Effects of Disorder in the Fibonacci Quasicrystal
EEffects of Disorder in the Fibonacci Quasicrystal
Anouar Moustaj, Sander Kempkes, and Cristiane Morais Smith
Institute of Theoretical Physics, Utrecht University. (Dated: November 24, 2020)We study the properties of the one-dimensional Fibonacci chain, subjected to the placement ofon-site impurities. The resulting disruption of quasiperiodicity can be classified in terms of therenormalization path of the site at which the impurity is placed, which greatly reduces the possibleamount of disordered behavior that impurities can induce. Moreover, it is found that the additionof multiple weak impurities can be treated by superposing the individual contributions together.In that case, a transition regime between quasiperiodic order and disorder exists, in which someparts of the system still exhibit quasiperiodicity, while other parts start to be characterized bythe localisation of the wavefunctions. This behavior is manifested through a symmetry in thewavefunction amplitude map, expressed in terms of conumbers.
I. INTRODUCTION
Since their discovery by Dan Shechtman [1], qua-sicrystals have attracted much attention. Their unusualproperties, such as low thermal conductivity, low fric-tion coefficients, high hardness, corrosion resistance andsuperplasticity have made them attractive for applica-tions. Their utility ranges from heat-insulating mate-rials, through coating that increases hardness, all theway to medical implants, where prosthetics made fromquasicrystalline material have shown very little cytotox-icity effects [2–6]. Many studies have been conductedon the nature and properties of quasicrystals. Not be-ing identified as classical crystals, they give rise to for-bidden discrete symmetries, such as the fivefold rota-tion group, with the archetypal example being the AlMgsample created by Dan Shechtman. One way to under-stand the emergence of this symmetry is by tiling a 2Dplane, as devised by Penrose [7]. On the other hand, amore structural way of understanding all quasiperiodicarrangements is to view them as projections from higherdimensional lattice spaces, which have a perfectly pe-riodic structure [8, 9]. Although mainly manufacturedin laboratories, quasicrystals have also been observed tooccur in nature, where the structure was found to existin a Siberian meteorite sample [10]. A mechanism forthe formation of quasicrystalline phases, both artificialand in nature, has recently been proposed. It consistsof the superposition of two 1D periodic subsystems withincomensurate periods, where charge-density waves fa-vor the emergence of a quasiperiodic tiling of the atomiclattice [11].The fact that quasicrystals are not periodic in theirmicroscopic structure makes them a more complicatedproblem to study than their periodic counterpart. Oneway to simplify the problem and still obtain relevant re-sults is to study a 1D abstraction of the real system.The most popular toy model in that case is given by the1D Fibonacci quasicrystal [12]. This is a tight-bindingmodel for a particle subject to a lattice potential. Ei-ther the on-site potential or the hopping parameter, de-pending on the model chosen, is modulated by the Fi-bonacci sequence and takes on two discrete values, as will be explained in more detail later on. This modelexhibits very interesting properties: its energy spectrumis singular continuous, which makes its semi-infinite ver-sion a proper fractal set, and its wavefunctions possessmultifractal properties [13]. A renormalization schemewas introduced to explain the features of the spectrumand its scaling symmetries [12]. This scheme was subse-quently used to understand the gap labeling theorem [14],applied to the purely hopping Fibonacci chain. Togetherwith the conumbering scheme [15], it offered an insightfulway to characterize the wavefunctions in terms of theirrenormalization paths [16]. Nowadays, new insights arestill being provided and interestingly, it has been shownthat this system also possesses topological characteristics[16–20].In this paper, we aim at understanding how impuri-ties disrupt the quasiperiodic order. We start by in-vestigating the effect of a single impurity on the wave-function canvas using the aforementioned renormaliza-tion scheme. We find that for weak impurities, a tran-sition regime exists in which the quasiperiodic order re-mains intact in parts of the system. This is manifestedby the preservation of a symmetry in the wavefunctionamplitude as a function of conumbered sites. As thestrength of the impurity is raised, a disordered phaseappears, which is characterized by the