Chaos makes isolated systems of many interacting particles quickly thermalize and forget about their past. Here, we show that quantum mechanics hinders chaos in many-body systems: although the quantum eigenstates are thermal and strongly entangled, exponentially many of them are scarred, that is, have an enlarged weight along underlying classical unstable periodic orbits. Scarring makes the system more likely to be found on an orbit it was initialized on, retaining a memory of its past and thus weakly breaking ergodicity, even at long times and despite the system being fully thermal and the eigenstate thermalization hypothesis fulfilled. We demonstrate the ubiquity of quantum scarring in many-body systems by considering a large family of spin models, including some of the most popular ones from condensed matter physics. Our findings, at hand for modern quantum simulators, prove structure in spite of chaos in many-body quantum systems.
Chaos makes isolated systems of many interacting particles quickly thermalize and forget about their past. Here, we show that quantum mechanics hinders chaos in many-body systems: although the quantum eigenstates are thermal and strongly entangled, exponentially many of them are scarred, that is, have an enlarged weight along underlying classical unstable periodic orbits. Scarring makes the system more likely to be found on an orbit it was initialized on, retaining a memory of its past and thus weakly breaking ergodicity, even at long times and despite the system being fully thermal and the eigenstate thermalization hypothesis fulfilled. We demonstrate the ubiquity of quantum scarring in many-body systems by considering a large family of spin models, including some of the most popular ones from condensed matter physics. Our findings, at hand for modern quantum simulators, prove structure in spite of chaos in many-body quantum systems.
Quantum many-body scars consist of a few low-entropy eigenstates in an otherwise chaotic many-body spectrum, and can weakly break ergodicity resulting in robust oscillatory dynamics. The notion of quantum many-body scars follows the original single-particle scars introduced within the context of quantum billiards, where scarring manifests in the form of a quantum eigenstate concentrating around an underlying classical unstable periodic orbit. A direct connection between these notions remains an outstanding problem. Here, we study a many-body spinor condensate that, owing to its collective interactions, is amenable to the diagnostics of scars. We characterize the system's rich dynamics, spectrum, and phase space, consisting of both regular and chaotic states. The former are low in entropy, violate the eigenstate thermalization hypothesis, and can be traced back to integrable effective Hamiltonians, whereas most of the latter are scarred by the underlying semiclassical unstable periodic orbits, while satisfying the eigenstate thermalization hypothesis. We outline an experimental proposal to probe our theory in trapped spin-1 Bose-Einstein condensates.
Quantum many-body scars (QMBSs) represent a weak ergodicity-breaking phenomenon that defies the common scenario of thermalization in closed quantum systems. They are often regarded as a many-body analog of quantum scars (QSs)—a single-particle phenomenon in quantum chaos—due to their superficial similarities. However, unlike QSs, a clear connection between QMBSs and classical chaos has remained elusive. It has nevertheless been speculated that in an appropriate semiclassical limit, QMBSs should have a correspondence to weakly unstable periodic orbits. In this paper, I present a counterexample to this conjecture by studying a bosonic model with a large number of flavors. The dynamics of out-of-time-ordered correlators (OTOCs) suggest that QMBS do not display chaotic behavior in the semiclassical limit. In contrast, chaotic dynamics are expected for initial states not associated with QMBS. Interestingly, the anomalous OTOC dynamics persist even under weak perturbations that eliminate the scarred eigenstates, suggesting a certain robustness in the phenomenon.
Unstable periodic orbits (UPOs) play a key role in the theory of chaos, constituting the ``skeleton'' of classical chaotic systems and ``scarring'' the eigenstates of the corresponding quantum system. Recently, nonthermal many-body eigenstates embedded in an otherwise thermal spectrum have been identified as a many-body generalization of quantum scars. The latter, however, are not clearly associated to a chaotic phase space, and the connection between the single- and many-body notions of quantum scars remains therefore incomplete. Here, we find the first quantum many-body scars originating from UPOs of a chaotic phase space. Remarkably, these states verify the eigenstate thermalization hypothesis, and we thus refer to them as thermal quantum many-body scars. While they do not preclude thermalization, their spectral structure featuring approximately equispaced towers of states yields an anomalous oscillatory dynamics preceding thermalization for wavepackets initialized on an UPO. Remarkably, our model hosts both types of scars, thermal and nonthermal, and allows us to study the crossover between the two. Our work illustrates the fundamental principle of classical-quantum correspondence in a many-body system and its limitations.
Unstable periodic orbits act as organizing structures for classical chaotic systems and underpin quantum scarring. Long known in single-particle systems, genuine quantum scars based on unstable periodic orbits have been recently extended to isolated many-body systems for time-independent Hamiltonians. Their fate under periodic driving, however, remains largely uncharted, challenged by the expectation that these systems should in general heat to infinite temperature. Here, we investigate how genuine scarring competes with the drive in a Floquet many-body system. Using chaotic spin chains, we demonstrate that Floquet states remain scarred in the high-frequency limit. Beyond this static correspondence, we uncover additional, driving-induced Floquet scars with no static analog. We construct a rich dynamical stability diagram with intermediate-frequency regimes of enhanced and quenched scarring, which we understand with a classical analysis of the Lyapunov exponent. Our results position Floquet systems as a natural platform for tuning the scarring behavior of quantum many-body systems.
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.
We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the ‘quantum leakage’ of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of ‘out-of-time-ordered correlators’.