Higher-order topological systems and their subsymmetry-protected topology

Abstract

Symmetry and topology are essential principles in topological physics. Recently, the idea of subsymmetryprotected topology-where some of the original symmetries are broken while a remaining subset, called subsymmetries, continues to protect specific boundary states-has been developed. Here, we extend subsymmetry-protected topology to higher-order topological systems from second-order topological insulators to semimetals. By introducing a subsymmetry-protecting perturbation that acts on a single sublattice and selectively preserves specific topological boundary states, we track the evolution of these states and their topological features using numerical and analytical methods, and we show that state-resolved quadrupole moments diagnose which corner or hinge modes remain topological. As a representative example of a second-order topological insulator, we begin with the Benalcazar-Bernevig-Hughes model. We demonstrate that, under a subsymmetry-protecting perturbation, subsymmetry-protected corner states remain pinned at zero energy and maintain quantized stateresolved quadrupole moments. In contrast, corner states on subsymmetry-broken boundaries shift away from zero energy and lose their quantized character. We further extend this framework to a three-dimensional secondorder topological semimetal, constructed by stacking second-order topological insulator layers, and analyze how second-order Fermi arc states-hinge-localized modes that link the projections of bulk Dirac points, in contrast to conventional surface Fermi arcs-evolve under a subsymmetry-protecting perturbation. While one second-order Fermi arc becomes dispersive and loses its quadrupolar character under a subsymmetry-breaking perturbation, the remaining second-order Fermi arcs retain chiral symmetry and preserve quantized quadrupolar characters. These findings demonstrate that subsymmetry-protected topology can manifest in both insulating and gapless phases, offering routes to engineering symmetry-resilient topological phases in electronic, photonic, and synthetic systems.