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

TL;DR

This paper evaluates the effectiveness of SVD-based diagnostics in detecting many-body localization transitions in non-Hermitian systems with time-reversal symmetry, revealing limitations in their quantitative reliability compared to exact methods.

Contribution

It benchmarks SVD diagnostics against exact diagonalization in TRS-preserving non-Hermitian models, highlighting their qualitative but not quantitative accuracy.

Findings

SVD shifts critical disorder strength estimates to larger values in some models

SVD can lead to different phase assignments compared to ED

SVD captures qualitative trends but lacks reliability for precise transition detection

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.