Wave Packet Dynamics, Ergodicity, and Localization in Quasiperiodic Chains
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Wave Packet Dynamics, Ergodicity, and Localization in Quasiperiodic Chains
Stefanie Thiem and Michael Schreiber
Institut f¨ur Physik, Technische Universit¨at Chemnitz, D-09107 Chemnitz, Germany
Uwe Grimm
Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United Kingdom (Dated: November 7, 2018)In this paper, we report results for the wave packet dynamics in a class of quasiperiodic chainsconsisting of two types of weakly coupled clusters. The dynamics are studied by means of the returnprobability and the mean square displacement. The wave packets show anomalous diffusion in astepwise process of fast expansion followed by time intervals of confined wave packet width. Applyingperturbation theory, where the coupling parameter v is treated as perturbation, the properties of theeigenstates of the system are investigated and related to the structure of the chains. The results showthe appearance of non-localized states only in sufficiently high orders of the perturbation expansions.Further, we compare these results to the exact solutions obtained by numerical diagonalization. Thisshows that eigenstates spread across the entire chain for v >
0, while in the limit v → v = 0. Caused by this ergodicity breaking, the wave packet dynamics changesignificantly in the presence of an impurity offering the possibility to control its long-term dynamics. PACS numbers: 71.23.Ft, 71.15.-m, 72.15.-v
I. INTRODUCTION
Understanding the relations between the spectral prop-erties of a given Hamiltonian and the dynamics of wavepackets which are governed by it, remains one of the ele-mentary questions of quantum mechanics that still posessignificant challenges, further emphasized by the discov-ery of quasicrystals.
While spectra of many Hamilto-nians decompose into a point-like part and an absolutelycontinuous part accompanied by bounded (localized) andunbounded (delocalized or extended) eigenstates, thereexists a large variety of Hamiltonians whose spectra, forcertain values of the parameters, are neither pure point-like nor absolutely continuous, nor a combination of both.In this case, the spectrum contains a singular continuouspart, and eigenstates are often found to be multifractal.Examples are Harper’s model of an electron in the mag-netic field, the kicked rotator as well as the Andersonmodel of an electron in a disordered medium. In the caseof an electron in a one-dimensional quasiperiodic system,as studied in this paper, many examples lead to spectrawhich are purely singular continuous as well.
To address the above mentioned challenge, we investi-gate wave packet dynamics in one-dimensional quasiperi-odic chains by numerical simulation as well as pertur-bation theory, and relate the results to the hierarchicalproperties of these chains. The content of this paper isorganized as follows: At first we introduce the construc-tion rule and structure of the chains in Sec. II. Section IIIthen focuses on numerical results for the time evolutionof wave packets. To obtain a better understanding of theproperties of wave packet spreading and localization, weapply perturbation theory in Sec. IV and further studythe influence of an impurity on the wave packet dynamicsin Sec. V, followed by a brief summary of our results.
II. QUASIPERIODIC CHAINS WITH GOLDEN,SILVER OR BRONZE MEANS
In this paper we study one-dimensional quasiperiodicsystems constructed by the inflation rule P = ( w → ss → sws n − , (1)iterated a times starting from the symbol w , where theletter w denotes a weak bond and s a strong bond. We re-fer to the resulting sequence after a iterations as the a thorder approximant C a with the length f a given by therecursive equation f a = f a − + nf a − and f = f = 1.Depending on the parameter n the inflation rule gener-ates different so-called metallic means, i.e. the lengths oftwo successive sequences satisfy the relationlim a →∞ f a f a − = λ , where λ is an irrational number with the continued frac-tion representation [¯ n ] = [ n, n, n, ... ]. For example, n = 1yields the well-known Fibonacci sequence related to thegolden mean λ Au = [¯1] = (1 + √ /
2, the case n = 2 cor-responds to the octonacci sequence with the silver mean λ Ag = [¯2] = 1 + √
