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Phononic topological states in 1D quasicrystals
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Page 1 of 8 AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1
1
2 Phononic topological states in one-dimensional quasicrystals
3
4 J. R. M. Silva and D. H. A. L. Anselmo
Departamento de F́ısica Teórica e Experimental,
5 Universidade Federal do Rio Grande do Norte,59078-900, Natal–RN Brazil
6
7 M. S. Vasconcelos∗
8 Department of Physics and Astronomy, Western University, London–Ontario, N6A 3K7, Canada
9
10 V. D. Mello
11 Departamento de F́ısica, Universidade do Estado do Rio Grande do Norte, Mossoró–RN, 59625-620, Brazil
12 (Dated: June 24, 2019)
13 In this work, we address the study of phonons propagating on a one-dimensional quasiperiodic lat-
14 tice, where the atoms are considered bounded by springs whose strength are modulated by equivalent
15 Aubry-André hoppings. As an example, from the equations of motion, we obtained the equivalent
16 phonon spectrum of the well known Hofstadter butterfly. We have also obtained extended, critical,
17 and localized regimes in this spectrum. By introducing the equivalent Aubry-André model through
18 the variation of the initial phase φ, we have shown that border states for phonons are allowed to
19 exist. These states can be classified as topologically protected states (topological states). By Cal-
culating the Inverse Participation Rate, we describe the localization of phonons and verify a phase
20 transition, characterized by the critical value of λ = 1.0, where the states of the system change from
21 extended to localized, precisely like in a metal-insulator phase transition.
22
23
24 I. INTRODUCTION optical [10, 12, 13] localization. The growing studies on
25 quasicrystalline materials made it possible to obtain sys-
26 The discovery of a new class of materials by Shecht- tems with very thin layers arranged in a quasiperiodic se-
27 man et al. in 1984 [1], when they were studying the quence [14] or wires with quantum wells of width around
28 diffraction figures for an alloy of Aluminum and Man- 7 nm [15]. Levine et al. works [16], with the synthesis
29 ganese (which give him the Nobel Prize in Chemistry of a quasicrystal defined by the Fibonacci sequence in-
30 of 2011 [2]), had started up a new and vibrant research spired Merlin et al. to create, in the laboratory, the first
31 area. At the first view, this system was defined as an in-
one-dimensional quasicrystal [17]. Since that, the way
termediate structure between crystalline and amorphous of studying one-dimensional quasicrystal structures has32
33 solids, but today it is well recognized that quasicrystals
become standard. In this procedure, we define two dis-
are interpreted as a natural extension of the notion of a tinct building blocks, each of which carries the necessary34
35 crystal to structures with quasiperiodic (QP), instead of
physical information, and then they are arranged accord-
periodic, arrangements of atoms [3, 4]. A more recent ing to a particular sequence. For example, they can be
36
updated definition of quasicrystals with dimensionality described in terms of a series of generations that obey a
37 n (n = 1, 2 or 3) is that they can also be defined as a relation of particular recursion [13]. The term quasicrys-
38 projection of a periodic structure into a higher dimen- tal is indeed limited to a subdivision of a larger group
39 sional space mD, where m > n [5]. The diffraction figure of deterministic aperiodic morphological sets. There are
40 found by Schectman and collaborators has a long-range artificial systems which are part of other deterministic
41 order but has no translational periodicity as the crys- structures that cannot be given this nomenclature (for
42 tals, but rather the self-similarity property by scaling [1]. an up-to-date reference see Ref. [18]). Examples of ape-
43 In the icosahedral and decagonal quasicrystal√s, the self- riodic structures that differ from quasicrystals are sys-
44 similarity is related to the Golden Ratio ((1 + 5)/2), so tems that obey the Rudin-Shapiro and Thue-Morse [13]
45 that the atoms are separated by distances which repre- sequences. The researches have shown fractal properties
46 sent the Fibonacci sequence. These new materials have in their spectra and the existence of a non-trivial phase
47 great potential for applications, as some researches show transition, such as the metal-insulating phase [19, 20],
48 that they are rigid and brittle with unique transport char- only by adjusting some parameters of the generation se-
49 acteristics [6] and have very low surface energies that quence. Recently, it was reported that it is possible to
50 make them good thermal insulators with photonic and have a topological phase in photonic quasicrystals (in 2D)
51 thermoelectric properties [7–9]. Their spectra features a without any magnetic field applied, but instead introduc-
fractal structure and aspects of electronic [10, 11] and ing an artificial gauge field via dynamic modulation [21].52
The idea that photonic crystals could exhibit an analog
53 like the quantum Hall edge states was initially proposed
54
∗ by Haldane and Raghu [22] in tri-dimensional photonic55 Corresponding author: mvasconcelos@ect.ufrn.br; Permanent crystals.
