Gain saturation-induced delocalization of the skin zero mode in Hatano-Nelson laser arrays

We investigate how nonlinear gain saturation affects topological lasing in the Hatano-Nelson (HN) array, a non-Hermitian lattice with asymmetric coupling and nonreciprocal transport. Building on our previous results for Su-Schrieffer-Heeger (SSH) laser arrays, where the lasing zero mode was found to deviate from its linear counterpart in the weak-coupling regime, we show that this behavior is not unique to the SSH model. Despite the localization of the zero mode in the HN array due to the non-Hermitian skin effect, we find that nonlinear gain saturation can induce delocalization, causing the mode to spread across the array under certain conditions. These results demonstrate that the loss of topological mode localization is a generic nonlinear effect, revealing fundamental limits to the notion of topological lasing in active photonic systems.