Staggered and extreme localization of electron states in fractal space

TL;DR

This paper analytically demonstrates a countably infinite set of localized electron states with varying localization lengths in a fractal Vicsek network, revealing unique spectral properties and the effects of magnetic flux.

Contribution

It provides an exact analytical framework for understanding multiple localization lengths and states in a fractal lattice without magnetic fields, extending previous cage concepts.

Findings

Existence of infinitely many localized states with different localization lengths.

Exact energy eigenvalues for localized states are determined.

Magnetic flux at half the quantum induces extreme localization.

Abstract

We present exact analytical results revealing the existence of a countable infinity of unusual single particle states, which are localized with a multitude of localization lengths in a Vicsek fractal network with diamond shaped loops as the 'unit cells'. The family of localized states form clusters of increasing size, much in the sense of Aharonov-Bohm cages [J. Vidal et al., Phys. Rev. Lett. 81, 5888 (1998)], but now without a magnetic field. The length scale at which the localization effect for each of these states sets in can be uniquely predicted following a well defined prescription developed within the framework of real space renormalization group. The scheme allows an exact evaluation of the energy eigenvalue for every such state which is ensured to remain in the spectrum of the system even in the thermodynamic limit. In addition, we discuss the existence of a perfectly conducting state at the band center of this geometry and the influence of a uniform magnetic field threading each elementary plaquette of the lattice on its spectral properties. Of particular interest is the case of extreme localization of single particle states when the magnetic flux equals half the fundamental flux quantum.