Localization properties of transmission lines with ...

Eur. Phys. J. B (2015) 88: 216

DOI: 10.1140/epjb/e2015-60080-y

Localization properties of transmission lines with generalized Thue-Morse distribution of inductances Edmundo Lazo, Eduardo Saavedra, Fernando Humire, Cristobal Castro and Francisco Cortés-Cortés

Eur. Phys. J. B (2015) 88: 216 DOI: 10.1140/epjb/e2015-60080-y

THE EUROPEAN PHYSICAL JOURNAL B

Regular Article

Localization properties of transmission lines with generalized Thue-Morse distribution of inductances Edmundo Lazo1,a , Eduardo Saavedra2, Fernando Humire1 , Cristobal Castro3, and Francisco Cortés-Cortés1 1 2 3

Departamento de F´ısica, Facultad de Ciencias, Universidad de Tarapac´ a, Arica, Chile Departamento de F´ısica, Universidad de Santiago de Chile, Santiago, Chile Escuela Universitaria de Ingenier´ıa Mec´ anica, Universidad de Tarapac´ a, Arica, Chile Received 30 January 2015 / Received in final form 10 June 2015 c EDP Sciences, Societ` Published online 2 September 2015 – a Italiana di Fisica, Springer-Verlag 2015 Abstract. We study the localization properties of direct transmission lines when we distribute two values of inductances LA and LB according to a generalized Thue-Morse aperiodic sequence generated by the inflation rule: A → AB m−1 , B → BAm−1 , m ≥ 2 and integer. We regain the usual Thue-Morse sequence for m = 2. We numerically study the changes produced in the localization properties of the I (ω) electric current function with increasing m values. We demonstrate that the m = 2 case does not belong to the family m ≥ 3, because when m changes from m = 2 to m = 3, the number of extended states decreases significantly. However, for m 3, the localization properties become similar to the m = 2 case. Also, the T frequency averaged transmission coefficient shows a strong dependence from the N system size and from the m value which characterize each m-tupling sequence. In addition, for all m value studied, using the scaling behavior of the ξ (ω) normalized participation number, the Rq (ω) Rényi entropies and the μq (ω) moments, we have demonstrated the existence of extended states for certain specific frequencies.

1 Introduction The localization properties of color noise quantum systems have been the subject of intense researches because these systems have revealed rich electronic properties [1–15]. Lately, the localization properties of quantum systems and classical electric systems have been studied using the dilution procedure [9–11,16–24]. Besides, a very interesting behavior has been found during the study of quasi-periodic and non-periodic systems [16,25–34]. Among the aperiodic models, one of the most studied is the Thue-Morse model. This aperiodic sequence is defined by the inflation rule A → AB and B → BA. From the localization point of view, the Thue-Morse aperiodic systems presents an intermediate behavior between periodic and quasiperiodic systems [34–46]. Some generalizations of the Thue-Morse sequences [38,41,43,46–48] have been recently studied, which can be defined by the general inflation rule A → An B l and B → B l An , where n and l are integers, and An represents n adjacent repetitions of the string A. Other possible generalization is the following, A → An B l and B → B n Al , where the n and l exponents are not exchanged. A special case of this generalization is the so-called m-tupling sequence defined for n = 1 and l = m − 1. In this case the inflation rule becomes A → AB m−1 and B → BAm−1 , with m ≥ 2. For m = 2 we regain the usual Thue-Morse sequence. Until now, the changes in the localization properties of the electric current function, as a function of the a

e-mail: [email protected]

