Topological structure of corner states protected by reflection symmetry in higher-order topological insulators

Higher-order topological systems have attracted significant attention in condensed-matter physics and quantum simulation due to their unique multidimensional topological characteristics. For higher-order topological insulators, in addition to the nontrivial topological structure of the system in momentum space, our study reveals that the real space of corner states also exhibit nontrivial topological geometric phases, when the reflection symmetries of the system satisfy noncommutation relations. In the particular case with anti-commutation relations, the effective space of corner states possesses a π phase, which leads to an anomalous multifold degeneracy in the corner states. Crucially, such degeneracy cannot be lifted without breaking the reflection symmetries of the system, highlighting its robustness under symmetry-preserving perturbations. The findings not only deepen the fundamental understanding of real-space topology of corner states in higher-order systems, but also establish a symmetry-protected design principle for engineering robust degeneracy in artificial topological materials.