Nonlinear topological laser based on multipole insulators

Two-dimensional higher-order topological insulators (HOTIs), characterized by distinctive one-dimensional edge states and zero-dimensional corner states, provide an ideal platform for developing higher-order topological lasers. In this work, we systematically investigate the two-dimensional Benalcazar-Bernevig-Hughes (BBH) model, which hosts quantized quadrupole moments and topologically protected corner and edge states. By confining the lasing mode to selected topological corner or edge states under controlled gain, we demonstrate that the stable light excitation achieved after long-time evolution is predominantly determined by the topological properties of the model Hamiltonian. To characterize the system's topological features, we introduce several diagnostic ratios: the corner decay ratio $\tau_{1}$ and edge-to-corner ratio $\tau_{2}$ quantify the localization degree and spatial extent of corner states, respectively, while the inter-corner transfer ratio $\chi$ measures the intensity transfer efficiency mediated by coherent edge-state dynamics. The abrupt changes in $\tau_{1}$ and $\tau_{2}$ as functions of the hopping parameter $\gamma/\lambda$ directly reveal topological phase transitions, providing a comprehensive toolkit for extracting topological signatures from the system's dynamical evolution. Additionally, modulating the lattice site parity enables flexible tuning of corner state localization positions, offering insights for device engineering. Our calculations reveal that achieving bistability between corner states and edge states is relatively challenging.