Relationship between localization-delocalization transition and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetry breaking

We investigate the interplay between parity-time-symmetry (PT-symmetry) breaking and localization transitions in non-Hermitian spin chains with an Aubry-André potential and competing Dzyaloshinskii-Moriya (DM) interactions. While prior studies (e.g., [S. Longhi, Phys. Rev. Lett. 122, 237601 (2019)0031-900710.1103/PhysRevLett.122.237601]) suggest a universal alignment of PT transitions and delocalization, our analysis of next-nearest-neighbor DM couplings reveals a critical deviation: the two phenomena decouple in certain parameter regimes, despite coinciding for nearest-neighbor interactions. Through eigenvalue analysis and inverse/normalized participation ratios (IPR/NPR), we identify three distinct phases: (i) extended states (weak potential), (ii) localized states (strong potential), and (iii) an intermediate hybrid phase unique to next-nearest-neighbor DM couplings. To unravel the dynamical fingerprints of these phases, we track the time-dependent density propagation, long-time survival probability, and phase-sensitive Loschmidt echo. The density evolution exhibits ballistic spreading in extended phases versus localized confinement, while the survival probability's decay rate, exponential in localized regimes versus algebraic in extended ones, serves as a clear phase discriminator. Notably, quenches between phases induce distinct dynamical signatures, with the Loschmidt echo sharply reflecting phase boundaries through its decay profile. These results establish dynamical metrics as indispensable tools for characterizing non-Hermitian phase transitions beyond spectral or localization measures.