Effect of Local Perturbations on Plasmons in Topological Insulators
EEffect of Local Perturbations on Plasmons in Topological Insulators
Yuling Guan, ∗ Zhihao Jiang, and Stephan Haas
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 91361 (Dated: February 5, 2021)We use a fully quantum mechanical approach to analyze the effects of molecule-scale perturbationson the plasmonic excitations in prototype models of topological insulators. Strongly localized surfaceplasmons are present in the host systems, arising from the topologically non-trivial single-particleedge states. A numerical evaluation of the RPA equations for the perturbed systems reveals howthe position and the internal electronic structure of the added molecules affect the degeneracy ofthe locally confined collective excitations, i.e., shifting the plasmonic energies of the host systemand changing their spatial charge density profile. In particular, we identify conditions under whichsignificant charge transfer from the host system to the added molecules occurs. Furthermore, theinduced energy of the perturbed topological systems due to external electric fields is determined.
I. INTRODUCTION
Topological insulators are characterized by a gappedbulk energy spectrum and symmetry-protected conduct-ing states on their surfaces. Collective excitations in suchsystems are generally known to be very sensitive to crys-tal defects and external perturbations, such as scanningtunneling microscope (STM) tips[1]. Here, we focus onthe effects of molecular-scale perturbations on plasmons[2, 3] in low-dimensional topological structures, domi-nated by weakly screened Coulomb interactions, such aslateral multi-wire superlattices [4], multi-sub-band com-pounds [5], and quasi-one-dimensional electron systems(1DES) [6]. Low-dimensional plasmons have alreadybeen experimentally realized in different settings, includ-ing
GaAs quantum wires [7] and trapped atoms [8].In topological insulators (TIs), the mobile charge car-riers are confined to non-trivial edge modes, such assymmetry-protected Dirac fermion states [9]. In theirbulk, the Fermi level lies in the gap between the con-duction band and the valence band, whereas on theirsurface, metallic conduction exists, e.g., via edge stateswith their spin locked to their momentum. For ex-ample, the quantum Hall effect has recently been ex-perimentally observed in magnetic topological insula-tors, such as (BiSb) Te [10], which in turn has led tothe discovery of new quantum phases in the topologi-cally nontrivial regime. Furthermore, one-dimensionalTIs have been realized by helical modes of surface elec-trons in nanowires[11], Haldane-like phases and Kondobreakdown [12], decoupled bilayers edge states [13],and lasing edge states in insulator arrays of micro-ringresonators[14].Nonetheless, topological insulator materials are typi-cally not pure, and local perturbations can severely affectthe topological states, which is the focus of this study.Such perturbations come in different forms. For exam-ple, impurities and vacancies exist both on the surfaceand in the bulk, producing resonance states with ener- ∗ [email protected] gies inversely proportional to the impurity strength[15].Magnetic impurities can open up local gaps, suppressingthe local density of states[16]. In the specific topologi-cal insulator film Bi Te , nonmagnetic Au impurities donot affect weak anti-localization (WAL), but magnetic Feimpurities quench it[17]. On the edges of 2D topologicalinsulators, spin impurities have been shown to lead toAnderson localization of the edge states[18].In this work, we analyze the effects of impurities oncollective plasmonic excitations in TIs, focusing on 1DHamiltonians for simplicity. As a benchmark, we firststudy simple metallic chains with propagating plasmonsin their bulk, and then two types of topological insula-tors, i.e. the Su-Schrieffer-Heeger (SSH) model and itsmirror analogue (m-SSH), which - in their topologicallynon-trivial regimes - host localized plasmons on the edgesand on domain walls of topological character. Specifi-cally, we determine the plasmonic excitation spectrumof these systems along with the real space charge mod-ulation of every resonance. From an analysis of thesecharge modulations we learn that plasmonic edge modesonly exist in the topologically non-trivial phases. Next,we introduce impurities in the form of molecular per-turbations, and examine their effects on the propagatingplasmons and on the localized edge plasmons.This paper is organized in the following way. In Sec. II,we discuss the mathematical and physical details of thereal-space RPA approach. In Sec. III, we introduce themodels under consideration and address the effects ofmolecular perturbations on homogeneous and topologi-cal chains, as well as the induced energy in these systemsdue to external electric fields. We conclude in Sec. IV. II. METHOD
In order to study the collective excitations in topologi-cal systems, we use the real-space random phase approxi-mation (RPA) [2, 3, 19]. Here, the non-interacting chargesusceptibility function is first evaluated in the atomic ba- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b sis,[ χ ( ω )] ab = 2 (cid:88) i,j f ( E i ) − f ( E j ) E i − E j − ω − iγ ψ ∗ ia ψ ib ψ ∗ jb ψ ja , (1)where a and b label atomic sites. E i and φ i are the elec-tronic eigenenergy and eigenstate of the i − th level, whichis obtained by diagonalizing the model Hamiltonian. f ( · )is the Fermi function, and the factor 2 accounts for thespin degeneracy. We then calculate the bare Coulombinteraction in the same basis, V ab = (cid:26) e / ( (cid:15) env | (cid:126)r a − (cid:126)r b | ) , if a (cid:54) = b,U /(cid:15) env , if a = b, (2)where (cid:15) env is the background dielectric constant from theenvironment, and U = (cid:82) d r d r (cid:48) e | φ ( r ) | | φ ( r (cid:48) ) | / | r − r (cid:48) | is the on-site Coulomb interaction parameter in the vac-uum, which we set to 14 .