56 address:Escola de Ciências e Tecnologia, Universidade Federal
do Rio Grande do Norte, 59078-900, Natal–RN, Brazil
57 On the other hand, phononic crystals are intensively
58
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60
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AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1 Page 2 of 8
2
1
2 studied as means to manipulate sound or elastic waves terfly [43]. After, we present the topological states of
3 (for a review see [23]) in the same way like in photonic phonons, considering an equivalent Aubry-André hop-
4 crystals. Following the same idea of Haldane and Ragh, ping [55], indicating the presence of border states for
researches have given attention to the search of the ex- phonons. Also, we study the phase transition through
5
istence of topologically protected edge states [24, 25] in the Inverse Participation Rate (IPR), where the system
6 those systems, which could be beneficial for many prac- changes from extended to localized. Finally, in section 4,
7 tical applications[26–28]. The topological effects in the we present the conclusions of this paper.
8 band structure in one-dimensional phononic crystal can
9 be characterized through topological invariants such as
10 Berry phases [29, 30] and Zak phases [31–33]. Recently, II. THEORETICAL MODEL
11 it was shown that these edge modes exist in the band
12 gap of 1D phononic crystals composed of two different The simple model for the phonon system can be defined
13 crystals, with distinct topological properties [34]. It also through the motion equation that represents the atoms
14 has been found in other types of phononic cystals [35]. as a spring-bound system with force constant Kn.
15 In this work, we investigate a system where it is possible
2
16 to have these edge modes in 1D, i.e., we study the edge ω un = −Kn+1un+1 −Knun−1 + Vnun (1)
17 modes in 1D phononic quasicrystals. Here, ω matches the vibration frequency, un are the in-18 Indeed, 1D quasicrystals were studied through the dividual displacements around the equilibrium position,
19 Harper model [36]. Many works on one-dimensional qua- and Vn = Kn+1 +Kn.
20 sicrystals showed that the localization property of the Considering a system of N atoms the motion equation
21 Harper model could be found in a quasicrystal through has the following matrix form:
22 the Hamiltonian of Aubry-André, considering the poten-
23 tial incommensurable with the lattice parameter [37–39].  
24 This system proved to present itself as a topological insu-
 
V −K 0 · · · 0 u00 1
25 lator which exhibits border states and non-trivial phases,  .  u1 
26 experimentally verified in the works of Kraus et al. [40],
−K V −K 0 · · · .. 1 1 2  
27 which used waveguides to obtain the frequency spectrum
 . . .  . . .. . . .  .. . . .  . 
28 in a quasicrystal, indicating the existence of a photonic

gap [41, 42]. The vast majority of published papers deal 
0 · · · . .  . un  =
29 0 · · · −Kn Vn −Kn+1 0 .
. 
with superlattices, exhibiting a fragmented energy spec-
30 .. .. . . . . . .

. .  .. .
trum of the famous Hofstadter butterfly [43] at the elec-  . . . . . . 0  31 tronic level, as well as for the optical case [44]. · · · 0 −  KN−1 VN− 1 −KN  
32 u· · · − N−1
33 Theoretical models for predicting the properties of
0 0 KN VN uN
quasiperiodic systems have been of considerable interest  
34 u0
to the scientific community, resulting in many theoreti-
35  u1 cal and experimental studies [40, 45–48]. However, some  
36  properties, like edge modes and topological states, in  
37  . one-dimensional quasicrystals, remain unexplored. The
38  .. localization of phonons in one-dimensional lattices has ω2  un  (2)39 already been studied for the Frankel-Kontorova model  .. 
