Helical Quasiperiodic Chains with Engineered Dissipation: Liouvillian Rapidity Diagnostics of Transport and Localization

Abstract: We study relaxation spectra of a quadratic spinless--fermion helical chain with an Aubry--Andre--type quasiperiodic potential and a single N--th neighbor (helical) hopping. Dissipation and pumping are introduced via local linear Lindblad jump operators and treated exactly using the third--quantization / Majorana covariance formalism. Focusing on periodic boundary conditions (to avoid edge artefacts) we compute the Liouvillian rapidities and their smallest nonzero real part (the rapidity gap) for several spatial dissipation patterns: uniform (all), single--site (one--site) and two--site (two--site) placement, plus pairwise gain/loss on helical partner sites. We show that uniform dissipation yields large, weakly lambda--dependent gaps, while sparse local dissipation produces gaps that shrink rapidly as the quasiperiodic potential lambda induces localization. Increasing t_N enhances relaxation by improving mode overlap with dissipative channels. Finite--size scaling, rapidity level statistics (Poisson vs Wigner--Dyson), and spatial profiles of slow modes provide a consistent picture linking Liouvillian spectral structure to transport and localization. Our results highlight Liouvillian rapidities as compact, experimentally relevant diagnostics of relaxation and sensitivity in engineered open quantum lattices.