Landau Levels of the Euler Class Topology

Abstract

Two-dimensional systems with $C_2\mathcal{T}$ ($P\mathcal{T}$) symmetry exhibit the Euler class topology $E\in \mathbb{Z}$ in each two-band subspace realizing a fragile topology beyond the symmetry indicators. By systematically studying the energy levels of Euler insulating phases in the presence of an external magnetic field, we reveal the robust gaplessness of the Hofstadter butterfly spectrum in the flat-band limit, while for the dispersive bands the gapping of the Landau levels is controlled by a hidden symmetry. We also find that the Euler class $E$ of a two-band subspace gives a lower bound for the Chern numbers of the magnetic subgaps. Our study provides new fundamental insights into the fragile topology of flat-band systems going beyond the special case of $E=1$ as e.g.~in twisted bilayer graphene, thus opening the way to a very rich, still mainly unexplored, topological landscape with higher Euler classes.

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Cited by in corpus (11)

• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Quantum Valley Hall effect without Berry curvature
• Disorder-induced topological quantum phase transitions in multi-gap Euler semimetals
• Quantum optics measurement scheme for quantum geometry and topological invariants
• Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene
• Multiplicative topological semimetals
• Atomistic theory of moiré Hofstadter's butterfly in magic-angle graphene
• Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence
• Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux
• Orbital magnetization reveals multiband topology
• Hinge modes of three-dimensional Euler insulators