Staggered and extreme localization of electron states in fractal space

Biplab Pal, Arunava Chakrabarti

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13 Scopus citations


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 forms clusters of increasing size, much in the sense of Aharonov-Bohm cages, 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 a 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.

Original languageEnglish
Article number214203
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number21
StatePublished - 13 Jun 2012
Externally publishedYes

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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