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arXiv:1911.05637 · Physical review. B./Physical review. B

Revivals imply quantum many-body scars

Alvaro Martin Alhambra, Anurag Anshu, Henrik Wilming

Published May 5, 2020• 51 citations
arXivHTMLPDF

Abstract

We derive general rigorous results relating revivals in the dynamics of quantum many-body systems to the entanglement properties of energy eigenstates. For a $D$-dimensional lattice system of $N$ sites initialized in a low-entangled and short-range correlated state, our results show that a perfect revival of the state after a time of at most $O(\mathrm{poly}(N))$ implies the existence of at least $O(\sqrt{N}/{log}^{2D}(N))$ ``quantum many-body scars'': energy eigenstates with energies placed in an equally spaced ladder and with R\'enyi entanglement entropy of at most $O(log(N))+O(|\ensuremath{\partial}A|)$ for any region $A$ of the lattice. This shows that quantum many-body scars are a necessary consequence of revivals, independent of particularities of the Hamiltonian leading to them. We also present results for approximate revivals and for revivals of expectation values of observables and prove that the duration of revivals of states has to become vanishingly short with increasing system size.

DOI
10.1103/physrevb.101.205107

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arXiv:2105.04567•Physical review. B./Physical review. B•Sep 7, 2021•20 citations
We derive general rigorous results relating revivals in the dynamics of quantum many-body systems to the entanglement properties of energy eigenstates. For a DDD-dimensional lattice system of NNN sites initialized in a low-entangled and short-range correlated state, our results show that a perfect revival of the state after a time of at most O(poly(N))O(\mathrm{poly}(N))O(poly(N)) implies the existence of at least O(N/log2D(N))O(\sqrt{N}/{log}^{2D}(N))O(N​/log2D(N)) ``quantum many-body scars'': energy eigenstates with energies placed in an equally spaced ladder and with R'enyi entanglement entropy of at most O(log(N))+O(∣\ensuremath∂A∣)O(log(N))+O(|\ensuremath{\partial}A|)O(log(N))+O(∣\ensuremath∂A∣) for any region AAA of the lattice. This shows that quantum many-body scars are a necessary consequence of revivals, independent of particularities of the Hamiltonian leading to them. We also present results for approximate revivals and for revivals of expectation values of observables and prove that the duration of revivals of states has to become vanishingly short with increasing system size.
The observations of persistent revivals in the Rydberg atom chain have revealed a weak ergodicity breaking mechanism known as quantum many-body scars, which is typically a collection of states with low entanglement embedded in otherwise thermal spectra. Here, by applying a generic formalism, we reveal a direct evolution from the quantum many-body scars to the multiple inverted quantum many-body scars, i.e., different sets of excited states with volume-law entanglement entropy embedded in a sea of states with area-law entanglement. When increasing the disorder strength, a series of exact eigenstates, acting as conventional quantum many-body scars in a regime of weak disorder, remain unchanged. Around each of these states, the inverted quantum many-body scars are introduced by the increased disorder. Moreover, the strong disorder also gives rise to additional sets of inverted quantum many-body scars with their energies concentrating in the middle of the exact eigenstates. As a result, all the multiple inverted quantum many-body scars are approximately equidistant in energy, reminiscent of conventional quantum many-body scarred states. Despite being the measure-zero states in the whole spectrum, these inverted quantum many-body scarred states significantly influence nonequilibrium dynamics in the large disorder regime. Random thermal states in a specific subspace show periodic revivals in the fidelity dynamics, while the typical charge-density-wave states exhibit persistent imbalance dynamics. We further examine the stability of the conventional and the inverted quantum many-body scars against the external random field. Our findings expand the variety of nonthermal systems and draw a connection between the weak violation of ergodicity and that of nonergodicity.
Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasilocal deformation of the Rydberg-blockaded Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying nonintegrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates-quantum many-body scars. Furthermore, using these insights, we construct a toy model that hosts exact quantum many-body scars, providing an intuitive explanation of their origin. Our results offer specific routes to enhancing coherent many-body revivals and provide a step toward establishing the stability of quantum many-body scars in the thermodynamic limit.
