Persistence of chirality in the Su-Schrieffer-Heeger model in the presence of on-site disorder
PPersistence of chirality in the Su-Schrieffer-Heeger model in the presence of on-sitedisorder
Myles Scollon and Malcolm P. Kennett
Department of Physics, Simon Fraser University8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada (Dated: May 1, 2020)We consider the effects of on-site and hopping disorder on zero modes in the Su-Schrieffer-Heegermodel. In the absence of disorder a domain wall gives rise to two chiral fractionalized bound states,one at the edge and one bound to the domain wall. On-site disorder breaks the chiral symmetry, incontrast to hopping disorder. By using the polarization we find that on-site disorder has little effecton the chiral nature of the bound states for weak to moderate disorder. We explore the behaviourof these bound states for strong disorder, contrasting on-site and hopping disorder and connect ourresults to the localization properties of the bound states and to recent experiments.
I. INTRODUCTION
The Su-Schrieffer-Heeger (SSH) model [1] was intro-duced in the context of polyacetylene but has attractedmuch interest as a model of non-interacting fermions inone dimension that displays charge fractionalization [2].The SSH model also gives a simple example of a modelwith topologically distinct states which arise for oppositehopping dimerization patterns. Fractionalization ariseswhen domain walls are introduced that separate the twodimerization patterns and give rise to zero energy modeswith specific chiralities bound at the domain walls.Recently there have been several experimental realiza-tions of the SSH model: in cold atom systems [3–6] andgraphene nanoribbons [7–9]. Condensed matter imple-mentations of the SSH model generically break the chi-ral (sublattice) symmetry that gives rise to zero modesthrough e.g. next-nearest-neighbour hopping [10] (as oc-curs in polyacetylene) or disorder. In this work we ex-plore the effect of broken chiral symmetry due to disorderon the states that are zero modes in the absence of disor-der. Specifically, this is important for the interpretationof experimental results on signatures of chirality in realsystems where disorder is inevitable and chiral symmetryis broken [7–9].Much of the previous work on disorder in the SSHmodel has focused on the case in which there is disorder inhopping amplitudes [11–13]. For this special class of dis-order, the chiral symmetry of the model is preserved, andhence zero modes in the clean model remain zero modesin the disordered model for weak disorder, only disap-pearing at a critical disorder value [ ? ]. On-site disorderexplicitly breaks chiral symmetry so that the zero modesin the clean limit are no longer topologically protectedand have a non-zero energy for infinitesimal disorder. Inthe limit of infinitesimal disorder we expect there to bestates that closely resemble the zero energy modes in theclean limit. Previous work [10, 14] has investigated howon-site disorder affects the localization properties of edgestates, but not their chiral properties. In particular, fora SSH model with a domain wall, if the system is largeenough, we might expect localized states at the wall and the edge to retain some of their chiral properties.In this work we study the disordered SSH model with adomain wall and explore the extent to which the localizedstates retain their chiral nature even though chiral sym-metry has been broken by on-site disorder. Our main toolto do this is the polarization – it has been shown that ina system with chiral symmetry there is a relationship be-tween the winding number and the polarization [15–21].Our main result is that we find that for even quite size-able disorder strengths, the bound states can be viewedas being chiral from a practical point of view, even if notperfectly so. We relate the changes in polarization as afunction of on-site disorder strength to changes in thelocalization properties of the electronic states.The structure of this paper is as follows: in Sec. II weintroduce the disordered SSH model and show numericalcalculations of its spectrum and of localized states. InSec. III we discuss our results and conclude. II. THE DISORDERED SSH MODEL
