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arXiv:2204.08047 · Physical review. B./Physical review. B

Weak ergodicity breaking in Josephson-junction arrays

Angelo Russomanno, Michele Fava, Rosario Fazio

Published Jul 15, 2022• 10 citations
arXivHTMLPDF

Abstract

We study the quantum dynamics of Josephson-junction arrays. We find isolated groups of low-entanglement eigenstates that persist even when the Josephson interaction is strong enough to destroy the overall organization of the spectrum in multiplets, and a perturbative description is no longer possible. These eigenstates lie in the inner part of the spectrum, far from the spectral edge, and provide a weak ergodicity breaking, reminiscent of the quantum scars. Due to the presence of these eigenstates, initializing with a charge-density-wave state, the system does not thermalize and the charge-density-wave order persists for long times. Considering global ergodicity probes, we find that the system tends toward more ergodicity for increasing system size: The parameter range where the bulk of the eigenstates look nonergodic shrinks for increasing system size. We study two geometries, a one-dimensional chain and a two-leg ladder. In the latter case, adding a magnetic flux makes the system more ergodic.

DOI
10.1103/physrevb.106.035123

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We study the quantum dynamics of Josephson-junction arrays. We find isolated groups of low-entanglement eigenstates that persist even when the Josephson interaction is strong enough to destroy the overall organization of the spectrum in multiplets, and a perturbative description is no longer possible. These eigenstates lie in the inner part of the spectrum, far from the spectral edge, and provide a weak ergodicity breaking, reminiscent of the quantum scars. Due to the presence of these eigenstates, initializing with a charge-density-wave state, the system does not thermalize and the charge-density-wave order persists for long times. Considering global ergodicity probes, we find that the system tends toward more ergodicity for increasing system size: The parameter range where the bulk of the eigenstates look nonergodic shrinks for increasing system size. We study two geometries, a one-dimensional chain and a two-leg ladder. In the latter case, adding a magnetic flux makes the system more ergodic.
We study the spin-1 XY model on a hypercubic lattice in d dimensions and show that this well-known nonintegrable model hosts an extensive set of anomalous finite-energy-density eigenstates with remarkable properties. Namely, they exhibit subextensive entanglement entropy and spatiotemporal long-range order, both believed to be impossible in typical highly excited eigenstates of nonintegrable quantum many-body systems. While generic initial states are expected to thermalize, we show analytically that the eigenstates we construct lead to weak ergodicity breaking in the form of persistent oscillations of local observables following certain quantum quenches-in other words, these eigenstates provide an archetypal example of so-called quantum many-body scars. This Letter opens the door to the analytical study of the microscopic origin, dynamical signatures, and stability of such phenomena.
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.
We study ergodicity breaking in the clean Bose-Hubbard chain for small hopping strength. We see the existence of a nonergodic regime by means of indicators as the half-chain entanglement entropy of the eigenstates, the average level spacing ratio, the properties of the eigenstate-expectation distribution of the correlation and the scaling of the inverse participation ratio averages. We find that this ergodicity breaking is different from many-body localization because the average half-chain entanglement entropy of the eigenstates obeys volume law. This ergodicity breaking appears unrelated to the spectrum being organized in quasidegenerate multiplets at small hopping and finite system sizes, so in principle, it can survive also for larger system sizes. We find that some imbalance oscillations in time which could mark the existence of glassy behavior in space are well described by the dynamics of a single symmetry-breaking doublet and quantitatively captured by a perturbative effective XXZ model. We show that the amplitude of these oscillations vanishes in the large-size limit. Our findings are numerically obtained for systems for L<12L<12L<12. Extrapolations of our scalings to larger system sizes should be taken with care, as discussed in the paper.
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.
The success of statistical mechanics in describing complex quantum systems rests upon typicality properties such as ergodicity. Both integrable systems and the recently discovered many-body localisation show that these assumptions can be strongly violated in either finely tuned cases, or in the presence of quenched disorder. In this thesis, we uncover a qualitatively different form of ergodicity breaking, wherein a small number of atypical eigenstates are embedded throughout an otherwise thermalising spectrum. We call this a many-body quantum scar, in analogy to quantum scars in single-particle quantum chaos, where quantum scarred eigenfunctions concentrate around associated periodic classical trajectories. \nWe demonstrate that many-body quantum scars can be found in an unusual model recently realised in a 51 Rydberg atom quantum simulator. The observed coherent oscillations following in a certain quench experiment are a consequence of the quantum scar. At the same time, the level statistics rules out conventional explanations such as integrability and many-body localisation. We develop an approximate method to construct scarred eigenstates, in order to describe their structure and physical properties. Additionally, we find a local perturbation which makes these non-equilibrium properties much more pronounced, with near perfect quantum revivals. At the same time the other eigenstates remain thermal. Our results suggest that many-body quantum scars forms a new class of quantum dynamics with unusual properties, which are realisable in current experiments.
Thermalization and its breakdown in interacting quantum many-body systems are governed by mid-spectrum eigenstates, which are typically accessible only in small system sizes amenable to exact diagonalization. Here we demonstrate that the density-matrix renormalization group (DMRG) effective Hamiltonian, an object routinely used to variationally approximate ground states, encodes detailed information about the dynamics far from equilibrium. In the random-field XXZ spin chain, the spectrum of the effective Hamiltonian is shown to capture the transition from thermal to many-body localized regimes, including spatially resolved probes of ergodic bubbles. Furthermore, the same approach also captures weak ergodicity breaking associated with quantum many-body scars. Our results establish the DMRG effective Hamiltonian as a versatile spectral probe of quantum thermalization and its breakdown in large systems beyond exact diagonalization.