localization of thewavefunctions. This is compatible with previous resultsin Ref. [21], where the authors found the existence of “res-onant states” in the presence of an impurity, and quanti-fied it in terms of an analog of a transmission coefficientfor the scattering off the impurity. Moreover, we find thatone can always label the type of transition regime by therenormalization path of the site at which the impurityhas been placed, no matter how strong it is. This alsoholds when adding multiple weak impurities, where theindividual contributions can just be superposed to obtainthe full disruptive pattern. Since the quasiperiodic orderis gradually lost, the transition regime can be character-ized by a classification of the kind of disorder induced,and we find that disorder develops in an organized way.The paper is structured as follows. In Sec. II, we give abrief description of the Fibonacci chain in a tight-bindingapproximation. We then provide an overview of the un- a r X i v : . [ c ond - m a t . d i s - nn ] N ov FIG. 1. (Color online) Projection of square lattice points onto the “Fibonacci line”. (a)
The distance between two projectedpoints is labeled S when it is short, and L when it is long. The window of acceptance, of width δ , is also shown. The sequenceof L and S is precisely the Fibonacci sequence. We colored the vertices in green and blue because they correspond to molecularand atomic sites, respectively, in the hopping model. We call a site “atomic” whenever it is bound to the next sites by two longbonds, and “molecular” whenever it has a short bond on one side and a long bond on the other. (b) Orthogonal projection ofthose same sites, onto a line of slope φ . This illustrates the conumbering scheme [15], which clusters atomic sites at the centerand molecular sites at the edges of the conumbered chain. derstanding of its spectrum through a renormalizationprocedure. In Sec. III, we introduce disorder by addingone impurity to the system and show that we can classifythe kind of disorder by the renormalization path label ofthe site at which the impurity has been placed. Finally,in Sec. IV, we present our conclusions and outlook. II. THE MODEL
We start by briefly introducing the Fibonacci chain inthe tight-binding approximation, followed by an analysisof its properties through a renormalization procedure.
A. Fibonacci Tight-Binding Model
1. Fibonacci Sequence a. Algebraic representation
The Fibonacci se-quence, represented by a binary alphabet { L, S } , can begenerated iteratively through the inflation rule S → L,L → LS, starting with the “zeroth” letter S . The N th iteration, W N , will be referred to as the N th approximant of theFibonacci word, the size of which will be denoted by | W N | = F N . This sequence has the propertylim N →∞ F N +1 F N = 1 + √ ≡ φ, (1) where φ is called the golden ratio . The Fibonacci se-quence is often represented in terms of word-size andtakes the form { F N } ∞ N =0 = { , , , , , , , . . . } . An-other property is that each term can be generated recur-sively through: F N +2 = F N +1 + F N , (2)with F = F = 1. In terms of Fibonacci words, therecursion relation can be written as W N +2 = W N +1 W N . b. Geometric representation The Fibonacci se-quence is known to be quasiperiodic . That is, it can beobtained as a projection of a higher-dimensional periodicsequence. This is the so called cut-and-project method ofgenerating quasiperiodic lattices. In this case, points on Z are projected onto a line that has a slope 1 /φ . Thewindow of acceptance is such that only the points withinthe region of width δ are projected. The procedure is de-picted in Fig. 1 (a) . The orthogonal projection of thesesites, that is onto a line of slope φ , will be very usefullater, as it defines the conumbering scheme. Introducedin Ref. [15], this scheme rearranges the sites accordingto their local environment. It can be obtained by theorthogonal projection of the same Z points within thewindow of acceptance, as shown in Fig. 1 (b) . On thisorthogonal projection, all atomic sites are placed in themiddle region of the line, while all molecular sites areplaced on the sides, as depicted in Fig. 1 (b) . An atomicsite is connected by two long bonds, while a molecularsite is connected by a long bond on one side, and a shortone on the other. This nomenclature is such that atomicsites are isolated in the strong modulation limit, whilemolecular sites stay bonded, as described in Sec. IIB. Therepresentation of the sites using this numbering methodwill turn out to be useful when analysing the behavior ofthe system in the presence of an impurity (see Sec. III).