The Hamiltonian for the SSH model on a N site chainmay be written as H SSH = − t N − (cid:88) n =1 [1 + ( − n u ] (cid:110) c † n +1 c n + c † n c n +1 (cid:111) , (1)where c n and c † n are annihilation and creation opera-tors for fermions on site n respectively, t is the hoppingstrength and u is the dimensionless strength of the stag-ger in the hopping. Fractionalized states arise if a domainwall is introduced into the parameter u [1, 2]. Here weconsider domain walls of the form: u = u tanh (cid:20) ( n − n ) ξ/a (cid:21) , (2)where u specifies the amplitude of the domain wall, n is the centre of the domain wall and ξ is the width, with a the lattice spacing. In the presence of such a domain wall, a r X i v : . [ c ond - m a t . d i s - nn ] A p r the SSH model develops fractionalized zero modes whichhave support on a single sublattice. In a finite chain withopen boundary conditions, and a domain wall of the formEq. (2), one of these zero modes will be localized at anedge, and the other will be localized at the domain wall,as illustrated in Fig. 1. | Y | FIG. 1. Wavefunction for the zero mode localized on thedomain wall in the centre of a N = 500 site chain in the cleanSSH model for ξ/a = 10 and u = 0 . We introduce disorder in the form of a random on-sitepotential with Hamiltonian H dis = N (cid:88) n =1 (cid:15) n c † n c n , (3)where (cid:15) n is a random variable drawn from a uniform dis-tribution on [ − W, W ]. Such a potential breaks chiralsymmetry and hence the fractionalization seen at W = 0will no longer be present. However, it is still of interestto study how the chirality of the bound states that formthe W = 0 zero modes evolve with increasing disorder.In particular the question we want to investigate is howthey lose their chirality with increasing disorder.We diagonalized the Hamiltonian H = H SSH + H dis fora N = 500 site chain and found the ordered list of energyeigenvalues. We averaged over 50000 disorder configura-tions to obtain Fig. 2. Once chiral symmetry is brokenby a disorder potential ( W (cid:54) = 0), the zero modes seen at W = 0 move away from being exactly at zero energy, butthey are clearly identifiable in the gap out to disorderstrengths of W/t ∼ .
3. Several other bound states arevisible in the gap out to
W/t ∼ . W/t = 0 . , . , and 1.5 in Fig. 3. Peakscorresponding to the bound states are clearly visible upto moderate disorder ( W/t (cid:46)
1) but for stronger dis-order the bands broaden sufficiently to obscure them.While the W = 0 zero modes do not continue to have -3-2-10123 E /t FIG. 2. Ordered energy eigenvalues of a N = 500 site chainin the SSH model as they evolve with increasing W/t , for ξ/a = 10 and u = 0 . zero energy for W (cid:54) = 0, we can ask whether they can betreated as chiral for practical purposes as the disorder isincreased.In the case of disorder that preserves chiral symmetry(e.g. hopping disorder) [11] one can consider a real-spacecalculation of a topological invariant which is closely re-lated to the polarization [21]. In the case of on-site dis-order that we consider here, there is no strict topologi-cal protection, so we instead focus on the polarization ofbound states, which can change continuously as disorderincreases. Specifically, we introduce projection operatorsˆ P A and ˆ P B , which project a bound state | ψ (cid:105) on to ei-ther the A or B sublattices respectively. We can usethese projection operators to calculate the polarization,i.e. the density imbalance between A and B sublattices P = (cid:104) ψ | ˆ P A − ˆ P B | ψ (cid:105) , (4)for bound states | ψ (cid:105) localized at the domain wall andthe edge. When W (cid:54) = 0 we select the bound statesby projecting disordered bound states | ψ ( W (cid:54) = 0) (cid:105) ontothe W = 0 bound states | φ ( W = 0) (cid:105) . The results weshow are for the states that have the maximum overlapwith the clean bound states i.e. those that maximize |(cid:104) ψ ( W (cid:54) = 0) | φ ( W = 0) (cid:105)| .We calculate the polarization P for these bound statesin the presence of both chiral symmetry preserving andchiral symmetry breaking disorder. We introduce chiralsymmetry preserving disorder via the Hamiltonian H chiral dis = N − (cid:88) n =1 τ n (cid:110) c † n +1 c n + c † n c n +1 (cid:111) , (5)where τ n is a random variable drawn from a uniform dis-tribution on [ − W, W ]. Similarly to Ref. [11], we find that D O S -3.5-2.5-1.5-0.5 0.5 1.5 2.5 3.5Energy (t )0.0050.0040.0030.0020.0010.000 D O S -3.5-2.5-1.5-0.5 0.5 1.5 2.5 3.5Energy (t )0.0030.0020.0010.000 D O S -3.5-2.5-1.5-0.5 0.5 1.5 2.5 3.5Energy (t ) FIG. 3. Disorder averaged density of states (DOS) for a SSHchain with N = 500 sites for a domain wall centred at site250 with width ξ/a = 100, strength u = 0 . N = 500for disorder strengths a) W/t = 0 .
