Spatial structure of correlations around a quantum impurity at the edge of a two-dimensional topological insulator
SSpatial structure of correlations around a quantum impurity at the edge of a two-dimensionaltopological insulator
Andrew Allerdt, A. E. Feiguin, and G. B. Martins
2, 3, ∗ Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA Department of Physics, Oakland University, Rochester, MI 48309, USA Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi, RJ, Brazil
We calculate exact zero-temperature real space properties of a substitutional magnetic impurity coupled tothe edge of a zigzag silicene-like nanoribbon. Using a Lanczos transformation [Phys. Rev. B , 085101(2015)] and the density matrix renormalization group method, we obtain a realistic description of stanene andgermanene that includes the bulk and the edges as boundary one-dimensional helical metallic states. Our re-sults for substitutional impurities indicate that the development of a Kondo state and the structure of the spincorrelations between the impurity and the electron spins in the metallic edge state depend considerably on thelocation of the impurity. More specifically, our real space resolution allows us to conclude that there is a sharpdistinction between the impurity being located at a crest or a trough site at the zigzag edge. We also observe,as expected, that the spin correlations are anisotropic due to an emerging Dzyaloshinskii-Moriya interactionwith the conduction electrons, and that the edges scatter from the impurity and “snake” or circle around it. Ourestimates for the Kondo temperature indicate that there is a very weak enhancement due to the presence ofspin-orbit coupling. PACS numbers: 72.10.Fk,72.15.Qm,73.22.-f,75.30.Hx,75.70.Tj
I. INTRODUCTION
A revolution is underway in the study of two-dimensional(2d) materials. Since the mechanical exfoliation of graphenefrom graphite was achieved in 2004 [see Ref. 2 for a detailedreview on graphene properties], many resources have been in-vested in the synthesis of other monolayer systems. These ef-forts have been rewarded through the discovery of many new2d compounds that are either stable in free-standing form orgrown in a substrate platform [for a comprehensive review,see Ref. 3]. Among these new 2d materials, a few raise addi-tional possibilities linked to the presence of non-trivial topo-logical phases such as silicene, germanene, and stanene. Asa result of the spin-orbit interaction (SOI), they present bulk-gapped phases with metallic helical (spin-momentum locked)edge states [for a review, see Ref. 4]. The locking of spinand momentum leads to suppression of elastic backscatteringof the helical electrons in the absence of spin-flip processes.This, in turn, leads to the possibility of dissipationless spinpolarized currents (see Refs. 5–7 for comprehensive reviews).Besides the interest in the basic physics associated to topo-logical phases in condensed matter materials (either 2d or 3d),the recent explosion of work in this field is also due to possibleapplications of topological insulators (TIs) in spintronics .For instance, strong spin-transfer-torque effects have recentlybeen observed at room temperature in a 3d TI .The search for new paradigms in electronics has spurredinterest in other quantum phenomena at the nanoscale: elec-tron correlations offer the promise of new functionality in na-noelectronics related to the possible manipulation of many-body ground states in suitably produced nanostructures suchas quantum dots and nanoribbons . An important exam-ple that has attracted a great deal of attention is the Kondostate, realized by coupling a magnetic impurity to conduc-tion electrons [see, for example, section VII of Ref. 14 fora recent review of the Kondo effect in carbon nanotubes]. In addition, Kondo physics offers the possibility of probing thespin-texture surrounding the magnetic moment, and to gainvaluable information about the effect of magnetic interactionsover the metallic surface state .A brief review of the Kondo effect in TIs (with a focus onquantum critical behavior) can be found in Ref. 16. Initialexperimental work concentrated on bulk-doped 3d TIs ,but since many TI-based devices require the surface of theTI to be in contact with ferromagnets, experiments were alsodone involving surface deposition of magnetic impurities in3d TIs .In a theoretical study of the Kondo effect at the surface of a3d TI, ˇZitko found that the Hamiltonian of a