Coherent wave transmission in quasi-one-dimensional systems with L\'evy disorder

Coherent wave transmission in quasi-one-dimensional systems with Lévy disorder

Ilias Amanatidis, Ioannis Kleftogiannis, Fernando Falceto, and Víctor A. Gopar

Phys. Rev. E 96, 062141 – Published 26 December 2017

Abstract

We study the random fluctuations of the transmission in disordered quasi-one-dimensional systems such as disordered waveguides and/or quantum wires whose random configurations of disorder are characterized by density distributions with a long tail known as Lévy distributions. The presence of Lévy disorder leads to large fluctuations of the transmission and anomalous localization, in relation to the standard exponential localization (Anderson localization). We calculate the complete distribution of the transmission fluctuations for a different number of transmission channels in the presence and absence of time-reversal symmetry. Significant differences in the transmission statistics between disordered systems with Anderson and anomalous localizations are revealed. The theoretical predictions are independently confirmed by tight-binding numerical simulations.

Received 28 September 2017

DOI:https://doi.org/10.1103/PhysRevE.96.062141

Physics Subject Headings (PhySH)

Complex systems Mesoscopics Open quantum systems Quantum interference effects Quantum transport

2-dimensional systems Disordered systems

Green's function methods Landauer formula Tight-binding model

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsInterdisciplinary Physics

Authors & Affiliations

Ilias Amanatidis

Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Republic of Korea

Ioannis Kleftogiannis^*

Department of Physics, National Taiwan University, Taipei 10617, Taiwan

Fernando Falceto and Víctor A. Gopar

Departamento de Física Teórica and BIFI, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

^*Present address: Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan.

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Vol. 96, Iss. 6 — December 2017

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Figure 1
Transmission results for 1D systems with standard localization. (a) Linear (main frame) and exponential (inset) dependence of $〈 ln T 〉$ and $〈 T 〉$ , respectively, with the system length $L$ . The solid lines correspond to the theoretical results, while the closed circles are obtained from the numerical simulations. (b) Complete distribution of the transmission $P (T)$ for a standard disordered system with average $〈 T 〉 = 0.5$ . The solid line corresponds to the theoretical distribution, while the histogram is extracted from the numerical simulations.
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Figure 2
Transmission results for systems with Lévy disorder ( $α = 1 / 2$ ). (a) Power-law dependence of $〈 ln T 〉$ (main frame) and $〈 T 〉$ (inset) for a 1D system with Lévy disorder with $α = 1 / 2$ . See Eqs. (10) and (11). Closed circles correspond to the numerical simulation results. (b) Complete distribution $P (T)$ for a Lévy disordered system with $α = 1 / 2$ and $〈 T 〉 = 0.5$ . The solid line is obtained from Eq. (13), while the histogram is obtained from the numerical simulations.
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Figure 3
Transmission distribution for $N = 2$ transmission channels with Lévy disorder characterized by (a) and (b) $α = 1 / 2$ and (c) and (d) $α = 1 / 3$ . The solid lines are obtained according to the theory described in the main text, while the histograms are extracted from the tight-binding numerical simulations. The values of the standard deviation $δ T$ and the constant $b$ in $s (α, μ, N, z)$ are (a) $δ T = 0.54$ and $b = 2.5$ , (b) $δ T = 0.52$ and $b = 1.35$ , (c) $δ T = 0.56$ and $b = 2.6$ , and (d) $δ T = 0.54$ and $b = 1.2$ . Good agreement between theory and numerical simulation can be seen in all panels. For comparison with Lévy disordered systems, (a) shows $P (T)$ (green-dashed-line histogram) for systems with standard disorder and ensemble average $〈 T 〉 = 0.7$ .
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Figure 4
Transmission distribution $P (T)$ for $N = 3$ transmission channels with Lévy disorder characterized by (a) and (b) $α = 1 / 2$ and (c) and (d) $α = 1 / 3$ . The solid lines are obtained according to the theory described in the main text, while the histograms are extracted from the tight-binding numerical simulations. The values of the standard deviation $δ T$ and the constant $b$ in $s (α, μ, N, z)$ are (a) $δ T = 0.73$ and $b = 2.8$ , (b) $δ T = 0.70$ and $b = 1.36$ , (c) $δ T = 0.75$ and $b = 2.6$ , and (d) $δ T = 0.62$ and $b = 0.5$ . Good agreement between theory and numerical simulation can be seen in all panels. The (green-dashed-line) histogram in (a) shows the distribution $P (T)$ for systems with standard disorder with ensemble average $〈 T 〉 = 1.0$ .
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Figure 5
Applied magnetic field ( $ϕ = 0.15$ ). Transmission distribution $P (T)$ with broken time-reversal symmetry for $N = 2$ transmission channels with Lévy disorder characterized by (a) and (b) $α = 1 / 2$ and (c) and (d) $α = 1 / 3$ . The solid lines are obtained according to the theory described in the main text, while the histograms are extracted from the tight-binding numerical simulations. The values of the standard deviation $δ T$ and the constant $b$ in $s (α, μ, N, z)$ are (a) $δ T = 0.5$ and $b = 3.3$ , (b) $δ T = 0.46$ and $b = 1.3$ , (c) $δ T = 0.52$ and $b = 2.5$ , and (d) $δ T = 0.46$ and $b = 0.8$ . Theory and numerical simulations are in agreement in all panels. The (green-dashed-line) histogram in (a) shows the distribution $P (T)$ for systems with standard disorder and broken time-reversal symmetry with ensemble average $〈 T 〉 = 0.7$ .
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Figure 6
Applied magnetic field ( $ϕ = 0.1$ ). Transmission distribution $P (T)$ with broken time-reversal symmetry for $N = 3$ transmission channels with Lévy disorder characterized by (a) and (b) $α = 1 / 2$ and (c) and (d) $α = 1 / 3$ . The solid lines are obtained according to the theory described in the main text, while the histograms are extracted from the tight-binding numerical simulations. The values of the standard deviation $δ T$ and the constant $b$ in $s (α, μ, N, z)$ are (a) $δ T = 0.50$ and $b = 1.6$ , (b) $δ T = 0.51$ and $b = 0.7$ , (c) $δ T = 0.70$ and $b = 1.9$ , and (d) $δ T = 0.67$ and $b = 1.6$ . Theory and numerical simulations are in agreement in all panels. The (green-dashed-line) histogram in (a) shows the distribution $P (T)$ for systems with standard disorder and broken time-reversal symmetry with ensemble average $〈 T 〉 = 1.5$ .
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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Coherent wave transmission in quasi-one-dimensional systems with Lévy disorder

Ilias Amanatidis, Ioannis Kleftogiannis, Fernando Falceto, and Víctor A. Gopar

Phys. Rev. E 96, 062141 – Published 26 December 2017

Abstract

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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Coherent wave transmission in quasi-one-dimensional systems with Lévy disorder

Ilias Amanatidis, Ioannis Kleftogiannis, Fernando Falceto, and Víctor A. Gopar

Phys. Rev. E 96, 062141 – Published 26 December 2017

Abstract

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