Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain

Domain Wall Dynamics in Classical Spin Chains: Free Propagation, Subdiffusive Spreading, and Soliton Emission

The nonequilibrium dynamics of domain wall initial states in a classical anisotropic Heisenberg chain exhibits a striking coexistence of apparently linear and nonlinear behaviors: the propagation and spreading of the domain wall can be captured quantitatively by linear, i.e., noninteracting, spin wave theory absent its usual justifications; while, simultaneously, for a wide range of easy-plane anisotropies, emission can take the place of stable solitons-a process and objects intrinsically associated with interactions and nonlinearities. The easy-axis domain wall only has transient dynamics, the isotropic one broadens diffusively, while the easy-plane one yields a pair of ballistically counterpropagating domain walls which, unusually, broaden subdiffusively, their width scaling as t^{1/3}.

Classical route to ergodicity and scarring in collective quantum systems

Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems.

Semiclassical Limit of a Measurement-Induced Transition in Many-Body Chaos in Integrable and Nonintegrable Oscillator Chains

We discuss the dynamics of integrable and nonintegrable chains of coupled oscillators under continuous weak position measurements in the semiclassical limit. We show that, in this limit, the dynamics is described by a standard stochastic Langevin equation, and a measurement-induced transition appears as a noise- and dissipation-induced chaotic-to-nonchaotic transition akin to stochastic synchronization. In the nonintegrable chain of anharmonically coupled oscillators, we show that the temporal growth and the ballistic light-cone spread of a classical out-of-time correlator characterized by the Lyapunov exponent and the butterfly velocity are halted above a noise or below an interaction strength. The Lyapunov exponent and the butterfly velocity both act like order parameter, vanishing in the nonchaotic phase. In addition, the butterfly velocity exhibits a critical finite-size scaling. For the integrable model, we consider the classical Toda chain and show that the Lyapunov exponent changes nonmonotonically with the noise strength, vanishing at the zero noise limit and above a critical noise, with a maximum at an intermediate noise strength. The butterfly velocity in the Toda chain shows a singular behavior approaching the integrable limit of zero noise strength.

Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations

The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its "light cone" in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equation exhibits a qualitatively similar OTOC and light cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ, which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity, and dissipation in partial differential equations (deterministic or stochastic) through the lens of OTOC.

Constrained Dynamics and Directed Percolation

In a recent work [A. Deger et al., Phys. Rev. Lett. 129, 160601 (2022).PRLTAO0031-900710.1103/PhysRevLett.129.160601] we have shown that kinetic constraints can completely arrest many-body chaos in the dynamics of a classical, deterministic, translationally invariant spin system with the strength of the constraint driving a dynamical phase transition. Using extensive numerical simulations and scaling analyses we demonstrate here that this constraint-induced phase transition lies in the directed percolation universality class in both one and two spatial dimensions.

Arresting Classical Many-Body Chaos by Kinetic Constraints

We investigate the effect of kinetic constraints on classical many-body chaos in a translationally invariant Heisenberg spin chain using a classical counterpart of the out-of-time-ordered correlator (OTOC). The strength of the constraint drives a "dynamical phase transition" separating a delocalized phase, where the classical OTOC propagates ballistically, from a localized phase, where the OTOC does not propagate at all and the entire system freezes. This is unexpected given that all spin configurations are dynamically connected to each other. We show that localization arises due to the dynamical formation of frozen islands, contiguous segments of spins immobile due to the constraints, dominating over the melting of such islands.

Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators

We present exact results for the classical version of the out-of-time-order commutator (OTOC) for a family of power-law models consisting of N particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form ∝∑_{i,j=1(i≠j)}^{N}|x_{i}-x_{j}|^{-k}∀k>1 where x_{i} is the position of the ith particle. We present numerical results for the OTOC for finite N at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many-body ground state. In the large-N limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone." We find remarkably distinct features in OTOC below and above k=3 in terms of going from non-Airy behavior (1<k<3) to an Airy universality class (k>3). We present certain additional rich features for the case k=2 that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatiotemporal spread of perturbations in long-range interacting systems.

Dynamical regimes of finite-temperature discrete nonlinear Schrödinger chain

We show that the one-dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultralow-, low-, and high-temperature regimes). This has been established via (i) one-point macroscopic thermodynamic observables (temperature T, energy density ε, and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference), and (iii) classical out-of-time-ordered correlators and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures T_{l-ul} (between low- and ultra-low-temperature regimes) and T_{h-l} (between the high- and low-temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward toward the understanding of DNLS which is ubiquitous in many fields and has a nonseparable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many-body systems.

Many-Body Chaos in Thermalized Fluids

Linking thermodynamic variables like temperature T and the measure of chaos, the Lyapunov exponents λ, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we show that in thermalized flows λ∝sqrt[T], in agreement with results from frustrated spin systems. This suggests an underlying universality and provides evidence for recent conjectures on the thermal scaling of λ. We also reconcile seemingly disparate effects-equilibration on one hand and pushing systems out of equilibrium on the other-of many-body chaos by relating λ to T through the dynamical structures of the flow.

Fingerprint of chaos and quantum scars in kicked Dicke model: an out-of-time-order correlator study

We investigate the onset of chaos in a periodically kicked Dicke model (KDM), using the out-of-time-order correlator (OTOC) as a diagnostic tool, in both the oscillator and the spin subspaces. In the large spin limit, the classical Hamiltonian map is constructed, which allows us to investigate the corresponding phase space dynamics and to compute the Lyapunov exponent. We show that the growth rate of the OTOC for the canonically conjugate coordinates of the oscillator is able to capture the Lyapunov exponent in the chaotic regime. The onset of chaos is further investigated using the saturation value of the OTOC, that can serve as an alternate indicator of chaos in a generic interacting quantum system. This is also supported by a system independent effective random matrix model. We further identify the quantum scars in KDM and detect their dynamical signature by using the OTOC dynamics. The relevance of the present study in the context of ongoing cold atom experiments is also discussed.

