Robustness of helical hinge states three-dimensional weak second-order topological insulators (WSOTIs) against disorders is studied. The pure WSOTI obtained from a ${\mathbb{Z}}_{2}$ first-order insulator through surface band inversion. Both bulk and in the are gapped, in-gap valley-momentum locked topologically protected by valley-Chern number. In presence disorders, robust while quantized con...

In [3] and [6] the concepts of smooth closure, smooth interior, smooth α-closure and smooth α-interior of a fuzzy set were introduced and some of their properties were obtained. In this paper, we introduce the concepts of several types of weak smooth compactness and weak smooth α-compactness in terms of these concepts introduced in [3] and [6] and investigate some of their properties.

Abstract A quantum spin Hall (QSH) insulator hosts topological states at the one-dimensional (1D) edge, along which backscattering by nonmagnetic impurities is strictly prohibited. Its 3D analogue, a weak (WTI), possesses similar quasi-1D confined side surfaces. The enhanced confinement could provide route for dissipationless current and better advantages applications relative to strong insulat...