Tunable arbitrarily located quasi-corner states in multiphase weak topological insulator.

In the established symmetry-based higher-order topological insulator (HOTI) phase classification, zero-dimensional (0D) topological states are treated as a pure bulk topology phenomenon determined only by symmetry indicators. Here we report unconventional compactly supported 0D topological states termed as quasi-corner states located protected by two-dimensional (2D) weak topology at desirable vacancy-defects in 2D anisotropic honeycomb lattices (AHLs). Two weak topological indexes are derived to classify four topological phases and the corresponding positions of Wannier charge center. The emergence of quasi-corner state at the boundary site requires that the Wannier center is located at the site's broken bond. Quasi-corner states do not exist under strong topological phase. At single vacancy defect, the three weak topological phases correspond to three quasi-corner states localized at different sites, which enables localization position tuning by uniaxial strain. Topology-protected long-range coupling with dynamic oscillation of coupled quasi-corner states is also observed and allows flexibly tunable light localization.