Landau level broadening, hyperuniformity, and discrete scale invariance

Abstract

We study the energy spectrum of a two-dimensional electron in the presence of both a perpendicular magnetic field and a potential. In the limit where the potential is small compared to the Landau level spacing, we show that the broadening of Landau levels is simply expressed in terms of the structure factor of the potential. For potentials that are either periodic or random, we recover known results. Interestingly, for potentials with a dense Fourier spectrum made of Bragg peaks (as found, e.g., in quasicrystals), we find an algebraic broadening with the magnetic field characterized by the hyperuniformity exponent of the potential. Furthermore, if the potential is self-similar such that its structure factor has a discrete scale invariance, the broadening displays log-periodic oscillations together with an algebraic envelope.

arXiv comment13 pages, 3 figures, published version

References in corpus (8)

• Scaling Theory of the Integer Quantum Hall Effect
• Approaching the Dirac point in high mobility multi-layer epitaxial graphene
• Discrete scale invariance and complex dimensions
• Hyperuniform States of Matter
• Hyperuniformity of Quasicrystals
• Landau Level Mixing and Levitation of Extended States in Two Dimensions
• Landau levels in quasicrystals
• Generalized quasiperiodic Rauzy tilings

Cited by in corpus (7)

• Local order metrics for many-particle systems across length scales
• Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability
• Hyperuniformity in two-dimensional periodic and quasiperiodic point patterns
• Multifractality and Hyperuniformity in Quasicrystalline Bose-Hubbard Models with and without Disorder
• Hyperuniform properties of the square-triangle tilings
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Critical behavior of the Ising model on square-triangle tilings