2, and for n = 3 one obtains the bronzemean λ Bz = [¯3] = (3 + √ / Due to the recursive inflation rule (1), these quasiperi-odic chains possess a hierarchical structure, which is moreclearly visible by using the alternative construction rule C a = C a − C a − ( C a − ) n − , yielding the same quasiperi-odic sequences for a ≥ C = w and C = s . Fur-ther, for given n , the structure of these chains consists ofonly two types of clusters with strong interactions, s n and s n +1 , which are separated by a single weak bond. Thelatter property can be related to the eigenstates of thesystem, whereas the hierarchical property has a crucialinfluence on the transport properties for weak coupling.We discuss both aspects in detail later.To obtain the quantum mechanical eigenstates of thesesystems, we only consider nearest-neighbor hopping andhence obtain the tight-binding Hamiltonian H = f a X l =0 | l i t l,l +1 h l + 1 | + f a X l =0 | l i ε l h l | , (2)represented in the orthogonal basis states | l i associatedto a vertex l . The off-diagonal matrix elements representthe kinetic energy in the tight-binding model. The diago-nal elements of the Hamiltonian matrix, which representthe potential energy of the sites, are set to zero ( ε l = 0)because no energetic disorder is taken into account forthe studied systems.The hopping parameters t are chosen according to theletters s (strong) and w (weak) of the quasiperiodic se-quence C with t s = 1 and t w = v (0 ≤ v ≤ a th order approximant is givenby f a + 1. This model can be interpreted as describingan electron hopping from one vertex of the quasiperiodicchain to a neighboring one, and the aperiodicity is givenby the underlying quasiperiodic sequence of couplings.We then have to solve the discrete time-independentSchr¨odinger equation H | Ψ i i = E i | Ψ i i by diagonalization of the Hamiltonian matrix of Eq. 2.Applying free boundary conditions, we obtain f a + 1eigenstates | Ψ i i = P f a l =0 Ψ il | l i and the corresponding en-ergy values E i . III. TIME EVOLUTION OF A WAVE PACKETON QUASIPERIODIC CHAINS
As outlined in the introduction, the dependency oftransport properties on the spectral properties of theHamiltonian is not yet fully understood, and thus theinvestigation of transport properties in quasiperiodic sys-tems continues to be of special interest. In this section,we study the wave-packet dynamics in such systems, inparticular in the limit of weak coupling where v ≪ | Φ i = P f a l =0 Φ l | l i ,which is initially localized at the center of the quasiperi-odic chain, i.e. Φ l ( t = 0) = δ ll with l = ⌈ f a / ⌉ . Itis represented in the basis of the orthonormal eigen-states Φ l ( t ) = P i Ψ il Ψ il ( t ). The solutions of the time-dependent Schr¨odinger equation then follow by the sep-aration approach with Ψ il ( t ) = Ψ il e − i E i t . -1 m ean s qua r e d i s p l a c e m en t d ( t ) time t v = 0.3v = 0.2v = 0.1v = 0.05v = 0.03 FIG. 1: Evolution of the width d ( t ) of a wave packet initiallylocalized in the middle of the silver mean chain C Ag10 with3364 sites, for several small values of v . The insets show twomagnified steps for v = 0 . Besides calculating the expansion of the wave packetin space, a more detailed analysis can be obtained bythe computation of the temporal autocorrelation function(also known as return probability) C ( t ) = 1 t Z t | Φ l ( t ′ ) | d t ′ and the mean square displacement (also called the width) d ( t ) = " f a X l =0 | l − l | | Φ l ( t ) | of the wave packet. It is known that a particle’s returnprobability decays with a power law C ( t ) ∼ t − δ , where δ is equivalent to the scaling exponent of the local densityof states, and δ = 1 refers to ballistic motion. Itis further known that the spreading of the width d ( t ) ofthe wave packet shows anomalous diffusion, i.e. d ( t ) ∼ t β with 0 < β < Here β = 0 corresponds tothe absence of diffusion, β = 1 / β = 1 to ballistic spreading.In addition, the wave packet dynamics exhibit mul-tiscaling, where different moments of the wave packetscale with different, nontrivially related exponents β . While wave packet localization impliesa pure point spectrum, the converse is not true, andthe more refined notion of semi-uniform localization isnecessary. However, the exact relations between parti-cle dynamics and singular or absolutely continuous spec-tra are less well understood. As a rule of thumb, systemswith singular continuous spectra exhibit anomalous dif-fusion, while absolutely continuous spectra may lead toeither anomalously diffusive or ballistic dynamics.
