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Information dynamics in a model with Hilbert space fragmentation

Dominik Hahn

Published Oct 1, 2021• 43 citations
Publisher

Abstract

The fully frustrated ladder - a quasi-1D geometrically frustrated spin one half Heisenberg model - is non-integrable with local conserved quantities on rungs of the ladder, inducing the fragmentation of the Hilbert space into sectors composed of singlets and triplets on rungs. We explore the far-from-equilibrium dynamics of this model through the entanglement entropy and out-of-time-ordered correlators (OTOC). The post-quench dynamics of the entanglement entropy is highly anomalous as it shows clear non-damped revivals that emerge from short connected chunks of triplets and whose persistence is therefore a consequence of fragmentation. We find that the maximum value of the entropy follows from a picture where coherences between different fragments co-exist with perfect thermalization within each fragment. This means that the eigenstate thermalization hypothesis holds within all sufficiently large Hilbert space fragments. The OTOC shows short distance oscillations arising from short coupled fragments, which become decoherent at longer distances, and a sub-ballistic spreading and long distance exponential decay stemming from an emergent length scale tied to fragmentation.

DOI
10.21468/scipostphys.11.4.074

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The fully frustrated ladder - a quasi-1D geometrically frustrated spin one half Heisenberg model - is non-integrable with local conserved quantities on rungs of the ladder, inducing the fragmentation of the Hilbert space into sectors composed of singlets and triplets on rungs. We explore the far-from-equilibrium dynamics of this model through the entanglement entropy and out-of-time-ordered correlators (OTOC). The post-quench dynamics of the entanglement entropy is highly anomalous as it shows clear non-damped revivals that emerge from short connected chunks of triplets and whose persistence is therefore a consequence of fragmentation. We find that the maximum value of the entropy follows from a picture where coherences between different fragments co-exist with perfect thermalization within each fragment. This means that the eigenstate thermalization hypothesis holds within all sufficiently large Hilbert space fragments. The OTOC shows short distance oscillations arising from short coupled fragments, which become decoherent at longer distances, and a sub-ballistic spreading and long distance exponential decay stemming from an emergent length scale tied to fragmentation.
The fully frustrated ladder – a quasi-1D geometrically frustrated spin one half Heisenberg model – is non-integrable with local conserved quantities on rungs of the ladder, inducing the local fragmentation of the Hilbert space into sectors composed of singlets and triplets on rungs. We explore the far-from-equilibrium dynamics of this model through the entanglement entropy and out-of-time-ordered correlators (OTOC). The post-quench dynamics of the entanglement entropy is highly anomalous as it shows clear non-damped revivals that emerge from short connected chunks of triplets. We find that the maximum value of the entropy follows from a picture where coherences between different fragments co-exist with perfect thermalization within each fragment. This means that the eigenstate thermalization hypothesis holds within all sufficiently large Hilbert space fragments. The OTOC shows short distance oscillations arising from short coupled fragments, which become decoherent at longer distances, and a sub-ballistic spreading and long distance exponential decay stemming from an emergent length scale tied to fragmentation.
Hilbert space fragmentation is an intriguing paradigm of ergodicity breaking in interacting quantum many-body systems with applications to quantum information technology, but it is usually adversely compromised in the presence of perturbations. In this work, we demonstrate the protection of constrained dynamics arising due to a combination of mirror symmetry and Hilbert space fragmentation by employing the concept of quantum Zeno dynamics. We focus on an Ising spin ladder with carefully chosen quantum fluctuations, which in the ideal case guarantee a perfect disentanglement under Hamiltonian dynamics for a large class of initial conditions. This is known to be a consequence of the interplay of Hilbert space fragmentation with a mirror symmetry, and we show numerically the effect of breaking the latter. To evince the power of this perfect disentanglement, we study the effect of generic perturbations around the fine-tuned model and show that we can protect against the undesirable growth of entanglement entropy by using a local Ising interaction on the rungs of the ladder. This allows us to suppress the entanglement entropy to an arbitrarily small value for an arbitrarily long time by controlling the strength of the rung interaction. Our work demonstrates the experimentally feasible viability of quantum Zeno dynamics in the protection of quantum information against thermalization.
