Non-abelian symmetries are thought to be incompatible with many-body localization, but have been argued to produce in certain disordered systems a broad non-ergodic regime distinct from many-body localization. In this context, we present a numerical study of properties of highly-excited eigenstates of disordered chains with SU(3) symmetry. We find that while weakly disordered systems rapidly thermalize, strongly-disordered systems indeed exhibit non-thermal signatures over a large range of system sizes, similar to the one found in previously studied SU(2) systems. Our analysis is based on the spectral, entanglement, and thermalization properties of eigenstates obtained through large-scale exact diagonalization exploiting the full SU(3) symmetry.
SU(N) symmetry is incompatible with the many-body localized (MBL) phase, even when strong disorder is present. However, recent studies have shown that finite-size
SU(2) systems exhibit nonergodic, subthermal behavior, characterized by the breakdown of the eigenstate thermalization hypothesis, and by the excited eigenstates entanglement entropy that is intermediate between area and volume law. In this paper, we extend previous studies of the
SU(2)-symmetric disordered Heisenberg model to larger systems, using the time-dependent density matrix renormalization group (tDMRG) method. We simulate quench dynamics from weakly entangled initial states up to long times, finding robust subthermal behavior at stronger disorder. Although we find an increased tendency towards thermalization at larger system sizes, the subthermal regime persists at intermediate time scales, nevertheless, and therefore should be accessible experimentally. At weaker disorder, we observe signatures of thermalization; however, entanglement entropy exhibits slow sublinear growth, in contrast to conventional thermalizing systems. Furthermore, we study dynamics of the
SU(3)-symmetric disordered Heisenberg model. Similarly, strong disorder drives the system into subthermal regime, albeit thermalizing phase is broader compared to the
SU(2) case. Our findings demonstrate the robustness of the subthermal regime in spin chains with non-Abelian continuous symmetry, and are consistent with eventual thermalization at large system sizes and long time scales, suggested by previous studies.
Previous studies reveal a crucial effect of symmetries on the properties of a single particle moving in a disorder potential.More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest.MBL systems are characterized by the emergence of quasilocal integrals of motion and by the area-law entanglement entropy scaling of its eigenstates.In this paper, we investigate the effect of a non-Abelian SUð2Þ symmetry on the dynamical properties of a disordered Heisenberg chain.While SUð2Þ symmetry is inconsistent with conventional MBL, a new nonergodic regime is possible.In this regime, the eigenstates exhibit faster than area-law, but still strongly subthermal, scaling of the entanglement entropy.Using extensive exact diagonalization simulations, we establish that this nonergodic regime is indeed realized in the strongly disordered Heisenberg chains.We use the realspace renormalization group (RSRG) to construct approximate excited eigenstates by tree tensor networks and demonstrate the accuracy of this procedure for systems of sizes up to L ¼ 26.As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs.To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by the RSRG.This approach is capable of describing systems of size L ≈ 2000.We characterize the resonances that arise due to such processes, finding that they involve an ever-growing number of spins as the system size is increased.Crucially, the probability of finding resonances grows with the system's size.Even at strong disorder, we can identify a large length scale beyond which resonances proliferate.Presumably, this proliferation would eventually drive the system to a thermalizing phase.However, the extremely long thermalization timescales indicate that a broad nonergodic regime will be observable experimentally.Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems.The approach introduced here provides a versatile tool for describing a broad range of disordered manybody systems, well beyond sizes accessible in previous studies.
Many-body localization (MBL) enables new nonergodic dynamical phases of matter that avoid the fate of thermalization. Do other ergodicity-breaking phases beyond the MBL paradigm exist? Here, the authors introduce a method for constructing and probing the stability of candidate nonergodic phases. As an example, they consider disordered spin systems, in which a non-Abelian SU(2) symmetry precludes the conventional MBL phase that is characterized by quasilocal integrals of motion and area-law entanglement entropy of eigenstates. Instead, the authors construct a candidate nonergodic phase characterized by eigenstates having logarithmic scaling of entanglement entropy. They find that the phase is only marginally unstable and, therefore, nonthermal yet non-MBL dynamics can be observed in such disordered systems.
Within the Thouless energy scale, many-body Hamiltonians behave like random matrices. We study the Thouless energy of the disordered Heisenberg quantum spin chain. This model has become the standard model for the study of many-body localization and the transition to an ergodic phase for weak disorder. We focus on the ergodic side of the Many-Body Localization transition. The Thouless energy extracted from long range spectral statistics and from the power spectrum of the fluctuations of the full momentum distribution coincide. This scale behaves non-monotonically and is peaked at an intermediate value of the disorder. An analysis of the scaling behavior of the Thouless energy and its comparison with the number of populated eigenstates after a quench suggests that this scale is not large enough to guarantee thermalization in the thermodynamic limit.
We explore thermalization and quantum dynamics in a one-dimensional disordered SU(2)-symmetric Floquet model, where a many-body localized phase is prohibited by the non-Abelian symmetry. Despite the absence of localization, we find an extended nonergodic regime at strong disorder where the system exhibits nonthermal behaviors. In the strong disorder regime, the level spacing statistics exhibit neither a Wigner-Dyson nor a Poisson distribution, and the spectral form factor does not show a linear-in-time growth at early times characteristic of random matrix theory. The average entanglement entropy of the Floquet eigenstates is subthermal, although violating an area-law scaling with system sizes. We further compute the expectation value of local observables and find strong deviations from the eigenstate thermalization hypothesis. The infinite-temperature spin autocorrelation function decays at long times as ${t}^{\ensuremath{-}\ensuremath{\beta}}$ with $\ensuremath{\beta}<0.5$, indicating subdiffusive transport at strong disorders.
An important challenge in the field of many-body quantum dynamics is to identify nonergodic states of matter beyond many-body localization (MBL). Strongly disordered spin chains with non-Abelian symmetry and chains of non-Abelian anyons are natural candidates, as they are incompatible with standard MBL. In such chains, real space renormalization group methods predict a partially localized, nonergodic regime known as a quantum critical glass (a critical variant of MBL). This regime features a treelike hierarchy of integrals of motion and symmetric eigenstates with entanglement entropy that scales as a logarithmically enhanced area law. We argue that such tentative nonergodic states are perturbatively unstable using an analytic computation of the scaling of off-diagonal matrix elements and accessible level spacing of local perturbations. Our results indicate that strongly disordered chains with non-Abelian symmetry display either spontaneous symmetry breaking or ergodic thermal behavior at long times. We identify the relevant length and timescales for thermalization: Even if such chains eventually thermalize, they can exhibit nonergodic dynamics up to parametrically long timescales with a nonanalytic dependence on disorder strength.