40 [49, 50] and for quasiperiodic systems by the transfer  . 
41 matrix formalism [51, 52]. Some works compare the  
42 frequency spectrum with the energy bands obtained for uN−1
43 a quasicrystal defined by the transfer matrix formalism uN
44 [53, 54], in which the results retain fractality properties. The force constant K obeys a quasiperiodic modula-
45 However, these studies do not consider the effects of the ntion, undergoing small changes dependent on each site, as
46 initial phase φ. Therefore, in this work, we have studied shown in the Figure 1. Each atom has a mass mn = 1.0,
47 the frequency spectrum, and localization of phonons in a connected by springs with force constant Kn, given by:
48 quasiperiodic lattice through modulation of the Aubry-
49 André [55] model in order to characterize these edge Kn = C(1 + λ cos(2πbn+ φ)) (3)
50 states as phononic topological states, and also we studied
51 the analogous to metal-insulator phase transition for this The cosine term of Kn comes from the interaction with
52 system. the external potential of amplitude λ on the force con-
53 This paper is organized in the following way. In sec- stant C. The φ variable corresponds to the initial phase
54 tion 2, we present the theoretical model for the aperiodic when n = 0. b is the inverse of the period of the co-
55 (incommensurate) 1D system studied here. In section 3, sine function, and, it controls the periodicity of the mod-
56 we show our numerical results and discussion. First, we ulation of our spring-bound model. This type of sys-
57 present a profile spectra similar to the Hofstadter but- tem presents different results, depending on whether b
58
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3
1
2 brary (GSL)[58], implemented in C++ routines to find
3 the properties of the phonon spectrum in quasiperiodic
4 media from a sine-type modulation in the force constant
between neighboring atoms, with a varying φ phase.
5
6
7
III. RESULTS AND DISCUSSION
8
9
10 FIG. 1. Quasiperiodic lattice with the interaction of the po- The phonon spectrum, ruled by the motion equation
tential of Aubry-André. Each atom is subject to a force con-
11 with force constant modeled in (3) presents a profile simi-stant Kn, incommensurate with the lattice parameter.
12 lar to the Hofstadter butterfly, when plotted as a function
13 of the parameter b [43].
14 is rational or irrational [56]. Whenever b is irrational,  
15 the spring modulation is incommensurate, resulting in a ) 4 . 0
16 a qua√siperiodic pattern. Specifically, if b is equal to b )
17 (1 + 5)/2, we recover the so-called Fibonacci sequence 2 . 0 3 . 0
18 [57]. 2 . 0
19 From the eigenvalues of the motion equation (2) we 1 . 0
20 can obtain the frequency spectrum, while the eigenvec- 1 . 0
21 tors give us the individual displacements for each site.
22 The nature of these displacements un, similar to what 0 . 00 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 00 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
23 occurs with the wave-function, can show itself as dis- b b
24 tributed over all sites or located in just a few ones [51]. c )
25 If this location is concentrated at the edge of the system,
26 these correspond to the topological states of the border.
2 . 0
The location of the displacements can be obtained by
27
the inverse of the participation rate (IPR) [37]. The IPR
28 1 . 0of the eigenvector k can be obtained by the following
29 relation:
30
31 IPR(k) = ∑∑ | 0 . 0u 4k,l| 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0l 2 , (4)32 ( |u 2 bl k,l| )
33
34 where the l index represents the sum over all sites on FIG. 2. Hofstadter butterfly for the frequency spectrum in
35 the lattice. The IPR indicates the inverse of the number three different λ values, namelly (a) λ = 0.5, (b) λ = 3.0,
36 of occupied sites L, so when the oscillations are equally and (c) λ = 1.0, for the aperiodic potential with 100 atoms
37 distributed, the IPR ≈ 1/L, whereas on the opposite (N =100) and φ = 0. We can notice that in λ = 1.0 the
situation of extreme localization, we have only one site spectrum presents a considerable amount of bands separated
38
vibrating with the respective frequency, which results in by increasingly narrow gaps characterizing a multifractal be-
