Suppressing backscattering of helical edge modes with a spin bath
SSuppressing backscattering of helical edge modes with a spin bath
Andrey A. Bagrov, ∗ Francisco Guinea,
2, 3, † and Mikhail I. Katsnelson ‡ Institute for Molecules and Materials, Radboud University,Heijndaalseweg 135, 6525 AJ, Nijmegen, The Netherlands IMDEA-Nanoscience, Calle de Faraday 9, E-28049 Madrid, Spain School of Physics and Astronomy, University of Manchester, Manchester M13 9PY
In this paper, we address the question of stability of protected chiral modes (e.g., helical edgestates at the boundary of two-dimensional topological insulators) upon interactions with the externalbath. Namely, we study how backscattering amplitude changes when different interaction channelsbetween the system and the environment are present. Depending on the relative strength of theCoulomb and spin-spin channels, we discover three different possible regimes. While the Coulombinteraction on its own naturally amplifies the backscattering and destroys the protection of chiralmodes, and the spin-spin channel marginally suppresses backscattering, their interplay can makethe backscattering process strictly irrelevant, opening the possibility to use the external spin bathas a stabilizer that alleviates destructive effects and restores the chirality protection.
Topological insulators (TI) are characterized by exis-tence of protected helical edge states, - one-dimensionalchiral modes at the edges of two-dimensional TI, andtwo-dimensional massless Dirac fermions at the surfacesof three-dimensional TI [1–7]. This is a manifestation ofa very general “bulk-edge correspondence” principle [8–11] which is probably one of the brightest applications oftopological and geometrical concepts in condensed mat-ter physics. Importantly, topological protection of theedge states is not absolute: they can be broken by spin-dependent scattering mechanisms such as scattering onmagnetic impurities [6, 12–14] or electron-electron inter-actions [15]. These factors result in the backscatteringand destruction of the helical modes, due to the inti-mate relation between their propagation direction anddirection of spins: if one flips the spin, one reverses themomentum. This effect has been considered from manyperspectives, and a variety of its possible physical conse-quences on the transport and spectral properties of heli-cal channels have been studied (see e.g. [16, 17]).Because of the importance of practical implications ofchiral edge modes, it is interesting to think of possibleways to reduce (or even eliminate) backscattering andmake the edge modes more stable. To achieve this goal,we suggest to couple the channel to a spin environment.While environment consisting of static spin degrees offreedom acts as a set of magnetic impurities that induceand amplify backscattering [18, 19], the physics of fastitinerant spins can be very different, as known in thetheory of magnetice resonance [20, 21]. According to thepopular decoherence program in quantum physics [22, 23](for the recent critical discussion of this program, see[24]), instant interactions between the environment andthe channel can be thought of as effective von Neumannprojective measurements that tend to make the spinsclassical via the “orthogonality catastrophe”: the envi-ronment degrees of freedom get entangled with spin-upand spin-down states of the system, and the small over-lap of corresponding wavefunctions suppresses the am- plitude of spin-flip processes [25–27]. A related model,where dissipation induces decoherence in a Luttinger liq-uid, has been studied in [28]. For the edge modes, thiswould mean stabilization of the states with definite mo-menta; in terminology of Zurek [22], they appear to be“pointer states” robust with respect to the interactionwith the environment. This situation looks unusual: inmost of the cases the interactions between the central sys-tem and the environment are much stronger dependenton the coordinates than on the momenta, which tendsto stabilize the states with definite coordinates, i.e. lo-calized in real space, rather than the states with defi-nite momenta [23]. Here