m values which characterize the m-tupling family, have not been studied. On the other hand, given the coincidence between the equations describing the diagonal tight-binding quantum model and the equations describing the diagonal transmission lines, various authors have used transmission lines to model quantum effects [6,49–52]. Motivated by this analogy, the localization properties of the I (ω) electric current function of direct and dual electric transmission lines (TL) have been recently studied by using different methods to distribute the Ln inductances and the Cn capacitances [17,18,49,53–57]. In this paper, we study the localization properties of the direct TL with constant capacitances, i.e., Cn = C0 , ∀n, when we distribute two values of inductances LA and LB according to the Thue-Morse m-tupling aperiodic sequence. We use the Hamiltonian map approach [15,57,58] to calculate the I (ω) electric current function and any other quantity, like the participation number P (ω) , the normalized participation number ξ (ω) = N1 P (ω), the transmission coefficient T (ω) , the μq moments and the Rq Rényi entropies. Also, using the scaling behavior of these quantities we have studied the I (ω) localization behavior as a function of the m value which defines the m-tupling sequence. As a result, we found that the m = 2 case does not belong to the m ≥ 3 case, because when we go from m = 2 to m = 3, the extended states number decreases sharply. Nevertheless, inside the m ≥ 3 family, the number of extended states increases with increasing m. In addition, using the scaling

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Eur. Phys. J. B (2015) 88: 216

to the direct electric transmission line, let us consider the following definition of two new variables xn and pn as a function of the electric current functions In and In+1 : xn = In pn+1 = In+1 − In .

After some algebra, we find the Hamiltonian map for our problem, namely,

Fig. 1. The direct electrical transmission line.

behavior of P (ω) , ξ (ω) , μq (ω) and Rq (ω), we demonstrate the existence of extended states for some specific frequencies for any m studied. This paper is organized as follows. Section 2 describes the model and methods. Section 3 shows the most important numerical results and Section 4 gives the conclusions of our work.

2 Model and methods 2.1 Direct electrical transmission lines In this work we study de localization behavior of the direct electrical transmission lines shown in Figure 1. Kirchhoff’s Loops Rule application to three successive cells of this TL permits us to write the dynamic equation 1 1 1 1 + − ω 2 Ln In − In−1 − In+1 = 0 Cn−1 Cn Cn−1 Cn (1) where ω is the frequency. We distribute two different values of inductance, LA and LB , using the Thue-Morse m-tupling aperiodic sequence, which can be obtained using the inflation rule LA → LA Lm−1 ; B

LB → LB Lm−1 ; A

m ≥ 2.

(2)

The m = 2 case corresponds to the usual Thue-Morse sequence, i.e., LA → LA LB , LB → LB LA . In this paper we use constant capacitances Cn = C0 , ∀n, therefore equation (1) becomes (3) 2 − ω 2 C0 Ln In − In−1 − In+1 = 0. The I (ω) electric current function and the localization properties of the aperiodic TL are calculated using the Hamiltonian map approach [15,57,58]. 2.2 Hamiltonian map approach Using the substitution αn = ω 2 C0 Ln , the dynamic equation (1) can be written as: (2 − αn ) In = In−1 + In+1 .

(5) (6)

(4)

We will study the localization properties of the disordered TL using the Hamiltonian map approach. To generate a classical two-dimensional Hamiltonian map corresponding

xn+1 = βn xn + pn ;

pn+1 = −αn xn + pn

(7)

where βn = (1 − αn ) . The extended character of the electric current function In = xn is represented by bounded trajectories of this map in the plane (p, x); on the contrary, the localized states are represented by unbounded trajectories. This relationship between localization properties and bounded and unbounded trajectories of the Hamiltonian map, can be analyzed considering the behavior of the Lyapunov exponent and the transmission coefficient. There are two Lyapunov exponents, corresponding to growing and decaying solutions. In numerical experiment only the growing solution survives. Its decaying counterpart cannot be seen. Therefore, presence of an unbounded trajectory serves as evidence of a localized state in the spectrum. In addition, the transmission co4 efficient T (ω) is defined as T (ω) = (1) 2 (2) 2 (see (1,2)

2+(rN ) +(rN )

Eq. (15) below), where rN are the radii of two trajectories at “time” n = N, that start from two perpendicular initial points. For fully extended state we have T (ω) → 1. (k) This condition can be met if, and only if, each rN ful(k) fills the condition (rN )2 → 1. This condition defines a bounded trajectory for each initial condition of the map. On the contrary, for a localized state we have T (ω) → 0. (k) We obtain this value if, and only if, each rN fulfills the (k) 2 condition (rN ) → ∞, corresponding to an unbounded trajectory. Let us now consider the transformation of the map (7) to the canonical variables (r, θ) in the usual way, i.e., x = r sin θ p = r cos θ.