39 IPR ≈ 1. havior.
40 The numerical results were obtained from the diago-
41 nalization of equation (2), with unitary values for both The equation (3)√depicts a spectrum of the quasicrys-42 the mass (mn = 1.0) and the force constant amplitude tal when b = (1 + 5)/2 ≈ 1.618. In Fig. 2 we have
43 (C = 1.0). In this way, we can study the influence of the phonon spectrum for three different values of the po-
44 modulation of the force constant Kn, on the frequency tential amplitude λ as a function of b. For λ = 0.5, in
45 spectrum of the phonons in an aperiodic 1D system. Fig. 2a, the spectrum presents bands very close while
46 From the diagonalization of the phonon system matrix for λ = 1.0, in Fig. 2c, we have the parameter which
47 (2), we found the frequency spectrum, in order to analyze represents a critical system [59]. For this value of λ (Fig.
48 the propagation of phonons in this quasiperiodic media. 2c), the allowed frequencies are defined by several bands
49 In the works of You, J.Q. et al, this model was studied composed by increasingly narrow gaps located between
50 for the case in which the masses of successive atoms obey the four larger gaps, characterizing a multifractal spec-
51 a Fibonacci sequence, using the formalism of the trans- trum, as we will see further in Fig. 3. As λ grows, there
52 fer matrix [54]. They demonstrated that the spectrum is a deformation between the gaps and the spectrum loses
53 is truncated in a fractal to a larger amount of atoms, this characteristic, and we can see the deformation of the
54 also evidenced in the works of Kohomoto et al [53] and larger gaps for λ = 3.0 due to intense variations in the
55 in the works of F. Salazar et al [52], which proposed a force constant (Fig. 2b).
56 modulation in the equation of motion. In our model, On the other hand, the states that cross larger gaps are
57 we used the computational package of Gnu Scientific Li- sensitive to the number of atoms in the lattice. Indeed,
58
59
60
Acc pted Man ω
ω us
ω cript
AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1 Page 4 of 8
4
1
2 for N =100 sites, we have obtained a spectrum with cer- values of N and λ. For N = 100 (Figs. 4a and 4c) the
3 tain bands circumventing the larger gaps. In Figure 3 bands cross only the smaller gaps, regardless of the two
4 we show the frequency spectrum as a function of b, for values of λ, 0.5 and 1.0, while increasing the number of
400 sites and setting the phase φ = π/2 to obtain higher sites up to 206 (Figs. 4b and 4d) the bands are narrowed,
5
definition in the frequency spectrum of phonons. and these states cross all gaps. When λ = 1.0 and N =
6 100 from Figure 4c we verify the presence of for
7 √
bidden
a ) bands for any approximation value for b = (1 + 5)/2,8 3 . 0 within the analyzed range (1.615 up to 1.62), similar to
9 what occurs in the electronic case, where they arise only
10 for the finite system, and the origin of this effect is due
11 to the conservation of the number of particles [60].
12
13   2 . 0 a ) b )
14 2 . 0 2 . 0
15
16 1 . 5 1 . 5
17 1 . 0 1 . 0
18 1 . 0
19 0 . 5 0 . 5
20
21 01. 0. 6 115. 6 116. 6 b117. 6 118. 6 119. 6 2 0
01. 0. 6 115. 6 116. 6 b117. 6 118. 6 119. 6 2 022   
23 0 . 0 d )0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 c )2 . 0
24 2 . 0b
25 1 . 5 1 . 5
26 b )
27 1 . 0 1 . 0
28 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 5 0 . 5
29 ω
30 c ) 01. 0. 6 115. 6 116. 6 b117. 6 118. 6 119. 6 2 0
01. 031 . 6 115. 6 116. 6 b117. 6 118. 6 119. 6 2 0
32 0 . 0 0 . 5 1 . 0 1 . 5
33 ω FIG. 4. Frequency spectrum around b = 1.618 for λ = 0.5
34 and λ = 1.0, varying the number of sites on the network for
35 FIG. 3. (a) Hofstadter butterfly for the frequency spectrum in four cases: a) N = 100 and λ = 0.5; b) N = 206 and λ = 0.5;
36 a quasicrystal as a function of parameter b. We use N = 400, c) N = 100 and λ = 1.0; d)N = 206 and λ = 1.0. We can see
37 φ = π/2 and λ = 1.0. (b) We highlight in the region char- that the location of the states that cross the gaps is modified
acterized by the dashed vertical line the frequencies in the
38 by the number of sites, and the bands are narrowed as weregion where b ≈ 0.618. (c) To evince the frequency replica- change λ.
