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arXiv:2011.04700 · Physical review. B./Physical review. B

Classical many-body chaos with and without quasiparticles

Thomas Bilitewski, Subhro Bhattacharjee, Roderich Moessner

Published May 7, 2021• 43 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.174302

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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=0T=0T=0, there is no dynamics at all. In the limit of low temperature, T=0+T={0}^{+}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 TTT. 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 TTT, 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.
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 find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.
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.
Truncating weak interactions is a natural simplification of a many-body Hamiltonian, but the validity of this approximation need not be uniform in distance and time. In a chaotic system, a weak long-range coupling can seed a small disturbance far from its source, after which the local dynamics can amplify that disturbance exponentially. We study this mechanism in an infinite-temperature classical Heisenberg spin chain with a dominant nearest-neighbor exchange and a weak interaction tail. Using a classical decorrelator, we compare exponentially and power-law decaying tails over several decades in interaction strength. For exponentially decaying interactions, the long-distance front remains ballistic when the tail is sufficiently shallow, with a velocity controlled by the competition between the spatial decay of the seed and the local chaotic growth. For sufficiently rapid exponential decay, the nearest-neighbor front, which is defined by an intrinsic butterfly velocity vbv_bvb​, is recovered within the accessible space-time window. For power-law interactions, the arrival time is logarithmic in distance for all exponents studied, corresponding to an exponentially expanding spatial front; the power-law exponent controls the coefficient of this logarithm. We interpret both behaviors using a seeding-and-amplification picture in which long-range couplings generate weak disturbances throughout the chain whose growth is governed primarily by the local nearest-neighbor dynamics. In this picture, the overall strength of the tail primarily controls when the long-range front becomes visible, while its spatial decay controls the propagation law. Thus a weak interaction tail can leave conventional spin-dynamical observables essentially unchanged while qualitatively altering the long-distance propagation of decorrelation.
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, $\ensuremath{\mu}$, and the butterfly speed, vb{v}_{b}vb​, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, DDD, and spin-autocorrelation time, $\ensuremath{\tau}$. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form $D\ensuremath{\sim}{v}_{b}^{2}/\ensuremath{\mu}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\sim}T$. The vanishing of $\ensuremath{\mu}\ensuremath{\sim}{T}^{0.48}$ is parametrically slower than that of the corresponding quantum bound, $\ensuremath{\mu}\ensuremath{\sim}T$, raising interesting questions regarding the semiclassical limit of such spin systems.
The maximum Lyapunov exponent (MLE) is evaluated as a function of temperature in the isotropic Heisenberg chain. At low temperatures the MLE varies almost quadratically with temperature; it corresponds closely with ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$, the rate at which the local self-correlation decays to half its initial value. At higher temperatures, the MLE saturates at a value close to 1/2 in the limit of large chain lengths. The strong stochasticity threshold (defined by the change of slope of the MLE) parallels closely the transition from predominantly ballistic to predominantly diffusive behavior of the self-correlation and the concomitant steep increase in ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$. The complete Lyapunov spectrum has been derived for a chain of 18 spins; deviations from linearity occurring at infinite temperature suggest that the chaoticity of the system is incomplete. Finally, it is suggested that a systematic study of finite-size effects might be useful in deciding the issue of anomalous versus conventional spin diffusion.