Landau Levels of the Euler Class Topology

Two-dimensional systems with $C_{2}mathcal{T}$ ($Pmathcal{T}$) symmetry exhibit the Euler class topology $Einmathbb{N}$ 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.