The Landau levels of Euler insulator

A fragile topology in 2-band subspaces protected by either PT or C₂T symmetries in 2D is characterized by a Z Euler class, which would degenerate to a Z₂ second Stiefel-Whitney class if the bands are connected to extra bands. We investigate the energy levels of such Euler fragile topological materials in the presence of an external magnetic field. As a consequence of the topology, a gap closing in the Hofstadter spectrum occurs at nonzero Euler class. Furthermore, larger Euler numbers lead to gap closing at smaller flux, even if the total Hall conductance is always zero. We propose an interpretation of the Landau level behaviour under the framework of Chern basis.