quantum mag-netic impurity coupled to metallic topological surface statesmaps into a conventional pseudogap single-channel Andersonimpurity model with SU(2) symmetry. It was also pointedout that despite the relatively trivial nature of the low-energyHamiltonian, the screening Kondo cloud should possibly dis-play a rather complex structure, which would be reflected innon-trivial spatial dependencies of the spin correlations be-tween the magnetic impurity and the topological surface statesinvolved in the Kondo-singlet formation.Motivated by Fourier-transform scanning tunneling spec-troscopy measurements done in Iron-doped Bi Te , whichanalyzed the energy-dependent spatial variations of the localdensity of surface states in terms of quasiparticle interference(QPI), Mitchell et al. conducted numerical simulations of asingle Kondo impurity on the surface of a 3d TI to identify thesignatures in QPI of the Kondo interaction between the heli-cal metal and the impurity. The QPI simulation results werefound to be markedly different from those obtained for non-magnetic or static magnetic impurities.The influence of the SOI on the Kondo effect in 2d TIs hasbeen addressed in several numerical works , particularlyfocusing on the behavior of the Kondo temperature. In overallagreement with Ref. 25, Isaev et al. demonstrated that strong a r X i v : . [ c ond - m a t . s t r- e l ] J un SOI leads to an unconventional Kondo effect (despite beingof the SU(2) kind) with an impurity spin screened by purelyorbital motion of surface electrons. At low energies, the im-purity spin forms a singlet state with the total electron angularmomentum, and the system exhibits an emergent SU(2) sym-metry, which is responsible for the Kondo resonance.Quantum Monte Carlo (QMC) simulations applied to asingle impurity Anderson model in a zigzag graphene edgeanalyzed the influence of spin-momentum locking over theKondo state and found indications of a broken spin rotationsymmetry in spin-spin correlation functions. A limitationof this effort, stemming from constraints associated to theQMC method, is the high value of the minimum temperatureachieved ( ≈ for graphene).In this work, we present results for the spin correlationsaround a substitutional Anderson impurity at the edge of asilicene-like topological insulator (more specifically for ger-manene and stanene, see Table I), with full spatial resolutionof the lattice. To the best of our knowledge, a detailed studyof these correlations in the Kondo ground state has not beenattempted so far.The structure of this article is as follows: In Sec. II wepresent the model and a very brief description of the numeri-cal method used (for details the reader is referred to previouspapers by the authors ). In Sec. III A we present the bandstructure for a zigzag nanoribbon (ZNRB) in the TI phase, fo-cusing on the edge states. Section III B presents local densityof states (LDOS) results for sites at the edge of the ZNRB,showing how it changes with SOI compared to the LDOS ofbulk sites. Section III C shows results for spin correlationsbetween the localized magnetic moment and the conductionspins. We analyze the cases of a substitutional impurity sit-ting at either crest or trough sites (as defined is Sec. II), andstudy the effects of spin orbit coupling. An analysis of the in-fluence of the SOI on the Kondo temperature is performed inSec. III D. The paper closes with a summary and conclusions. II. MODEL
The independent electron Hamiltonian H band describingthe 2-dimensional topological insulator corresponds to a tight-binding band structure that is appropriate for silicene, ger-manene, and stanene : H band = − t (cid:88) (cid:104) i,j (cid:105) σ c † iσ c jσ + i λ SO √ (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) σ σν ij c † iσ c jσ , (1)where c † iσ creates an electron in site i with spin σ (note thatthe letter σ stands for σ = ↑↓ when used as a subindex,and for σ = ± when used within equations). In addition, (cid:104) i, j (cid:105) runs over nearest-neighbor sites and (cid:104)(cid:104) i, j (cid:105)(cid:105) runs overnext-nearest-neighbor sites. The first term describes nearest-neighbor hoppings with transfer integral t . The second term isthe effective spin-orbit interaction with coupling λ SO , where ν ij = +1 if the next-nearest-neighbor hopping is anticlock-wise and ν ij = − if it is clockwise (in relation to the positive z -axis). The parameter values for silicene, germanene, and t (eV) v F a ( ˚A ) λ SO λ R θ Graphene 2.8 9.8 2.46 10 − . v F is in units of 10 m/s, λ SO and λ R (Rashba SOI) in meV. θ is the bond angle. Adapted fromRef. 