Higher-Order Topological Systems and Their Sub-Symmetry-Protected Topology

TL;DR

This paper explores how sub-symmetry-protected topology extends to higher-order topological systems, demonstrating the robustness of certain boundary states and quadrupole moments under specific perturbations in insulators and semimetals.

Contribution

It introduces a framework for analyzing sub-symmetry protection in higher-order topological phases, including insulators and semimetals, with both numerical and analytical validation.

Findings

Sub-symmetry-protected corner states remain at zero energy under perturbation.

Quantized quadrupole moments diagnose topological boundary states.

Certain Fermi arc states preserve topological features under sub-symmetry protection.

Abstract

Symmetry and topology are essential principles in topological physics. Recently, the idea of sub-symmetry-protected topology -- where some of the original symmetries are broken while a remaining subset, called sub-symmetries, continues to protect specific boundary states -- has been developed. Here, we extend sub-symmetry-protected topology to higher-order topological systems from second-order topological insulators to semimetals. By introducing a sub-symmetry-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 sub-symmetry-protecting perturbation, sub-symmetry-protected corner states remain pinned at zero energy and maintain quantized state-resolved quadrupole moments. In contrast, corner states on sub-symmetry-broken boundaries shift away from zero energy and lose their quantized character. We further extend this framework to a three-dimensional second-order 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 sub-symmetry-protecting perturbation. While one second-order Fermi arc becomes dispersive and loses its quadrupolar character under a sub-symmetry-breaking perturbation, the remaining second-order Fermi arcs retain chiral symmetry and preserve quantized quadrupolar characters.