39 tion pattern, we make an amplification in the dotted square
40 in (b), where the spectrum repeats in a self-similar fashion. The φ phase also modifies the shape of the spectrum,
41 as shown in Figure 5a, where we consider a lattice with
42 We can see that the figure (Fig. 3) is similar to the 100 atoms and λ = 0.5. We can notice that the frequen-
43 Hofstadter butterfly obtained for the electronic case of cies are distributed in four separate intervals with larger
44 the Hamiltonian of Aubry-André [55, 60]. The varia- gaps. Only a few modes cross the forbidden frequency
45 tions of b present a characteristic of self-similarity for the gaps, with an almost sinusoidal dispersion. Below of the
46 frequencies, maintaining the structure composed by the main panel, we can see the displacements un (eq. 2)
47 larger gaps and some crossed modes. In the highlighted against the index n, calculated for three case, character-
48 region (Fig. 3b), we see that the three major gaps are ized by the (red in color online version) cross in main
49 replicated, and the frequency spectrum follows the same panel on the Figure 5, namely u1 (Fig. 5b), u2 (Fig.
50 pattern (Fig. 3c). The limit for this replication is ruled 5c) and to the right u3 (Fig. 5d). We can see that u1
51 by the precision of step, b, where we consider an incre- and u3 modes are strongly localized at the edges of the
52 ment of 10
−3 for a fixed phase φ = π/2 in the equation system, while the calculated mode for the center, labeled
53 (3). by u2, it is extended mode through all sites. The modes
54 The phonon equivalent for the Hofstadter Butterfly is u1 and u3 represent the topological states of phonons in
55 subject to a strong influence from the number of sites this system. They are formed by states at the edge of the
56 and the λ parameter. In√Figure 4, we show the frequency quasiperiodic lattice, for a set of well-specified parame-
57 modes around b = (1 + 5)/2 ≈ 1.618, for two diferrents ters φ, λ and N . On another hand, on the limit when
58
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Ac ωb =  0 . 6 1 8 b  =  0 . 6 1 8c pted Maω n ωus
ω ω cript
  
Page 5 of 8 AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1
5
1 a ) b )
2 a ) 2 . 0 2 . 0
3 2 . 0
4 1 . 5 1 . 5
5 x
6 1 . 5 x 1 . 0 1 . 0
7 x
8 0 . 5 0 . 5
9 1 . 0
10 0 .00. 0 0 . 2 0 . 4 0 . 6 0 0. 8.00. 0 1 . 00 . 2 0 . 4 0 . 6 0 . 8 1 . 0
11 c ) ϕ/2π d ) ϕ/2π
12 0 . 5 2 . 0 2 . 0
13
14 1 . 5 1 . 5
15 0 . 0
16 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0ϕ/2π 1 . 0 1 . 0
17 0 . 5 0 . 5 0 . 5
18 b ) c ) d ) 0 . 5 0 . 5
19 0 . 0 0 . 0 0 . 0
20 0 . 0 0 .
- 0 . 5 - 0 . 5 0 . 0 0 . 2 0 . 4 0 . 6 0 . 80
0. 0
- 0 . 5 1 . 0
0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
21 0 5 0 1 0 0 5 0 1 0 00 5 0 1 0 0 ϕ/2π ϕ/2π
22 n n n
23 FIG. 6. Frequencies as functions of the φ phase, obtained
24 for changes in the value of b in the third decimal place with
25 FIG. 5. (a) Phonon frequency spectrum, as a function of the φ N = 100 and λ = 0.5. a) b = 1.615; b) b = 1.617; c) b = 1.618;
26 phase. The parameters used were: N = 100; λ = 0.5. Below, d) b = 1.620.
27 in (b-d) plots, we present the eigenvectors as function of n,
28 for the un points marked with a cross (×) on the spectrum,
namelly u (b), u (c) and u (d).