4 (see also Ref. 38). stanene are given in Table I, where the corresponding valuesfor graphene are given for comparison (note that as graphenedoes not buckle, its λ SO value effectively vanishes, thus it isgapless and has no measurable topological properties, con-trary to silicene, germanene, and stanene). In Table I we alsolist the Rashba spin-orbit interaction for each material. Forthe sake of simplicity (and given its small value) we omitted itfrom our calculations. The unit of energy for all results shownhere is t .In accordance with Table I, the ratios λ SO /t for silicene,germanene, and stanene are . , . , and . . Wewill use in our calculations (unless stated otherwise) a valuethree times larger than the one for germanene, i.e., λ SO =0 . . Therefore, our results should describe the Kondo effect ingermanene and stanene. Due to its much smaller SOI, siliceneis expected to have a behavior (not shown) very similar to thatof graphene.The total Hamiltonian H T = H band + H imp + H hyb in-cludes, besides H band , also the impurity and its hybridizationwith the lattice: H imp = (cid:15) ( n imp , ↑ + n imp , ↓ ) + U n imp , ↑ n imp , ↓ (2) H hyb = V (cid:88) σ ( c † imp ,σ c kσ + h . c . ) , (3)where n imp ,σ = c † imp ,σ c imp ,σ and c † imp ,σ ( c imp ,σ ) creates (an-nihilates) an electron at the impurity, which is described by asingle impurity Anderson model (SIAM) with Coulomb inter-action U and orbital energy (cid:15) .As for the hybridization between the impurity and the lat-tice, H hyb , we will focus on a substitutional impurity, whichreplaces an atom from either sublattice and therefore has over-lap integrals with more than one lattice site (from the oppo-site sublattice). Therefore, c kσ stands for a symmetric linearcombination of two (or three, depending on which edge sitewe are considering) nearest-neighbors to the lattice site oc-cupied by the impurity. In the case of a zigzag nanoribbon(studied in this work), with an edge geometry schematicallyrepresented as · · · / \ / \ · · · , which we denote as a sequenceof sites · · · ABABA · · · , the choice is between A sites (withcoordination three) and B sites (with coordination two). Ourcalculations show a wide variation in the results (mainly forthe spin correlations) depending on what edge site the impu-rity replaces. In the following, we will refer to A and B sitesin the zigzag profile as ‘trough’ and ‘crest’ sites, respectively. FIG. 1. (a) Sketch of the graphene geometry studied through theLanczos transformation for one impurity sitting at the edge (circle)and the measurement site in the bulk at a distance R (diamond). In(b) we show the geometry of the equivalent problem, with the twoseed orbitals coupled to non-interacting tight-binding chains. TheLanczos orbitals only start interfering at a distance R/ , introduc-ing a hopping term between the chains. There is an exact canonicaltransformation connecting the two problems. Our results clearly show that the metallic topologically non-trivial states that screen the magnetic impurity reside in thecrest sites, while the trough sites and their immediate neigh-borhood, as our spin correlation results have shown, behave assmall metallic “puddles”, which leak from the metallic edges,and are surrounded by the bulk.In order to perform unbiased numerical simulations of themodel just described to calculate spin-spin correlations, weuse the so-called block Lanczos method recently introducedby the authors (see also Ref. 39 for an independent devel-opment of the same ideas). This approach enables one to studyquantum impurity problems with a real space representationof the lattice, and in arbitrary dimensions, using the densitymatrix renormalization group method (DMRG) . By gen-eralizing the ideas introduced in Ref. 35 for single impurityproblems, we reduce a complex lattice geometry to a singlechain, or a multi-leg ladder in the case of multiple impurities.In order to measure spin-spin correlations in real space, weneed to employ the multi-impurity formulation described indetail in Refs. 36 and 39. In this approach, we use two seedstates corresponding to the impurity site and the orbital wherethe correlations will be measured. A block Lanczos recursionwill generate a block tridiagonal matrix that can be interpretedas a single-particle Hamiltonian on a ladder geometry. This isillustrated schematically in Fig. 1(a), showing the case of oneseed state at the edge, and the second one somewhere in thebulk. The typical equivalent problem we need to solve numer-ically is depicted in the panel Fig. 1(b). Due to the presence ofspin-orbit coupling in the bulk, this geometry cannot be fur-ther simplified. Still, the Hamiltonian in the new basis will beone-dimensional, and local, i.e. , the many-body terms are stillthe same as in the original Anderson impurity coupled to theoriginal lattice.We want to emphasize that this mapping (see Fig. 1) is ex-
