Limitations of SVD-Based Diagnostics for Non-Hermitian Many-Body Localization with Time-Reversal Symmetry

Abstract: Singular value decomposition (SVD) has been used to construct Hermitian-like diagnostics for non-Hermitian many-body systems, but its reliability for identifying many-body localization (MBL) transitions -- particularly in time-reversal-symmetry (TRS) preserving settings -- remains unclear. Here we benchmark SVD-based diagnostics against exact diagonalization (ED) in TRS-preserving non-Hermitian hard-core-boson chains with nonreciprocal hopping, considering three representative potentials: a quasiperiodic potential, random disorder, and a Stark potential. We compare spectral statistics, half-chain entanglement entropy, inverse participation ratio, and spectral form factors. For the quasiperiodic and random-disorder models, ED yields mutually consistent transition estimates, whereas SVD systematically shifts the inferred critical disorder strength to larger values and can lead to different phase assignments. In contrast, for the clean Stark model ED and SVD locate a consistent critical tilt. Our results show that while SVD-based diagnostics capture qualitative trends, they are not generically reliable for quantitatively locating the MBL transition in TRS-preserving non-Hermitian many-body systems.