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arXiv:1904.05182 · Journal of Statistical Mechanics Theory and Experiment

Connection between quantum-many-body scars and the Affleck–Kennedy–Lieb–Tasaki model from the viewpoint of embedded Hamiltonians

Naoto Shiraishi

Published Aug 1, 2019• 87 citations
arXivHTMLPDF

Abstract

Abstract We elucidate the deep connection between the PXP model, which is a standard model of quantum many-body scars, and the Affleck–Kennedy–Lieb–Tasaki (AKLT) Hamiltonian. Using the framework of embedded Hamiltonians, we establish the connection between the PXP Hamiltonian and the AKLT Hamiltonian, which clarifies the reason why the PXP Hamiltonian has nonthermal energy eigenstates similar to the AKLT state. Through this analysis, we find that the presence of such nonthermal energy eigenstates reflects the symmetry in the AKLT Hamiltonian.

DOI
10.1088/1742-5468/ab342e

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Physics and mathematics of quantum many-body scars and beyond

Keita Omiya

This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.

•Jan 1, 2026•0 citations
Abstract We elucidate the deep connection between the PXP model, which is a standard model of quantum many-body scars, and the Affleck–Kennedy–Lieb–Tasaki (AKLT) Hamiltonian. Using the framework of embedded Hamiltonians, we establish the connection between the PXP Hamiltonian and the AKLT Hamiltonian, which clarifies the reason why the PXP Hamiltonian has nonthermal energy eigenstates similar to the AKLT state. Through this analysis, we find that the presence of such nonthermal energy eigenstates reflects the symmetry in the AKLT Hamiltonian.
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.
Here, the authors develop a general formalism to analytically compute the entanglement spectra of exact excited states and use it to obtain the entanglement spectra of the infinite tower of states of the spin-SSS AKLT models, which are the first analytical examples of what are now called quantum many-body scars. Located in the bulk of the spectrum, these states have a subthermal entanglement entropy, showing for the first time the violation of the strong eigenstate thermalization hypothesis (ETH) in a well-known translation-invariant nonintegrable model.
Clifford augmented methods have emerged as a powerful technique to simulate quantum circuits and many-body systems by exploiting the stabiliser structure found in said systems, whilst additionally enabling one to readily quantify the non-stabilizerness (i.e.\ magic) present. In this work, we utilise a qudit Clifford augmented simulation method to probe the presence of magic in the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, finding that its ground state and scar states are highly magical as quantified by their Stabiliser Rényi Entropy (SRE). Nevertheless, owing to its low entanglement structure, these states are still readily expressible as a Clifford augmented Matrix Product State with low bond-dimension and moreover may be time-evolved using the Clifford augmented time-dependent variational principle (TDVP) algorithm. We utilise Clifford augmented TDVP to study the string order dynamics in the scar states of the AKLT model after a quench within the Haldane phase to the anti-ferromagnetic Heisenberg Hamiltonian. We observe that there is a decay of string order following the quench, which is more pronounced with increasing bimagnon number. This finding highlights the sensitivity of string order in the AKLT scar states to symmetry-preserving quenches that break the underlying restricted spectrum generating algebra, even when the initial and final Hamiltonians belong to the same topological phase.
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 propose a framework that extends the notion of symmetry-protected topological properties beyond the ground-state paradigm to dynamically isolated subspaces formed by exceptional non-thermal energy eigenstates of non-integrable systems, known as quantum many-body scars (QMBS). We introduce the concept of a symmetry-protected topological (SPT) scar subspace -- a Hilbert subspace stabilized by a restricted spectrum-generating algebra (rSGA) while being protected by on-site, inversion, and time-reversal symmetries. QMBS often admit a non-interacting quasiparticle description, which enables matrix-product representations with small bond dimension. Although individual QMBS do not necessarily retain the protecting symmetries of the Hamiltonian, we show that the subspace formed by the symmetry-connected QMBS does retain them, giving rise to consistently emerging topological properties across the entire scar subspace. Using the spin-111 Affleck--Kennedy--Lieb--Tasaki (AKLT) model, we demonstrate that its bimagnon scar subspace reflects the topological properties of the SPT ground state, as evidenced by the appropriate bond-space symmetry representations, the expected topological response, and the numerically verified long-range string order. Our findings indicate that scar subspaces can inherit -- and in inhomogeneous cases systematically modify -- the topological character of the SPT ground state, offering a new and experimentally accessible platform for probing symmetry-protected topology beyond the ground-state regime.
This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.