Consequently, the systems considered here should showanomalous diffusion with the mentioned power-law de-pendency d ( t ) ∼ t β due to their singular continuous -1 m ean s qua r e d i s p l a c e m en t d ( t ) time t a v = 0.3v = 0.2v = 0.1v = 0.05v = 0.03 10 -1 m ean s qua r e d i s p l a c e m en t d ( t ) time t b v = 0.3v = 0.2v = 0.1v = 0.05v = 0.03 FIG. 2: Same as Fig. 1, but for (a) the golden mean model C Au18 with 4182 sites and (b) the bronze mean model C Bz8 with 5117sites. -2 -1 t e m po r a l au t o c o rr e l a t i on f un c t i on C ( t ) time t a v = 0.03v = 0.05v = 0.1v = 0.2v = 0.3 10 -2 -1 t e m po r a l au t o c o rr e l a t i on f un c t i on C ( t ) time t b v = 0.03v = 0.05v = 0.1v = 0.2v = 0.3 10 -2 -1 t e m po r a l au t o c o rr e l a t i on f un c t i on C ( t ) time t c v = 0.03v = 0.05v = 0.1v = 0.2v = 0.3 FIG. 3: Temporal autocorrelation function C ( t ) of a wave packet initially localized at the center of (a) a golden mean chain C Au18 with 4182 sites, (b) a silver mean chain C Ag10 with 3364 sites, and (c) a bronze mean chain C Bz8 with 5117 sites. Results areshown for small coupling parameters v . spectra, which is confirmed by our numerical re-sults. Figures 1 and 2 show the development of the meansquare displacement of the wave packet over time for thegolden, silver, and bronze mean models in the regimeof strong quasiperiodic modulation ( v ≤ . β Au ≈ . β Ag ≈ .
39, and β Bz ≈ .
40 obtained for v = 0 . v and the associated val-ues of β see Yuan et al. ), there are intervals where d ( t )grows according to a power law d ( t ) ∼ t β ′ with a constant( v -independent) exponent β ′ which are intercepted byflat regimes. As demonstrated by the insets in Fig. 1, inthese flat regimes d ( t ) strongly oscillates in a self-similarmanner, reflecting the hierarchical structure of the sys-tem. Nevertheless, the width d ( t ) remains bounded fromabove by a constant.While for the silver mean chain the steps in log d ( t ) have about the same size for a particular value of v , weobserve small as well as large steps for the golden andbronze mean models. This may be caused by the dif-ferent underlying inflation rule, which in the case of theoctonacci chain leads to a symmetric sequence (the se-quence is palindromic for this case) and to asymmetricones for the other two systems. However, as a generaltrend we observe an increase of the logarithm of the timedistance between successive steps with decreasing cou-pling constants v in all three models.The same behavior can also be observed for the au-tocorrelation function C ( t ), as shown in Fig. 3. Allthree quasiperiodic chains show a stepwise behavior ofthe return probability C ( t ), where the minimum value of C ( t ) depends on the system size and on the number of w bonds. Again the stepwise process is clearly visible,where a step here consists of the decrease of C ( t ) with apower law with v -independent exponent δ ′ , followed bya time interval of constant return probability. The timeintervals for the power-law behavior and the flat parts ofboth quantities C ( t ) and d ( t ) are in correspondence witheach other. Further, for large time the return probabilityand wave-packet width are bounded by a constant dueto the finite size f a of the system.A more detailed inspection of the wave packet dynam-ics reveals that breathing modes are responsible for theoscillations, while the wave-packet spreading itself is lim-ited to low-amplitude leaking out of the region in whichit is confined. Eventually, the wave packet expands fastto reach the next level of the hierarchy, before the wholeprocess repeats. This behavior is even more evident inthe time evolution of the probability density of such wavepackets, as shown in Fig. 4. At first up to t = e in Fig. 4the wave packet is confined in a narrow range around itsinitial position in the environment of the approximant C Ag7 and oscillates back and forth in this range corre-sponding to the strong oscillations of d ( t ). Then, between t = e and t = e the wave packet expands almost bal-listically and a significant probability density is found inthe neighboring C Ag7 sequences (cp. t ≥ e ). A similarbehavior can be already seen for the previous levels ofthe hierarchy (cp. the panels up to t = e with those for t ≥ e in Fig. 4), where the wave packet can be observedto spread from the central C Ag5 structure to a three-fold C Ag5 sequence occurring in the middle of the central C Ag7 chain.This indicates that the values of the flat regimes in d ( t ) and C ( t ) are directly related to the chain structureand do not depend on the coupling parameter v . In par-ticular, we can give a rough estimate of the correspond-ing values of d ( t ) for the plateaus by assuming that thewave packet is uniformly distributed in a confined regionat the center of the chains. For an approximant C Ag a the center of the octonacci chain is made up from thethree-fold sequences C Ag o C Ag o C Ag o with o = a − b ( b ∈ N ,0 < o < a ). Confining the wave packet to these re-gions we obtain for the approximant C Ag10 of Fig. 1 thevalues d ( t ) = 2 . , , ,
500 for o = 2 , , , d ( t ) ∼ t β ′ withexponents β ′ Au ≈ . β ′ Ag ≈ .