(8) (9)

Using these equations in the Hamiltonian map (7), we can calculate the relationship consecutive rn values, between rn+1 defining the term Γn = rn , which can be obtained in the following form Γn = 2 − (1 + 2αn βn ) sin2 θn + (1 − 2αn ) sin 2θn . (10) At the same time we obtain the recurrence equation for the phase map θn , cos θn + βn sin θn cos θn − αn sin θn where βn = (1 − αn ) and αn = ω 2 C0 Ln . tan θn+1 =

(11)

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We define the Lyapunov exponent λ as a function of the electric currents In and In+1 , in the following form,

N 1

In+1

. ln

N →∞ N In

n=1

λ = lim Giving that

In+1 In

=

(12)

xn+1 xn ,

In+1 xn+1 = In xn

from relation (8) we obtain rn+1 sin θn+1 = . (13) rn sin θn

Using Γn = rn+1 rn , the Lyapunov exponent λ can finally be written as a function of the Hamiltonian map elements,

N N 1 1

sin θn+1

. (14) λ = lim ln |Γn |+ lim ln

N →∞ N N →∞ N sin θn

n=1 n=1 Accordingly, λ only depends on the phase θn of the map, which, in turn, is a function of the kind of disorder contained in the distribution of capacitances Ln . In addition, using the Hamiltonian map formalism, we can calculate the T (ω) transmission coefficient putting the disordered system in between two semi-infinite ordered chains. T (ω) can be obtained from the following expression [15,59] T (ω) =

4 2 2 , (1) (2) 2 + rN + rN

(15)

(1,2)

are the radii of two trajectories at “time” where rN n = N, that start from two perpendicular initial points, namely, π (1) (1) (2) (2) = (1, 0) ; r0 , θ0 = 1, . (16) r0 , θ0 2 (1)

(2)

The values (rN )2 and (rN )2 are entirely determined by the relation rn+1 = Γn rn . As a consequence, we can cal(k) culate (rN )2 in the following form: 2 2 (k) N = Πn=1 (17) Γn(k) , k = 1, 2. rN In this way, we have an explicit expression for the transmission coefficient T of the diagonal disorder (ω) in terms contained in αn = ω 2 C0 Ln .

The μq moments associated to the electric current function I (ω) can be defined as: μq (ω) =

2q

|In |

.

μq (N ) = N (1−q)Dq ,

q = 1.

(19)

For extended states, we obtain a linear relationship between ln(μq ) and ln(N ). The slope sμq of the straight line determines the Dq fractal dimension associated to a sμq specific frequency, i.e., Dq = (1−q) , q = 1. When Dq equals the spatial dimension, the states under study are extended states, because they uniformly spread over the whole system. In addition, the μq moments can be related to the Rq Rényi entropies in the following way Rq =

ln (μq ) , (1 − q)

q = 1.

(20)

For the limit case q → 1, we obtain the S Shannon entropy, i.e., limq→1 Rq = − N n=1 |In | ln |In | = S. For a fully extended electric current μq (N ) = N (1−q) , which implies that Rq (N ) = ln N, ∀q and for fully localized electric current μq (N ) = 1, ∀q, suggesting that Rq (N ) = 0. The Dq fractal dimension can also be defined through the scaling of the Rq Rényi entropies, i.e., Rq = ln N Dq .

(21)

For extended states, we obtain a linear relationship between Rq and ln(N ). The slope sRq of the straight line determines the Dq fractal dimension associated to a specific frequency, i.e., Dq = sRq . As a partial summary we can say that we can recognize the localization degree of the I (ω) electric current function using the scaling behavior of the participation number P (ω) , the normalized participation number ξ (ω) , the Rq Rényi entropies and the μq moments.

3 Numerical results

2.3 The µq moments and the Rq R´ enyi entropies

N

4 IP R (ω) = P −1 (ω) = N n=1 |In | . Here, P (ω) is the participation number. Besides, we can calculate the normal(ω) . For a fully localized participation number ξ (ω) = PN ized electric current, such that In = 1 and Il=n = 0, ∀l, we obtain μq (N ) = 1, ∀q, implying that P (N ) = 1 and ξ (N ) = N1 → 0. For a fully extended electric current state where In = √1N , ∀n, we obtain μq (N ) = N (1−q) , in turn P (N ) = N and ξ (ω) = 1. The fractal dimension Dq can be defined through the scaling of the μq moments.

