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.
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.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale τdev, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.
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^, rather than the intrinsic right-eigenstate structure of H^; 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.
We investigate disordered oneand two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators such as the number of principal components is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spinand fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.
We investigate the phase transitions from chaotic to nonchaotic dynamics in a quantum spin chain with a local non-Hermitian disorder, which can be realized with a Rydberg atom array setting. As the disorder strength increases, the emergence of nonchaotic dynamics is qualitatively captured through the suppressed growth of Krylov complexity and quantitatively identified through the reciprocity breaking of Krylov space. We further find that the localization in Krylov space generates another transition in the weak disorder regime, suggesting a weak ergodicity breaking. Our results closely align with conventional methods, such as the entanglement entropy and complex level spacing statistics, and pave the way to explore non-Hermitian phase transitions using Krylov complexity and associated metrics.
The explorations of non-Hermiticity have been devoted to investigate the disorder-induced many-body localization (MBL). However, the sensitivity of the spatial boundary conditions and the interplay of the non-Hermitian skin effect with many-body phenomena are not yet clear. For a MBL system in the presence of nonreciprocal tunnelings and random disorder potential, we identify two different complex-real spectral transitions, one is present for both open and periodic boundaries while the other is present only for open boundaries of coupled non-Hermitian chains. The later is driven due to the interchain coupling at weak disorder where the level statistics of the real eigenenergy phase follows a Gaussian orthogonal ensemble. We further characterize wave functions through the (biorthogonal) inverse participation ratio and fractal dimension, which reveal the suppression of skin effect in the non-Hermitian MBL phase. Finally, we demonstrate that the quench dynamics of the local particle density, spin imbalance, and entanglement entropy also signify the hallmark of the boundary effects and nonergodic character of many-body localization.