85, and β ′ Bz ≈ .
98 de-termined for the smallest coupling constant v = 0 .
03 con-sidered here. Computing the results also for all 6 otherqualitatively different initial positions of the wave packetfor the octonacci chain we obtained about the same scal-ing exponents β ′ . For instance for the octonacci chain wefound β ′ = 0 . − .
85, where some of these differencesmight be caused by fluctuations of the width d ( t ) whichare present in the regime of strong expansion and makefitting difficult. These scaling exponents tend towards
0 200 400 600 800 1000 1200 site | Φ (ln t = 22) || Φ (ln t = 20) || Φ (ln t = 18) || Φ (ln t = 16) || Φ (ln t = 14) || Φ (ln t = 12) || Φ (ln t = 10) || Φ (ln t = 8) || Φ (ln t = 6) | C C C C C C C FIG. 4: Snapshots for the evolution of a wave packet initiallylocalized at the center of the octonacci chain C Ag9 with v = 0 . C Ag9 is visualized by showing the occurrence of the patterns of C Ag8 and C Ag7 in the chain. the exponents obtained for v → which indicates thatthe fast expansion is not governed by the weak coupling,but rather a kind of resonance between the different lev-els of the hierarchy. Further, for the return probabil-ity we find the exponents δ ′ Au ≈ . δ ′ Ag ≈ .
71, and δ ′ Bz ≈ .
76, which are again relatively close to the expo-nents for v → However, these values of the scal-ing exponent δ ′ might differ from the exact result, be-cause in one dimension we cannot rule out the influenceof subdominant logarithmic contributions for the consid-ered short time intervals in the step-like process. Similar behaviors for d ( t ) and C ( t ) have been reportedbefore for the Fibonacci chain with strong quasiperiodicoscillations. Wilkinson and Austin found the samestep-like process when studying the spreading of a wavepacket for Harper’s equation of an electron in a mag-netic field. Based on a qualitative model of the wave-packet spreading in the semiclassical approximation andon numerical simulations, they argued that a hierarchi-cal splitting of the energy spectrum into constant-widthbands leads to a step-like behavior with β ′ = 1, which issmoothed due to the (broad) distribution of band widths.The value β ′ < v ≪
1. The energy levels of the singular continuousspectrum for 0 < v < v → This suggests that the self-similar spreading of thewave packet is only an approximate description of a moregeneral multiscale dynamics.Further, the self-similarity of quasiperiodic sequenceswas previously used in a renormalization-group perturba-tive expansion that provided a great deal of insight intothe eigenstate properties, and showed multiscaling ofwave packet dynamics. In the following section we fo-cus on ergodic rather than hierarchical properties. Byan elementary analysis of the perturbation theory of de-generate levels at v = 0 for small coupling constants v ,we show that, on the one hand, eigenstates delocalize forany v >
0, in contradistinction to (trivial) localization at v = 0. On the other hand, in the limit as v →
0, eigen-states delocalize across only one set of clusters containingthe same number of atoms, i.e. we obtain a subcluster lo-calization due to the breaking of ergodicity.