(18)

n=1

Here we are considering normalized electric current, i.e., N 2 n=1 |In | = 1. For q = 2, the μ2 (ω) moment coincides with the IP R (ω) inverse participation ratio, namely,

In this paper we study the localization properties of the normalized I (ω) electric current function of direct elecN 2 trical TL ( n=1 |In | = 1). We use constant capacitances Cn = C0 = 0.5, ∀n, and we distribute two values of the inductances, LA = 1.6 and LB = 1.5, according to the Thue-Morse m-tupling aperiodic sequence defined by m−1 the inflation rule LA → LA Lm−1 and LB → LB LA . For B the numerical calculations, we choose the LA and LB values in such a way that they generate a greater number of states with the property that Λ (ω) ≥ 1, were Λ (ω) is the

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Fig. 2. ξ normalized participation number as a function of ω frequency, for three different cases: a) the periodic case LA = LB = 1.6 (blue dashed line), b) m-tupling sequence with m = 3 (red line), and c) the random distribution of the two values LA = 1.6 and LB = 1.5 (thick black line).

normalized localization length. This condition ensures the existence of extended states, at least for finite system size. In addition, we have found that all the numerical results do not depend on the (r0 , θ0 ) initial point of the Hamiltonian map, used to calculate the localization properties of the m-tupling sequences. On the other hand, we know that the localization properties of the usual Thue-Morse sequence are intermediate between the periodic case and the random case, and we suppose that the m-tupling sequence behaves in the same way. To display this behavior, in Figure 2 we compare the ξ normalized participation number as a function of ω frequency, for three different cases: a) the periodic case LA = LB = 1.6 (blue dashed line), b) the m-tupling sequence with m = 3 (red line), and c) the random distribution of the two values LA = 1.6 and LB = 1.5 (thick black line). There we can see that for the m = 3 case, and for certain frequencies, the ξ (ω) normalized participation number behaves like a extended states ξ (ω) → 23 = 0.667 , however, for other frequencies, ξ (ω) → 0, indicating a localized behavior. This is an indication that localization behavior of the m-tupling sequences presents an intermediate behavior. For m = 2 we obtain the usual Thue-Morse sequence. For m ≥ 3, we obtain the m-tupling sequences. For a specific m value, the system size N is given by the expression N = miter , where iter indicates the iteration of the inflation rule. In Figure 3 we show the behavior of the ξ normalized participation number versus ω for four m values, i.e., m = {2, 3, 5, 9}. For each m we use a specific Nm system size value, namely, N2 = 221 , N3 = 313 , N5 = 59 and N9 = 97 . In this figure we can observe that the m = 2 case (the usual Thue-Morse case) is very different of the m ≥ 3 case, because for m = 2 there are many regions where ξ (ω) → 23 , that characterizes the fully extended state, but for m = 3 almost all extended states disappear. However, when m increases (m > 3),

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Fig. 3. ξ versus ω for m = {2, 3, 5, 9} for fixed Nm system size value, i.e., N2 = 221 , N3 = 313 , N5 = 59 and N9 = 97 . The case m = 2 (the Thue-Morse case) is very different of the case m ≥ 3, because for m = 3, almost all extended states which exist in the m = 2 case have disappeared. However, when m increases (m 3), the TL begins to regain its extended states.

Fig. 4. T versus N in a log-log scale for m = {2, 3, 5, 9}. When m changes from m = 2 to m = 3, for a fixed and high N value, T m=3 T m=2 . For the case m 3, the behavior of the m-tupling sequence is similar to the m = 2 case.