FIG. 2. (Color online) (a) The energy spectra of a zigzag stanenenanoribbon for λ SO = 0 . . (b) Zoom on the (red) dashed square inpanel (a) showing details of the energy dispersion of the edge states.(c) Similar to panel (b), but for λ SO = 0 . , which is appropriatefor graphene. (d) Value of | C iσ | for the four metallic edge statesassociated to the wave vectors k x and k x in panel (b). Note that,as discussed in more detail in the text, they are located exclusivelyat crest sites, and states with different spin polarizations propagate inopposite directions at each edge.FIG. 3. (Color online) (a) LDOS for two edge sites in the TI phase( λ SO = 0 . ): the crest site LDOS [dark (red) curve] displays a pro-nounced spectral density peak at the Fermi energy, while the troughsite LDOS [gray (blue) curve] has very small, but finite spectral den-sity at the Fermi energy. (b) Color contour plot showing the LDOSacross the width of a nanoribbon N = 80 sites wide ( λ SO = 0 . ).Notice how the spectral density at the Fermi energy is located mostlyat the edges [and primarily at crest sites, as shown in panel (a)], whilethe bulk remains gapped. act and both geometries are connected by a unitary transfor-mation. The combination with the DMRG method allows usto obtain exact results with real space resolution and uncoverthe marked difference between crest and trough sites, as de-scribed above, which is inaccessible to the majority of othermethods traditionally used to study magnetic impurity mod-els. Results in this work were obtained by keeping up to 3000DMRG states, which grants an accuracy of the order of − or better for the energy and correlations. III. RESULTSA. Zigzag nanoribbon band structure
Fig. 2(a) shows the band structure obtained for H band ina ZNRB with periodic boundary conditions (PBC) in the x -direction and open boundary conditions (OBC) in the y -direction. The SOI used was λ SO = 0 . and the nanorib-bon is sites across in the OBC direction. Each band isdouble degenerate, besides the Kramers ω ( k x ) = ω ( − k x ) de-generacy, and the energy spectrum is particle-hole symmetricaround ω = 0 (half-filling). In panel (b) we show a zoom intothe dashed (red) box shown in panel (a), focusing on the bandsthat straddle across the valence and conduction bands. Thesebands host the 4 helical edge states (two in each edge). Panel(c) shows the same zoom, but for λ SO = 0 . (appropriate forgraphene), showing the well studied flat bands associated tothe edge states present in a graphene ZNRB (see Ref. 42 fordetails).In Fig. 2(d) we see a plot of the coefficient squared, for eachsite across the ZNRB, for the four edge states that are locatedsymmetrically around k x = π , i.e. , k x = π − δ and k x = π + δ ≡ − π + k x = − k x . Note that k x and k x are timereversed momenta (Kramers related), indicating propagationin opposite directions. The location of k x and k x , and theassociated edge states, are schematically indicated in panel(b). The results are for λ SO = 0 . and δ = π/ . For aZNRB with width of N = 80 sites, each state in the 160 bands(including spin) for an arbitrary value of k x can be written as | E j ( k x ) (cid:105) = (cid:88) i,σ C iσ,j ( k x ) | iσ (cid:105) (4)where i = 1 , · · · , runs through the sites across the OBCdirection, σ = ↑↓ indicates spin, E j ( k x ) (for j = 1 , · · · , )runs through all the eigen-energies for a specific k x value, and | iσ (cid:105) is a localized orbital state on site i with spin σ . Taking k x = k x , k x as defined above and j values corresponding tothe double-degenerate bands connecting the valence and con-duction bands ( j = 79 to , i.e. , the 4 edge states), we plot | C iσ,j ( k x ) | for each site iσ . In reality, we plot only the val-ues for the edge sites (crest sites at each edge), since all theother coefficients vanish ( | C ↑ | = 0 . , for example). Asindicated by the labels in Fig. 2(d) ( k x ↑ and k x ↓ ), states inthe i = 1 edge propagate in opposite directions, with oppositespin polarizations, and the reverse occurs ( k x ↓ and k x ↑ ) onthe i = 80 edge. Therefore, these helical edge states are verylocalized, propagating through what we dubbed as crest sites,presenting the characteristic locking of spin and momentumthat is associated to the non-trivial topological phase createdby the SOI . In addition, narrower ZNRB have less localizedstates, the localization varying slightly with k x as we moveaway from k x = π , and the states become (discontinuously)completely delocalized (becoming bulk states) once these twoedge bands merge with either the conduction or the valenceband.To close this section, we make a few remarks on the effectsof electronic interactions. These are typically too weak to in-troduce noticeable spin correlations in the bulk, especially due FIG. 4. Detail of the LDOS at a trough site, using a large numberof poles and a small broadening η = 0 . of the Green’s functionimaginary part. The spectral weight in the pesudogap-like regionaround the Fermi level is small but finite . to the vanishing density of states at the Dirac points. However,theory has predicted that electronic repulsion causes a hy-bridization between edge states on opposite sides of a ZNRB,inducing edge ferromagnetism . These effects depend onboth the magnitude of the interactions and the width W of theribbon, and decay very rapidly as W − .An analysis of the literature indicates that there is a greatdeal of controversy regarding the experimental verification ofthe theoretical prediction of ferromagnetic edge states at theedges of a ZNRB : for example, all the measurements in-tended to uncover them are constrained to either charge trans-port or scanning tunneling microscopy experiments, whoseresults are open to alternative interpretations. Therefore, thecurrent consensus seems to be that the existence of the mag-netic edge states has not been settled yet. Clearly, magnetismwould break time reversal symmetry, compromising the exis-tence of the topological edge states. Recent work by Lado andFern´andez-Rossier shows, through mean-field calculations,that a large enough SOI can suppress the magnetic momentof the edge states. Nonetheless, this problem falls beyond thescope of our work, which focuses on understanding the prop-erties of the Kondo state in the absence of magnetism. B. Zigzag nanoribbon local density of states
Figure 3(a) shows the LDOS for a crest [(red) dark graycurve] and for a trough [(blue) light gray] site at the edge ofthe ZNRB. In agreement with the results shown in Fig. 1(d),the conducting edge states [with spectral density at the Fermienergy ( (cid:15) F = 0 )] are mostly located at crest sites. Panel (b)shows a contour plot of the LDOS for all sites in the OBC di-rection. The spectral density at the Fermi energy is restrictedto the edge [primarily to a crest site, as shown in panel (a)]while all the sites away from the edge present an insulatingspectra, where the size of the gap partially derives from thespin-orbit interaction and mostly from the confinement along FIG. 5. (Color online) Variation of the LDOS with the value of λ SO for a crest (solid lines) and a trough (dashed lines) site. By vary-ing λ SO by almost two orders of magnitude, from − (most pro-nounced peak at ω = 0 ) to . (almost flat at ω = 0 ), we see thatthe main change in the LDOS of a crest site is the continuous spreadof the spectral weight (located at the Fermi energy peak) into an al-most flat distribution that covers the gap. The LDOS of a trough sitealmost does not vary with λ SO . the OBC direction (the wider the nanoribbon, the smaller isthe energy gap). It is important to notice that, unlike the insu-lating bulk, the LDOS at the trough sites displays a small butfinite LDOS at the Fermi energy (for a detailed view, with in-creased resolution, see Fig. 4), leaving room for the formationof a Kondo state.In Fig. 5 we report the effects of SOI in the LDOS. By in-creasing λ SO from . to . the spectral weight peak at theFermi energy gradually broadens until a flat distribution closesthe gap. The results for a trough site (dashed curves) show thataside from a slight increase of the LDOS at the edge of the gapand a slight increase of the gap itself, the LDOS changes onlymarginally. A contour plot for the LDOS across the nanorib-bon for λ SO = 0 . (not shown), similar to Fig. 3(a), shows thatthe bulk sites do not qualitatively change their LDOS, i.e. , thegap in the bulk remains intact. C. Spin correlations