IV. RALEIGH-SCHR ¨ODINGER THEORY
Raleigh-Schr¨odinger theory allows the recursive con-struction of matrices in subspaces for a degenerateeigenenergy to a given order p , whose diagonalizations(the secular problem) yield corrections to the unper-turbed eigenenergies. Within this approach, the Hamil-tonian is decomposed into an unperturbed system H (0) and a perturbation H (1) with H = H (0) + λH (1) , wherethe hopping parameter v is treated as the perturbationyielding H (0) = H ( v = 0) and H (1) = H ( v ) − H ( v = 0).Although the accuracy of O ( v p +1 ) of the expansion tothe p th order is not guaranteed, the theory yields goodresults for small perturbations and preferably large sep-arations between the degenerate energy levels. The firstcondition is met since we only consider small values of v and the latter one is satisfied since we found that forthe unperturbed system | E i − E j | > c with c Au = 0 . c Ag = 0 .
20 and c Bz = 0 . p and n due to the chain structure mentioned in Sec. II.The reason is that the chain consists of strongly coupledclusters with n +1 and n +2 atoms, which are weakly con-nected via the hopping parameter v . For v = 0 we havean unperturbed system with 2 n + 3 highly degeneratelevels, where all eigenstates are localized on individualclusters. In higher orders of perturbation theory, theselocalized states then spread first over neighboring clus-ters of the same type as the coupling among the clustersis taken into consideration, and, for a sufficiently highorder, delocalize across the whole chain.Since the maximal number of letters w between twoconsecutive clusters of length s n +1 is also n +1, the eigen-states of these clusters delocalize only in order n +1 of the perturbation theory. However, small clusters of length s n are connected by at the most 2 ( n >
1) or 3 ( n = 1) weakbonds. More precisely, the dimension of the secular prob-lem for each type of cluster separated by not more than q weak bonds, changes from O ( q ) at most for p < q to O ( f a ) for p ≥ q . Only the latter case allows for multifrac-tal and/or extended states to be present in the solutionsof the perturbed system.As an illustrating example we analyze the octonaccichain with n = 2. In the unperturbed system there are 7levels, given by E (0) sss = ± ( √ ± / E (0) ss = ±√ , ss and sss clusters, the eigen-states delocalize only in the 2nd and 3rd order of theperturbation theory. In first-order expansion ( p = 1), theonly correction are 6 levels linear in v , splitting off thethree E (0) ss levels, because · ] sswss [ · is the only possibil-ity for which two clusters of the same type are connectedby a single w bond. There are also remaining separate ss clusters, so that the unperturbed E (0) ss levels are stillpresent in the spectrum. For p = 2, all ss clusters be-come connected while sss states are still not extendeddue to the existence of · ] ssswsswsswsss [ · sequences inthe chain. For p = 3, states belonging to sss levels delo-calize as well. Figure 5 shows this splitting of the energylevels, where the solutions obtained by different ordersof the perturbation theory are shown in comparison tothe energy levels obtained by numerical diagonalizationof the Hamiltonian H . The results of the perturbationapproach and the exact energy values are close, althoughonly 2nd-order corrections are taken into account. In thecase of E (0) ss the 2nd-order corrections seem to overcom-pensate the error leading to a stronger splitting of theenergy levels compared to the exact numerical results.To investigate the issue of convergence further, we no-tice that, even when calculated to all orders of v , thesecular problems for the two types of clusters give solu-tions that are inevitably restricted to the clusters of thegiven type, with zero component on the clusters of theother type. For various values of v we check by numericaldiagonalization whether this can be confirmed. Figure 6shows the total probability that the particle in an eigen-state Ψ i with an energy E i will be on a large cluster s n +1 P s n +1 ( E i ) = X l ∈ s n +1 | Ψ il | (3)for the golden, silver, and bronze mean model. Figure6 illustrates how the degenerate eigenvalues for v = 0spread into wider and wider bands with increasing v . Theresults imply that P s n +1 strongly depends on the energyof the corresponding eigenstate for small v , because it iseither large for the states belonging to the s n +1 bandsand vanishes for the states of the s n bands in the limit v → n + 1 bands with highprobabilities and n bands with low probabilities, in cor-respondence with the number of eigenstates generated by numerical diagonalizationperturbation theory p = 0 p = 1 p = 2(1+ √ √ √ a numerical diagonalizationperturbation theory p = 0 p = 1 p = 2(1+ √ √ √ b FIG. 5: Comparison of energy levels obtained by the perturbation theory approach and by numerical diagonalization of theperturbed Hamiltonian H . Results are shown for the octonacci chain with n = 2, a = 6, and (a) v = 0 . v = 0 .