395 eV . The RPA dielectricresponse function (a matrix) is then calculated from (cid:15) RPA ( ω ) = I − V χ ( ω ) . (3)We identify the plasmonic excitations from the electronenergy loss spectrum (EELS), which is defined by [20]EELS( ω ) = max n (cid:26) − Im (cid:20) (cid:15) n ( ω ) (cid:21) (cid:27) , (4)for each single frequency ω , where (cid:15) n is the n -th eigen-value of (cid:15) RPA ( ω ). The EELS is peaked at plasmon fre-quencies, defined by (cid:15) n ( ω ) = 0. We simultaneously ob-tain the real-space charge distribution patterns from thecorresponding eigenvectors of the dielectric matrix. Inthe analysis below, the maximum of − Im[1 /(cid:15) n ( ω )] is de-fined as the 1st EELS, to be distinguished from the 2ndEELS, which is obtained from the second maximum valueof − Im[1 /(cid:15) n ( ω )]. The second EELS is also peaked whena plasmon mode is degenerate. In this paper, we focus on1D models, for which the 1st and 2nd EELS are sufficientfor the analysis.While the EELS yields the plasmonic eigenmodes in asystem, it does not include any information of the sys-tem’s response to external fields that are applied in ex-periments to excite the system. We calculate the inducedcharge distribution due to a specific external field φ ext via ρ ind ( ω ) = χ RPA ( ω ) φ ext ( ω ) , (5)using now the (interacting) RPA charge susceptibilityfunction, χ RPA ( ω ) = [I − χ ( ω ) V ] − χ ( ω ) . (6) φ ( r ) = σ √ π − e − r / σ is the Gaussian orbitals. The on-site Coulomb interaction is mathematically divergent.Here we set an artificial on-site separation 1 ˚ A , which is smallerthan the lattice spacing a of our models. From this, we determine the induced potential and in-duced electric field in real space. By integrating theinduced electrical field over real space, we obtain thefrequency-dependent induced energy spectrum, U ind ( ω ),which peaks at the plasmon frequencies [21]. III. RESULTS AND DISCUSSIONA. Plasmonic excitations in a decoratedone-dimensional metallic chain - non-topological case
FIG. 1. (a) Illustration of a decorated metallic chain. (b)Energy dispersion of a homogeneous metallic chain in mo-mentum space. (c) Blue: EELS of a finite homogeneousmetallic chain with open boundary conditions. Red: EELS ofthe same open-ended metallic chain with additional diatomicmolecules. (d)-(g) Real-space charge density modulation for(d) a low-energy plasmon in the pure metallic chain. (e) ahigh-energy plasmon in the pure metallic chain. (f) a low-energy plasmon in the decorated chain. (g) a confined low-energy plasmon in the bulk at ω = 3 .
878 eV of the decoratedchain.
