Many-body chaos is a default property of many-body systems; at the same time, near-integrable behavior due to weakly interacting quasiparticles is ubiquitous throughout condensed matter at low temperature. There must therefore be a, possibly generic, crossover between these very different regimes. Here, we develop a theory encapsulating the notion of a cascade of lightcones seeded by sequences of scattering of weakly interacting harmonic modes as witnessed by a suitably defined chaos diagnostic (classical decorrelator) that measures the spatiotemporal profile of many-body chaos. Our numerics deals with the concrete prototypical case of a classical Heisenberg chain, for either sign of the interaction, at low temperatures where the short-time dynamics are well captured in terms of noninteracting spin waves—characteristic to continuous symmetry breaking. To model low-temperature dynamics, we use ensembles of initial states with randomly embedded point defects in an ordered background, which provides a controlled setting for studying the scattering events. The decorrelator exhibits a short-time integrable regime followed by an intermediate “scarred” regime of the cascade of lightcones in progress; these then overlap, leading to an avalanche of scattering events which finally yields the standard long-time signature of many-body chaos.
Many-body chaos is a default property of many-body systems; at the same time, near-integrable behavior due to weakly interacting quasiparticles is ubiquitous throughout condensed matter at low temperature. There must therefore be a, possibly generic, crossover between these very different regimes. Here, we develop a theory encapsulating the notion of a cascade of lightcones seeded by sequences of scattering of weakly interacting harmonic modes as witnessed by a suitably defined chaos diagnostic (classical decorrelator) that measures the spatiotemporal profile of many-body chaos. Our numerics deals with the concrete prototypical case of a classical Heisenberg chain, for either sign of the interaction, at low temperatures where the short-time dynamics are well captured in terms of noninteracting spin waves—characteristic to continuous symmetry breaking. To model low-temperature dynamics, we use ensembles of initial states with randomly embedded point defects in an ordered background, which provides a controlled setting for studying the scattering events. The decorrelator exhibits a short-time integrable regime followed by an intermediate “scarred” regime of the cascade of lightcones in progress; these then overlap, leading to an avalanche of scattering events which finally yields the standard long-time signature of many-body chaos.
We study correlations, transport, and chaos in a Heisenberg magnet as a classical model many-body system. By varying temperature and dimensionality, we can tune between settings with and without symmetry breaking and accompanying collective modes or quasiparticles (spin waves) which in the limit of low temperatures become increasingly long-lived. Changing the sign of the exchange interaction from a ferro- to an antiferromagnetic one varies the spin-wave spectrum, and hence the low-energy spectral density. We analyze both conventional and out-of-time-ordered spin correlators (decorrelators) to track the spreading of a spatiotemporally localized perturbation---the wing beat of the butterfly---as well as transport coefficients and Lyapunov exponents. We identify a number of qualitatively different regimes. Trivially, at T=0, there is no dynamics at all. In the limit of low temperature, T=0+, integrability emerges, with infinitely long-lived magnons; here the wave packet created by the perturbation propagates ballistically, yielding a light cone at the spin-wave velocity which thus subsumes the butterfly velocity; inside the light cone, a pattern characteristic of the free spin-wave spectrum is visible at short times. On top of this, residual interactions (nonlinearities in the equations of motion) lead to spin-wave lifetimes which, while divergent in this limit, remain finite at any nonzero T. At the longest times, this leads to a standard chaotic regime; for this regime, we show that the Lyapunov exponent is simply proportional to the (inverse) spin-wave lifetime. Visibly strikingly, between this and the short-time integrable regimes, a scarred regime emerges: Here, the decorrelator is spatiotemporally highly nonuniform, being dominated by rare and random scattering events seeding secondary light cones. As the spin correlation length decreases with increasing T, the distinction between these regimes disappears and at high temperature the previously studied chaotic paramagnetic regime emerges. For this, we elucidate how, somewhat counterintuitively, the ballistic butterfly velocity arises from a diffusive spin dynamics.