2. Forreasons of symmetry only energy values
E > p r obab ili t y on ss c l u s t e r s energy E a v = 0.0v = 0.1v = 0.2v = 0.3v = 0.4v = 0.5v = 0.6v = 0.7v = 0.8v = 0.9 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 p r obab ili t y on sss c l u s t e r s energy E b v = 0.0v = 0.1v = 0.2v = 0.3v = 0.4v = 0.5v = 0.6v = 0.7v = 0.8v = 0.9 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 p r obab ili t y on ssss c l u s t e r s energy E c v = 0.0v = 0.1v = 0.2v = 0.3v = 0.4v = 0.5v = 0.6v = 0.7v = 0.8v = 0.9 FIG. 6: Probability P s n +1 ( E ) that a particle in an eigenstate with a given energy E is on the s n +1 sublattice, for v =0 . , . , . . . , .
9. Only states with
E < E = 0.Results are given for (a) the golden mean model C Au19 with 6765 sites, (b) the silver mean model C Ag11 with 8120 sites, and (c)the bronze mean model C Bz8 with 5117 sites. Note that there are a few additional states not included in the perturbation theoryapproach, which are caused by the boundary conditions. the large and small clusters as obtained by our perturba-tion theory approach. However, for v > P s n +1 is greater than 0 for all energy bands, which im-plies that the eigenstates spread over both types of clus-ters and thus are ergodic. Here ergodicity denotes thespreading of eigenstates over both types of clusters.While Fig. 6 shows results for only one iterant of thethree considered models, we have also compared the re-sults for P s n +1 of smaller and higher approximants tosee whether there is any systematic deviation from theshown values. We found that, although there are muchless states for smaller systems or many more states forlarger approximants, the values of P s n +1 do not shift sys-tematically for almost all of the states. Instead, the ad-ditional points cluster in the same way as in Fig. 6.This supports the possibility that eigenstates for an in-finite quasicrystalline chain remain ergodic even for verysmall values of v . This is in disgareement with the re-sults of the Raleigh-Schr¨odinger perturbation expansionof degenerate levels, which might, nevertheless, still beaccurate in the limit as v → P s n +1 ( E i ) v → −−−→ ( E i is caused by s n +1 clusters0 if E i is caused by s n clusters , which is a necessary condition but not a sufficient one. V. ERGODICITY AND INFLUENCE OF ANIMPURITY
Although the eigenstates remain ergodic even for weakcoupling v , the wave packet dynamics are strongly influ-enced, as outlined in Sec. III. This leads to interest-ing consequences when a single impurity of strength u is placed at a site l ′ by changing a diagonal element ofthe Hamiltonian, i.e. H ′ = | l ′ i u h l ′ | . At first we studythe spreading of wave packets in the presence of such animpurity at different types of clusters and then presentresults for the maximum wave packet width when theimpurity is placed at or near the initial position of thewave packet.In the first situation we performed several numericalexperiments for various initial positions of the impurityand of the wave packet for different values of u . We foundthat for large u the impurity acts as a barrier, effectivelycutting the chain into two halves. The consequence isthat the wave packet is reflected at the impurity indepen-dently of its initial site, even if the coupling v is small.For u →
0, on the other hand, the unperturbed wavepacket propagation of the case u = 0 is restored.Understanding the wave packet propagation in theregime of intermediate values of u , however, poses sig-nificant challenges and surprising results. A commonsituation is shown in Fig. 7 for the silver mean model,where the wave packet is initially localized at an sss clus-ter and the impurity u is placed either on an ss or sss cluster near the center of chain. In this case the evolu-tion of the wave packet exhibits high sensitivity on the position of the impurity, approaching two quite differ-ent stationary states. In particular, while in Fig. 7(a)the final state is just slightly perturbed from the finalstate for u = 0, in Fig. 7(b) most parts of the wavepacket are reflected and only a small amplitude can leakthrough the barrier. This kind of wave packet dynamicsis a consequence of the nearly non-ergodic spreading