Before discussing plasmons in topological insulators,let us first consider - as a benchmark - a homogeneousone-dimensional (1D) metallic chain (MC) with openboundaries, described by the real-space tight-bindingHamiltonian,ˆ H = t M − (cid:88) n =1 ( | n + 1 (cid:105) (cid:104) n | + H.c. ) + µ M (cid:88) n =1 | n (cid:105) (cid:104) n | , (7)where M is the number of the atoms in the chain, t is thenearest-neighbor hopping parameter, and µ is the chem-ical potential. We explore the plasmonic excitations inthis simple model, calculating the EELS for M = 54atomic sites, t = 1 . µ =-1.0 eV, which is shownby the blue line in Fig. 1(c). The EELS is made ofa continuum of plasmonic excitations in the frequencyrange between ω = 0 eV and ω = 7 .
28 eV. Furthermore,we observe a pseudo-gap at around ω ≈ , ,
34 and 49 of the open-ended 54-site host chain, re-spectively. They are aligned vertically, with an internalhopping ˜ t between the two atoms of the molecule, and asmall tunneling hopping t (cid:48) between the molecule and thechain. The red line in Fig. 1(c) shows the correspondingEELS of the DMC. Compared with the EELS of the pureMC, we see that for frequencies below ≈ ω > ω ∈ { . , } is strongly affected dueto the added molecules. Moreover, the molecules are ex-cited as well, leading to interesting charge transfer phe-nomena due to the interactions between the decoratingmolecules and the host 1D MC (see e.g. Fig. 1(f)). Fur-thermore, a pattern of oscillating charges between nearbydecorating molecules is observed. In Fig. 1(g), we find an-other interesting mode, where the charge density in the1D MC is confined due to the addition of the molecules. These two modes are not observed in the host MC, andthey both occur in the most affected energy regime forthe chosen parameter set. B. Plasmonic excitation in the SSH model
Let us now turn to a prototype symmetry-protectedtopological insulator, i.e., the SSH model [22], describedby a one-dimensional lattice Hamiltonian of spinlessfermions with staggered hopping parameters, as illus-trated in Fig. 2(a). There are two atoms in each primitivecell, labeled by A and B .The real-space Hamiltonian of the SSH model is givenby ˆ H = t N (cid:88) m =1 ( | m, B (cid:105) (cid:104) m, A | + H.c. )+ t N − (cid:88) m =1 ( | m + 1 , A (cid:105) (cid:104) m, B | + H.c. )+ µ N (cid:88) m =1 ( | m, A (cid:105) (cid:104) m, A | + | m, B (cid:105) (cid:104) m, B | ) , (8)where N is the number of unit cells. t and t arethe intra-cell and inter-cell hopping parameters, respec-tively. The open-ended SSH chain has two distinct topo-logical sectors whose topological invariants can be rep-resented by the number of zero-energy single-particleedge states N es in the energy gap. In Fig. 3(a), weshow the electronic energy spectra of the SSH chainwith 52 sites in both the topologically non-trivial sec-tor ( t = 0 .
75 eV < .
25 eV = t ) and in the trivialsector ( t = 1 .
25 eV > .
75 eV = t ). In the formercase we observe two zero-energy edge states in the gap,whereas in the latter case there are no zero-energy edgestate. Due to the bulk-boundary correspondence, thesetopological properties can also be identified via the bulkwinding number W .We now focus on the SSH model in the topologicallynon-trivial sector ( t = 0 .
75 and t = 1 .
25) and ana-lyze its two-particle excitations. The EELS is shown inFig. 4, where we observe two continua (mainly consist-ing of bulk modes) separated by an energy gap, as wellas two isolated modes in the gap at ω = 4 .
471 eV and ω = 7 .
