Hofstadter Topology: Non-crystalline Topological Materials at High Flux

Abstract

The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding $\sim 10^4$ T magnetic fields to a trivial band structure. In this work, we show that when a magnetic field is added to an initially topological band structure, a wealth of remarkable possible phases emerges. Remarkably, we find topological phases which cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with nonzero Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with nontrivial Kane-Mele invariant produces a 3D TI or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of two-fold rotation and time-reversal and show that there exists a 3D higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group which exists at rational values of the flux. The advent of Moiré lattices also renders our work relevant experimentally. In Moiré lattices, it is possible for fields of order $1-30$ T to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

arXiv commentMain Text (5 pages, 2 figures) and supplementary material (53 pages and 22 figures)

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• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Multiple Flat Bands and Topological Hofstadter Butterfly in Twisted Bilayer Graphene Close to the Second Magic Angle
• Chiral Magic-Angle Twisted Bilayer Graphene in a Magnetic Field: Landau Level Correspondence, Exact Wavefunctions and Fractional Chern Insulators
• Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25T ($2\pi$ Flux)
• Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
• Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene
• Dynamical correlations and order in magic-angle twisted bilayer graphene
• Spinful topological phases in acoustic crystals with projective PT symmetry
• Geometric characterization of anomalous Landau levels of isolated flat bands
• Fragile topology in line-graph lattices with two, three, or four gapped flat bands
• Fragile topologically flat band in the checkerboard antiferromagnetic monolayer FeSe
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Bulk and Edge Properties of Twisted Double-Bilayer Graphene
• Implementing a synthetic magnetic vector potential in a 2D superconducting qubit array
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• Magnetic Bloch Theorem and Reentrant Flat Bands in Twisted Bilayer Graphene at $2\pi$ Flux
• Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices
• Moiré lattice effects on the orbital magnetic response of twisted bilayer graphene and Condon instability
• Landau Levels of the Euler Class Topology
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• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
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• Spectroscopy of the Fractal Hofstadter Energy Spectrum
• Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands
• Generalized Peierls substitution for the tight-binding model of twisted multilayer graphene in a magnetic field
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
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