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

TL;DR

The paper evaluates whether SVD-based, Hermitian-like diagnostics can reliably locate the MBL transition in TRS-preserving non-Hermitian many-body systems by benchmarking against exact diagonalization across quasi-periodic, random-disorder, and Stark hard-core-boson chains. It uses a suite of indicators, including complex level-spacing ratios, entanglement entropy, IPR, and dissipative spectral form factors, to compare ED and SVD results. The main finding is that ED and SVD agree only in the clean Stark model (critical tilt $\gamma_c\approx 3$), while in the disordered cases SVD systematically shifts the transition to larger disorder (e.g., $W_c^{\mathrm{E}}\approx 6$ vs $W_c^{\mathrm{S}}\approx 8$ for quasi-periodicity, and $\chi_c^{\mathrm{E}}\approx 7.7$ vs $\chi_c^{\mathrm{S}}}\approx 8.5$–$9.5$ for random disorder) and can yield inconsistent dynamical signatures (DSFF vs $\sigma$FF). This demonstrates that SVD-based diagnostics, though capturing qualitative trends, are not generally reliable for quantitatively locating the MBL transition in TRS-preserving non-Hermitian systems, and ED-based diagnostics remain essential. The Stark model’s agreement suggests the mechanism behind ED–SVD concordance warrants further investigation, and future work should explore robustness to analysis choices and scaling to larger sizes.

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.

Figures (9)

• Figure 1: Disorder-averaged spacing ratios (a) $\langle r^{\mathrm{E}}\rangle$ and (b) $\langle r^{\mathrm{S}}\rangle$ versus quasiperiodic potential strength $W$ for different system sizes. Dashed lines indicate the corresponding RMT and Poisson reference values.
• Figure 2: Half-chain entanglement entropy and IPR versus quasiperiodic strength $W$ for different system sizes. Entanglement: (a) $\langle S^{\mathrm{E}}\rangle$ and (b) $\langle S^{\mathrm{S}}\rangle$. IPR: (c) $\langle \mathrm{IPR}^{\mathrm{E}}\rangle$ and (d) $\langle \mathrm{IPR}^{\mathrm{S}}\rangle$. Dashed lines mark the critical points from finite-size scaling.
• Figure 3: Phase diagram of the quasiperiodic model in the $(g,W)$ plane. (a) ED boundary extracted from $\langle S^{\mathrm{E}}\rangle$. (b) SVD boundary extracted from $\langle S^{\mathrm{S}}\rangle$. The marked point indicates the representative parameters $(g,W)=(0.5,7)$.
• Figure 4: Dynamics of the DSFF and $\sigma$FF for the quasiperiodic model at $L=14$, averaged over 200 realizations. (a) DSFF for $W=7$ and $15$; the dashed line marks the Poisson behavior. (b) $\sigma$FF for $W=4$ and $7$; dashed lines show the GOE predictions.
• Figure 5: Disorder-averaged spacing ratios (a) $\langle r^{\mathrm{E}}\rangle$ and (b) $\langle r^{\mathrm{S}}\rangle$ versus random disorder strength $\chi$ for different system sizes $L$. Dashed lines denote the corresponding RMT and Poisson reference values.