344 eV, labelled by p and p . The real-spacecharge density modulations of these two modes are shownin Figs. 4(b) and (c). We observe that they are both lo-calized at the ends of the chain. Furthermore, the chargedistribution of the higher frequency mode [Fig. 4(c)] ismore strongly localized than the lower frequency mode[Fig. 4(d)]. We also point out that these two modes areboth two-fold degenerate in the pure SSH model, as isindicated by singularities in the second EELS. We willstudy the effects of molecular perturbations on mode de-generacy further below. FIG. 2. (a) Illustration of the SSH model on a bipartite tight-binding chain. (b) Illustration of the mirror SSH model onthe tight-binding chain with a mirror inversion at its center. (c) Illustration of the SSH model with an additional diatomicmolecule, connected with one site of the chain. The position of the connected site is varied from the edge to the bulk. (d)Illustration of the mirror SSH model with an additional diatomic molecule, connected with one site of the chain. The positionof the connected site is varied from the center of the bulk to the edge.FIG. 3. (a) Energy spectrum of the SSH model on a 52-siteopen-ended chain. There are two zero-energy edge states inthe topologically non-trivial sector ( t = 0 . eV < . eV = t ), but no edge state in the trivial sector ( t = 1 . eV > . eV = t ). (b) Energy spectrum of the mirror SSH modelon a 55-site open-ended chain. There is one zero-energy lo-calized edge state for a strongly coupled inversion center andthree zero-energy localized states for a weakly coupled inver-sion center. In previous work, localized plasmons in open-ended TIshave been shown to originate from the topological elec-tronic edge states. This was demonstrated by decom-posing the full charge susceptibility χ full0 into bulk andtopological surface contributions[21], which allows us tofocus on χ topo0 instead of χ full0 for an isolated examinationof these modes. Using this decomposition, (cid:88) i,j . . . (cid:124) (cid:123)(cid:122) (cid:125) χ full0 = (cid:88) i ∈ TS (cid:88) j / ∈ TS · · · + (cid:88) i/ ∈ TS (cid:88) j ∈ TS (cid:124) (cid:123)(cid:122) (cid:125) χ topo0 · · · + (cid:88) i,j / ∈ TS . . . (cid:124) (cid:123)(cid:122) (cid:125) χ bulk0 , (9)TS is the set of the topological zero-energy edge statesin the bulk gap. The spectrum of χ topo0 preserves theplasmonic edge modes, along with their degeneraciesand their real-space modulations, compared to the χ full0 FIG. 4. (a) EELS of the SSH model in the topologically non-trivial sector (M = 52 sites). (b) and (c): charge densitymodulations of the two localized plasmons in the first EELSat an intermediate and at a high frequency, only observed inthe topological sector. (see Figs. 4 and 5). Below we calculate the EELS ofthe SSH model, using only χ topo0 , and denote the re-sulting spectrum by EELS topo ( ω ), which is shown inFig. 5(a). As expected, there is no bulk plasmonic con-tinuum in the spectrum because of removal of the bulkcontributions, χ bulk0 . EELS topo ( ω ) shows three peaks at ω = 2 .
038 eV , .
526 eV and 3 .
846 eV. Each mode has atwo-fold degeneracy of different parities: odd parity andeven parity, and they all have localized charge distribu-tions [see Fig. 5(b) to (g), up: even parity; down: oddparity]. Also, similar to the EELS full ( ω ) with full sus-ceptibility, the edge plasmons are more strongly localizedat higher frequencies, resembling the patterns shown inFig. 4. Furthermore, note that even though the bulk sus-ceptibility does not contribute to the localization of theedge plasmons, it still affects the excitation energies ofthese modes.We also calculate the induced energy spectrum U ind( ω ) of the SSH model in response to specific external elec-tromagnetic fields, which can be directly compared to FIG. 5. (a) EELS using only the topological charge suscepti-bility χ topo0 in the topological sector of the SSH model on anopen-ended 52-site chain. (b) to (g): charge density modula-tions of the three degenerate localized plasmonic excitationsat different frequencies. The edge modes observed in χ full0 arepreserved when only χ topo0 is considered.FIG. 6. (a) Induced energy spectrum in the 52-site open-ended SSH chain in the topological sector, subject to a linearexternal electrical field. (b), (c) and (d): charge density mod-ulation of the mode corresponding to the three excitationshighlighted in (a). experiments. Here we consider a linear external electricpotential applied to a finite SSH chain in the topologi-cally non-trivial sector. The resulting spectrum shownin Fig. 6(a) contains three main peaks at the exactlysame frequencies as those obtained from EELS topo ( ω )[Fig. 5(a)]. The induced charge distributions of thesemodes in Figs. 6(b-d) also show great resemblance to theeigenmode patterns in Figs. 5(e-g). We also note that inthe presence of an applied linear external potential thesemodes are no longer degenerate. This is expected be-cause, under the linear potential, only modes with oddparity are excited. The eigenmodes shown in Figs. 5(b-d)are not excited in this case. FIG. 7. (a) to (d): Topological EELSs in the SSH chain withone added diatomic molecule at different positions, X m , whichmeans the molecule is connected with the xth site on thechain. (e) and (f): charge density modulation of the modescorresponding to the two high-frequency excitations in (a). C. Effects of added diatomic molecules onplasmons in the SSH chain
