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arXiv:2307.06755 · Physical Review Letters

Quantum Many-Body Scars in Dual-Unitary Circuits

Leonard Logarić, Shane Dooley, Silvia Pappalardi, John Goold

Published Jan 3, 2024• 21 citations
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Abstract

Dual-unitary circuits are a class of quantum systems for which exact calculations of various quantities are possible, even for circuits that are nonintegrable. The array of known exact results paints a compelling picture of dual-unitary circuits as rapidly thermalizing systems. However, in this Letter, we present a method to construct dual-unitary circuits for which some simple initial states fail to thermalize, despite the circuits being "maximally chaotic," ergodic, and mixing. This is achieved by embedding quantum many-body scars in a circuit of arbitrary size and local Hilbert space dimension. We support our analytic results with numerical simulations showing the stark contrast in the rate of entanglement growth from an initial scar state compared to nonscar initial states. Our results are well suited to an experimental test, due to the compatibility of the circuit layout with the native structure of current digital quantum simulators.

DOI
10.1103/physrevlett.132.010401

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Dual-unitary circuits are a class of quantum systems for which exact calculations of various quantities are possible, even for circuits that are nonintegrable. The array of known exact results paints a compelling picture of dual-unitary circuits as rapidly thermalizing systems. However, in this Letter, we present a method to construct dual-unitary circuits for which some simple initial states fail to thermalize, despite the circuits being "maximally chaotic," ergodic, and mixing. This is achieved by embedding quantum many-body scars in a circuit of arbitrary size and local Hilbert space dimension. We support our analytic results with numerical simulations showing the stark contrast in the rate of entanglement growth from an initial scar state compared to nonscar initial states. Our results are well suited to an experimental test, due to the compatibility of the circuit layout with the native structure of current digital quantum simulators.
Dual-unitary circuits are a class of locally interacting quantum many-body systems displaying unitary dynamics also when the roles of space and time are exchanged. These systems have recently emerged as a remarkable framework where certain features of many-body quantum chaos can be studied exactly. In particular, they admit a class of ``solvable'' initial states for which, in the thermodynamic limit, one can access the full nonequilibrium dynamics. This reveals a surprising property: when a dual-unitary circuit is prepared in a solvable state the quantum entanglement between two complementary spatial regions grows at the maximal speed allowed by the local structure of the evolution. Here we investigate the fate of this property when the system is prepared in a generic pair-product state. We show that in this case, the entanglement increment during a time step is submaximal for finite times, however, it approaches the maximal value in the infinite-time limit. This statement is proven rigorously for dual-unitary circuits generating high enough entanglement, while it is argued to hold for the entire class.
Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are ``as random as a coin toss.'' Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power ep(U){e}_{p}(U)ep​(U) of the basic two-particle unitary building block, UUU, of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries ui{u}_{i}ui​ and vi{v}_{i}vi​ such that the building block is ${U}^{\ensuremath{'}}=({u}_{1}\ensuremath{\bigotimes}{u}_{2})U({v}_{1}\ensuremath{\bigotimes}{v}_{2})$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power ep(U){e}_{p}(U)ep​(U). Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.
Quantum many-body scars are rare exceptions to thermalization: they sustain non-thermal stationary states without the protection of any local conservation law, and are generally expected to be fragile. Here we construct an analytically tractable random unitary circuit hosting a single scar, and derive from first principles the thermalization mechanism governing perturbations thereof - described by a picture of fluctuating interfaces. Surprisingly, despite being thermodynamically irrelevant for local observables, the scar leaves a sharp fingerprint in the entanglement dynamics, driving a transition as a function of perturbation strength that is not probed by any local measurement.
Dual-unitary quantum circuits can be used to construct 1+1 dimensional lattice models for which dynamical correlations of local observables can be explicitly calculated. We show how to analytically construct classes of dual-unitary circuits with any desired level of (non-)ergodicity for any dimension of the local Hilbert space, and present analytical results for thermalization to an infinite-temperature Gibbs state (ergodic) and a generalized Gibbs ensemble (nonergodic). It is shown how a tunable ergodicity-inducing perturbation can be added to a nonergodic circuit without breaking dual unitarity, leading to the appearance of prethermalization plateaux for local observables.
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
Nonergodic features of chaotic quantum many-body systems are commonly characterized through local observables, fidelity, and entanglement entropy. Here, using global and nonlocal measures of nonstabilizerness (magic), we study quantum many-body scars near infinite temperature in (1+1)(1+1)(1+1)-dimensional Abelian Z2\mathbb{Z}_2Z2​ and U(1)(1)(1) lattice gauge theories, both analytically and numerically. We derive an exact closed-form expression for the nonlocal trace distance magic solely from the Schmidt spectrum, reducing the optimization over local unitaries to a finite maximization over stabilizer-compatible Schmidt ranks. We find that scar eigenstates can exhibit extensive global magic despite their anomalously low entanglement, while retaining anomalously large nonlocal magic compared with ergodic states. We further show that constrained Hilbert spaces can generate irreducible nonlocal magic through stabilizer-incompatible Schmidt ranks, even for a flat entanglement spectrum.