Möbius transformations and electronic transport properties of large disorderless networks

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 Möbius 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.

Typical length scales in conducting disorderless networks

“…Very recently, it has been found that the electronic scattering properties of a layered linear periodic structure and those of a regular nonlinear network model are described exactly by the dynamics of intermittent low-dimensional nonlinear maps [1][2][3] . The presence of these maps is a consequence of the combination rule of scattering matrices when the scattering systems are built via consecutive replication of an element or motif.…”

Manifestations of the onset of chaos in condensed matter and complex systems

“…The transitions from localized to delocalized states correspond to transitions from regularity to chaos [32,33,34,35] and the chaotic regimes hold an ergodic property [1] that can be used to obtain scattering ensemble averages. In the language of the quantum case the conductance in the crystalline limit reflects the periodic or chaotic nature of the attractors (see Fig.…”

“…This property is reminiscent of the drastic reduction in state variables displayed by large arrays of coupled limit-cycle oscillators, for which their macroscopic time evolution has been shown [36] to be governed by underlying low-dimensional nonlinear maps of the Möbius form. The parallelism between the transport properties of the networks studied in [33] and the dynamical properties of arrays of oscillators is made more visible by noticing that in both problems the variables of interest are similar: phase shift of the scattered states and phase change with time of coupled oscillators, both determined by the action of the Möbius group. In the scattering problem we arrive at the basic nonlinear map by first constructing a family of self-similar networks, and then relating the scattering matrices that represent two members of the family.…”

“…(c) Bifurcation diagram formed by period one (localized states) and chaotic (delocalized states) attractors[32]. (d) The model scattering matrix performs Möbius group transformations in the unit complex circle as the system size grows[33]. Families of Horizontal Visibility (HV) networks obtained along the routes to chaos.…”

A zodiac of studies on complex systems

“…Localizations. A vital recursion relation for size growth of a basic wave scattering model was recognized as a nonlinear iteration map with a bifurcation diagram where tangent bifurcations separate periodic (insulating) and chaotic (conducting) attractors [17][18][19][20]. Sums.…”