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arXiv:2001.03839 · Physical review. B./Physical review. B

Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models

Daniel K. Mark, Cheng-Ju Lin, Olexei I. Motrunich

Published May 18, 2020• 176 citations
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Abstract

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.

DOI
10.1103/physrevb.101.195131

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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.
We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model, a celebrated nonintegrable model. The states thus obtained for the AKLT model can be interpreted as from one to an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong eigenstate thermalization hypothesis. We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another nonintegrable model, and extend our construction to the generalized integer spin AKLT models.
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.
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.
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 introduce and study several classes of quantum spin models with multibody interactions that exhibit quantum many-body scars. The models are constructed by two different methods: one exploiting boundary states in integrable spin chains and the other based on a variant of existing methods such as restricted spectrum generating algebras. The first method allows us to construct deformations of the Majumdar-Ghosh and Affleck-Kennedy-Lieb-Tasaki models---prototypes of frustration-free systems. With the second method, we construct a large class of spin-1 models involving scalar spin chirality in both one and two dimensions. Interestingly, in some cases, the models so constructed have towers of scar states of different character. For each example, we show that the scar states behave differently from thermal states by comparing their spectral and dynamical properties with those of other states. We also show that a superposition of the scar states constructed by the second method exhibits perfectly periodic revivals in the dynamics.