Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1104.5679

Strange eigenstates and anomalous transport in a Koch fractal with hierarchical interaction

Arunava Chakrabarti
View original source ↗Read source document ↗Export metadata ↓

Abstract

Stationary states of non-interacting electrons on a Koch fractal are investigated within a tight binding approach. It is observed that if a hierarchically long range hopping is allowed, a suitable correlation between the parameters defining the Hamiltonian leads to spectacular changes in the transport properties of finite, but arbitrarily large fractals. Topologically identical structures, that are found to support the same distribution of the amplitudes of eigenstates, are conducting in some cases and insulating in the others, depending on the choice of the hierarchy parameter. The values of the hierarchical parameter themselves display a self-similar, fractal character.

Explore related subjects

cond-mat.dis-nn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Arunava Chakrabarti. 2011-04-29. Strange eigenstates and anomalous transport in a Koch fractal with hierarchical interaction. https://doi.org/10.1016/j.physleta.2011.07.061

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

R-transforms for non-Hermitian matrices: a spherical integral approach

In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.

cond-mat.dis-nn↗
arXiv2026

Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, we consider matrices of the form $\mathbf{A} + \mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A} + \mathbf{B}$ satisfies remarkably simple boundary equations that depend on the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.

cond-mat.dis-nn↗
arXiv2026

Electrical conductivity of crack-template-based transparent conducting films: mean-field approximation, effective-medium theory, and simulation

In this work, crack-template-based transparent conducting films were modeled as networks corresponding to the edges of a two-dimensional Poisson--Voronoi diagram. Two types of networks were considered: the original one, in which the conductance of each edge was inversely proportional to its length, and the effective one, in which all edges had the same conductance obtained from the effective-medium theory. The mean-field approximation was used for analytical evaluation of the electrical conductivity. Direct numerical calculations for the Poisson--Voronoi diagram showed that the mean-field approximation overestimated the effective conductivity of the original network by approximately 13\%, and of the effective network by 79\%. In addition, a honeycomb network with an edge conductance distribution corresponding to the Poisson--Voronoi diagram was studied: for it, the predictions of the effective-medium theory turned out to be more accurate than for the Poisson--Voronoi diagram, which was explained by the greater structural homogeneity of the periodic honeycomb lattice. The results indicate that, when modeling crack-template-based transparent conducting films, the application of the mean-field approximation may lead to significant errors if the resistance of individual conductors is not simply proportional to their length. This possibility is discussed as a motivation for future studies of hierarchical cracks with variable width, which are not directly investigated here.

cond-mat.dis-nn↗