we provide a formal analysis ofthe effect the environment has on the backscattering ofhelical states, using the renormalization group approachsimilar to the one used in [26, 29–32]. It turns out that,depending on the ratio of the exchange and direct inter-actions, the environment can both suppress and enhancethe backscattering.We start with the following one-dimensional s − d model, which, albeit simple, captures all the relevant as-pects of more complicated and peculiar systems: H = (cid:88) k c † ( k ) H c ( k ) c ( k ) + (cid:88) k ; i =1 , d † i ( k ) H di ( k ) d i ( k ) − (1) J (cid:88) q (cid:32)(cid:88) k c † ( k ) (cid:126)σc ( k + q ) (cid:33) (cid:88) p ; i =1 , d † i ( p ) (cid:126)σd i ( p − q ) , where (cid:126)σ are the Pauli matrices, k is the one-dimensionalspatial momentum, and the standard notation is used: (cid:88) k = π/a (cid:90) − π/a adk π , (2)where a is the lattice constant. Here c ( k ) and d , ( k ) arethe chiral edge modes of topological insulator and the a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l environment degrees of freedom, respectively: c ( k ) = ( c ↑ ( k ) , c ↓ ( k )) , (3) d i ( k ) = ( d ↑ i ( k ) , d ↓ i ( k )) , and the Hamiltonians of each sector are given by H c ( k ) = (cid:18) (cid:126) v F k h h − (cid:126) v F k (cid:19) , (4) H d , ( k ) = (cid:18) ± (cid:126) ck ± (cid:126) ck (cid:19) . (5)Since there is no preferred helicity in the environment,we take into account both right-moving ( i = 1) and left-moving ( i = 2) particles, and represent them for sim-plicity as two independent fermionic flavors. The barebackscattering is introduced via the off-diagonal term h of the edge modes Hamiltonian. A concrete mechanismthat induces backscattering is not important for our con-siderations.In what follows, we will analyze how the parametercontrolling backscattering changes due to the interac-tions with the spin environment, relying upon pertur-bative renormalization group approach [26, 29–32]. As itwill be evident, other interaction channels will be inducedon top of the isotropic spin-spin interaction introducedin the Hamiltonian (1), and it turns out to be conve-nient to include them into the original Hamiltonian as ageneralized vertex: H int = Γ ( i ) αβγδ (cid:88) q,p,k c † α ( k ) c β ( k + q ) d † i,γ ( p ) d i,δ ( p − q ) , (6)Γ ( i ) αβγδ = J ( i )00 I αβ ⊗ I γδ + J ( i ) zz σ zαβ ⊗ σ zγδ + (7) J ( i ) (cid:16) σ xαβ ⊗ σ xγδ + σ yαβ ⊗ σ yγδ (cid:17) + J ( i )0 z I αβ ⊗ σ zγδ + J ( i ) z σ zαβ ⊗ I γδ , where we also added the Coulomb channel J , the spin-charge channels J z and J z , and the possible anisotropybetween Z and XY spin couplings. This reduces to theisotropic spin interaction of (1) if J ( i )00 = J ( i )0 z = J ( i ) z = 0 , J ( i ) zz = J ( i ) = J. (8)To make the notations more handy and reduce the num-ber of indices, hereinafter we denote the coupling con-stants J (1) as plain J , and J (2) as ˜ J .As we elaborated in the introduction, we expect thespin-spin interactions between the edge of the topologi-cal insulator and the bath to make pointer states of thesystem to be states with well-defined spin, and thus sta-bilize the helical modes. In terms of the renormalizationgroup flow for the model (1), (6), it means that the mode-mixing parameter h is expected to become irrelevant inthe infrared.The leading order quantum correction to h is givenby off-diagonal part of the two-loop self-energy diagram FIG. 1. Self-energy correction to the helical edge modes.Wavy lines denote the propagators of the environment modes.Latin letters stand for x, y, z , and the Greek ones denote thespin indices. Sum over all combinations of
A, B, C, D allowedby the structure of vertex (7) has to be taken. shown in Fig. 1 (from now on all calculations will beconducted for the Matsubara Green’s functions): G − c ( iω, k ) = G − c, ( iω, k ) − Σ( iω, k ) , (9) h ( iω, k ) = h + Σ ( iω, k ) , where the bare Green’s function of edge fermions is re-lated to their Hamiltonian (4) as G − c, ( iω, k ) = iω · I − H c ( k ) . (10)The polarization loop is given by a simple integral:Π AC , ( p ) = π/a (cid:90) − π/a adq π ∞ (cid:90) −∞ dω q π Tr (cid:2) σ A G d , ( iω p + iω q , p + q ) · (11) σ C G d , ( − iω q , − q ) (cid:3) = apπ ( ∓ iω p + (cid:126) cp ) δ AC To obtain the self-energy correction, we need to sum overall possible combinations of
