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arXiv:2305.06371 · Physical review. B./Physical review. B

Protecting Hilbert space fragmentation through quantum Zeno dynamics

Pranay Patil, Ayushi Singhania, Jad C. Halimeh

Published Nov 7, 2023• 0 citations

Abstract

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.

DOI: 10.1103/physrevb.108.195109

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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.

In quantum many-body systems with kinetically constrained dynamics, the Hilbert space can split into exponentially many disconnected subsectors, a phenomenon known as Hilbert-space fragmentation. These subsectors can be viewed as protecting classical information about the initial state. We show that the interplay of fragmentation with symmetries enables the protection of quantum information. Focusing on charge conserving systems with charge conjugation and translation symmetries, we show that the nontrivial action of these symmetries on the fragmented subsectors generates exponentially many logical qubits encoded in degenerate pairs of eigenstates, which can be highly entangled. This symmetry action also provides necessary conditions for experimental signatures of Hilbert-space fragmentation, such as the persistence of density imbalances at late times.

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.

We propose an entanglement-enhanced sensing scheme that is robust against spatially inhomogeneous always-on Ising interactions. Our strategy is to tailor coherent quantum dynamics employing the Hilbert-space fragmentation (HSF), a recently recognized mechanism that evades thermalization in kinetically constrained many-body systems. Specifically, we analytically show that the emergent HSF caused by strong Ising interactions enables us to design a stable state where part of the spins is effectively decoupled from the rest of the system. Using the decoupled spins as a probe to measure a transverse field, we demonstrate that the Heisenberg limited sensitivity is achieved without suffering from thermalization.

In quantum many-body systems with kinetically constrained dynamics, the Hilbert space can split into exponentially many disconnected subsectors, a phenomenon known as Hilbert-space fragmentation. We study the interplay of such fragmentation with symmetries, focusing on charge conserving systems with charge conjugation and translation symmetries as a concrete example. The non-Abelian algebra of these symmetries and the projectors onto the fragmented subsectors leads to the emergence of exponentially many logical qubits encoded in degenerate pairs of eigenstates, which can be highly entangled. This algebra also provides necessary conditions for experimental signatures of Hilbert-space fragmentation, such as the persistence of density imbalances at late times.

Quantum kinetically constrained models have recently attracted significant attention due to their anomalous dynamics and thermalization. In this work, we introduce a hitherto unexplored family of kinetically constrained models featuring conserved particle number and strong inversion-symmetry breaking due to facilitated hopping. We demonstrate that these models provide a generic example of so-called quantum Hilbert space fragmentation, that is manifested in disconnected sectors in the Hilbert space that are not apparent in the computational basis. Quantum Hilbert space fragmentation leads to an exponential in system size number of eigenstates with exactly zero entanglement entropy across several bipartite cuts. These eigenstates can be probed dynamically using quenches from simple initial product states. In addition, we study the particle spreading under unitary dynamics launched from the domain wall state, and find faster than diffusive dynamics at high particle densities, that crosses over into logarithmically slow relaxation at smaller densities. Using a classically simulable cellular automaton, we reproduce the logarithmic dynamics observed in the quantum case. Our work suggests that particle conserving constrained models with inversion symmetry breaking realize so far unexplored dynamical behavior and invite their further theoretical and experimental studies.