ofthe eigenstates for the two different types of clusters forsmall v as discussed above (cp. Fig. 6). The explanationis that the wave packet is constructed by a superpositionof all eigenstates and because it is initially localized onan s n +1 cluster, eigenstates caused by this type of clusterare contributing with a much higher probability than theeigenstates of the s n clusters. Consequently, the impu-rity is felt as a barrier by the wave packet when placedon the same type of cluster as the initial position of thewave packet.In the second situation we address the influence of animpurity placed at or near the initial site of the wavepacket on the dynamics by studying the dependence ofthe final wave packet width on the impurity strength u .Figure 8 shows the maximum value of d ( t ) attained inthe course of the evolution of a wave packet in the oc-tonacci chain, which was initially localized at the site l = ⌈ f a / ⌉ . Here only systems C Ag a with odd a and withthe impurity placed at the initial site of the wave packetor its left neighbor site are shown. To obtain these max-imum values of the width we perform the calculations asin Fig. 1, but now for the perturbed system and differentvalues of the impurity strength u up to very large times,where the system is governed by finite size effects and d ( t ) becomes constant.The results show that, for small u , the width of thewave packet equals the results for the unperturbed sys-tem for both positions of the impurity. For large u , weobtain a strongly localized final wave packet when theimpurity is added at the initial site l of the wave packetand a constant width of the wave packet when placingthe impurity at the left neighbor site l −
1. In the firstcase, the expansion of the wave packet in the eigenstatebasis is dominated by strongly localized wave functionscaused by the large impurity u and the wave packet canno longer spread across the chain. In the latter case,the impurity acts as a barrier placed at the center of thechain, and consequently the wave packet is reflected andonly spreads across one half of the system as in Fig. 7(b),and d ( t ) reaches a plateau in Fig. 8. Nevertheless, thewidth d ( t ) is reduced compared to an unperturbed sys-tem with half the system size because in the presenceof an impurity always some localized eigenstates, whichdo not spread across the quasiperiodic chain, occur andcontribute to the expansion of the wave packet.However, in between these two extremes there is a widerange of values of u for which the final width of the wavepacket is significantly reduced even for u ≪ v , signal-ing dynamical localization. There are nevertheless sev-eral well-defined peaks in Fig. 8 for some values of u atwhich the maximum wave-packet width is significantly
0 100 200 300 400 500 site a | Φ (ln t = 24) || Φ (ln t = 21) || Φ (ln t = 18) || Φ (ln t = 15) || Φ (ln t = 12) || Φ (ln t = 9) || Φ (ln t = 6) |
0 100 200 300 400 500 site b | Φ (ln t = 24) || Φ (ln t = 21) || Φ (ln t = 18) || Φ (ln t = 15) || Φ (ln t = 12) || Φ (ln t = 9) || Φ (ln t = 6) | FIG. 7: Snapshots of the evolution of two wave packets in the presence of an impurity for the octonacci chain with n = 2, a = 8,and v = u = 0 .
1. The wave packet is initially localized on a large cluster in a local environment · ] ws x ssw [ · at the site x . In thetwo panels (a) and (b) the impurity is located on a small cluster or a large cluster in a local environment · ] ws b ssws a swsssw [ · at the sites indicated by a and b , respectively, as visualized by a vertical line in each panel. For easier comparison, the verticaldashed line in each panel marks the position of the impurity in the other panel. The long-time wave packet dynamics exhibithigh sensitivity on whether the impurity is located on the same type of cluster or not as the one on which the wave packet wasinitially localized. -6 -5 -4 -3 -2 -1 m a x i m u m w i d t h d ( t ) strength of impurity u a = 7, impurity at initial site u a = 9, impurity at initial site u a = 7, impurity at site u a = 9, impurity at site u FIG. 8: Maximum width of a wave packet attained during itsevolution in the presence of a single impurity of strength u for the octonacci chain with n = 2 and v = 0 .
