Universal spectral moment theorem and its applications in non-Hermitian systems

The high sensitivity of the spectrum and wavefunctions to boundary conditions, termed the non-Hermitian skin effect, represents a fundamental aspect of non-Hermitian systems. While it endows non-Hermitian systems with unprecedented physical properties, it presents notable obstacles in grasping universal properties that are robust against microscopic details and boundary conditions. In this Letter, we introduce a pivotal theorem: in the thermodynamic limit, for any non-Hermitian systems with finite-range interactions, all spectral moments are invariant quantities, independent of boundary conditions, posing strong constraints on the spectrum. Utilizing this invariance, we propose an analytic criterion for parity-time (PT) exactness that is applicable in any dimensions with any boundary conditions. we find that the PT transition identified using this method captures dynamical signatures observable in experiments, aligns with the exceptional point, and is distinct from the real-to-complex spectral transition, contrary to traditional expectations. We verify these findings in 1D and 2D lattice models.