Our model takes into account a real-space description ofthe lattice. Due to the spatial dependence of the density ofstates, the physics of the magnetic impurity will vary accord-ingly, displaying important differences determined by its lo-cation. This will be clearly visible in the spin correlationsresults. Fig. 6 shows the spin correlations between the impu-rity and the surrounding conduction electrons for a substitu-tional impurity (replacing a silicon atom) sitting at crest andtrough sites in panels (a) and (b), respectively. The spin cor-relations (cid:104) (cid:126)S imp · (cid:126)s j (cid:105) are calculated for sites j along the edge,as indicated. Results correspond to three different values of U = 1 . [(red) squares], . [(green) circles], and . [(blue)triangles]. Correlations are strong at short distances, their FIG. 6. (Color online) Spin correlations for a substitutional impuritylocated at a crest (a) and trough (b) edge site, respectively. Resultsshow correlations along edge sites j for U = 1 . [(red) squares], U = 2 . [(green) circles], and U = 4 . [(blue) triangles], V = 0 . ,and λ SO = 0 . .FIG. 7. (Color online) Comparison of spin correlations between crest[(red) squares] and trough [(blue) triangles] sites for a substitutionalimpurity. The parameter values are V = 0 . , λ SO = 0 . , and U = 1 . . There is a clear qualitative difference between impuritiessitting at crest and trough sites. magnitude falls away from the impurity, and their range in-creases with U . In addition, in both cases, correlations withsites in the opposite sublattice to the one where the impurity islocated (odd sites) are antiferromagnetic, while they are fer-romagnetic for same sublattice sites (even sites). These re-sults are typical of those obtained for the Kondo effect for an S = 1 / impurity connected to a non-interacting chain (see,for example, Ref. 53). Ferromagnetic correlations dominatefor crest site impurities, as opposed to trough site impurities, FIG. 8. (Color online) Spin correlations for a substitutional impu-rity located five lattice spacings, counting from the edge, into thebulk, for U = 1 . [(red) squares], U = 2 . [(blue) triangles], and λ SO = 0 . . The impurity forms a localized singlet with the threeneighboring spins, differently from a trough site, whose correlationsextend along the edge [see Fig. 6(b)]. Note that the line of j -sitesprobed is parallel to the edge. where antiferromagnetic correlations dominate. In addition,crest site correlations have larger magnitude and are longerranged than those for trough sites.It is also interesting to note that an impurity located eitherin a crest or a trough site correlates much more strongly tocrest sites along the edge. In contrast, the dominant (antifer-romagnetic) correlations for a trough site impurity have veryshort range and are essentially independent of U , while thedominant (ferromagnetic) correlations for a crest site impu-rity decay slowly and increase slightly with U . The presenceof ferro or antiferromagnetic correlations stems naturally fromthe expected structure of the RKKY interaction on a bi-partitelattice . However, the vanishing of the correlations at evendistances from the trough sites can be attributed to the nodalstructure of the electronic wavefunctions near the Fermi level,that interfere destructively to yield a very small amplitude onthose sites .As seen in Fig. 7, impurities at crest sites induce large andslowly decaying hybridization clouds with dominant ferro-magnetic correlations that decay algebraically with distance[(red) squares in Fig. 7]. This can be understood in terms ofcrest sites forming an effective one-dimensional channel .On the other hand, impurities at trough sites form very smallclouds [(blue) triangles in Fig. 7], as if sitting in small metal-lic “puddles” that leak from the metallic edges, surrounded bythe bulk. To illustrate that the results for a trough site are qual-itatively different from those for a bulk site, Fig. 8 shows thecorrelations around a substitutional impurity in the bulk, fivelattice spacings from the edge (note that the vertical axis scalein Fig. 8 is considerably smaller than that in Fig. 7). Whenlocated in a bulk site, the impurity forms a localized singletwith its three neighbors, completely decoupled from the