2. Tight-Binding Hamiltonians
The Fibonacci chain is constructed by considering thenearest-neighbor tight-binding Hamiltonian H = ∞ (cid:88) i =1 (cid:20) V i | i (cid:105) (cid:104) i | + t i | i (cid:105) (cid:104) i + 1 | + h.c. (cid:21) . (3)We can either modulate the on-site potential V i or thehopping parameter t i . We shall refer to the two casesas the “on-site model” and the “hopping model”, respec-tively. The modulation is applied as follows: V i = (cid:40) V w , if i th letter is LV s , if i th letter is S, (4)for the on-site model and t i = (cid:40) t w , if i th letter is Lt s , if i th letter is S, (5)for the hopping model. The subscripts w and s have beenchosen to reflect that L , standing for long, would corre-spond to a weak bond (hopping strength) and S (short)for a stronger bond. The two models have been studiedextensively in Refs. [12, 18, 22], where the renormaliza-tion scheme [12] was applied to understand the multifrac-tal properties of the model [16, 22]. The on-site modelwas also studied in Ref. [18] through the perspective oflocal symmetries, where a systematic way to control theedge modes of a finite chain was devised.In the remainder of this paper, we focus on the hoppingmodel, in which we apply periodic boundary conditionsto properly renormalize it. The resulting spectrum hasmany interesting properties, which are characteristic ofquasiperiodic systems. The semi-infinite chain is singularcontinuous and is also a fractal [16, 23]. An example ofthe spectrum of a N = 16 chain is shown in Fig. 2. There,we observe a trifurcarting structure that is self-similar,which led Nori et al. [12] to devise the renormalizationprocedure to explain these features. There is also a directmapping between the on-site and hopping models underthe perturbative renormalization scheme (see Ref. [12]). B. Renormalization of the Chains
The spectrum of the Fibonacci chain can be under-stood by performing a perturbative renormalization pro-cedure to the hopping model, which is exact in the limit ρ ≡ t w /t s →
0. Note that this can also be applied to the
FIG. 2. Energy spectrum of the N=16 approximant Fi-bonacci chain, with 1597 sites, in the hopping model. Thetrifurcating structure can be seen at different energy scales.The self-similarity of this structure is also visible. The mod-ulation strength has been set to ρ = 0 . on-site Fibonacci chain, which after one renormalizationstep becomes a hopping Fibonacci chain. We start fromthe original Hamiltonian and without loss of generality,set the on-site energy to be a constant, V i ≡ V = 0.Then, we split it into an unperturbed part H and a per-turbation H , with H = (cid:88) j t s | j (cid:105) (cid:104) j + 1 | + h.c, if j mod( φ ) < φ − ,H = (cid:88) j t w | j (cid:105) (cid:104) j + 1 | + h.c, if j mod( φ ) ≥ φ − . (6)The unperturbed Hamiltonian has three levels with avery large degeneracy, namely E = 0 , ± t s . This sets thestarting point of the renormalization procedure, which isapplied to each of the three unperturbed levels indepen-dently. Following the nomenclature proposed by Mac´e etal. [16], we have one atomic deflation, corresponding tothe atomic level ( E = 0) and two molecular deflationscorresponding to the bonding and anti-bonding molec-ular levels ( E = ± t s ). The atomic deflation takes theoriginal chain of size F N to a smaller Fibonacci chain ofsize F N − , while the molecular deflations map the origi-nal chain to one of size F N − . The renormalized hoppingstrengths, in each case, are given by: { t (cid:48) w , t (cid:48) s } = (cid:26) t w t s , t w (cid:27) = ρ { t w , t s } , (bonding) (cid:26) t w t s , − t w t s (cid:27) = ρ { t w , − t s } , (atomic) (cid:26) t w t s , − t w (cid:27) = ρ { t w , − t s } , (anti-bonding) . We can thus write the original N th approximant Hamil-tonian, H N , as a direct sum of three sub-Hamiltoniansup to and including O ( ρ ) [22]: H N = (cid:16) ρ H N − + t s (cid:17) ⊕ ρ H N − ⊕ (cid:16) ρ H N − − t s (cid:17) . (7)Since each of the chains are themselves Fibonacci approx-imants, we can apply this procedure iteratively until onecannot decimate any generation further. The procedurefor each type of cluster is depicted in Figs. 3 and 4. FIG. 3. (Color online) Decimation of the atomic-leveled chain.We