Physical interpretation of the spectral approach to delocalization in infinite disordered systems

Kostadinova, Eva; Liaw, Constanze; Matthews, L. S.; Hyde, Truell

doi:10.1088/2053-1591/3/12/125904

Cited by 9 publications

(20 citation statements)

References 37 publications

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“…Since D n s, is a positive, monotonically decreasing sequence, one can construct approximate lower bounds (with respect to the probability distribution) of the limit D s, . This is exactly the approach taken in [13,[18][19][20]22]. In these works, it is noted that the distance values may decay logarithmically, so the authors performed a careful re-scaling of the horizontal axis by an exponent a < 0.…”

Section: Numerical Simulations In a One-dimensional Disordered Systemmentioning

confidence: 79%

“…The spectral method was introduced in [13], where it was also used to numerically confirm the existence of extended states for the two-dimensional discrete random Schrödinger operator for weak disorder. Applications to other underlying geometries, such as the square, hexagonal, triangular lattice in two-dimensions, and the three-dimensional square lattice, were explored in [18][19][20][21][22]. The unperturbed operator used in these papers was the classical Laplacian (s = 1).…”

Section: This Phenomenon Is Known As Anderson Localization and Has Momentioning

confidence: 99%

“…We have successfully employed the spectral approach in numerous settings and have demonstrated it to be quite effective in the study of transport behavior -see, for instance, [13,[18][19][20][21].…”

Section: Spectral Approach To Transport In Disordered Systemsmentioning

confidence: 99%

See 2 more Smart Citations

Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method

Padgett¹,

Kostadinova²,

Liaw³

et al. 2020

J. Phys. A: Math. Theor.

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This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a onedimensional disordered system that employs the discrete fractional Laplacian, (−∆) s , s ∈ (0, 2), in combination with results from spectral and measure theory. It is a classical mathematical result that the standard Anderson model exhibits localization of energy states for all nonzero disorder in one-dimensional systems. Numerical simulations utilizing our proposed model demonstrate that this localization effect is enhanced for sub-diffusive realizations of the operator, s ∈ (1, 2), and that the super-diffusive realizations of the operator, s ∈ (0, 1), can exhibit energy states with less localized features. These results suggest that the proposed method can be used to examine anomalous diffusion in physical systems where strong interactions, structural defects, and correlated effects are present.

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Section: Numerical Simulations In a One-dimensional Disordered Systemmentioning

confidence: 79%

Section: This Phenomenon Is Known As Anderson Localization and Has Momentioning

confidence: 99%

See 1 more Smart Citation

Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method

Padgett¹,

Kostadinova²,

Liaw³

et al. 2020

J. Phys. A: Math. Theor.

Self Cite

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show abstract

“…It is important to note, however, that the spectral approach can be applied to the entire class of Anderson-type Hamiltonians (first introduced in [35]) and generalized to any dimension or geometry. Definitions of mathematical concepts, such as cyclicity and spectral decomposition, can be found in [13], [14], [36]. These references also provide detailed proofs and a physical interpretation of the model.…”

Section: B Spectral Approachmentioning

confidence: 99%

Spectral approach to transport in a two-dimensional honeycomb lattice with substitutional disorder

et al. 2019

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The transport properties of a disordered two-dimensional (2D) honeycomb lattice are examined numerically using the spectral approach to the quantum percolation problem, characterized by an Andersontype Hamiltonian. In our simulations, substitutional disorder (or doping) is represented by a modified bimodal probability distribution of the on-site energies. The results indicate the existence of extended energy states for nonzero disorder and the emergence of a transition towards localized behavior for critical doping concentration > 0.3%, in agreement with the experimentally observed metal-to-insulator transition in graphene sheet doped with hydrogen.

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“…We now outline the setup of the Anderson localization problem and the basic logic of the spectral method. Detailed proofs and physical interpretation of the model can be found in Liaw et al [11][12][13] Consider the static lattice, zero temperature transport problem, where a wave propagates through a disordered medium. The 2D dust crystal employed in the present numerical simulations exhibits hexagonal symmetry.…”

Section: Spectral Approach To the Anderson-type Problemmentioning

confidence: 99%

Transport properties of disordered two‐dimensional complex plasma crystal

Kostadinova

Guyton

Cameron

et al. 2018

Contributions to Plasma Physics

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SummaryIn this work, we investigate numerically the transport properties of a 2D complex plasma crystal using diffusion of coplanar dust lattice waves. In the limit where the Hamiltonian interactions can be decoupled from the non-Hamiltonian effects, we identify two distinct types of wave transport: Anderson-type delocalization and long-distance excitation. We use a recently-developed spectral approach to evaluate the contribution of the Anderson problem and compare it to the results of the simulation. The benefit of our approach to transport problems is twofold. First, we employ a highly tunable macroscopic hexagonal crystal, which exhibits many-body interactions and allows for the investigation of transport properties at the kinetic level. Second, the analysis of the transport problem in 2D is provided using an innovative spectral approach, which avoids the use of scaling and boundary conditions. The comparison between the analytically predicted and numerically observed wave dynamics allows for the study of important characteristics of this open system. In our simulations, we observe long-distance lattice excitation, which occurs around lattice defects even when the initial perturbation does not spread from the center to the exterior of the crystal. In the decoupled Hamiltonian regime, this many-body effect can be contributed to the dust lattice interaction with the plasma environment.

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Physical interpretation of the spectral approach to delocalization in infinite disordered systems

Cited by 9 publications

References 37 publications

Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method

Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method

Spectral approach to transport in a two-dimensional honeycomb lattice with substitutional disorder

Transport properties of disordered two‐dimensional complex plasma crystal

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