the TL begins to regain its extended states. This behavior can be explained in the following way. The gap structure of the I (ω) electric current function of the aperiodic TL depends on the m value which defines the m-tupling sequence, and at the same time depends on the N system size. For fixed m, when N increases, more gaps open up. As a consequence, the T frequency averaged transmission coefficient diminishes because T → 0 in the gaps. Nω T (ω) where Nω Here T is calculated as T = N1ω j=1 is the number of frequencies in the spectrum. Figure 4 shows T versus N in a log-log scale for m = {2, 3, 5, 9}. In the first place we can see that T → 0 for N → ∞, for every m value. We can also see that when m changes from m = 2 to m = 3, for a fixed and high N value, T m = 3 presents very small values in comparison with T m = 2 . At the same time, we can see that for the

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Fig. 5. ξ (ω) versus ω for m = 2, for three different N = 2iter values, with iter = {19} (black points), iter = {20} (red points) and iter = {21} (blue points). There exists a large number of frequencies for which the values of ξ (ω) coalesce in a single value independent of the N system size, namely, ξ (ω) → const., indicating an extended behavior.

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Fig. 6. ξ (ω) versus ω for m = 2. A partial view of Figure 5 showing gaps and extended states. The vertical arrows show regions with extended states (ξ (ω) → const.).

m ≥ 3 class, T m increases with increasing m values, in such a way that for m 3, the localization behavior of the m-tupling sequence becomes similar to the localization behavior of the m = 2 case. However, for low N values, we find that T m = 2 is lower than T m = 9 and also find that T m = 2 behaves in a similar way to T m = 5 . It is important to note that the results shown in Figure 3 were calculated for large values of N . Next we will study in details the m = 2 case (the usual Thue-Morse sequence) and the m = 5 case (intermediate between the m = 3 and m = 9 cases). 3.1 The Thue-Morse case (m = 2) The Thue-Morse aperiodic sequence can be obtained using the inflation rule: LA → LA LB and LB → LB LA . In Figure 5 we show the behavior of the normalized participation number ξ (ω) as a function of the ω frequency for the case m = 2, for three N values, where N is obtained as N = 2iter , where iter = {19, 20, 21}. In this figure we can see a large number of frequencies for which the values of ξ (ω) coalesce in a single value, independently from the N system size, namely, ξ (ω) → const. This behavior indicates that the participation number P (ω) grows proportionally to the N system size and, as a consequence, the normalized participation number ξ (ω) = N1 P (ω) tends to a constant number. This is a clear indication of the existence of extended states. To see in details this scaling behavior, in Figure 6 we show four regions of Figure 5, in the neighborhood of some of the ω frequencies where ξ (ω) tends to a constant value (existence of extended states). In each picture of Figure 6, the vertical arrows show regions where extended states can be found. We also indicate the presence of gaps in regions where ξ (ω) increases with increasing values of N . Instead,

Fig. 7. ln(P ) versus ln(N ) and T versus N, for N ranging from Nmin = 213 = 8192 to Nmax = 223 = 8, 388, 608, for three ω frequencies contained in Figure 6d, namely, ω = 2.117428 (gap, full circle), ω = 2.117503 (localized state, full triangle) and ω = 2.117753 (extended state, open diamond). (a) Only for ω = 2.117753 we find a straight line with slope s = 0.9974. For this frequency I (ω) is an extended state. (b) T (N ) oscillates around a value greater than zero for the extended state (ω = 2.117753), but for gaps and localized states the transmission coefficient T (N ) goes to zero.

the localized states can be found for the frequencies for which ξ (ω) diminishes with increasing N . To study the behavior of the P (ω) participation number as a function of N , in Figure 7a we show ln(P ) versus ln(N ), for N ranging from Nmin = 213 = 8192 to Nmax = 223 = 8, 388, 608, for three ω frequencies contained in Figure 6d, namely, ω = 2.117428 (gap, full circle), ω = 2.117503 (localized state, full triangle) and ω = 2.117753 (extended state, open diamond). Only for ω = 2.117753 we find a linear relationship. The straight line with slope s = 0.9974 is an indication of the extended character of the I (ω) electric current function.

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Fig. 8. ξ (ω) versus ω for m = 5, for three different N = 2iter values, with iter = {7} (red points), iter = {8} (blue points), iter = {9} (green points). When we increase the values of N , we can find many specific frequencies for which the three values of ξ (ω) collapse in a single value, namely, ξ (ω) → const. This behavior is an indication of a extended state of the electric current function. The arrow indicates a region which will be studied in Figure 9.

