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arXiv:2305.00827 · Physical review. B./Physical review. B

Fractionalization paves the way to local projector embeddings of quantum many-body scars

Keita Omiya, Markus Müller

Published Aug 8, 2023• 20 citations
arXivHTMLPDF

Abstract

Many systems that host exact quantum many-body scars (towers of energy-equidistant low entanglement eigenstates) are governed by a Hamiltonian that splits into a Zeeman term and a sum of local terms that annihilate the scar subspace. We show that this unifying structure also applies to models, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model or the PXP model of Rydberg-blockaded atoms, which were previously believed to evade this characterization. To fit these models into the local annihilator framework, we need to fractionalize their degrees of freedom and enlarge the associated Hilbert space. The embedding of the original system in a larger space elucidates the structure of their scar states, which are characterized as the common kernel of local annihilators. This simplifies their construction and reveals close analogies with lattice gauge theories.

DOI
10.1103/physrevb.108.054412

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Many systems that host exact quantum many-body scars (towers of energy-equidistant low entanglement eigenstates) are governed by a Hamiltonian that splits into a Zeeman term and a sum of local terms that annihilate the scar subspace. We show that this unifying structure also applies to models, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model or the PXP model of Rydberg-blockaded atoms, which were previously believed to evade this characterization. To fit these models into the local annihilator framework, we need to fractionalize their degrees of freedom and enlarge the associated Hilbert space. The embedding of the original system in a larger space elucidates the structure of their scar states, which are characterized as the common kernel of local annihilators. This simplifies their construction and reveals close analogies with lattice gauge theories.
We study the nature of the ergodicity-breaking ``quantum many-body scar'' states that appear in the PXP model describing constrained Rabi oscillations. For a wide class of bipartite lattices of Rydberg atoms, we reveal that the nearly energy-equidistant tower of these states arises from the Hamiltonian's close proximity to a generalized projector-embedding form in Phys. Rev. Lett. 119, 030601 (2017), a structure common to many models hosting quantum many-body scars. We construct a non-Hermitian but strictly local extension of the PXP model hosting exact quantum scars and show how various Hermitian scar-stabilizing extensions from the literature can be naturally understood within this framework. The exact scar states are obtained analytically as large spin states of explicitly constructed pseudospins. The quasiperiodic motion ensuing from the N\'eel state is finally shown to be the projection onto the Rydberg-constrained subspace of the precession of the large pseudospin.
Exact ``towers'' of quantum many-body scar states are recently discovered sets of states in some many-body nonintegrable systems. They are evenly spaced in energy and do not obey the eigenstate thermalization hypothesis. Here, the authors provide a unified framework of proving several towers of states known in the literature. This sheds light on the general structure of exact scar towers, and on the specific models themselves: the 1D AKLT models, spin-1 XY model, and a spin-\textonehalf{} domain wall conserving model. In particular, they provide a simple proof for the scars in the AKLT models and discover a ``pyramid'' of exact states in the spin-\textonehalf{} model.
Quantum many-body scars (QMBS) are nonthermal eigenstates embedded in otherwise thermal spectra. A broad class of exact QMBS is realized as fixed-momentum magnon states above a ferromagnetic reference state. Here we prove a structural theorem for this class. Specifically, we show that any local Hamiltonian hosting such ``ferromagnetic scar states'' necessarily admits a decomposition into a Zeeman term and terms containing local projectors that annihilate the scar states locally. This result establishes that an appropriate generalization of the Shiraishi--Mori construction is essentially exhaustive for ferromagnetic QMBS and provides a unified structural explanation for the recurrent appearance of projector-based interactions and equally spaced scar towers across a broad family of exact scar Hamiltonians.
Quantum many-body scars are nonthermal excited eigenstates of nonintegrable Hamiltonians, which could support coherent revival dynamics from special initial states when scars form an equally spaced tower in the energy spectrum. For open quantum systems, engineering many-body scarred dynamics by a controlled coupling to the environment remains largely unexplored. Here, we provide a general framework to exactly embed quantum many-body scars into the decoherence-free subspaces of Lindblad master equations. The dissipative scarred dynamics manifest persistent periodic oscillations for generic initial states, and can be practically utilized to prepare scar states with potential quantum metrology applications. We construct the Liouvillian dissipators with the local projectors that annihilate the whole scar towers, and utilize the Hamiltonian part to rotate the undesired states out of the null space of dissipators. We demonstrate our protocol through several typical models hosting many-body scar towers and propose an experimental scheme to observe the dissipative scarred dynamics based on digital quantum simulations and resetting ancilla qubits.
We discover a variety of exact quantum many-body scars in higher-spin kinetically constrained models through the recently developed DMRG-S algorithm [Zhang et al., Phys. Rev. Lett. 131, 020402 (2023)]. Specifically, for the higher-spin PXP model on arbitrary bipartite lattices of any spatial dimension, we find exact many-body scars that are equidistantly spaced in the energy spectrum and exhibit similar structures to the ground state of the Affleck-Kennedy-Lieb-Tasaki model. For the one-dimensional Fermi-Hubbard model with a tilted potential in a certain parameter regime, whose effective model is equivalent to a kinetically constrained spin model with four degrees of freedom on each site, we find several many-body scars at energy E=0E=0E=0 and $E=\ifmmode\pm\else\textpm\fi{}\sqrt{2}$ that can be exactly represented as matrix product states with finite bond dimensions. Our results demonstrate that larger local degrees of freedom in the kinetically constrained models provide a much broader space for the emergence of quantum many-body scars and weak ergodicity breaking.
Quantum many-body scars are non-thermal eigenstates embedded in the spectra of otherwise non-integrable Hamiltonians. Paradigmatic examples often appear as quasiparticle towers of states, such as the maximally ferromagnetic spin-1/2 states, also known as Dicke states. A distinguishing feature of quantum many-body scars is that they admit multiple local "parent" Hamiltonians for which they are exact eigenstates. In this work, we show that the locality of such parent Hamiltonians strongly constrains the relative placement of these states within the energy spectrum. In particular, we prove that if the full set of Dicke states are exact eigenstates of an extensive local Hamiltonian, then their energies must necessarily be equally spaced. Our proof builds on recent results concerning parent Hamiltonians of the WWW state, together with general algebraic structures underlying such quasiparticle towers. We further demonstrate that this equal spacing property extends to local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and expander graphs. Hamiltonians with kkk-local interactions and a bounded number of interaction terms per site are also encompassed by our proof. On the same classes of graphs, we additionally establish equal spacing for towers constructed from multi-site quasiparticles on top of product states. For the towers considered here, an immediate corollary of the equal spacing property is that any state initialized entirely within the quantum many-body scar manifold exhibits completely frozen entanglement dynamics under any local Hamiltonian for which those scars are exact eigenstates. Overall, our results reveal a stringent interplay between locality and the structure of quantum many-body scars.