29 1 2 3 tion. In order to study this phase transition, in Figure
30 7 we present the inverse of the participation rate for the
31 frequency values located within the upper gap (frequency
N goes to infinity, with b incommensurate, the bands values greater than 1.7). We can see that, depending on32
in the phonon spectrum showed in figure 5 should not the λ parameter, the IPR can present a phase transition
33
depend on the phase φ. In this case we should have a in the displacements un of the lattice. For values smaller34 banded fragmented spectrum, like those in the reference than λ = 1.0 the displacements are scattered across all
35 [61], and the border states should disappear. In this case, sites representing extended states of the system, but as
36 we can solve this model using the transfer matrix model we increase the value of λ, the IPR shows an intense
37 to find the eigenvalues of Eq. (1) [62]. localization (high IPR), representing a transition in the
38 The numerical precision for the inverse of the frequency system’s oscillations, from extended displacements to lo-
39 (b) in the force constant Kn considerably alters the al- calized oscillations.
40 lowed eigenvalues in the spectrum as a function of the The location of the displacements in the phonon spec-
41 potential phase, as shown in Figure 6. In this way, we trum is also modified, as a function of the φ phase, as
42 can see that the phonon states remain crossing the gaps; we can see in Figure 8, where we added the gray (color,
43 however, a translation occurs in the modes, and also a in online version) scale for a system with 100 sites and
44 gentle deformation can be observed. All the eigenvalues λ = 0.5. The states that cross the second largest gap
45 were obtained from the equation (2), and they were cal- present a more intense localization between the gaps in
46 culated for four different b approximations, for φ values this figure (higher IPR, darker color), while the remain-
47 between 0 and 2π. The border states that emerge in the der is fully extended (grayish, or red in color version, i.e.,
48 larger gaps move, arising for different phase values. lower IPR; see Fig. 5). When we vary the λ parameter up
49 The location of the frequency ranges within the upper to the critical value (λ = 1.0), the set of extended bands
50 gap depends heavily on the λ parameter, which repre- are narrowed and the states that cross the larger gaps are
51 sents the amplitude of the cosine modulation in eq. (3). more localized, as seen in Figure 9. The frequency range
52 Clearly, from Fig. 2 we can see that there are two behav- where the gaps exist are very close to that ones in Figure
53 iors in the energy (frequency) spectra: banded (where the 8, mainly in the upper gap, not altering, therefore, the
54 bands are well defined) or unbanded (where the states are frequency range for emergence of the topological states.
55 very narrow sets so that it is impossible to define them as In Figure 10, we present the frequency dispersion as
56 a band) spectrum, depending on the parameter λ. This a function of the amplitude of the potential, incommen-
57 will play an essential role in the referred phase transi- surate with the IPR, in color scale. We can see that for
58
59
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Ac  u ω1c
u2
ted u3 Man ω ωus
ωcri ω pt
AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1 Page 6 of 8
6
1
2 1 . 0
3 2 . 0
4
5 0 . 8
6 1 . 5
7
8 0 . 6
9 1 . 0 0 . 5 0 0 0
10 0 . 4 0 . 3 0 0 0
11 0 . 5
12 0 . 1 0 0 0
13 0 . 2 I P R
14 0 . 0
15 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
16 0 . 0 ϕ/2π0 . 0 1 . 0 2 . 0 3 . 0 4 . 0
17 λ
18 FIG. 9. Frequency spectrum as a function of the φ phase of
19 FIG. 7. Inverse of the participation rate for displacements the potential, with IPR in color scale for λ = 1.0, in a lattice
20 with N = 100 and frequencies in the gap (1.7 up to 3.0), with 100 atoms.
21 displaying a transition between displacements for values of
22 λ > 1.0
23 3 . 0
24
25 2 . 5
26
27 2 . 0
28 2 . 0
29 1 . 5 0 . 5 0 0 0
30
31 1 . 5 1 . 0
0 . 3 0 0 0
32
0 . 5 0 0 0 0 . 1 0 0 0
33 0 . 5
1 . 0 I P R34 0 . 3 0 0 0
35 0 . 00 . 0 1 . 0 2 . 0 3 . 0 4 . 0
36 0 . 5 0 . 1 0 0 0 λ
37
38 I IPPRR
39 0 . 0 FIG. 10. Frequency dispersion as function of the amplitude
40 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 parameter λ, with 100 atoms in the lattice, considering the
ϕ/2π initial phase φ = 0. We can see that for values greater than41 λ = 1.0 the spectrum presents many narrow bands spread
42 with higher band gaps, and for λ < 1.0, we have larger and
43 FIG. 8. Frequency spectrum as a function of the φ phase of well organized bands, characterizing λ = 1.0 as a critical value
44 the potential with IPR in color scale for λ = 0.5 in a lattice for a phase transition.
