Phase singularity HOTI quantum cascade laser

We propose a 1D topological phase-twist laser, in which each air hole in a triangular lattice is defined by the phase angle of its position vector with respect to the lattice center, designated as the origin. The Kekulé phase is then modulated to introduce a branch cut with twisted local phases, resulting in a phase-singularity point at the center. This singularity helps to create a topologically protected defect in the lattice, supporting a robust bound mode that physically originates from the Jackiw-Rebbi solution. Unlike previous TLs reported in the literature, the proposed TL is implemented on an amorphous lattice and does not follow the conventional bulk-boundary correspondence.