The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.
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.
We study the disordered Heisenberg spin chain, which exhibits many-body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}}$, whereas the conductivity exhibits a low-frequency power law $\ensuremath{\sigma}\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\alpha}}$. The exponents depict subdiffusive behavior $\ensuremath{\beta}<1/2,\phantom{\rule{0.28em}{0ex}}\ensuremath{\alpha}>0$ at all finite disorders and convergence to the scaling result $\ensuremath{\alpha}+2\ensuremath{\beta}=1$ at large disorders.
The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analog of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.
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.
The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasiparticles. We here consider as an archetypical example for such a system the spin-
21 XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization
m=0. At a finite magnetization density
m, the spin-autocorrelation function $\mathrm{\ensuremath{\Pi}}(x,t)$ (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasiparticle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a subdiffusive
t1/3 scaling at large magnetization densities $m\ensuremath{\rightarrow}\frac{1}{2}$, familiar from noninteracting models; it crosses over into a diffusive scaling
t1/2 upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient superdiffusion to ballistic relaxation at long times. In the limit $m\ensuremath{\rightarrow}0$, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent
z close to the value
23 familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator $\mathrm{\ensuremath{\Pi}}$ is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.