Quantum chaos and the eigenstate thermalization hypothesis are based on the assumption of the validity of random matrix theory description on the spectrum and eigenstates. They provide the foundation and descriptions for the typical dynamics and thermalization in generic closed quantum systems. In this thesis, we investigate situations where the systems show atypical dynamics or anomalous thermalization, conflicting with the usual expectations from quantum chaos and eigenstate thermalization hypothesis. We first examine weak thermalization in a nonintegrable spin chain. The system shows long-lived strong oscillations and relaxes to the thermal equilibrium weakly. We identify the dynamics describable by quasiparticles and recognize the oscillation frequency to be the quasiparticle mass gap. We also estimate the damping time for the oscillations. Next, we study prethermalization, a phenomenon where a system relaxes to an intermediate almost-equilibrium stage before reaching the true thermal equilibrium. We study a nonintegrable spin chain in the strong coupling limit, where an almost-conserved quantity emerges and gives rise to the prethermalization. We also study a newly proposed diagnostic for quantum chaos: out-of-time-ordered correlators. Contrasting to the chaotic systems, we inspect their behaviors in various noninteracting integrable models. Finally, we dig into the quantum many-body scar states in the PXP model which describes a Rydberg atom chain. These special states do not satisfy the random matrix theory description nor the eigenstate thermalization hypothesis, therefore defying quantum chaos.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the Fock states. The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant Fock-space hopping, delocalization can occur either through a crossover or via a transition that becomes sharp in the appropriate large-system dynamic limit. In either case, the perturbation strength at the onset of chaos scales to zero in the usual thermodynamic limit, with a size dependence that we estimate analytically and compute numerically for a few specific models. We also identify two intermediate finite-size-dependent regimes: There is generally an intermediate nonchaotic regime in which integrability is broken strongly enough to produce some systemwide many-body resonances but not enough to thermalize the system. In spatially extended systems (but not in quantum dots) there is also a crossover or transition between chaotic regimes where the ratio of the system size to the mean free path of the quasiparticles of the integrable system is small versus large compared to unity.
To probe the limits of dynamic thermalization, we numerically investigate the stability of exceptional periodic classical trajectories in chaotic many-spin systems and explore a possible connection between these trajectories and exceptional nonthermal quantum eigenstates known as "quantum many-body scars." The systems considered are chaotic spin chains with short-range interactions, both classical and quantum. On the classical side, the chosen periodic trajectories are such that all spins instantaneously point in the same direction, which evolves as a function of time. We find that the largest Lyapunov exponents characterizing the stability of these trajectories have surprisingly strong and nontrivial dependencies on the interaction constants and chain lengths. In particular, we identify rather long spin chains, where the above periodic trajectories are Lyapunov stable on many-body energy shells overwhelmingly dominated by chaotic motion. We show that the above phenomenology can be quantitatively described by connecting Lyapunov instabilities of translationally invariant periodic trajectories to irreducible representations of the translational symmetry group with well-defined wave vectors. We also find that instabilities around periodic trajectories in modestly large spin chains develop into a transient nearly quasiperiodic nonergodic regime. In some cases, the lifetime of this regime is extremely long, which we interpret as a manifestation of Arnold diffusion in the vicinity of integrable dynamics. On the quantum side, we numerically investigate the dynamics of quantum states starting with all spins initially pointing in the same direction: These are the quantum counterparts of the initial conditions for the above periodic classical trajectories. Our investigation reveals the existence of quantum many-body scars for numerically accessible finite chains of spins-3/2 and higher. No evidence of quantum scars was observed for spin-1/2 chains, while spin-1 chains were found to be transitional in this respect. The dynamic thermalization process dominated by quantum scars is shown to exhibit a slowdown in comparison with generic thermalization at the same energy. Finally, we identify quantum signatures of the proximity to a classical separatrix of the periodic motion.