We study in detail an open quantum generalization of a classical kinetically constrained model-the East model-known to exhibit slow glassy dynamics stemming from a complex hierarchy of metastable states with distinct lifetimes. Using the recently introduced theory of classical metastability for open quantum systems, we show that the driven open quantum East model features a hierarchy of classical metastabilities at low temperature and weak driving field. We find that the effective long-time description of its dynamics not only is classical, but shares many properties with the classical East model, such as obeying an effective detailed balance condition and lacking static interactions between excitations, but with this occurring within a modified set of metastable phases which are coherent, and with an effective temperature that is dependent on the coherent drive.
We study in detail an open quantum generalization of a classical kinetically constrained model-the East model-known to exhibit slow glassy dynamics stemming from a complex hierarchy of metastable states with distinct lifetimes. Using the recently introduced theory of classical metastability for open quantum systems, we show that the driven open quantum East model features a hierarchy of classical metastabilities at low temperature and weak driving field. We find that the effective long-time description of its dynamics not only is classical, but shares many properties with the classical East model, such as obeying an effective detailed balance condition and lacking static interactions between excitations, but with this occurring within a modified set of metastable phases which are coherent, and with an effective temperature that is dependent on the coherent drive.
In this thesis we apply, and build upon, a comprehensive theory of metastability in open quantum systems. We gain a full understanding of the long time states and the dynamics between them in a 1D open quantum Ising model, and a glassy, dissipative quantum East model: in order to do this, we develop algorithmic approaches to conducting such analyses in generic models exhibiting a metastability which can be considered classical. In addition, we provide an extensive discussion of the use of symmetries in the numerical study of open quantum systems, containing results of substantial use throughout this work. We finish with a general set of results about the spectra of operators describing the evolution of classical or quantum models undergoing stochastic resetting, allowing a precise analysis of the effect of resets on metastability in the 1D open quantum Ising model.
We show how graph theory concepts can provide an insight into the origin of slow dynamics in systems with kinetic constraints. In particular, we observe that slow dynamics is related to the presence of strong hierarchies between nodes on the Fock-space graph in the particle occupation basis, which encodes configurations connected by a given Hamiltonian. To quantify hierarchical structures, we develop a measure of centrality of the nodes, which is applicable to generic Hamiltonian matrices and inspired by established centrality measures from graph theory. We illustrate these ideas in the quantum East (QE) model. We introduce several ways of detuning nodes in the corresponding graph that alter the hierarchical structure, defining a family of QE models. We numerically demonstrate how these detunings affect the degree of nonergodicity on finite systems, as evinced by both the time dependence of density autocorrelations and eigenstate properties in the detuned QE models.
We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions---which occur in the large size limit---through the metastable behavior of many-body systems of moderate sizes accessible to numerics.
One of the general mechanisms that give rise to the slow cooperative relaxation characteristic of classical glasses is the presence of kinetic constraints in the dynamics. Here we show that dynamical constraints can similarly lead to slow thermalization and metastability in translationally invariant quantum many-body systems. We illustrate this general idea by considering two simple models: (i) a one-dimensional quantum analogue to classical constrained lattice gases where excitation hopping is constrained by the state of neighboring sites, mimicking excluded-volume interactions of dense fluids; and (ii) fully packed quantum dimers on the square lattice. Both models have a Rokhsar-Kivelson (RK) point at which kinetic and potential energy constants are equal. To one side of the RK point, where kinetic energy dominates, thermalization is fast. To the other, where potential energy dominates, thermalization is slow, memory of initial conditions persists for long times, and separation of timescales leads to pronounced metastability before eventual thermalization. Furthermore, in analogy with what occurs in the relaxation of classical glasses, the slow-thermalization regime displays dynamical heterogeneity as manifested by spatially segregated growth of entanglement.
Metastability in many-body quantum systems and its associated exponentially-long timescales have been the subject of considerable recent interest. Here, we focus on a class of driven-dissipative many-body open quantum systems described by a Lindbladian having hidden time-reversal symmetry (a form of quantum detailed balance). Examples include boundary-driven interacting spin chains, bosonic lattice models and driven-dissipative collective spin models. We suggest that for such systems, slow timescales in the vicinity of a dissipative first-order phase transition can be analytically predicted using a special purification of the non-equilibrium steady state. We show the accuracy of our conjecture through detailed studies of a dissipative transverse-field Ising model with collective and local decay, and a driven-dissipative nonlinear cavity model. Our results allow quantitative insights into metastability and slow dynamics for a range of systems, including cases where semiclassical or path-integral instanton approaches are intractable.
We study in detail the properties of the quantum East model, an interacting quantum spin chain inspired by simple kinetically-constrained models of classical glasses. Through a combination of analytics, exact diagonalization and tensor-network methods we show the existence of a transition, from a fast to a slow thermalization regime, which manifests itself throughout the spectrum. On the slow side, by exploiting the localization of the ground state and the form of the Hamiltonian, we explicitly construct a large (exponential in size) number of non-thermal states which become exact finite-energy-density eigenstates in the large-size limit, as expected for a true phase transition. A ``super-spin'' generalization allows us to find a further large class of area-law states proved to display very slow relaxation. These states retain memory of their initial conditions for extremely long times. Our numerical analysis reveals that the localization properties are not limited to the ground state and that} many eigenstates have large overlap with product states and can be approximated well by matrix product states at arbitrary energy densities. The mechanism that induces localization to the ground state, and hence the non-thermal behavior of the system, can be extended to a wide range of models including a number of simple spin chains. We discuss implications of our results for slow thermalization and non-ergodicity more generally in disorder-free systems with constraints and we give numerical evidence that these results may be extended to two dimensional systems.