TY - JOUR

T1 - Hofstadter Topology

T2 - Noncrystalline Topological Materials at High Flux

AU - Herzog-Arbeitman, Jonah

AU - Song, Zhi Da

AU - Regnault, Nicolas

AU - Bernevig, B. Andrei

N1 - Funding Information:
We thank Fang Xie, Biao Lian, Christopher Mora, and Benjamin Wieder for helpful discussions. We also thank one of our referees for pointing out Ref. . B. A. B., N. R., and S. Z.-D. were supported by the Department of Energy Grant No. desc0016239, the Schmidt Fund for Innovative Research, Simons Investigator Grant No. 404513, and the Packard Foundation. Further support was provided by the National Science Foundation EAGER Grant No. DMR-1643312, NSF-MRSEC DMR-1420541, BSF Israel US foundation No. 2018226, and ONR No. N00014-20-1-2303.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/12/2

Y1 - 2020/12/2

N2 - 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 ∼104 T magnetic fields to a trivial band structure. In this Letter, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane-Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a 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 that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

AB - 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 ∼104 T magnetic fields to a trivial band structure. In this Letter, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane-Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a 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 that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

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U2 - 10.1103/PhysRevLett.125.236804

DO - 10.1103/PhysRevLett.125.236804

M3 - Article

C2 - 33337182

AN - SCOPUS:85097566994

VL - 125

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 23

M1 - 236804

ER -