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arXiv:1806.09624 · Physical review. B./Physical review. B

Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis

Sanjay Moudgalya, Nicolas Regnault, Bogdan Andrei Bernevig

Published Dec 27, 2018• 359 citations
arXivHTMLPDF

Abstract

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-$S$ 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.

DOI
10.1103/physrevb.98.235156

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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.
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.
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.
We revisit the $\ensuremath{\eta}$-pairing states in Hubbard models and explore their connections to quantum many-body scars to discover a universal scars mechanism. $\ensuremath{\eta}$-pairing occurs due to an algebraic structure known as a spectrum generating algebra (SGA), giving rise to equally spaced towers of eigenstates in the spectrum. We generalize the original $\ensuremath{\eta}$-pairing construction and show that several Hubbard-like models on arbitrary graphs exhibit SGAs, including ones with disorder and spin-orbit coupling. We further define a restricted spectrum generating algebra (RSGA) and give examples of perturbations to the Hubbard-like models that preserve an equally spaced tower of the original model as eigenstates. The states of the surviving tower exhibit a subthermal entanglement entropy, and we analytically obtain parameter regimes for which they lie in the bulk of the spectrum, showing that they are exact quantum many-body scars. The RSGA framework also explains the equally spaced towers of eigenstates in several well-known models of quantum scars, including the Affleck-Kennedy-Lieb-Tasaki model.
We construct a 1D model Hamiltonian of spinless fermions for which the spinless analog of $\ensuremath{\eta}$-pairing states are quantum many-body scars of the model. These states are excited states and display subvolume entanglement entropy scaling; they form a tower of states that are equally spaced in energy (resulting in periodic oscillations in the Loschmidt echo and in the time evolution of observables for initial states prepared in a superposition of them) and are atypical in the sense that they weakly break the eigenstate thermalization hypothesis, while the other excited states are thermal. We extend the approach by presenting models with a tower of scar states generated by multimers located at the edge of the Brillouin zone.
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.
A recent experiment in the Rydberg atom chain observed unusual oscillatory quench dynamics with a charge density wave initial state, and theoretical works identified a set of many-body "scar states" showing nonthermal behavior in the Hamiltonian as potentially responsible for the atypical dynamics. In the same nonintegrable Hamiltonian, we discover several eigenstates at an infinite temperature that can be represented exactly as matrix product states with a finite bond dimension, for both periodic boundary conditions (two exact E=0 states) and open boundary conditions (two E=0 states and one each E=±sqrt[2]). This discovery explicitly demonstrates the violation of the strong eigenstate thermalization hypothesis in this model and uncovers exact quantum many-body scar states. These states show signatures of translational symmetry breaking with a period-2 bond-centered pattern, despite being in one dimension at an infinite temperature. We show that the nearby many-body scar states can be well approximated as "quasiparticle excitations" on top of our exact E=0 scar states and propose a quasiparticle explanation of the strong oscillations observed in experiments.