We present a class of quantum dimer models on the kagome lattice with full translational invariance that feature a quantum many-body scar state of analytically known entanglement properties within their spectra. Using exact diagonalization on lattices of up to 60 sites, we show that nonscar states conform to the eigenstate thermalization hypothesis. Specifically, we show that energies are distributed according to the Gaussian ensemble expected of their respective symmetry sector, illustrate the existence of the scar from bipartite entanglement properties, and demonstrate revival phenomena in studies of fidelity dynamics.
Quantum many-body scars (QMBSs) are a novel paradigm for the violation of the eigenstate thermalization hypothesis—Hamiltonians of these systems exhibit mid-spectrum eigenstates that are equidistant in energy and which possess low entanglement and evade thermalization for long times. We present a novel approach to understanding the anomalous dynamical behavior in these systems. Specifically, we postulate that QMBS Hamiltonians H can generically be partitioned into a set of terms O a which do not commute over the entire Hilbert space, but commute to all orders within the subspace of scar states. All states in the scar subspace thus evolve according to a “broken unitary” U s ( t ) = ∏ a e − i O a t , where H = ∑ a O a , which provides a simple interpretation of the anomalous dynamical features exhibited by quantum scars; they evolve in time according to a simpler unitary operator allowing for revivals. It is found that the observed dynamical decoupling is often a direct consequence of simple local conditions, which when satisfied lead to recurring aspects of QMBSs, including equidistant eigenvalues, many-body revivals, and sub-volume-law entanglement entropy. Two classes of scar models emerge in this picture—those with a finite set of O a , as pertaining of, for instance, scars in the AKLT model, and those with an extensive set of such operators, such as eta pairing scar states in the Hubbard model. Besides discussing how many well-known scar models fit into the above picture, we show how the broken unitary formalism captures many known scar construction methods, known in the literature and generalizes others such as quasiparticles matrix product state (MPS) based methods, and the Shiraishi-Mori approach. Published by the American Physical Society 2024
Quantum many-body scars are special eigenstates that violate the eigenstate thermalization hypothesis while residing at finite energy density along with thermalizing eigenstates. The spin-1 XY model is known to host a family of such exceptional states originating from long-lived quasiparticle excitations that exhibit anomalously low entanglement entropy and long-time periodic revivals, resulting in weak ergodicity breaking. We study the stability of these scarred states against typical U(1) symmetry preserving perturbation in the XY chain. While perturbation theory can describe the deformed scar states at small system sizes, finite-size scaling of the perturbation matrix elements indicate that the scars ultimately thermalize in larger chains. Nonetheless, we demonstrate that the long-range order associated with the scars decays under the perturbation, and we estimate the relaxation timescale of oscillatory dynamics in certain local observables to be of order λ−2λ^{-2}λ−2, where λλλ is the perturbation strength.
We generally expect quantum systems to thermalize and satisfy the eigenstate thermalization hypothesis (ETH), which states that finite-energy-density eigenstates are thermal. However, some systems, such as many-body localized systems and systems with quantum many-body scars, violate ETH and have high-energy athermal eigenstates. In systems with scars, most eigenstates thermalize, but a few atypical scar states do not. Scar states can give rise to a periodic revival when time-evolving particular initial product states, which can be detected experimentally. Recently, a family of spin Hamiltonians was found with magnetically ordered three-colored eigenstates that are quantum many-body scars [Lee et al., Phys. Rev. B 101, 241111(R) (2020)]. These models can be realized in any lattice that can be tiled by triangles, such as the triangular or kagome lattices, and have been shown to have close connections to the physics of quantum spin liquids in the Heisenberg kagome antiferromagnet. In this paper, we introduce a generalized family of nnn-colored Hamiltonians with ``spiral colored'' eigenstates made from nnn-spin motifs such as polygons or polyhedra. We show how these models can be realized in many different lattice geometries and provide numerical evidence that they can exhibit quantum many-body scars with periodic revivals that can be observed by time-evolving simple product states. The simple structure of these Hamiltonians makes them promising candidates for future experimental studies of quantum many-body scars.