45 with 100 atoms.
46
47 IV. CONCLUSIONS
48
49 The system studied here consists of an adaptation of
50 λ < 1.0 the frequencies present itself as bands separated the Hamiltonian of Aubry-André to deal with the ele-
51 by larger gaps, while for λ = 1.0 the spectrum comes mentary vibrations of the unidimensional quasicrystalline
52 down to vibration modes at specific and well-defined fre- lattice. The equation of the eigenvalues for our case con-
53 quencies. The λ parameter allows one to control the dis- sists of a coupled system, with the elementary oscillations
54 tribution of bands in the spectrum, representing a mul- in each site interacting by a force constant given in the
55 tifractal when they are close enough (λ = 1.0) and de- coupling with the neighboring atoms. We have used the
56 forming the spectrum for higher values of λ, organizing numerical diagonalization method to find the allowed fre-
57 itself in modes of vibration with wide gaps. quencies (computational package of GSL). We have found
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ω an ωuscript
Page 7 of 8 AUTHOR SUBMITTED MANUSCRIPT - JPCM-113970.R1
7
1
2 that the frequency spectrum is modified according to the phononic quasicrystal, on order to show that is possible
3 λ interaction parameter, presenting the phonon equiva- to have localized boundary states, which can manifest the
4 lent for the Hofstadter Butterfly’s for λ = 1.0, keeping same topological properties of a 2D photonic quasicrys-
the symmetry and gaps close to the one obtained in the tal [21]. Therefore, it is possible to define an equivalent
5
electronic case. Also, we have verified that the number of Chern number for our case, and to classify topologically
6 atoms in the network influences the number of gaps and the states of the system studied here [40, 64]. We will
7 edges modes that cross these gaps. For the very precise consider this in further works. Finally, by studying the
8 value of the b parameter (around b = 1.618), and choos- individual displacements un, for a given frequency, we
9 ing a given λ, it is possible to control these frequency demonstrate that it is possible to characterize an equiva-
10 bands. lent metal-insulator phase transition in this equivalent
11 The interaction with this quasicrystalline potential can Aubry-André model (Fig. 7). We hope that the re-
12 be modulated by the φ phase, causing the atoms on the sults presented here would stimulate other researchers
13 edge of the system to vibrate with one modulated fre- to search these topological states in other equivalent sys-
14 quency lying between the gaps, characterizing an edge tems, generating news applications, and future devices
15 mode. It is well known that topological phases are char- based on this fascinating theory.
16 acterized by edge states confined near the boundaries of
17 the system, whose modes are lying in a bulk energy (or
18 frequency) gap [63], and it is unaffected by disorder or
19 deforming, for example. Therefore, we can infer that,
ACKNOWLEDGMENTS
20 with an analogy with the photonic case [40], we have a
21 phononic topological phase exhibiting the so-called “topo- We want to thank CNPq (Conselho Nacional de Desen-
22 logical states”. Indeed, the topological properties of the volvimento Cient́ıfico e tecnológico) for the partial financ-
23 1D quasicrystals can emerge in two ways: considering ing. M. S. Vasconcelos thanks the Department of Physics
24 the existence of phase transitions when we have a con- and Astronomy at the University of Western Ontario
25 tinuously deforming between two topologically distinct for hospitality during his sabbatical as visiting profes-
quasicrystals or by the appearance of edge states which sor where this study was finished. This research also was
26
traverse the bulk gaps as a function of some control- financed in part by the Coordenação de Aperfeiçoamento
27 lable parameter (which in this case is the initial phase φ). de Pessoal de Nı́vel Superior (CAPES) of Brazil (Finance
28 Specifically, we have considered the second way in a 1D Code 88881.172293/2018-01).
29
30
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