Topological phases of the dimerized Hofstadter butterfly

In this work, we study the topological phases of dimerized square lattice in presence an external magnetic field. The dimerization pattern lattice's hopping amplitudes can induce a series bulk energy gap openings Hofstadter spectrum at certain fractional fillings, giving rise to various phases. particular, show that $\frac{1}{2}$-filling quadrupole insulator phase with moment quantized $\frac{e}{2}$ and associated corner-localized mid-gap states exists parameter regime for all fluxes. At $\frac{1}{4}$ filling, system host obstructed atomic limit or Chern For those configurations gapped fillings below $\frac{1}{4}$, is non-vanishing numbers. Across diagram, both bulk-obstructed boundary-obstructed transitions exist model.