Phase transitions in transmission lines with long-range fluctuating correlated disorder

doi:10.1016/j.physb.2013.02.043

Physica B: Condensed Matter

Volume 419, 15 June 2013, Pages 19-27

https://doi.org/10.1016/j.physb.2013.02.043 Get rights and content

Abstract

In this work we study the localization properties of the disordered classical dual transmission lines, when the values of capacitances ${C_{j}}$ and inductances ${L_{j}}$ fluctuate in phase in the form $C_{j} = C_{0} + b \sin (2 π x_{j})$ and $L_{j} = L_{0} + b \sin (2 π x_{j})$ , where b is the fluctuation amplitude. ${x_{j}}$ is a disordered long-range correlated sequence obtained using the Fourier filtering method which depends on the correlation exponent $α$ . To obtain the transition point in the thermodynamic limit, we study the critical behavior of the participation number D. To do so, we calculate the linear relationship between $\ln (D)$ versus $\ln (N)$ , the relative fluctuation $η_{D}$ and the Binder cumulant B_D. The critical value obtained with these three methods is totally coincident between them. In addition, we calculate the critical behavior of the normalized localization length $Λ (b)$ as a function of the system size N. With these data we build the phase diagram $(b, α)$ , which separates the extended states from the localized states. A final result of our work is the disappearance of the conduction bands when we introduce a finite number of impurities in random sites. This process can serve as a mechanism of secure communication, since a little alteration of the original sequence of capacitances and inductances, can destroy the band of extended states.

Introduction

The study of low-dimensional disordered systems with long-range and short-range correlations have attracted scientific interest in the last decades, because this type of disordered systems can support extended states or a transition from localized states to extended states [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]–14]. However, for uncorrelated disordered systems, the Anderson localization theory predicts the absence of extended states for low-dimensional systems [15], [16], [17]. A very interesting review on Anderson localization in low-dimensional systems with correlated disorder, can be found in the work of Izrailev et al. [18].

On the other hands, due to the analogy of the transmission lines to quantum (electrons and excitons) and classical (atomic vibrations) systems, recently, analytical and numerical studies of disordered electrical transmission lines (TL) have been proposed [19], [20], [21], [22]. In Refs. [20], [21], the localized-extended transition was studied considering a ternary map to distribute the disorder in the capacitances (diagonal disorder) of the TL. The phase diagram separating localized states from extended states was obtained using the finite-size scaling method. Given that the long-range correlated sequences were generated using two very different methods: the Fourier filtering method [23] and the Ornstein–Uhlenbeck process [24], respectively, the phase diagrams are very different. In addition, the behavior of the Rényi entropies [25] of TL with Fibonacci distribution of two values of inductances have been recently studied [22]. In this case, the diagonal and off-diagonal terms of the dynamic equation vary simultaneously. The electric current function $I (ω)$ of the TL found in this paper can be classified as extended, intermediate and localized, in complete accordance with the behavior of the quantum wave function of Fibonacci one-dimensional tight-binding systems.

In the present work we study the localization behavior of the classical dual electrical TL with long-range correlated disorder in the distribution of capacitances ${C_{j}}$ and inductances ${L_{j}}$ . We introduce the disorder from a random sequence with a power spectrum $S (k) \propto k^{- (2 α - 1)}$ , where $α \geq 0.5$ is the correlation exponent. From the resulting long-range correlated sequence ${x_{j}}$ we generate the disordered distribution of capacitances and inductances in the following form $C_{j} = C_{0} + b \sin (2 π x_{j})$ and $L_{j} = L_{0} + b \sin (2 π x_{j})$ . In this way, the capacitances C_j and inductances L_j vary in phase around C₀ and L₀, respectively. The parameter b measures the amplitude of the fluctuation and varies from b=0 (periodic case) to $b < \min (C_{0}, L_{0})$ , to avoid a negative value of capacitances and inductances.

If the correlation exponent $α$ take the value $α = 0.5$ , we obtain a random sequence (white noise) and the TL is in the localized state for every frequency $ω$ . For $α > 0.5$ and depending on the values of the fluctuation parameter b, the electric current function $I (ω)$ can be a localized function or an extended function, for fixed system size N. This implies the existence of a critical value of the correlation exponent $α_{c} (b)$ for each value b of the fluctuation amplitude. To determine the localization properties of the electric current $I (ω, b, α, N)$ in the thermodynamic limit, we study the critical behavior of the participation number $D (ω, b, α, N)$ using three different methods: (a) study of the linear relationship between $\ln (D)$ and $\ln (N)$ , (b) study of the relative fluctuation $η_{D} (ω, b, α, N)$ of the participation number D as a function of the system size N and (c) study of the Binder cumulant $B_{D} (ω, b, α, N)$ of the participation number D as a function of the system size N. In addition, we study the behavior of the normalized localization length $Λ (b)$ as a function of the system size N to determine the critical curve $b_{c} (α)$ . With the obtained results we build the phase diagram in the plane $(b, α)$ .

