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.
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.
Strong local disorder in interacting quantum spin chains can turn delocalized eigenmodes into localized eigenstates, giving rise to many-body localized phases. This is accompanied by distinct spectral statistics: chaotic for the delocalized phase and integrable for the localized phase. In isolated systems, localization and chaos are defined through a web of relations among eigenvalues, eigenvectors, and real-time dynamics. These may change as the system is made open. We ask whether random dissipation (without random disorder) can induce chaotic or localized behavior in an otherwise integrable system. The dissipation is described using non-Hermitian Hamiltonians, which can effectively be obtained from Markovian dynamics conditioned on null measurement. In this non-Hermitian setting, we argue in favor of the use of the singular value decomposition. We complement the singular value statistics with different diagnostic tools, namely, the singular form factor and the inverse participation ratio and entanglement entropy of singular vectors. We thus identify a crossover of the singular values from chaotic to integrable spectral features and of the singular vectors from delocalization to localization. Our method is illustrated in an XXZ Hamiltonian with random local dissipation.
Utilizing exact diagonalization techniques, we investigate a one-dimensional, nonreciprocal, interacting hard-core boson model under a Stark potential with tail curvature. By employing the nonzero imaginary eigenenergies ratio, half-chain entanglement entropy, and eigenstate instability, we numerically confirm that the critical points of spectral real-complex (RC) transition and many-body localization (MBL) phase transition are not identical, and an examination of the phase diagrams reveals that the spectral RC transition arises before the MBL phase transition, which suggests the existence of a non-MBL-driven spectral RC transition. These findings are quite unexpected, and they are entirely different from observations in disorder-driven interacting non-Hermitian systems. This work provides a useful reference for further research on phase transitions in disorder-free interacting non-Hermitian systems.
Recent studies on disorder-induced many-body localization (MBL) in non-Hermitian quantum systems have attracted great interest. However, the non-Hermitian disorder-free MBL still needs to be clarified. We consider a one-dimensional interacting Stark model with nonreciprocal hoppings having time-reversal symmetry, the properties of which are boundary dependent. Under periodic boundary conditions (PBCs), such a model exhibits three types of phase transitions: the real-complex transition of eigenenergies, the topological phase transition, and the non-Hermitian Stark MBL transition. The real-complex and topological phase transitions occur at the same point in the thermodynamic limit but do not coincide with the non-Hermitian Stark MBL transition, which is quite different from the non-Hermitian disordered cases. By the level statistics, the system transitions from the Ginibre ensemble (GE) to the Gaussian orthogonal ensemble (GOE) to the Possion ensemble with the increase of the linear tilt potential's strength. The real-complex transition of the eigenvalues is accompanied by the GE-to-GOE transition in the ergodic regime. Moreover, the second transition of the level statistics corresponds to the occurrence of non-Hermitian Stark MBL. We demonstrate that the non-Hermitian Stark MBL is robust and shares many similarities with disorder-induced MBL, which several existing characteristic quantities of the spectral statistics and eigenstate properties can confirm. The dynamical evolutions of the entanglement entropy and the density imbalance can distinguish the real-complex and Stark MBL transitions. Finally, we find that our system under open boundary conditions lacks a real-complex transition, and the transition of non-Hermitian Stark MBL is the same as that under PBCs.
In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversal symmetry (TRS), will exhibit three types of phase transition: Real-complex transition of eigenenergies, topological phase transition, and many-body localization (MBL) phase transition. For the real-complex transition, it is found that the imaginary parts of the eigenenergies are always suppressed by the MBL. Moreover, by calculating the winding number, a topological phase transition can be revealed with the increase of potential amplitude, and we find that the behavior is quite different from the single-particle systems. Based on our numerical results, we conjecture that these three types of phase transition occur at the same point in the thermodynamic limit, and the MBL transition of quasiperiodic system and disordered system should belong to different universality classes. Finally, we demonstrate that these phase transitions can profoundly affect the dynamics of the non-Hermitian many-body system.
Non-Hermitian many-body localization (NH MBL) has emerged as a possible scenario for stable localization in open systems, as suggested by spectral indicators identifying a putative transition for finite system sizes. In this work, we shift the focus to dynamical probes, specifically the steady-state spin current, to investigate transport properties in a disordered, non-Hermitian XXZ spin chain. Through exact diagonalization for small systems and tensor-network methods for larger chains, we demonstrate that the steady-state current remains finite and decays exponentially with disorder strength, showing no evidence of a transition up to disorder values far beyond the previously claimed critical point. Our results reveal a stark discrepancy between spectral indicators, which suggest localization, and transport behavior, which indicates delocalization. This highlights the importance of dynamical observables in characterizing NH MBL and suggests that traditional spectral measures may not fully capture the physics of non-Hermitian systems. Additionally, we observe a noncommutativity of limits in system size and time, further complicating the interpretation of finite-size studies. These findings challenge the existence of NH MBL in the studied model and underscore the need for alternative approaches to understanding localization in non-Hermitian settings.
We study the behavior of spread complexity in the context of non-Hermitian many-body localization Transition (MBLT). Our analysis has shown that the singular value spread complexity is capable of distinguishing the ergodic and many-body localization (MBL) phase from the presaturation peak height for the non-hermitian models having time-reversal symmetry (TRS) and without TRS. On the other hand, the saturation value of the thermofield double (TFD) state complexity can detect the real-complex transition of the eigenvalues on increasing disorder strength. From the saturation value, we also distinguish the model with TRS and without TRS. The charge density wave complexity shows lower saturation values in the MBL phase for the model with TRS. However, the model without TRS shows a completely different behavior, which is also captivated by our analysis. So, our investigation unravels the real-complex transition in the eigenvalues, the difference between the model having TRS and without TRS, and the effect of boundary conditions for the non-hermitian models having MBL transitions, from the Krylov spread complexity perspective.