Missing link between the 2D Quantum Hall problem and 1D quasicrystals

Abstract

This paper discusses a connection between two important classes of materials, namely quasicrystals and topological insulators as exemplified by the Quantum Hall problem. It has been remarked that the quasicrystal ``inherits" topological properties from the 2D Quantum Hall model. We show this explicitly by introducing the Fibonacci-Hall model as a link between a 1D quasicrystal and the magnetic problems. We show here how Chern numbers for bands in periodic approximants of quasicrystals can be computed, along with gap labels. The Chern numbers are thus seen as a consequence of a flux parameter ${\varphi }^S$ induced by the geometry of winding in 2D space of the quasicrystal. We show the existence of lines of Lifshitz transitions in the phase space of the model. These are marked by change of Chern number and disappearance of edge states. The proposed extrapolation method can be generalized to higher dimensional 2D and 3D quasicrystals, where higher order Chern numbers could be computed, and related to experimentally measurable transport quantities.

arXiv comment11 pages including Supplementary Information. To appear in Phys.Rev.B Letters

References in corpus (24)

• Topological States and Adiabatic Pumping in Quasicrystals
• Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump
• Observation of Topological Phase Transitions in Photonic Quasicrystals
• Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models
• Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model
• Higher-Order Topological Insulators in Quasicrystals
• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Fractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• K-Theory and Pseudospectra for Topological Insulators
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Measuring topological invariants from generalized edge states in polaritonic quasicrystals
• Revealing the topology of quasicrystals with a diffraction experiment
• Topological models in rotationally symmetric quasicrystals
• Analytical results for Scaling Properties of the Spectrum of the Fibonacci Chain
• Hofstadter butterfly of a quasicrystal
• Kohn-Luttinger mechanism driven exotic topological superconductivity on the Penrose lattice
• Electronic Energy Spectra of Square and Cubic Fibonacci Quasicrystals
• Nonmonotonic crossover and scaling behaviors in a disordered 1D quasicrystal
• Topological superconductivity in Fibonacci quasicrystals
• Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems
• Properties of the Ammann-Beenker tiling and its square approximants
• Closing of gaps and gap labeling and passage from molecular states to critical states in a 2D quasicrystal
• Gap structure of 1D cut and project Hamiltonians

Cited by in corpus (1)

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