1, b)
W/t = 0 .
7, and c)
W/t = 1 . P goes to zero for large W for the W (cid:54) = 0 bound states atthe domain wall and the edge, consistent with the transi-tion in winding number with W identified in Ref. [11], asillustrated in Fig. 4. Even though the states lose their po-larization, they remain localized for all disorder strengths[22].We performed similar calculations for the SSH modelwith on-site disorder and display the results in Fig. 5.We found that for N (cid:38)
300 that our results appear tobe independent of N . We also see that | P | decays morequickly with W/t than for hopping disorder, but that1 − | P | (cid:28) W/t that are an appreciablefraction of 1, demonstrating that small amounts of on-site disorder do not greatly alter the chiral nature of thestates.Unlike the situation in which there is hopping disorder, -1.0-0.50.00.51.0 P Wall state Edge state
FIG. 4. Polarization P for the states bound at the domainwall (Wall state) and the edge (Edge state) in a chain with N = 500 sites with ξ/a = 10 and u /t = 0 . | P | does not approach zero with increasing W/t andin fact increases towards 1 with increasing W/t . Thereason for this behaviour can be illuminated with theinverse participation ratio, defined byIPR = (cid:80) i | ψ ( r i ) | (cid:12)(cid:12)(cid:12)(cid:80) i | ψ ( r i ) | (cid:12)(cid:12)(cid:12) , (6)which gives a measure of localization. The value of theIPR differs significantly between localized and extendedstates. For localized states, the IPR takes a constantvalue, whereas for extended states, the IPR scales like1 /L d where d is the spatial dimension. The IPR is illus-trated in Fig. 6 and illustrates that the localization lengthincreases up to a disorder strength of W/t ∼ −
2, con-sistent with results obtained for edge states in smallersystems [10, 14]. The localization length decreases atlarger values of disorder, consistent with Anderson local-ization becoming more important. The states are alwayslocalized, as expected for a one dimensional disorderedfermion system [23], but the degree of localization varieswith disorder strength.The behaviour seen in P can be understood from apicture in which increasing on-site disorder breaks chiralsymmetry so that the zero disorder zero mode states startto have some support on both sublattices, but unlike thehopping disorder case, P does not go to zero, becausewith increasing on-site disorder strength, the states be-come sufficiently localized that most of their support is ona single site. Figure 6 illustrates that there is a crossoverfrom a localized state that retains much of its W = 0chiral character to a strongly Anderson localized state asa function of W/t . P u /t =0.1 u /t =0.2 u /t =0.3 u /t =0.4-1.0-0.9-0.8-0.7-0.6-0.5 P u /t =0.1 u /t =0.2 u /t =0.3 u /t =0.4 FIG. 5. Polarizations P for the states bound at the domainwall and the edge in a chain with 500 sites and domain wallwidth ξ/a = 10 for the SSH model with on site disorder forfour different domain wall strengths u /t = 0 . , . , . , and0.4. III. DISCUSSION
We studied the SSH model with on-site disorder andcompare our results to those obtained for hopping dis-order. Our results demonstrate that even though chiralsymmetry is broken by the introduction of on-site dis-order, the zero energy states at zero disorder evolve so that they continue to be strongly polarized for
W/t (cid:46) I P R Wall state Edge state
FIG. 6. Inverse Participation Ratio (IPR) for the statesbound at the domain wall (Wall state) and the edge (Edgestate) in a chain with 500 sites and a domain wall width of ξ/a = 10 and u = 0 . We note that our calculations here have direct rele-vance to recent experiments. In particular, two groupsused graphene nanoribbons [7–9] to engineer the SSHmodel and studied edge states in these systems. Ourresults here show that the edge states that are topologi-cally protected in the clean limit persist to large values ofdisorder. Hence, given the inevitability of some level ofon-site disorder in experiment, the edge states observedin experiment are still meaningful approximations to theclean case. From a theoretical perspective, the fraction-alization [24] seen in the SSH model in one dimension hasbeen generalized to two dimensions [25–28] and it wouldbe very interesting to see how disorder affects the zeroenergy modes in those models.
ACKNOWLEDGEMENTS
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