The topological SSH model studied above hosts bothbulk plasmons and localized plasmonic edge modes [21].While the former show no essential difference from thepropagating modes in the 1D MC in Sec. III A, the lo-calized edge plasmons respond differently to molecularperturbations. Here we introduce diatomic molecules inthe vicinity of the SSH chain and study their effects onthe plasmonic excitations. Specifically, we place a singlediatomic molecule above the SSH chain and graduallychange its position from the edge to the center of thechain, as illustrated in Fig. 2(c). The tunneling hoppingbetween the molecule and the SSH chain is denoted by ˜ t ,with ˜ t < t , t ). The internal hopping in the molecule isdenoted as t (cid:48) , with t (cid:48) > t , t . Such a situation could beexperimentally realized by atoms attached on an STM tipor by tip atoms themselves when scanning over a sample,such as described in Ref. [23]. Here, we focus on studyingthe effects of the perturbing molecule at various positionson the plasmonic edge modes observed in Sec. III B.We first consider the effect of a diatomic molecule inproximity to one of the ends of the SSH chain [Fig. 2(c)],which is expected to have maximum impact on the plas-monic edge modes. Fig. 7(a) shows the EELS of sucha perturbed SSH chain, together with the unperturbedcase for comparison. We find that the molecular per-turbation on the edge site removes the degeneracy of allthree modes observed in the pure host system. Due tothe additional molecule attached to one edge site, thetwo ends of the chain are no longer equivalent. There-fore, they now each host edge modes with slightly differ-ent energies. For instance, in Figs. 7(e) and (f) we showthe real-space charge distribution patterns for the twohighest energy modes [labeled as p and p in Fig. 7(a)].They originate from a degenerate pair at ω = 3 .
486 eVof the host model. In the presence of the molecular per-turbation applied to the left end of the open chain, themode localized on the right end of the chain remains atthe same frequency as before because of its far distanceaway from the local perturbation, whereas the edge modeat the left end of the chain now has a slightly shifted en-ergy. Naturally, plasmons that are localized close to theleft end of the chain are mostly affected by this localperturbation.As we gradually move the perturbing molecule fromthe left end to the center of the chain, the effects on theplasmonic edge modes become less pronounced. Quan-titatively, this depends on the localization length of theplasmonic edge modes. As mentioned above, the highest-energy mode is most localized. The two lower-energymodes are slightly more extended [Figs. 5(b-g)]. Whenthe molecule is being moved towards the center of thechain, the highest-energy mode first becomes unaffectedto the perturbation, then followed by the lower-energyones. In Figs. 7 (b-d) we show the variation of the EELS,as the molecular perturbation is gradually moved to thecenter. In detail, we can see that when the perturbingmolecule is moved onto the 3 rd site away from the thechain edge, the highest-energy mode is already not af-fected. However, the two lower-energy modes are stillaffected by the perturbation, as we can see from the splitpeaks. When the perturbing molecule is on the 7 th site ofthe chain, the second-highest-energy mode becomes un-affected as well [Fig. 7(c)]. Finally, when the moleculeis on the 15 th site of the chain, which is quite deep intothe bulk, all three modes are unaffected. In this case, thefull EELS is almost the same as for the unperturbed hostsystem.In the above calculations, the internal hopping t (cid:48) wasfixed to 2, which is larger than the hopping in the chain.Here, we would like to examine how the the excitationenergies change when we modify t (cid:48) . When we consideronly the topological susceptibility, the excitations are lo-calized at the two ends of the chain. In Fig. 8 we see thatthe excitation on the right end remains constant at about E R = 3.84 eV when the perturbing molecule is connectedto the left end. However, the excitation at the left end, E L , shifts to higher energies when t (cid:48) is increased becausea higher energy is required to excite the molecule with alarger internal energy gap t (cid:48) . Furthermore, we find thatthe charge transferred from the chain to the moleculealso changes with t (cid:48) . W m in Fig. 8 shows the percentageweight of the charge on the molecule compared to thecharge in the entire system (host chain plus perturbationmolecule). We observe that the relative weight on the FIG. 8. Dependence of the energy of the topological plasmonsand of the charge transferred from the host to the diatomicmolecule on the internal hopping t (cid:48) , when the molecule isconnected with an edge atom of the chain. E L (Red) is thehighest energy excitation on the left end of the chain (seeFig. 