Localization-delocalization transition and $$\mathscr{P}\mathscr{T}$$-symmetry breaking via second-neighbor coupling in a Heisenberg spin chain

The exploration of non-Hermitian quantum systems, particularly those governed by parity–time ($$\mathscr {P}\mathscr {T}$$)-symmetry, has revealed a rich landscape of unconventional phase transitions and localization phenomena. Spin chains with quasiperiodic potentials and competing Heisenberg interactions provide a versatile framework for probing these effects. In this study, we investigate the dynamical and spectral signatures of $$\mathscr {P}\mathscr {T}$$-symmetry breaking and localization-delocalization transitions in spin chains subjected to a non-Hermitian Aubry-André potential and tunable Heisenberg couplings across nearest ($$J_1$$) and next-nearest neighbors ($$J_2$$). Our analysis reveals that increasing $$J_1$$ monotonically shifts the critical point $$\lambda _C$$, while $$J_2$$ induces a non-linear response, generating a distinct minimum in $$\lambda _C$$. Notably, $$J_2$$ leads to the emergence of partially imaginary eigenvalues beyond $$\lambda _C$$, decoupling the onset of $$\mathscr {P}\mathscr {T}$$-symmetry breaking from the localization transition. Using inverse and normalized participation ratios, we identify three distinct phases: extended, localized, and an intermediate hybrid regime characterized by a mobility edge. To capture the dynamical fingerprints of these phases, we employ time-dependent density distribution, long-time survival probability $$\mathrm {P(r)}$$, analysis. Extended phases exhibit ballistic spreading and algebraic decay, while localized regimes show spatial confinement and exponential decay. These observables serve as robust indicators of non-Hermitian phase transitions, offering insights beyond static spectral measures. Overall, our findings underscore the critical role of dynamical metrics in characterizing phase structure in non-Hermitian spin systems. As interest in non-equilibrium quantum dynamics continues to grow, such models provide a compelling platform for understanding the interplay between $$\mathscr{P}\mathscr{T}$$-symmetry, localization, and quantum coherence in complex many-body systems.