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arXiv:2306.08864 · Physical review. B./Physical review. B

Stable real-energy spectral dynamics with topological transitions and non-Hermitian many-body localization

Shujie Cheng, Xixi Feng, Wen Chen, Niaz Ali Khan, Gao Xianlong

Published May 17, 2024• 6 citations
arXivHTMLPDF

Abstract

In this paper, the interplay between non-Hermiticity, quasidisorder, and repulsive interaction is studied for hard-core bosons confined in a one-dimensional optical lattice, where non-Hermiticity is induced by the nonreciprocal hoppings and the on-site gain and loss breaking the time-reversal symmetry. Although the energy spectra of the static system are fully complex, with the evolution of the initial state, the real part of the expectation value of the Hamiltonian under the time-evolved wave function changes stably. By means of the entanglement entropy and its dynamical evolution, as well as the inverse participation ratio, the many-body localization (MBL) is found to play the key role in the stability of the dynamical behavior of the real part of the expectation value, independent of whether the spectrum of the static Hamiltonian is real or complex. In the delocalization phase, the dynamical evolution of the real part of the expectation value is unstable. Meanwhile, the nearest-neighbor level spacings statistics shows the MBL transition accompanied by the transition from the Ginibre distribution to the complex Poisson distribution, different from the one in the time-reversal invariant system. In addition, the dynamical stability of the real part of the energy and the MBL transition can be characterized by the winding number, indicating that the MBL transition and the topological transition occur simultaneously, and the realization of the Hamiltonian is discussed.

DOI
10.1103/physrevb.109.174209

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cond-mat.dis-nnCross-listedarXiv:2602.07349•Physical review. B./Physical review. B•Feb 10, 2026•1 citation
In this paper, the interplay between non-Hermiticity, quasidisorder, and repulsive interaction is studied for hard-core bosons confined in a one-dimensional optical lattice, where non-Hermiticity is induced by the nonreciprocal hoppings and the on-site gain and loss breaking the time-reversal symmetry. Although the energy spectra of the static system are fully complex, with the evolution of the initial state, the real part of the expectation value of the Hamiltonian under the time-evolved wave function changes stably. By means of the entanglement entropy and its dynamical evolution, as well as the inverse participation ratio, the many-body localization (MBL) is found to play the key role in the stability of the dynamical behavior of the real part of the expectation value, independent of whether the spectrum of the static Hamiltonian is real or complex. In the delocalization phase, the dynamical evolution of the real part of the expectation value is unstable. Meanwhile, the nearest-neighbor level spacings statistics shows the MBL transition accompanied by the transition from the Ginibre distribution to the complex Poisson distribution, different from the one in the time-reversal invariant system. In addition, the dynamical stability of the real part of the energy and the MBL transition can be characterized by the winding number, indicating that the MBL transition and the topological transition occur simultaneously, and the realization of the Hamiltonian is discussed.
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.
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.
Many-body localization is shown to suppress the imaginary parts of complex eigenenergies for general non-Hermitian Hamiltonians having time-reversal symmetry. We demonstrate that a real-complex transition, which we conjecture occurs upon many-body localization, profoundly affects the dynamical stability of non-Hermitian interacting systems with asymmetric hopping that respects time-reversal symmetry. Moreover, the real-complex transition is shown to be absent in non-Hermitian many-body systems with gain and/or loss that breaks time-reversal symmetry, even though the many-body localization transition still persists.
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.
The non-Hermitian systems exhibit extreme sensitivity to the boundary conditions. The change in the eigenspectrum with tuning boundary parameter is intimately connected to the non-Hermitian skin effect. The single-particle systems are affected by the boundary perturbations; however, the interplay of a random disorder potential and nonreciprocal hopping under boundary perturbations of an interacting many-body system is not yet clear. In this work, we examine the boundary sensitivity of a non-Hermitian interacting fermionic system in the presence of a random disorder potential. A nonzero boundary parameter results in real-complex spectral transitions with nonreciprocal (or unidirectional) hopping at weak disorder, while the many-body localization at strong disorder washes away real-complex transitions leading to dynamical stability and real eigenvalue spectrum. We show that the boundary-driven real-complex spectral transitions of the non-Hermitian chain are accompanied by the corresponding changes in the level statistics and nearest level-spacing distributions. The intriguing features of nonreciprocity and boundary sensitivity are further revealed using the averaged inverse participation ratios. Finally, we find distinct behavior in the quench dynamics of local particle density, population imbalance, and entanglement entropy of charge-density-wave ordered state that corroborate the real-complex and localization transitions. Our results provide a route to understanding disordered many-body systems under a generalized boundary.
Singular value decomposition (SVD) provides a convenient way to construct Hermitian-like diagnostics for non-Hermitian many-body systems, but its reliability for locating many-body localization (MBL) transitions remains unclear, particularly in systems preserving time-reversal symmetry (TRS). We benchmark SVD-based diagnostics against exact diagonalization (ED) in TRS-preserving non-Hermitian hard-core-boson chains with nonreciprocal hopping, considering quasiperiodic, random-disorder, and Stark potentials. We compare level statistics, half-chain entanglement entropy, inverse participation ratio, and spectral form factors. For the quasiperiodic and random-disorder models, ED-based entanglement and IPR yield mutually consistent finite-size transition estimates, whereas the corresponding SVD-based estimates are systematically shifted to larger disorder strengths and can lead to different phase assignments. The discrepancy is also reflected in the spectral form factors, where the ED-based dissipative spectral form factor and the SVD-based singular form factor can indicate different spectral regimes at the same parameters. In contrast, for the clean Stark model, ED and SVD give consistent transition estimates. We identify the origin of this model dependence as the fact that SVD probes the auxiliary Hermitian operator H^†H^\hat H^\dagger\hat HH^†H^, rather than the intrinsic right-eigenstate structure of H^\hat HH^; consequently, SVD can be quantitatively reliable only when the corresponding bulk-state structures remain aligned. Our results show that SVD-based diagnostics can capture qualitative RMT-to-Poisson trends, but are not generically reliable quantitative probes of MBL transitions in TRS-preserving non-Hermitian many-body systems.