7(f)). E R (Blue) is the highest energy excitation on theright end of the chain (see Fig. 7(e)). W m is the percentageweight of the charge transferred from the SSH chain to themolecule. molecule drops from about 16% to about 8% when t (cid:48) isincreased, i.e. there is less charge transfer from the hostto the molecule. Hence, the internal electronic structureof the added molecule affects its ability to hybridize withthe host system. If the molecule has a larger internalenergy gap, i.e. larger t (cid:48) , then it is more protected fromhybridization with the host system.We conclude that a local perturbation can affect thetopologically originated plasmonic edge modes in theSSH model only significantly when it is sufficiently closeto the edge of the chain. In other words, the edge plas-mon modes in the topological SSH model are very robustagainst local perturbations, which is different from bulkplasmon modes. It has been shown before that thesemodes are also quite stable when subjected to global ran-dom noise [21]. D. Plasmonic excitations in the mirror-SSH model
In addition to the SSH model, let us also inspect plas-mons in the mirror-SSH model, which is a variant ofthe SSH model by reflecting the chain about its center[Fig. 2(b)]. This mirror-SSH (mSSH) model is inversionsymmetric with respect to the mirror interface located atthe middle point of the chain, which also hosts localizedzero-energy state(s) depending on the hopping character-istics at the interface. In Fig. 3(b), the energy spectra ofthe strong interface mSSH model ( t at the center is 1.25 eV ) and the weak interface mSSH model ( t at the centeris 0.75 eV ) are displayed, where we observe one zero-energy state and three zero-energy states, respectively.We calculate the EELS of the mSSH model plasmonsby using only the “topological part” of the susceptibil-ity χ topo0 , defined in the same manner as for the SSHmodel. Fig. 9(a) shows the EELS of the mSSH model forboth the strong mirror interface (hopping at the mirror FIG. 9. (a) EELS of the open-ended mirror-SSH chain(M=53) with weak mirror interface hopping and on an open-ended 55-site chain with strong mirror interface hopping. (b)and (c): charge density modulation of the modes correspond-ing to the excitations at the weak mirror interface. (d) and(e): charge density modulation of the modes correspondingto the excitations at the strong mirror interface. is t = 1 . eV ) and the weak mirror interface (hoppingat the mirror is t = 0 . eV ). In each case, we observe aset of (non-degenerate) excitations that are all stronglylocalized around the mirror interface. We show the real-space charge modulations of some typical excitations inFigs. 9 (b-e). As we can see here, the modes aroundthe strong interface and the weak interface have differentparities. Similar to what we observed before in the SSHmodel, the modes with higher energies are more stronglylocalized. E. Effects of added diatomic molecules in themirror-SSH chain
Here we introduce molecular perturbations into themSSH model, starting from the central site (the mir-ror interface) of the chain and moving towards one end,as shown in Fig. 2(d). We focus on studying its effectson the localized plasmons around the interface. Unlikethe edge plasmons observed in the SSH model, the inter-face plasmons here are non-degenerate even in the unper-turbed case. So, there is no degeneracy splitting effect.We will, however, observe other interesting effects due tothe perturbation.Fig. 10(a) shows the perturbed EELS of the weak-interface mSSH model, together with the unperturbedone (red line) for comparison. The spectra are similarto the SSH model discussed in section (c), in the sensethat the effects from the perturbation are weakened whenthe perturbing molecule is gradually shifted away fromthe charge concentration area of the localized plasmons.When the added molecule is connected to the 26 th site,which is the center site of the 53-site chain, all of theexcitations change their positions (green line) because FIG. 10. (a) Comparison of the EELS of the weak inter-face mirror-SSH model with M=53 sites in the presence of aperturbation diatomic molecule placed at different positions(green line: connected with the 26 th site; orange line: con-nected with the 20 th site) and the model without perturbation(red line). (b) - (e): charge density modulations of the modescorresponding to the excitations highlighted in (a). of significant charge transfer between the chain and themolecule. Figs. 10(b) and (c) show the charge distribu-tions of two typical excitations in (a). Here, we