A, B, C, D indices in Fig. 1that give non-trivial contributions, as well as over thetwo flavors of the environment modes. The resulting ex-pression at zero external momentum isΣ (0 ,
0) = − π/a (cid:90) − π/a adq π ∞ (cid:90) −∞ dω p π ahq ( h + ω q + (cid:126) v F q ) · (cid:32) α ( J ) iπω q − π (cid:126) cq − α ( ˜ J ) iπω q + π (cid:126) cq (cid:33) , (12)where we introduced α ( J ) = J + J z − J z − J zz . (13)Although there is a natural ultraviolet (UV) cut-off givenby the lattice constant a , it is convenient to formally con-sider the momentum integral over the second loop as log-arithmically divergent in the a → FIG. 2. Vertex correction to the coupling matrices Γ (1 , . extract the leading scaling that defines the renormaliza-tion group flow. Evaluating the integral over frequenciesvia residues, and then expanding the integrand around | q | → ∞ , we obtain the correction to backscattering am-plitude from a thin momentum shell | q | ∈ [Λ , Λ + d Λ]: h (Λ + d Λ) = h (Λ) + δ Σ = h (Λ) − (14)24 π d Λ (cid:90) Λ a h ( q )[ α ( J ) + α ( ˜ J )] dq (cid:112) h ( q ) + (cid:126) v F q (cid:16) (cid:126) cq + (cid:112) h ( q ) + (cid:126) v F q (cid:17) = h (Λ) − π h (Λ) Λ+ d Λ (cid:90) Λ a [ α ( J ) + α ( ˜ J )] dq (cid:126) v F ( c + v F ) q , where the additional overall factor of 2 is due to integra-tion over both positive and negative momenta. That is,we obtain the corresponding flow equation: dhd log Λ = − a h π (cid:126) v F ( c + v F ) (cid:104) α ( J ) + α ( ˜ J ) (cid:105) , (15)If we ignore for a moment renormalization of other pa-rameters of the model, we can readily conclude: h (Λ) = h · (cid:18) ΛΛ UV (cid:19) γ , (16)where for further convenience we introduce a notationfor the exponent, as it serves as a good measure of the“irrelevance” of the process: γ = − a π (cid:126) v F ( c + v F ) (cid:104) α ( J ) + α ( ˜ J ) (cid:105) , (17)If only spin-spin interactions are present α ( J ) + α ( ˜ J ) = − J zz − ˜ J zz , (18)and the mode mixing is clearly irrelevant in the infraredlimit Λ → γ > (1 , , and the Fermi-velocities v F and c .Renormalization of the couplings is given by one-loopvertex diagram shown in Fig. 2. The correspond-ing momentum integral is also logarithmically divergent,and, omitting the intermediate steps similar to what wehave done when computed the backscattering amplituderenormalization, we arrive at the following system of RGflow equations: dJd log Λ = aπ (cid:126) ( c + v F ) J ( J + J z − J z − J zz ) d ˜ Jd log Λ = aπ (cid:126) ( c + v F ) ˜ J (cid:16) ˜ J − ˜ J z + ˜ J z − ˜ J zz (cid:17) dJ d log Λ = a π (cid:126) ( c + v F ) (cid:0) J + ( J − J z ) + ( J z − J zz ) (cid:1) d ˜ J d log Λ = a π (cid:126) ( c + v F ) (cid:16) J + ( ˜ J + ˜ J z ) + ( ˜ J z + ˜ J zz ) (cid:17) dJ z d log Λ = − aπ (cid:126) ( c + v F ) (cid:0) J − ( J − J z )( J z − J zz ) (cid:1) d ˜ J z d log Λ = aπ (cid:126) ( c + v F ) (cid:16) ˜ J + ( ˜ J + ˜ J z )( ˜ J z + ˜ J zz ) (cid:17) dJ z d log Λ = a π (cid:126) ( c + v F ) (cid:0) J − ( J − J z ) − ( J z − J zz ) (cid:1) d ˜ J z d log Λ = a π (cid:126) ( c + v F ) (cid:16) − J + ( ˜ J + ˜ J z ) + ( ˜ J z + ˜ J zz ) (cid:17) dJ zz d log Λ = − aπ (cid:126) ( c + v F ) (cid:0) J + ( J − J z )( J z − J zz ) (cid:1) d ˜ J zz d log Λ = aπ (cid:126) ( c + v F ) (cid:16) − ˜ J + ( ˜ J + ˜ J z )( ˜ J z + J zz ) (cid:17) (19)Fermi velocity renormalization comes from the diag-onal part of the self-energy diagram Fig. 1. Formallyspeaking, there are two different Fermi-velocities for the two edge chiral modes that renormalize independently: dv F d log Λ = − a π (cid:126) ( c + v F ) v F J , (20) d ˜ v F d log Λ = − a π (cid:126) ( c + ˜ v F ) ˜ v F ˜ J , but we can consistently assume symmetry between them,and impose J = ˜ J , v F = ˜ v F at all scales.In principle, we also have to derive the renormalizationgroup flow for the Fermi velocity c , but since v F (cid:29) c inthe cases of interest (when the discussed