bulk.Correlations in this case are practically identical to the two-site problem with an Anderson impurity connected to a singlenon-interacting site (not shown).We proceed to compare spin correlations with and withoutSOI to understand how it changes the coupling of a magnetic FIG. 9. (Color online) Effect of SOI on spin correlations for sub-stitutional impurity at crest (a) and trough (b) sites. Results are for U = 2 . . The SOI causes an overall increase of the spin correlations. impurity to the edge states. Fig. 9 shows results for a substi-tutional impurity, at both crest and trough sites, demonstrat-ing an overall increase of the correlations when the SOI isintroduced. At first sight, these results seem to indicate thatthe SOI does not change qualitatively the results. However,it is important to notice that the SOI induces an anisotropyin the spin correlations ( (cid:104) S z s jz (cid:105) (cid:54) = (cid:104) S x s jx (cid:105) ) that stems froman emergent Dzyaloshinskii-Moriya interaction between theimpurity and the conduction electrons , as illustrated inFig. 10. This effect is clearly visible for the case of impuritiessitting at crest sites, with transverse XY correlations [(red)squares] and along the Z direction [(blue) circles] having op-posite sign, indicating helical order. However, for impuritieslocated at trough sites [Fig. 10(b)] this effect is very small orpractically nonexistent.As demonstrated above, the spatial resolution of our calcu-lations are instrumental at determining the full structure of thespin correlations and uncover the response of the edge statesto the presence of substitutional magnetic impurities. Resultsfor a crest-site impurity are summarized in Fig. 11. Panel (a)presents a color map of the impurity spin correlations in anextended region surrounding the substitutional atom. Detailsalong the bulk and edge directions are shown in panel (b) and(c), respectively. The actual position of the j sites for panels(b) and (c) are indicated in (a). From these results it emergesthat the edge state is scattered and “snakes”, or circles around FIG. 10. (Color online) Comparison between (cid:104) S imp x s jx (cid:105) and (cid:104) S imp z s jz (cid:105) for a substitutional impurity at a crest (a) and a trough(b) site for U = 4 . , V = 0 . , and λ SO = 0 . . The anisotropyis expected as a consequence of the spin-orbit interaction . Ascan be seen by comparing panels (a) and (b), the anisotropy is muchlarger for a crest site and it increases with U [see Fig. 6(a)]. the impurity . This effect implies that the problem may notbe trivially studied using a one-dimensional lead to representthe edge. This was already pointed out through a no-go theo-rem in Ref. 57 stating that a helical liquid with an odd num-ber of modes cannot emerge from a purely one-dimensionalmodel. In addition, even though the impurity is substitutional,it appears as though it is side coupled to the edge , and notembedded in it. D. Kondo temperature
Anderson showed , with his “Poor man’s scaling”, that theKondo problem can be treated perturbatively at energies largerthan the so-called Kondo temperature T K , which is the onlyrelevant energy scale, and does not depend on the high en-ergy details. A renormalization group analysis shows that thesystem flows toward an attractive strong coupling fixed point,described by a tightly bound state formed by the impurity andthe conduction electrons, the “Kondo singlet”. In this regimeone can show that many quantities satisfy a universal scalingcharacterized precisely by T K . This quantity has a strict uni-versal meaning in the thermodynamic limit (or rather, in theuniversal scaling regime). In finite systems, like the one wepresently discuss, one can define a similar energy scale as theenergy gained by the system by forming a Kondo singlet, the FIG. 11. (Color online) (a) Color map of the spin correlations for asubstitutional impurity at a crest site, for U = 2 . , V = 0 . , and λ SO = 0 . . (b) and (c) show results along the edge and perpendicularto it, respectively, for λ SO = 0 . [(red) squares] and λ SO = 0 . [(blue) triangles]. The actual positions of the j sites in (b) and (c) areindicated in panel (a). correlation energy: E corr = E − E proj , (5)where E is the ground state energy, and E proj = (cid:104) g . s . | S − imp HS + imp | g . s . (cid:105)(cid:104) g . s . | S − imp S + imp | g . s . (cid:105) , (6)with | g . s . (cid:105) being the ground state. The