M\"obius transformations and electronic transport properties of large disorderless networks

TL;DR

This paper demonstrates that the electronic transport states in large, disorderless networks can be characterized by Lyapunov exponents of M"obius maps, linking chaotic dynamics with conductive properties.

Contribution

It introduces a novel approach using M"obius transformations and Lyapunov exponents to describe transport phases in large networks, connecting chaos theory with mesoscopic physics.

Findings

Insulating and conducting states correspond to negative and zero Lyapunov exponents.

Conductive phase exhibits weak chaos and anomalous transport.

Results verified for various network structures, highlighting degree of freedom reduction.

Abstract

We show that the key transport states, insulating and conducting, of large regular networks of scatterers can be described generically by negative and zero Lyapunov exponents, respectively, of M\"obius maps that relate the scattering matrix of systems with successive sizes. The conductive phase is represented by weakly chaotic attractors that have been linked with anomalous transport and ergodicity breaking. Our conclusions, verified for serial as well as parallel stub and ring structures, reveal that mesoscopic behavior results from a drastic reduction of degrees of freedom.