renormalizationof backscattering amplitude is strong), and they appearin 1 / ( v F + c ) combination, renormalization of the bathFermi velocity can be neglected.Now we are ready to solve flow equations (15), (19),(20) numerically in different regimes, and identify howthe backscattering of chiral modes is affected by the envi-ronment. To make numerical estimates, we need to agreeon the values of bare physical quantities. Fermi-velocityof the edge degrees of freedom in two-dimensional Bi Te topological insulators is measured to be v F (cid:39) · cm/s[33]. The spin bath velocity c is a free parameter thatcan be tuned to any value by choosing a proper environ-ment material, and we find the effect of backscatteringsuppression to be stronger when c is small, ∼ cm/s,i.e. when the bath is insulating. The in-plane lattice con-stant for Bi Te is a = 6 . h = 0 . • The Coulomb interaction is dominant: J = ˜ J = 0 . , (21) J = ˜ J = J zz = ˜ J zz = 0 . (22)The energy gap in Bi T e is ∆ E (cid:39) . • Spin-spin channel is dominant: J = ˜ J = J zz = ˜ J zz = 0 . , (23) J = ˜ J = 0 . (24)While this case seems quite special since normallythe Coulomb interactions are stronger than the s − d exchange, it is instructive to consider this regimeas it shows a possibility to use the environment tosuppress backscattering and enhance protection ofthe chiral edge modes. • Spin and Coulomb interactions are comparable: J = ˜ J = J zz = ˜ J zz = J = ˜ J = 0 . − log(Λ / Λ UV ) h / h CoulombCompetingSpin − log(Λ / Λ UV ) γ FIG. 3. RG flows of the backscattering amplitude. The bluecurve depicts the Coulomb-interaction dominated case, theyellow one - the case of dominant spin-spin interaction chan-nel, and the green one - the regime of interplay. Inset: thecorresponding flows of the “irrelevance” parameter γ . other induced interaction channels is highly non-trivial.At intermediate energies, if the two competing channelsare present, Coulomb reduces the effect of suppressing.However, if one goes to lower energies, it assists the spininteractions in suppressing the process of backscatter-ing, and makes h flowing to zero even when capacity ofthe spin channel is exhausted, and renormalization of h stopped. Another way to see this is to look at the insetof Fig. 3, where renormalization of the exponent of (16)is shown. Though Coulomb interaction decreases the ini-tial value of the “irrelevance” exponent γ , deep in the in-frared it prevents γ from flowing to zero. The differencebetween the two regimes looks rather mild, but since thecoupling constants flow towards strong coupling in theinfrared, the leading order perturbative analysis tends tounderestimate renormalization of γ , and a stronger effectcan be expected.In this paper, by deriving the leading order pertur-bative renormalization group flow equations, we havestudied how interactions with environment affect thebackscattering of chiral modes in helical edge channels.We have discovered that the interplay of the Coulomband spin-spin interactions between the modes and theenvironment leads to a non-trivial phase diagram. Dom-inance of the Coulomb interaction expectedly leads toamplification of the backscattering, making chirality ofthe propagating modes poorly defined. If only the spin-spin interaction channel is present, the backscatteringgets marginally suppressed along the RG flow, receiving afinite negative correction to its bare amplitude. The mostinteresting situation is when both the interaction chan-nels are at work. Then the Coulomb interaction assiststhe spin-spin one in suppressing backscattering, makingit rather relevant than marginal. The conducted analysisallows us to conclude that the external bath of itinerantspins can be not only dangerous for the chirality of modesin the channel, but also, in certain regimes, can serve as astabilizer and alleviate the destructive effect of backscat-tering, restoring the protection of the chiral modes. Fora particular example of archetypical 2D topological in-sulator Bi Te , we have estimated that interactions withenvironment can reduce the backscattering amplitude to ∼
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