosonic fields obey π [φ (x) , θ (x )] = i sgn(x − x). 2 (D2) The chiral densities are given by ρR(L) (x) = − 1 (∂x φ 2π ∂x θ) . (D3) After bosonizing the low energy effective model and upon applying a unitary transformation generated by S = exp [iζθ0 ] (D4) with ζ = δF d† d − 1 , δF = KUF , and using the fac2 πv tor e−iζθ(0) ∂x φ (x) eiζθ(0) = ∂x φ (x) − iζ [θ (0) , ∂x φ (x)] = v 2π î ó 2 2 dx K (∂x θ) + K −1 (∂x φ) Å ã 1 † − ε0 d d − . 2 (D6) Here ε0 denotes the distance of the bound state from the Fermi energy of the edge channel, F . In what follows we focus on the resonant case for which 0 = 0. In addition, λ = 1 − δF , φ0 = φ(x = 0), θ0 = θ(x = 0), vB , yB , yt are dimensionless couplings, and K is the Luttinger parameter and v is the edge plasmon velocity. Using Cardy’s approach [53] and taking into account that ¨ ∂ e2iφ0 (τ ) e−2iφ0 (0) ∼ |τ |−2K , (D7) ¨ ∂ i[φ0 (τ )−λθ0 (τ )] −i[φ0 (0)−λθ0 (0)] −α(K,λ) e e ∼ |τ | , (D8) α(K, λ) = K λ2 K −1 + , 2 2 (D9) we arrive at the set of RG equations valid to second order in the couplings describing backscattering and tunneling in and out of the resonant level given in (32)-(35). The RG equations are similar to those derived in Ref. 48 for a model of a resonant level that is side-coupled to an interacting 1D electron system. As described in the main text, the equations show that for weak to moderate attractive interactions (i.e. K 1), the tunneling operator ∝ yt is flows to strong coupling. On the other hand, both the backscattering interaction (∝ yB ) and potential (∝ vB ) will be initially suppressed. 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