observethat there is little charge transfer from the chain to themolecule in the high energy excitation, but more signif-icant charge transfer to the molecule connected to thecenter site for the lower energy mode. The reason for thisis that the internal hopping magnitude of the moleculeis t (cid:48) = 2 eV, which is closer to the frequency of the lowenergy excitation. However, when we move the pertur-bation from the center towards the edge, the EELS curvewill coincide with the original curve (red line) in the highand intermediate energy regimes, since the charge is moreconcentrated at the center in the higher energy modes[Fig. 10(e)], so the interaction between the chain and themolecule substantially vanish. In this case, there is stillcharge transfer in lower energy modes, and the charge dis-tribution will not be symmetrical anymore [Fig. 10(d)].Next, we discuss the strong interface case, wherebythe hopping at the center mirror interface is strongerin magnitude. Fig. 11(a) shows a comparison of thestrong interface with (red and orange line) or without(blue line) the molecular perturbation. For the stronginterface, since the plasmon is also concentrated in thecenter we observe analogous spectra as for the weak inter-face. Specifically, the effect of the perturbation moleculeon topological surface plasmons in the mirror-chain de-creases when we move it from the chain center towardsthe chain ends, especially in the higher energy regime.When ω is larger than 2 . eV , the blue curve and orangecurve coincide with each other, which means there is littleeffect when the molecular perturbation is connected withthe 20 th site than connected with the 26 th site. In thelow energy region, however, we also observe an excita-tion shift caused by charge transfer (Fig. 11(d)) because FIG. 11. (a) Comparison of the EELS of the strong interfacemirror-SSH model in the presence of a diatomic perturbationmolecule placed at different positions (green line: connectedwith the 26 th site; orange line: connected with the 20 th site)and the model without perturbation (blue line). (b) - (e):charge density modulations of the modes corresponding tothe excitations indicated in (a). the plasmonic excitation expands to the 20 th site of thechain. However, when the molecular perturbation is con-nected with the 26 th of the chain (in the center), all ofthe excitations of the host system shift from low energyto high energy, and plasmonic charge is transferred to themolecule [Figs. 11(b) and (c)]. IV. CONCLUSIONS
In this work, we have examined the effects of added di-atomic molecules on the topological modes of 1D TIs, andcompared them to the benchmark of a simple metallicchain. By analysis of the electron energy loss excitationspectrum and of the real space charge distributions ofthe plasmon modes, we conclude that such local pertur-bations of TIs can significantly affect their plasmonic ex-citations in the topologically non-trivial regime, i.e., theirdegeneracy and their charge distribution. We also iden-tified conditions under which electrons transfer from thehost chain to the added molecules. Here, the position ofthe local perturbation is an important parameter, as wellas the internal hopping ˜ t within the diatomic molecule.It will be interesting to further analyze analogous effectsof perturbation molecules in higher-dimensional topolog-ical models which harbor dispersive surface modes, suchas the two-dimensional SSH model and graphene, whichmay become localized due to the impurities. ACKNOWLEDGMENTS
We wish to acknowledge useful discussions with Hen-ning Schl¨omer and Malte R¨osner. This work was sup- ported by the US Department of Energy under grant
FIG. 12. Energy spectrum, induced by a linear external elec-trical field, of a 52-site SSH chain with one diatomic perturba-tion molecule located at different positions, calculated usingonly the topological susceptibility χ topo0 . number DE-FG02-05ER46240. Appendix A: Induced energy of the perturbed SSHmodel
In Sec. III B, we discussed the induced energy spectrumof the SSH chain and analyzed the spectrum in Fig. 6. Inthe spectrum we observe three excitations at the exactlysame positions as in Fig. 5. In Sec. III C, the effects oflocal perturbation were considered. Here, we analyze theinduced energy spectrum of the topological SSH model inthe presence of a diatomic molecular perturbation, sub-ject to an external linear electrical field.as a reference, theblue dashed line in Fig. 12(a) shows the spectrum with-out perturbation that we have already discussed. Figs. 12(a)-(d) reveal that the effects of the perturbation to SSHmodel in the presence of an external electrical field aresimilar to the EELS spectrum, i.e., mostly affecting thedegeneracy with respect to the position. As the per-turbing molecule is gradually moved from the left end(0th side) towards the bulk (15th site), the effects onthe plasmonic edge modes become less pronounced. 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