have also colored the sites consistently with our previousdefinition for atomic and molecular sites. This figure gives asimple example of a renormalization path of a site in the N th approximant chain. In this case, a chain of length F = 8 isrenormalized to a chain of length F = 2. The renormalizationpath is just ‘am’. This figure was inspired by those made inRefs. [12, 16].FIG. 4. (Color online) Decimation of the molecular-leveledchain. The same coloring scheme has been used to label thesites. Here, we have another simple example of renormaliza-tion paths, namely ‘ma’ on the left and ‘mm’ in the mid-dle. The chain of length F = 8 is changed to a chain oflength of F = 3. This figure was inspired by those made inRefs. [12, 16]. To be more precise, the atomic decimation takes everyatomic site of the chain ( F N − of them) and connectsevery pair with a renormalized hopping strength that isstrong when they are close to each other, and weak whenthey are separated by longer distances (see Fig. 3). Themolecular decimation, on the other hand, takes every su-perposition of two molecular sites (eigenstate of energy ± t s , each of which has a F N − degeneracy), connectedby t s in the original Fibonacci chain, and creates a newchain made up by these superpositions. The hoppingsare connected in a similar way as in the atomic deflation,i.e. a strong hopping whenever these two renormalizedsites are separated by a short distance in the originalchain, and a weak hopping when the distance is larger(see Fig. 4).The notion of renormalization path will play an im-portant role later on, when we study the effect of im-purities in a Fibonacci chain. The renormalization pathis defined in two ways. The first one is a string of let-ters ‘a’ and ‘m’, which stand for the nature of the site ateach renormalization step. Starting from an N th approx-imant chain, two types of paths appear: ‘amma...’ for anatomic site at the top of the renormalization process, or‘mmam...’ for a molecular one. The other definition per-tains to energy eigenstates and the cluster to which theybelong. Since there are three distinct clusters, we definethe eigenstate renormalization path by a string of let-ters ‘t’, ‘c’ and ‘b’, describing the top cluster (bondingmolecular), the central cluster (atomic) and the bottomcluster (anti-bonding molecular), respectively. A partic-ular level can then be encoded by the symbolic string se-quence ‘tctbt...c’ for example. There exists a particularsymmetry between the two renormalization paths in theperturbative limit ( ρ (cid:28) C N ( j ) = jF N − mod( F N )[see Fig. 1 (b) ]. The fractal nature of the eigenstates be-comes manifest in this representation, which is used inSec. III to show how impurities disrupt this order. III. INTRODUCING DISORDERA. Localization
There are multiple ways of introducing disorder byadding impurities. It was already shown [21] that eventhe introduction of one impurity in a relatively long chainhas a drastic effect on the spectrum. In particular, itreduces the fractal dimension of the global density ofstates, while increasing that of the local wavefunctions.Moreover, since the spectrum of the Fibonacci chain issingular-continuous [23], i.e. it has an infinite number ofgaps, every state is affected by the presence of this impu-rity. The extent to which they are affected will howeverdepend on the strength of the impurity.The introduction of an impurity at some site m can beimplemented through the addition of H I = V I | m (cid:105) (cid:104) m | (8)to the original Hamiltonian, which we now call H = H + H . Using a Green’s-function formalism, one canstudy the effect of this impurity by considering it to be FIG. 5. Impurity placed on a generation 9 chain (55 sites) atsite number 11. The localized state is clearly visible with itswhite stripe extending over a wide range of sites, as indicatedby the black arrow on the right. For the sake of clarity, weused a strong impurity of order 10 t w , with the ratio ρ =0.2. a perturbation to the non-disordered Hamiltonian [24].The resulting self-consistent Dyson equation in the pres-ence of the impurity at site m is G ( i, j ; E ) = G ( i, j ; E ) + G ( i, m ; E ) V I G ( m, j ; E ) , (9)where G is the free Green’s function, given by G ( i, j ; E ) = (cid:104) i | ( E − H ) − | j (cid:105) = (cid:88) α (cid:104) i | α (cid:105) (cid:104) α | j (cid:105) E − E α , and | α (cid:105) is an eigenstate of H with energy E α . Eq. (9)can be solved in terms of the free Green’s function toyield G ( i, j ; E ) = G ( i, j ; E ) + G ( i, m ; E ) V I G ( m, j ; E )1 − V I G ( m, m ; E ) , showing that the pole of the total Green’s function isgiven by G ( m, m ; E ) = 1 V I . (10)The state resulting from this particular pole is shown tobe exponentially localized. Let E I be this state’s energy,and its amplitude at some site j be given by ψ ( j ) = (cid:104) j | E I (cid:105) . The overlap between the amplitude at site i and j is given by a contour integral of the Green’s functionaround the point z = E I : ψ ( i ) ψ ∗ ( j ) = 12 πi (cid:73) γ EI dz G ( i, j ; z )= Res[ G ( i, j ; E I ))]= − G ( i, m ; E I ) G ( m, j ; E I ) dG ( m,m ; z ) dz (cid:12)(cid:12) z = E I . Since it is known that G ( i, j ; E I ) decays exponentiallywhen E I is outside of the unperturbed Hamiltonian’sspectrum, we can conclude that this state has an ex-ponential localization around the site at which the im-purity is placed. The extent of this localization will nat-urally depend on the strength of the impurity. We plota numerically obtained eigenstate map in Fig. 5, wherethe localization can be visualised for one of the eigen-states (white stripe, extending over several sites, belowthe center of the figure). This kind of behavior marks adeparture from criticality, in which the wavefunctions ofquasiperiodic systems is neither extended nor localized.Naturally, since our one-particle system is scattering off asingle impurity, it is only one state that localizes strongly.In Fig. 5, we see that the localization is around the siteat which the impurity was placed (site number 11 onthe figure, where the darkest square marks the strongestamplitude). A proper transition to an insulating regimewould show much more localization than this and wouldrequire us to place many impurities, and/or increase theirstrength. B. Organizing Disorder: Renormalization Path
Before one departs completely from criticality, it is pos-sible to observe a structure in the way how disorder setsin. We found that it can be organized according to therenormalization path that the site, at which the impu-rity has been placed, follows. This is best exemplifiedthrough the concrete application of a weak impurity (at10% hopping strength) in a generation 9 chain, with 55sites. In that case, a total of 9 renormalization paths ex-ist. Each one of these is responsible for the generation ofone disordered graph, as shown in Fig. 6 and Fig. 7. Fromnow on, we will positioned the impurities at random sitesin the Fibonacci chain. Then, we will identify the corre-sponding impurity positions in terms of conumbers. Thisallows us to identify which positions affect the eigenstatemap the least. In Fig. 6 (a) , we present the unperturbedfractal pattern, as obtained in Ref. [16]. Its central partresults mainly from the atomic sites, while the four sur-rounding structures result from molecular sites. Then,we successively position an impurity at various atomicsites in Figs. 6 (b)-(f )
The resulting patterns are in analmost one-to-one correspondence with the renormaliza-tion paths of the sites at which the impurities are placed.Every site belonging to one path gives rise to the samedisruption pattern [with the exception of the last twoFigs. 6 (e),(f ) ]. The same is true in the case of impuri-ties placed on molecular sites, as shown in Figs. 7 (a)-(f ) .There is again a one-to-one correspondence between thepattern and the renormalization path, except for the lasttwo cases, Figs. 7 (e),(f ) . Starting from an atomic im-purity at the carefully selected AAA site, as shown inFig. 6 (b) , we see that it mostly affects one eigenstate.Noting that the eigenstates are ordered in terms of in-creasing energy, we see that the slight increase from the
