Higher-order topological insulators in a magnetic field

Two-dimensional (2D) generalization of the Su-Schriffer-Heeger (SSH) model serves as a platform for exploring higher-order topological insulators (HOTI). We investigate this model in a magnetic field which interpolates two models studied so far with zero flux and $\pi$ flux per plaquette. We show that in the Hofstadter butterfly there appears a wide gap around the $\pi$ flux, which belongs to the same HOTI discovered by Benalcazar-Bernevig-Hughes (BBH). It turns out that in a weak field regime HOTI could exist even within a small gap disconnected from the wider gap around $\pi$ flux. To characterize HOTI, we employ the entanglement polarization (eP) technique which is useful even if the basic four bands split into many Landau levels under a magnetic field.