In Figure 7b we show the behavior of the transmission coefficient T (N ) as a function of N , for the same three ω frequencies. There we can clearly see that T (N ) oscillates around a value greater than zero for the extended state (ω = 2.117753, open diamond), but for gaps and localized states the transmission coefficient T (N ) goes to zero. As a partial summary of the localization behavior of the aperiodic Thue-Morse distribution of inductances LA and LB (the m = 2 case), we can say that there are a lot of extended states, localized states and gaps. Using the scaling properties of the P (ω) participation number and the scaling properties of the ξ (ω) normalized participation number, we have demonstrated the existence of extended states for certain ω frequencies. 3.2 The m-tupling case (m ≥ 3) and For m ≥ 3, we use the inflation rule: LA → LA Lm−1 B LB → LB Lm−1 to generate the m-tupling aperiodic seA quence. We want to study the changes produced in the localization properties of the I (ω) electric current function as a function of the m-tupling number. In the above Figures 3 and 4, we have shown the influence of the m value in the localization behavior of the aperiodic TL. To observe in more details the localization properties of the TL as a function of the ω frequency for the m ≥ 3 case, we study the m = 5 case, which is intermediate between the cases m = 3 and m = 9, according to the results shown in Figures 3 and 4. In Figure 8 we show ξ (ω) versus ω for three different N values (N = 5iter , and iter = {7, 8, 9}). In this way we can observe the scaling behavior of the normalized participation number ξ (ω) . For this m = 5 case, we find many specific frequencies for which the three values of ξ (ω) collapse in a single value, i.e., ξ (ω) → const.

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Fig. 9. Scaling behavior of ξ (ω) in a region indicated by the arrow in Figure 8. We analyze three frequencies: ω = 2.02909, ω = 2.02961 and ω = 2.02966. For ω = 2.02909 the three values of ξ (ω) collapse in a single point, i.e., ξ (ω) → const. with increasing N values. This is an indication of the extended behavior of this state. For the other two frequencies, ξ (ω) → 0, indicating a localized behavior. For ω = 2.02961 we have a fast localization process, but for ω = 2.02966 we have a slow localization process.

This is an indication of the extended behavior of the I (ω) electric current function. In Figure 9 we show in detail the scaling behavior of ξ (ω) in a very restricted frequency region indicated by the arrow in Figure 8. In this figure we analyze three frequency values, namely, ω = 2.02909, ω = 2.02961 and ω = 2.02966. We can observe that for ω = 2.02909 the three values of ξ (ω) collapse in a single point, i.e., ξ (ω) → const. with increasing values of N . This is an indication of the extended behavior of this state. A very different situation occurs for the other two frequencies, where, ξ (ω) → 0, indicating a localized behavior of the electric current function. For ω = 2.02961, the three values of ξ (ω) rapidly approach to zero with increasing values of N , indicating a fast localization process. However, for ω = 2.02966, we have a slow localization process, because the three values of ξ (ω) slowly tend towards zero with increasing values of N . To see a new characterization of the localization properties of the three frequencies under study for the m = 5 case, in Figure 10 we show the behavior of the μq moments (Figs. 10a, 10c and 10e) and the Rq Rényi entropies (Figs. 10b, 10d and 10f) as a function of the N system size, for four q values, i.e., q = {2, 3, 4, 5} . Figures 10a and 10b correspond to the extended state with frequency ω = 2.02909. For this case and for each q value, we obtain a linear relationship for ln μq versus ln N (see Fig. 10a) and also we obtain a linear relationship for Rq versus ln N (see Fig. 10b). Using the scaling behavior of the μq moments (19) and the scaling behavior of the Rq Rényi entropies (21), we can obtain the Dq fractal dimension. Within the computational accuracy, the Dq values

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Fig. 10. ln μq versus ln N and Rq versus ln N for m = 5, for four q values, i.e., q = {2, 3, 4, 5}. (a) and (b) correspond to the extended state with frequency ω = 2.02909, because for each q value we obtain a linear relationship. For the frequencies ω = 2.02961 and ω = 2.02966 (c) to (f), the TL behaves as a localized system. However, these localized states can be considered extended state for finite N value, because a linear relationship can be observed.

