Staggered and extreme localization of electron states in fractal space

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.

9 pages, 8 figures

References in corpus (7)

• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Critical parameters from generalised multifractal analysis at the Anderson transition
• Metal-insulator transition in an aperiodic ladder network: an exact result
• Determining mean first-passage time on a class of treelike regular fractals
• Spin filtering by a periodic nanospintronic devices
• Adiabaticity and localization in one-dimensional incommensurate lattices
• Flux induced semiconducting behavior of a quantum network

Cited by in corpus (8)

• Flat band analogues and flux driven extended electronic states in a class of geometrically frustrated fractal networks
• Extended Vicsek fractals: Laplacian spectra and their applications
• Engineering light localization in a fractal waveguide network
• Loop current states and their stability in small fractal lattices of Bose-Einstein condensates
• Exotic electron states and tunable magneto-transport in a fractal Aharonov-Bohm interferometer
• Linearized spectral decimation in fractals
• Aharonov-Bohm caging of an electron in a quantum fractal
• A fractal geometry immersed in a hierarchical magnetic flux distribution