Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model

2015/04/30 by T. S. Jackson, Thomas Jackson, David Bauer +1 · 3 citations
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Electron #Fractional quantum Hall effect #Geometry #Landau quantization #Lattice (music) #Limit (mathematics) #Mathematical analysis #Mathematics #Physics #Quantum #Quantum Hall effect #Quantum and electron transport phenomena #Quantum mechanics #Quantum spin Hall effect #Theoretical physics #Topological Materials and Phenomena #cond-mat.mes-hall #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.93.235133

published as Phys. Rev. B 93, 235133 (2016) · [v2] improved presentation; 11 pages, 5 figures

arxiv created 2015/11/23 · openalex publication_date 2016/06/17 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.

Citations

• From fractional Chern insulators to a fractional quantum spin hall effect
• Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices
• Single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase
• Fractional Quantum Hall States of Atoms in Optical Lattices
• Quantal phase factors accompanying adiabatic changes
• Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
• Berry phase effects on electronic properties

Cited by

• Quaternion-Kahler geometry of time reversal symmetric crystals
• Quantum mechanics as an approximated model: A geometrodynamical approach
• Characterizing fractional topological phases of lattice bosons near the first Mott lobe

Related

• Fractional quantum Hall effect in the Hofstadter model of interacting fermions
• Geometric stability of topological lattice phases
• Composite Hofstadter bands with Dirac fermion spectrum of fractional quantum Hall states
• Fractional Quantum Hall Physics in Topological Flat Bands
• Interaction-driven plateau transition between integer and fractional Chern Insulators