1. The wavepacket is initially localized at the center of the chain x = l inthe local environment · ] wssws u s x swssw [ · and the impurityis either placed at the same site u = x or its left neighborsite u . Note that we only included approximants C Ag a withodd a because for even a the local environment of the initialposition of the wave packet is different. enhanced, compared to the cases of slightly smaller andslightly larger values of u . These peaks persist for differ-ent system sizes and for different positions of the impurity u , although the positions and structure of the peaks can change. Figure 8 shows that the peaks for the four sys-tems considered there occur at the same strength of theimpurity, especially for relatively large values of u .Having a closer look at the eigenstates caused by theimpurity, we found that those peaks appear at impu-rities u , where at least some of these perturbed statescoincide with the bands in the energy spectrum of thequasiperiodic approximants shown in Fig. 6. In this casethe states caused by the impurity hybridize with the un-perturbed states of this band and consequently the wavepacket is able to spread along the chain easier. This alsoexplains the differences of the peaks for a = 7 and a = 9in Fig. 8, because the structure of the energy spectrumof the 9th approximant is more complicated and thusadditional peaks occur.Further, in Fig. 9 we compare the maximum widthsof the wave packet for impurities placed at the 1st leftneighbor and the 3rd left neighbor in systems C Ag a witheven a . It is clearly visible that the maximum widths ofthe wave packet for these systems are almost identical forthe two different positions of the impurity. The reasonis that the impurities are in both cases located at theedges of the ss cluster, which is adjacent to the cluster,where the wave packet is initially localized. In contrastthe curve of the approximant C Ag9 shows a completely dif-ferent behavior, because here the wave packet is locatedin a different local environment in the beginning.Further, by repeating similar numerical experimentsfor various values of the coupling strength v , we foundthat the peak structure becomes less distinct with in- -6 -5 -4 -3 -2 -1 m a x i m u m w i d t h d ( t ) strength of impurity u a = 6, impurity at site u a = 8, impurity at site u a = 6, impurity at site u a = 8, impurity at site u a = 9, impurity at site u FIG. 9: Same as Fig. 8, but for even a with the impurity ofstrength u located at the 1st and 3rd left neighbor. The wavepacket is initially localized at the center x = l of the chainin the local environment · ] wsssw u ss u w x sswsssw [ · . creasing v . For the silver mean model we observed thatit persists up to v ≈ .
4. For this value of v the widthsof the energy bands become smaller than the gaps be-tween them, which in turn means that the impurity-related eigenstates coincide very often with the energybands and thus peaks merge so that almost no valleysoccur.These results show that in quasiperiodic quantumwires one can strongly influence the long-range electronictransport properties by inducing local perturbations atdifferent positions and of various strengths. The char-acteristics are related to the nature of the eigenstates,which spread only across one type of cluster in the limit v →
0. Knowing the structure of the energy bands and ofthe eigenstates allows one to design quasiperiodic chainswith impurities that can act as sort of control gates.
VI. CONCLUSION
In this paper, we considered the electronic transportin one-dimensional quasiperiodic chains consisting of twotypes of clusters which are weakly coupled by a hop-ping parameter v . The investigations of the wave-packetdynamics revealed the occurrence of a stepwise process with time intervals of power-law growth, followed by aregime with confined wave-packet width. Nevertheless,the average wave packet dynamics can be classified asanomalous diffusion. These results are consistent withthe literature. The stepwise behavior is caused bythe hierarchical structure of the chains, leading to theconfinement of the wave packet until it expands fast toreach the next level of the hierarchy.The perturbation theory approach allowed us to drawa connection between the structure of the weakly coupledclusters and the localization characteristics of the eigen-functions Ψ, which only become delocalized in sufficientlyhigh orders of the perturbation expansion. This happenswhen clusters of a specific type become connected, i.e. inorder n + 1 of the expansion for large clusters and in the2nd or 3rd order for small clusters.However, while we obtained that the eigenfunctionsspread ergodically over all clusters of the chain for v > v → Acknowledgments
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