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.

arXiv comment9 pages, 8 figures

References in corpus (22)

• Anderson localization of a non-interacting Bose-Einstein condensate
• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Aharonov-Bohm cages in two-dimensional structures
• Multifractal finite-size-scaling and universality at the Anderson transition
• Critical parameters from generalised multifractal analysis at the Anderson transition
• Physical nature of critical wave functions in Fibonacci systems
• Transmission eigenvalues and the bare conductance in the crossover to Anderson localization
• Metal-insulator transition in an aperiodic ladder network: an exact result
• On the role of a new type of correlated disorder in extended electronic states in the Thue-Morse lattice
• Determining mean first-passage time on a class of treelike regular fractals
• Spin filtering by a periodic nanospintronic devices
• Spin filtering due to quantum interference in periodic mesoscopic networks
• A renormalization group analysis of extended electronic states in 1d quasiperiodic lattices
• Adiabaticity and localization in one-dimensional incommensurate lattices
• Flux induced semiconducting behavior of a quantum network
• Nonreciprocal Anderson Localization in Magneto-Optical Random Structures
• A proposed signature of Anderson localization and correlation-induced delocalization in an N-leg optical lattice
• The three-dimensional Anderson model of localization with binary random potential
• Extended electron states and magneto-transport in a 3-simplex fractal
• Spin freezing by Anderson localization in one-dimensional semiconductors
• Decoherence-induced conductivity in the discrete 1D Anderson model: A novel approach to even-order generalized Lyapunov exponents
• Anomalous Transmission in a Hierarchical Lattice

Cited by in corpus (15)

• Topological states on fractal lattices
• Topology in the Sierpiński-Hofstadter problem
• Flat bands in fractal-like geometry
• Nontrivial topological flat bands in a diamond-octagon lattice geometry
• Flat band analogues and flux driven extended electronic states in a class of geometrically frustrated fractal networks
• Engineering slow light and mode crossover in a fractal-kagome waveguide network
• Engineering light localization in a fractal waveguide network
• Extended Vicsek fractals: Laplacian spectra and their applications
• 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
• Spin filtering and switching action in a diamond network with magnetic-nonmagnetic atomic distribution
• Linearized spectral decimation in fractals
• Aharonov-Bohm caging of an electron in a quantum fractal
• Fractalized Metals
• A fractal geometry immersed in a hierarchical magnetic flux distribution