Quantum chaos and the correspondence principle

The correspondence principle is a cornerstone in the entire construction of quantum mechanics. This principle has been recently challenged by the observation of an early-time exponential increase of the out-of-time-ordered correlator (OTOC) in classically nonchaotic systems [E. B. Rozenbaum et al., Phys. Rev. Lett. 125, 014101 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.014101]. Here, we show that the correspondence principle is restored after a proper treatment of the singular points. Furthermore, our results show that the OTOC maintains its role as a diagnostic of chaotic dynamics.

Absence of Fast Scrambling in Thermodynamically Stable Long-Range Interacting Systems

In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as R^{-α}, where R is the distance. In such systems, the fast scrambling of quantum information or the exponential growth of information propagation can potentially occur according to the decay rate α. In this regard, a crucial open challenge is to identify the optimal condition for α such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with α>D (D: spatial dimension), where the total energy is extensive in terms of system size and the thermodynamic limit is well defined. We rigorously demonstrate that the OTOC shows a polynomial growth over time as long as α>D and the necessary scrambling time over a distance R is larger than t≳R^{[(2α-2D)/(2α-D+1)]}.

Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach

Out-of-time-ordered correlators (OTOCs) have been extensively used as a major tool for exploring quantum chaos, and recently there has been a classical analog. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior: the sustained chaos, transient chaos, and nonchaotic region, as clearly exhibited by different geometrical shapes in the OTOC heat map. To quantify such differences, we look at instantaneous speed (IS), finite-time Lyapunov exponents (FTLEs), and velocity-dependent Lyapunov exponents (VDLEs) extracted from OTOCs. Introduction of these quantities turns out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation, and coupling, FTLEs and IS exhibit transition between sustained chaos and nonchaotic regimes with intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system, and we find that our analytical results can very well describe the nonchaotic regime as well as the late-time behavior in the transient regime of the Duffing chain. We believe that this analysis is an important step forward towards understanding nonlinear dynamics, chaos, and spatiotemporal spread of perturbations in many-particle open systems.

Transport, correlations, and chaos in a classical disordered anharmonic chain

We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators, and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size N, disorder strength Δ, and temperature T. The conductivity κ_{N}, obtained for finite length (N), saturates to a value κ_{∞}>0 in the large N limit, for all values of disorder strength Δ and temperature T>0. We show evidence that for any Δ>0 the conductivity goes to zero faster than any power of T in the (T/Δ)→0 limit, and find that the form κ_{∞}∼e^{-B|ln(CΔ/T)|^{3}} fits our data. This form has earlier been suggested by a theory based on the dynamics of multioscillator chaotic islands. The finite-size effect can be κ_{N}<κ_{∞} due to boundary resistance when the bulk conductivity is high (the weak disorder case), or κ_{N}>κ_{∞} due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the out-of-time-ordered commutator.

Universality Classes of Spin Transport in One-Dimensional Isotropic Magnets: The Onset of Logarithmic Anomalies

We report a systematic study of finite-temperature spin transport in quantum and classical one-dimensional magnets with isotropic spin interactions, including both integrable and nonintegrable models. Employing a phenomenological framework based on a generalized Burgers' equation in a time-dependent stochastic environment, we identify four different universality classes of spin fluctuations. These comprise, aside from normal spin diffusion, three types of superdiffusive transport: the Kardar-Parisi-Zhang universality class and two distinct types of anomalous diffusion with multiplicative logarithmic corrections. Our predictions are supported by extensive numerical simulations on various examples of quantum and classical chains. Contrary to common belief, we demonstrate that even nonintegrable spin chains can display a diverging spin diffusion constant at finite temperatures.

Quantum diffusion in spin chains with phase space methods

Connecting short-time microscopic dynamics with long-time hydrodynamics in strongly correlated quantum systems is one of the outstanding questions. In particular, it is hard to determine various hydrodynamic coefficients such as the diffusion constant or viscosity starting from a microscopic model: exact quantum simulations are limited to either small system sizes or to short times, which are insufficient to reach asymptotic behavior and so various approximations must be applied. We show that these difficulties, at least for particular models, can be circumvented by using the cluster truncated Wigner approximation (CTWA), which maps quantum Hamiltonian dynamics into classical Hamiltonian dynamics in auxiliary high-dimensional phase space. We apply CTWA to XXZ next-nearest-neighbor spin-1/2 chains and XY spin ladders, and find behavior consisting of short-time spin relaxation which gradually crosses over to emergent diffusive behavior at long times. For a random initial state, we show that CTWA correctly reproduces the whole spin spectral function. Necessary in this construction is sampling from properly fluctuating initial conditions: the Dirac mean-field (variational) ansatz, which neglects such fluctuations, leads to incorrect predictions.

Temperature Dependence of the Butterfly Effect in a Classical Many-Body System

We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, μ, and the butterfly speed, v_{b}, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, D, and spin-autocorrelation time, τ. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form D∼v_{b}^{2}/μ and τ^{-1}∼T. The vanishing of μ∼T^{0.48} is parametrically slower than that of the corresponding quantum bound, μ∼T, raising interesting questions regarding the semiclassical limit of such spin systems.