FIG. 6. (Color online) Atomic disorder classes listed in table I. On the left side, we have two color codes: first, the localwavefunction modulus is plotted in shades of red; on the lower part, we have represented the lattice sites by their nature.Each site is either molecular (green), atomic (blue), or has an impurity (yellow). The parameters chosen for the numericalcalculation are ρ = 0 . V I = 0 . t w . Note that because the impurity is weak, it mainly affects the atomic cluster and leavesthe molecular cluster almost intact. (a) No impurity; (b-d) impurity on an AAA, AMA and AAM site, respectively; (e) and (f ) impurity on an AMM site,TABLE I. List of renormalization paths and the amount of distinct graphs it generates. For reference we included the numberof sites belonging to a particular renormalization path. Note the almost one-to-one correspondence between the number ofgraphs and the renormalization paths.Renormalization Path Number of sites Number of distinct graphsMMMM 16 1MMMA 8 1MMA 8 1MAM 8 2MAA 2 1AMM 8 2AMA 2 1AAM 2 1AAA 1 1 impurity potential shifts this state’s energy (which is theclosest to the unperturbed atomic energy E = 0) up-wards. Since all of the renormalized chains correspondto atomic subclusters, the molecular clusters are left com-pletely intact. In fact, even the molecular subclusters ofthe original atomic cluster are relatively well preserved.In Fig. 6 (c) , the impurity is placed at a AMA site. We now have two states that are mostly affected by the pres-ence of the impurity. These are the atomic states ofthe two F chains resulting from the decimation proce-dure (one atomic site per chain, with atomic energies E = ± t (cid:48) s = ± ( ρ/ t s , respectively). The previous twoexamples were the ones that disturbed the spatial dis-trubtion of the wavefunctions the least. The next three FIG. 7. (Color online) Molecular disorder classes listed in table I. The color coding and parameters are the same as above. Inthis case, we find disorder mostly in the molecular cluster while the atomic one is left almost intact. (a-f )
Impurities on sitesbelonging to classes MAA, MMA, MMMA, MMMM and MAM (I and II), respectively. (AAM, AMM I and AMM II) show a higher level of dis-order in the structure. However, we still see that in theweak impurity regime, this disorder is mainly confined tothe atomic cluster.When placing an impurity on molecular sites, allresulting possiblities are shown in Fig. 7 (a)-(f ) . InFig. 7 (a) , we see the least amount of disorder, whichcan easily be explained, just as in the atomic case, bythe disruption of two particular states. These are theatomic sites of two F chains (one per chain, of energy E = ± t s , respectively), resulting from an MAA decima-tion procedure. This is qualitatively very similar to theAMA case, except that the atomic subclusters are differ-ent, and are centered around the unperturbed molecularenergy states. The next five graphs show the same kind ofbehavior, in the weak impurity regime, as in the atomiccase. That is, we mostly see a disruption of the sym-metric patterns within the molecular clusters, while theatomic cluster is mostly left intact.We have listed in table I all the renormalization pathsand number of graphs that they generate. The obser-vation is that there is an almost one-to-one correspon-dence between the amount of renormalization paths andthe possible type of disorder the system is subjected to. It is not exact, as there are two renormalization pathsthat each give rise to two disctinct graphs, namely AMMin the atomic case, and MAM in the molecular case[see Figs. 6 and 7 (e)-(f ) ]. Therefore, we can predicthow many types of disordered graphs can be obtained ifwe can calculate the amount of distinct renormalizationpaths.In fact, based on our numerical implementations, wesee that this classification even holds for strong impu-rity strengths, but as the latter grows, more states areaffected by its presence, and the disorder grows substan-tially. This is illustrated in Figs. 8 (a),(b) , where we plottwo graphs with a strong on site impurity placed on anatomic and a molecular site, respectively.Another interesting property is the additivity of thedisrupted patterns, when one adds more weak impuri-ties. One can just superpose the eigenstate maps of theindividual impurity disruptions to obtain the total disor-dered pattern. This is illustrated by the simple examplesin Fig. 9: there are two atomic impurities in Fig. 9 (a) ,characterized by the renormalization paths AAA+AMA,and in Fig. 9 (b) , we add a molecular impurity to the pre-vious two, such that the total disruption is characterizedby the paths AAA+AMA+MMMA. It is clear that they FIG. 8. (Color online) Disorder induced by a strong impurityof order V I = 10 t w . (a) Atomic impurity. (b)