are totally coincident between them, and coincides with the dimension of the linear space, namely, Dq ≈ 1, ∀q. This result is an indication that the m-tupling sequence is not a multifractal sequence. The absence of multifractality, together with the fact that Dq equals the spatial dimension, proves that these states are evenly spread over the entire system. This is a new way to show the extended behavior of the I (ω) electric current function for ω = 2.02909. On the other hand, from Figures 10c–10f, we can see the typical behavior of localized states for the frequencies ω = 2.02961 (see Figs. 10c and 10d) and ω = 2.02966 (see Figs. 10e and 10f), because we cannot find a linear relationship for ln μq and Rq versus ln N . However, for some finite N values, we can find a linear relationship indicating that these states can be considered as extended state. In this way we can observe that for ω = 2.02961 (see Figs. 10c and 10d) the linear relationship disappears for low N values (N ≈ 37 000), corresponding to a state of fast localization. However, for ω = 2.02966 (see Figs. 10e and 10f) we find a slow localization behavior, because the linear relationship holds for very long sequences (N ≈ 730 000). This behavior is totally coincident with the one observed in Figure 9 studying the ξ (ω) scaling behavior. The localization behavior of these three frequencies can be confirmed studying the scaling behavior of the participation number P (ω) and the behavior of the transmission coefficient T (ω) as a function of the N system size. In Figure 11a we show ln (P ) versus ln (N ) for the frequencies under study. Only for ω = 2.02909 (open square) we obtain a linear relationship with slope s = 0.99514, telling that this state is an extended state. The other curves show the fast (ω = 2.02961) and slow (ω = 2.02966) localization behavior. Figure 11b shows that the T (ω) transmission coefficient is practically constant as a function of N only

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Fig. 11. ln (P ) versus ln (N ) and T (ω) versus N, for m = 5, for the same frequencies of Figure 10. (a) Only for ω = 2.02909 we obtain a linear relationship with slope s = 0.99514, that tells that this state is an extended state. The other curves show the fast (ω = 2.02961) and slow (ω = 2.02966) localization behavior. (b) T (ω) is practically constant only for the extended state (ω = 2.02909). For the other frequencies, we can clearly see the fast (ω = 2.02961) and slow (ω = 2.02966) localization.

for the extended state ω = 2.02909 (open square). For the other frequencies, T (ω) → 0, indicating the fast and slow localization.

4 Conclusions Finally, we have studied the localization properties of the transmission lines when we distribute two different values of inductances LA and LB , according to the aperiodic Thue-Morse m-tupling sequence, defined by the inflation rule LA → LA Lm−1 and LB → LB Lm−1 , for integer B A m ≥ 2. For m = 2 we obtain the usual Thue-Morse aperiodic sequence. In this paper we use the Hamiltonian map approach to calculate the In (ω) amplitudes of the electric current function. The localization behavior was studied using various quantities, namely, the participation number P (ω) , the normalized participation number ξ (ω) = N1 P (ω), the μq moments and the Rq Rényi entropies. In addition, we have studied the behavior of the T (ω) transmission coefficient and the T frequency averaged transmission coefficient as a function of the N system size. The total number of extended states that can be found in the full frequency band, depends on the m number which determines the Thue-Morse m-tupling sequence. This is the reason why the m = 2 case does not belong to the m ≥ 3 class, because for m = 2 is easier to find extended states in comparison with the class m ≥ 3. However, inside the m ≥ 3 class we can find more and more extended states with increasing m values. Also, using the scaling properties of P (ω), ξ (ω) , μq (ω) and Rq (ω) , we have shown that for some specific ω frequencies we can find extended states. For the localized states, we can find slow and fast localized states. The slow localized states can appear as extended state for finite N system size.

Page 8 of 8 E. Lazo acknowledges the support of this research by the Direcci´ on de Investigaci´ on y Extensi´ on Académica de la Universidad de Tarapac´ a under project No. 4720-14. C.E. Castro acknowledges the Fondecyt program PAI and the contribution to the Project No. 7912010015. It is important to note that all authors contributed equally to the paper.

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