operators S ± imp act onthe impurity site and project the ground state singlet onto astate where the impurity and the bulk are disentangled, thusforming a product state. This clearly is a variational estimateof the correlation energy, and comparisons to the dynamicalspin correlations show that indeed it is an accurate measureof T K . Notice that, even though the Hamiltonian is gapless, E corr is finite.Our results for E corr , for an impurity at a crest site, areshown in Fig. 12(a) as a function of the interaction U for dif-ferent values of SOI ( . ≤ λ SO ≤ . ), and in panel (b) as afunction of λ SO for different values of interaction ≤ U ≤ (varying in steps of 1). Unlike prior work by Zarea et at. thatpredicts an exponential enhancement of the Kondo tempera-ture in the presence of SOI, we find that this enhancement isvery weak [see panel (b)], in agreement with numerical renor-malization group treatments of this problem . IV. SUMMARY AND CONCLUSIONS
We have applied the Lanczos transformation method com-bined with the DMRG , to study the many-body groundstate of a quantum ( S = 1 / ) impurity (modeled as an An-derson impurity) coupled to the edge of a zigzag nanoribbon FIG. 12. (Color online) Characteristic energy scale E corr (see textfor definition), which is a measure of the Kondo temperature, as afunction of (a) the Coulomb interaction U and (b) the spin orbit cou-pling λ SO . Results in (b) are for the same values of U as in (a), andthe strength of the interaction increases as indicated by the arrow. of stanene, a slightly buckled (non-planar) honeycomb latticeof Sn atoms, which hosts a topologically protected metallicedge state. The main motivation was to study the detailedspatial structure of the spin correlations between the quan-tum impurity and the electrons in the host, which characterizethe Kondo ground state. We identified marked differences be-tween the results for the two distinct sites in the zigzag edge,namely an outermost and an innermost one, which we dubbedcrest and trough sites, respectively.The behavior observed through the spin correlations is quitecomplex and rich. For substitutional impurities located at ei-ther crest or trough sites, the spin correlates primarily withelectrons along the edge, and decouples from the bulk. Fur-thermore, irrespective of the position of the impurity (crest ortrough site), spin correlations with conduction electrons lo-cated at crest sites are larger than for trough sites. The signof the spin correlations is determined by the spins belongingto the same or opposite sublattice. In addition, for impuritiesat crest sites, ferromagnetism dominates, while the opposite occurs for trough sites.The effects of SOI in the TI phase are mostly present forimpurities sitting at crest sites, increasing the range of thecorrelations and introducing helical order along the edge thatoriginates from an effective Dzyaloshinskii-Moriya interac-tion. Remarkably, the SOI does not affect the spin symmetryfor an impurity on a trough site, another indication that edgestates reside mostly on crest sites.Unlike previous calculations that consider the coupling ofthe impurity to one-dimensional effective modes , in ourformulation the helical liquid arises naturally as an edge ef-fect of the 2d bulk. It has been observed that helical liquidswith an odd number of modes cannot be obtained from one-dimensional lattice models . In our treatment, the edge hasunequivocally a single mode contribution, even away from theparticle-hole symmetric point.From the real space picture obtained from our method weare able to resolve the structure of the correlations at differentsites along the zigzag edge and into the bulk. We find thatsubstitutional impurities sitting at a trough-site form a local-ized bound state with conduction electrons in a small metallicpuddle that leaks out of the edge. On the other hand, a crest-site impurity scatters the edge state around it, resulting in theformation of a long range screening cloud along the edge.Finally, we have used a variational estimate of the corre-lation energy to obtain a measure of T K in our finite systemas a function of both U and λ SO . Our results show that theKondo temperature for a substitutional impurity at a crest siteis very weakly enhanced by the introduction of SOI, in agree-ment with numerical renormalizaton group calculations of asimilar problem . ACKNOWLEDGMENTS
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