Hofstadter Topology: Non-crystalline Topological Materials at High Flux

TL;DR

This paper explores novel topological phases induced by magnetic flux in non-crystalline materials, revealing phases unattainable in crystalline insulators and connecting magnetic flux with higher-order topological states.

Contribution

It demonstrates that applying magnetic flux to topological band structures induces new phases, including higher-order topological insulators, using magnetic translation groups and Moiré lattice systems.

Findings

Discovery of flux-induced topological phases not realizable in crystalline insulators

Identification of 3D higher-order topological insulator phases with corner modes

Relevance of Moiré lattices for experimental realization of Hofstadter topological phases

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 Moir\'e lattices also renders our work relevant experimentally. In Moir\'e 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.