The problem studied in this paper is similar to the problem studied by Shima et al. [11] in the electronic case. They studied the localization properties of electron eigenstates in one-dimensional systems with long-range correlated diagonal disorder. The phase diagram they get is similar to the phase diagram obtained in our work.

This paper is organized as follows. Section 2 describes the model and methods used to calculate the critical behavior of some quantities. In addition, we indicate the numerical procedure to calculate them. Section 3 presents the numerical results for the set of quantities under study as a function of the correlation exponent $α$ , the amplitude of the fluctuation b and the system size N. The critical behavior in the plane $(b, α)$ is presented. In Section 4 we discuss the secure communication. Finally, the conclusions of our work are presented in Section 5.

Section snippets

Model and method

We consider the classical electrical dual TL with horizontal capacitances C_j and vertical inductances L_j. Application of Kirchhoff's Loop Rule to three successive unit cells of the circuit leads to the following linear relation between the currents circulating in the (j−1)-th, j-th and (j+1)-th cells [19], [20], [21], [22] $d_{j} I_{j} - L_{j - 1} I_{j - 1} - L_{j} I_{j + 1} = 0$ where $d_{j} = (L_{j - 1} + L_{j} - (1 / ω^{2} C_{j}))$ and $ω$ are the frequencies. In this work we introduce the fluctuating correlated disorder in C_j and L_j in the following form: $C_{j}$

Numerical results

For the numerical calculation we consider a fixed frequency $ω = 3.6$ , and we use arbitrary values of the conductance and inductance: $C_{0} = 0.5$ , $L_{0} = 1.0$ . In addition, the average of every quantity is calculated using 10⁴ samples. To prevent a negative value of capacitances C_j and inductances L_j, the amplitude of the fluctuation b must be less than the minimum between $C_{0} = 0.5$ and $L_{0} = 1.0$ respectively. In this case we have the following condition: $0 \leq b < C_{0} = 0.5$ .

In first place, we study the behavior of the

Secure communication

In this section we study the effect produced in the localization properties of the transmission line when we change the values of capacitances and or inductances in one or more sites of the TL with random values R, with $R \in (0, 1)$ . This procedure introduces alterations in the original sequence given by the relationship (2), which determines the distribution of capacitances ${C_{k}}$ and inductances ${L_{k}}$ . We can introduce random values in four different cases: (a) we change only the capacitances, $C_{n} = R$ ,

Conclusions

In conclusion, we have studied the localization properties of the classical dual transmission lines when we introduce long-range correlated disorder in the distribution of capacitances and inductances. The specific disorder is given by the following relations: $C_{j} = C_{0} + b \sin (2 π x_{j})$ , $L_{j} = L_{0} + b \sin (2 π x_{j})$ , where ${x_{j}}$ is obtained from de Fourier filtering method. To study the localization behavior of the disordered dual transmission lines, we consider a set of quantities which depend of the frequency $ω$ ,

Acknowledgment

E. Lazo acknowledges the support of this research by the Dirección de Investigación y Extensión Académica de la Universidad de Tarapacá under project no. 4722-11. In addition, E. Lazo acknowledges the warm hospitality of the Departamento de Física Fundamental, Universidad de Salamanca, Spain, during a working stay. E. Diez acknowledges the financial support of this research by the Cariplo Foundation (project QUANTDEV) and by the Spanish Ministry of Science and Innovation (FIS2009-07880,

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Phase transitions in transmission lines with long-range fluctuating correlated disorder

Abstract

Introduction

Section snippets

Model and method

Numerical results

Secure communication

Conclusions

Acknowledgment

Phys. Rev. B

Phys. Rev. B

Phys. Rev. Lett.

Phys. Rev. B

Eur. Phys. J. B

Phys. Rev.

Phys. Rev.

Rev. Mod. Phys.

Rev. Mod. Phys.

Phys. Rev. E

Phys. Lett. A

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. B

J. Phys. Condens. Matter

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. Lett.

Physica A Amsterdam

Physica B

Phys. Lett. A

Phys. Rev. Lett.

Physica E

Phys. Rev. B

Phys. Rev. B