Understanding the boundaries between quantum thermalization and localization in many-body systems remains a central frontier of condensed matter and quantum information science. In this work, we investigate the dynamics and spectral properties of a generic model with long-range three-body-interaction, namely, a system with non-local three-wave-mixing. This model has been realized recently with a microwave Fabry-Perot cavity terminated on one end by a superconducting qubit mirror. Utilizing exact diagonalization techniques, we uncover a striking paradox: the global energy level spacing statistics show integrability, even though all dynamic observables and inverse participation ratios of the eigenstates indicate ergodicity and delocalization. We show that this behavior is a hallmark of strong Hilbert space fragmentation driven by kinematic constraints rather than an explicit global symmetry. Inside these sectors, dynamics scramble rapidly, as evidenced by the out-of-time-ordered correlator (OTOC), while global transport is heavily bottlenecked, resulting in a logarithmic relaxation to equilibrium. This picture is further confirmed by fluctuations in eigenstate entanglement entropy at the same energy. Finally, we demonstrate that the late time OTOC average scales with system size, providing a distinct experimentally accessible signature of the underlying three-body kinetic bottlenecks.
Quantum many-body models with both Hilbert space fragmentation and non-stationarity have recently been identified. Hilbert space fragmentation does not immediately imply non-stationarity. However, strictly local dynamical symmetries directly imply non-stationarity. It is demonstrated here that these symmetries are equivalent to local fragmentation into spatially localized blocks. Using strictly local dynamical symmetries, a lower bound is given here for persistent oscillations of generalised out-of-time-ordered correlation functions (OTOCs). A novel notion of genuinely many-body continuous time translation symmetry breaking is provided by demanding non-trivial spatial modulation of the Fourier transform of the OTOC. Such non-trivial spatial-temporal dynamics stems from a perpetual backflow of quantum scrambling. Here we call systems with time-translation symmetry breaking in the OTOC, OTO crystals. This breaking cannot be realised by systems with a single effective degree of freedom (e.g. spin precession). Furthermore, the breaking is stable to all local unitary and dissipative perturbations. An XYZ Creutz ladder is presented as an example.
Out-of-time-order correlators (OTOCs) have become established as a tool to characterise quantum information dynamics and thermalization in interacting quantum many-body systems. It was recently argued that the expected exponential growth of the OTOC is connected to the existence of correlations beyond those encoded in the standard Eigenstate Thermalization Hypothesis (ETH). We show explicitly, by an extensive numerical analysis of the statistics of operator matrix elements in conjunction with a detailed study of OTOC dynamics, that the OTOC is indeed a precise tool to explore the fine details of the ETH. In particular, while short-time dynamics is dominated by correlations, the long-time saturation behavior gives clear indications of an operator-dependent energy scale ω_{GOE} associated to the emergence of an effective Gaussian random matrix theory. We provide an estimation of the finite-size scaling of ω_{GOE} for the general class of observables composed of sums of local operators in the infinite-temperature regime and found linear behavior for the models considered.
Although most quantum systems thermalize locally on short timescales independent of initial conditions, recent developments have shown this is not always the case. Lattice geometry and quantum mechanics can conspire to produce constrained quantum dynamics and associated glassy behavior, a phenomenon that falls outside the rubric of the eigenstate thermalization hypothesis. Constraints ``fragment'' the many-body Hilbert space due to which some states remain insulated from others and the system fails to attain thermal equilibrium. Such fragmentation is a hallmark of geometrically frustrated magnets with low-energy ``icelike manifolds'' exhibiting a broad range of relaxation times for different initial states. Focusing on the highly frustrated kagome lattice, we demonstrate these phenomena in the Balents-Fisher-Girvin Hamiltonian (easy-axis regime), and a three-coloring model (easy-plane regime), both with constrained Hilbert spaces. We study their level statistics and relaxation dynamics to develop a coherent picture of fragmentation in various limits of the XXZ model on the kagome lattice.