Porous Haldane model: topological phase transitions and flat bands.

To investigate the influence of nanoholes on Chern insulators (CIs), we propose a porous Haldane model that considers the nearest-neighbor (NN) hoppings and next-NN (NNN) hoppings with staggered magnetic fluxes. This model supports multiple topological phases with different filling factors. At 2/5 filling, CI phases withC=±1,C = 2,C=±3,C=±4and higher-order topological insulator (HOTI) appear. At 9/20 filling, CI withC=±1,C = 2,C = 3, and HOTI phases are obtained. At half-filling, this model exhibits CI withC=±1,C = 2, andC=-3and HOTI phases. Unlike conventional HOTIs, these HOTI phases host gapless edge states and robust corner states which are characterized by a quantized quadrupole. Additionally, there is a topological flat band (TFB) with a flatness ratio about 13 with the NN and NNN hoppings. Based on the TFB model, we further investigate aν=1/2fractional CI state with hard-core bosons filling.