We investigated the corner-states characteristics of C ₄-symmetric higher-order topological crystalline insulators. By evaluating the charge polarization and fractional corner charge, we obtained the abundant topological properties for different intercell hoppings amplitudes and identified that the non-trivial fractional corner charges are accompanied by second-order topological corner states. Interestingly, there are two different types of corner states topologically protected by generalized chiral symmetry, which guarantees that both corner states are pinned to zero energy. One of the corner states is localized at the corners like the one protected by generalized chiral symmetry of three-atom sublattice. However, the other corner states are localized at the two sublattices nearest to the corners, which has no counterpart in a lattice with generalized chiral symmetry. Moreover, both of them have robustness against defects and disorder.

我们研究了C ₄的角态特性-对称的高阶拓扑晶体绝缘体。通过评估电荷极化和分数角电荷，我们获得了不同单元间跳跃幅度的丰富拓扑特性，并确定了非平凡分数角电荷伴随着二阶拓扑角态。有趣的是，有两种不同类型的角状态受到广义手性对称的拓扑保护，这保证了两个角状态都固定为零能量。角状态之一位于角处，就像受三原子亚晶格的广义手征对称性保护的状态一样。然而，其他角状态位于最靠近角的两个亚晶格处，这在具有广义手征对称性的晶格中没有对应物。而且，