Molecular im-purity. In both cases, all clusters are affected by its presence.FIG. 9. (Color online) Example of multiple weak impurities( V I = 0 . t w ) placed on the Fibonacci chain. (a) AAA-AMApattern. (b)
AAA-AMA-MMMA pattern. It is remarkablethat one can just add the individual contributions from eachof the classes to generate the (weak) multi-impurity classes. are obtained by the superposition of the graphs shown inFigs. 6 (b),(c) and Fig. 7 (c) .Naturally, there is a limitation to these observations,as the amount of paths grows substantially with the sizeof the chain. Moreover, as the impurities grow stronger,the additivity of the graphs breaks down and the im-purities start to influence each other, an effect that iseasily visible in the eigenstate map in Fig. 10. There weadded an impurity with ten times the previous strengthon a AAA site and another one on an MMMA site. Wecan clearly see that the two individual graphs are notsuperimposed. Nonetheless, it still remains that beforeone ends up in a completely insulating regime throughthe introduction of impurities, the existence of resonantstates [21], especially in the case of weak impurities (oforder 0 . t w ), allows for what can be called a “transitionregime”, in which some level of quasiperiodic order is stillpresent in parts of the system. FIG. 10. (Color online) Two relatively strong impuritiesplaced on an MAM site and a AMM site. Their strengthis set at V I = 10 t w . The result is not a superposition of theindividual MAM and AMM graphs (see Fig. 8 for compari-son). IV. CONCLUSION
To conclude, in this work we studied the effect of dis-order in the spectrum and wavefunction amplitude of aFibonacci quasicrystal. We first introduced the 1D modelin the tight-binding approximation and briefly explainedhow to understand the spectrum through a deflation pro-cedure. This was then followed by the introduction ofan impurity in the quasiperiodic lattice, which leads tothe appearance of at least one localized state. We havethen shown that in the weak impurity regime, disorder isintroduced in a very structured manner, following a la-belling provided by the renormalization path of the sitesat which the impurities have been placed. This pathstructure holds in fact for any impurity strength in thesingle impurity case. For weak impurities, disorder is re-stricted to subclusters of the system. This indicates thatthere exists a transition regime between a more insulatingstate where disorder is dominant, and the typical criticalstates present in a quasiperiodic lattice. We emphasizethat the viewpoint offered in this paper stems from thevery important notion of conumbers. They turn out tobe essential in bringing out the structure with which dis-order sets in and makes our observations very intuitive.Although the renormalization scheme is only exact inthe limit ρ (cid:28)
1, even ρ = 0 . ρ ≤ . (b) . Indeed, this site is thecentre of a very large region of palindromic symmetry.However, this naive interpretation was quickly ruled outas the MMA graph in Fig. 7 (a) offered very little dis-ruption, even though it had no locally symmetric regionsurrounding it. In recent work, the on-site model of theFibonacci chain has been studied using a local resonatormode framework [18]. It would be interesting to find ananalog version of this framework in the hopping model,which offers an intuitive understanding of the effect of im-purities on local symmetries. This could be helpful to fur-ther understand the topological features of the Fibonaccichain. Indeed, it is possible to study the robustness of thetopological phase by subjecting the system to impurities.Since the gap labelling theorem has been reinterpreted interms of renormalization paths [16], the presence of sta- ble gap states can be analyzed by the amount of disorderin the graphs. This in turn should be related to how theimpurity breaks the (local) palindromic symmetry or pre-serves it. Hence, one would be able to evaluate whetherthe topological phase is protected by this symmetry, orif it is of a different nature. ACKNOWLEDGMENTS