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arXiv:2304.04792 · SciPost Physics

Topologically stable ergodicity breaking from emergent higher-form symmetries in generalized quantum loop models

Charles Stahl, Rahul Nandkishore, Oliver Hart

Published Mar 8, 2024• 14 citations
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Abstract

We present a set of generalized quantum loop models that provably exhibit topologically stable ergodicity breaking. These results hold for both periodic and open boundary conditions, and derive from a one-form symmetry (notably not being restricted to sectors of extremal one-form charge). We identify simple models in which this one-form symmetry can be emergent, giving rise to the aforementioned ergodicity breaking as an exponentially long-lived prethermal phenomenon. We unveil a web of dualities that connects these models, in certain limits, to models that have previously been discussed in the literature. We also identify nonlocal conserved quantities in such models, which correspond to patterns of system-spanning domain walls and are robust to the addition of arbitrary k k -local perturbations.

DOI
10.21468/scipostphys.16.3.068

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We present a set of generalized quantum loop models that provably exhibit topologically stable ergodicity breaking. These results hold for both periodic and open boundary conditions, and derive from a one-form symmetry (notably not being restricted to sectors of extremal one-form charge). We identify simple models in which this one-form symmetry can be emergent, giving rise to the aforementioned ergodicity breaking as an exponentially long-lived prethermal phenomenon. We unveil a web of dualities that connects these models, in certain limits, to models that have previously been discussed in the literature. We also identify nonlocal conserved quantities in such models, which correspond to patterns of system-spanning domain walls and are robust to the addition of arbitrary k k -local perturbations.
We present a new route to ergodicity breaking via Hilbert space fragmentation that displays an unprecedented level of robustness. Our construction relies on a single emergent (prethermal) conservation law. In the limit when the conservation law is exact, we prove the emergence of Hilbert space fragmentation with an exponential number of frozen configurations. These configurations are low-entanglement states in the middle of the energy spectrum and therefore constitute examples of quantum many-body scars. We further prove that every frozen configuration is absolutely stable to arbitrary perturbations, to all finite orders in perturbation theory. In contrast to previous constructions, our proof is not limited to symmetric perturbations, or to perturbations with compact support, but also applies to perturbations with long-range tails, and even to arbitrary geometrically nonlocal k-body perturbations, as long as k/L→0 in the thermodynamic limit, where L is linear system size. Additionally, we identify one-form U(1) charges characterizing some nonfrozen sectors, and discuss the dynamics starting from typical initial conditions, which we argue is best interpreted in terms of the magnetohydrodynamics of the emergent one-form symmetry.
Obstruction to Ergodicity from Locality and U(1)U(1)U(1) Higher Symmetries on the Lattice
We argue that the presence of any exact U(1)U(1)U(1) higher-form symmetry, under mild assumptions, presents a fundamental obstruction to ergodicity under unitary dynamics in lattice systems with local interactions and finite on-site Hilbert space dimension. Focusing on the two-dimensional case, we show that such systems necessarily exhibit Hilbert space fragmentation and explicitly construct Krylov sectors whose number scales exponentially with system size. While these sectors cannot be distinguished by symmetry quantum numbers, we identify the emergent integrals of motion which characterize them. Our symmetry-based approach is insensitive to details of the Hamiltonian and the lattice, providing a systematic explanation for ergodicity-breaking in a range of systems, including quantum link models.
We show how combining a discrete symmetry with topological Hilbert space fragmentation can give rise to exponentially many topologically stable qubits protected by a single discrete symmetry. We illustrate this explicitly with the example of the CZp\mathsf{CZ}_pCZp​ model, where the encoded qubits are stable to arbitrary symmetry-respecting perturbations for parametrically long times, substantially enhancing the robustness of a recently proposed construction based on nontopological fragmentation. In this model, the encoded qubits naturally come in pairs for which a universal set of transversal logical gates can be performed, ruling out (by the Eastin-Knill theorem) the possibility of using them for quantum error correction. We also comment on the combination of symmetry enrichment and topological fragmentation more generally, and the implications for use of systems exhibiting Hilbert space fragmentation as quantum memories.
We propose physically reasonable systems capable of avoiding ergodicity at infinite time in the thermodynamic limit, even with generic perturbations and when coupled to a heat bath. In two dimensions, the rainbow loop soup has (stretched) exponentially numerous absolutely stable nonergodic states with diverging energy but vanishing energy density. In three dimensions the rainbow membrane soup has (stretched) exponentially numerous nonergodic states with diverging energy barriers, leading to infinite-time robust ergodicity breaking that even survives coupling to a nonzero temperature heat bath. We describe our results in the language of exact emergent symmetries and demonstrate how the systems avoid common instabilities. Our construction naturally connects to quantum dimer models, topologically ordered systems, the group word construction, and Hamiltonians whose low-energy eigenstates exhibit anomalous entanglement entropy.
Abstract Symmetry-protected topological phases 1–4 cannot be described by any local order parameter and are beyond the conventional symmetry-breaking model 5 . They are characterized by topological boundary modes that remain stable under symmetry respecting perturbations 1–4,6–8 . In clean, gapped systems without disorder, the stability of these edge modes is restricted to the zero-temperature manifold; at finite temperatures, interactions with mobile thermal excitations lead to their decay 9–11 . Here we report the observation of a distinct type of topological edge mode 12–14 , which is protected by emergent symmetries and persists across the entire spectrum, in an array of 100 programmable superconducting qubits. Through digital quantum simulation of a one-dimensional disorder-free stabilizer Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles for a wide range of initial states. We show that the interaction between these edge modes and bulk excitations can be suppressed by dimerizing the stabilizer strength, leading to an emergent U(1) × U(1) symmetry in the prethermal regime of the system. Furthermore, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence, despite the system being disorder-free and at finite temperature. Our results establish a viable digital simulation approach 15–18 to experimentally study topological matter at finite temperature and demonstrate a potential route to construct long-lived, robust boundary qubits in disorder-free systems.
We introduce a large class of models exhibiting robust ergodicity breaking in quantum dynamics. Our work is inspired by recent discussions of ``topologically robust Hilbert space fragmentation'' but massively generalizes in two directions: First from states describable as ``loop-soups'' to a broader class of states reminiscent of string-nets and sponges and, second, from models restricted to square or cubic lattices to models defined on arbitrary lattices and graphs, including those without translation invariance. Our constructions leverage a recently proposed group-theory framework [Balasubramanian et al., Phys. Rev. X 14, 021034 (2024)] and identify a host of new phenomena arising from the interplay of ``group-model dynamics'' and lattice structure. We make crisp connections to gauge theories, and our construction generalizes Kitaev's quantum double to infinite groups.