We study a nonreciprocal generalization [EPL 60, 418 (2002)] of the classical Heisenberg spin chain, in which the exchange coupling is nonsymmetric, and show that it displays a ballistic spreading of chaos as measured by the decorrelator. We show that the interactions are reciprocal in terms of transformed variables, with conserved quantities that can be identified as magnetization and energy, with a Poisson-bracket algebra and Hamiltonian dynamics. For strictly antisymmetric couplings in the original model the conserved quantities diffuse, the decorrelator spreads symmetrically, and a simple hydrodynamic theory emerges. The general case in which the interaction has symmetric and antisymmetric parts presents complexities in the limit of large scales. Ballistic propagation of chaos survives the inclusion of interactions beyond nearest neighbours, but the conservation laws in general do not.
We study a nonreciprocal generalization [EPL 60, 418 (2002)] of the classical Heisenberg spin chain, in which the exchange coupling is nonsymmetric, and show that it displays a ballistic spreading of chaos as measured by the decorrelator. We show that the interactions are reciprocal in terms of transformed variables, with conserved quantities that can be identified as magnetization and energy, with a Poisson-bracket algebra and Hamiltonian dynamics. For strictly antisymmetric couplings in the original model the conserved quantities diffuse, the decorrelator spreads symmetrically, and a simple hydrodynamic theory emerges. The general case in which the interaction has symmetric and antisymmetric parts presents complexities in the limit of large scales. Ballistic propagation of chaos survives the inclusion of interactions beyond nearest neighbours, but the conservation laws in general do not.
The Hamiltonian nature of the precessional dynamics of the classical Heisenberg model leads to reciprocal interactions amongst the spins. Heisenberg spins are reciprocal in nature. In this work, we study the dynamics of a nonequilibrium classical spin chain in which the neighbours interact through a purely non-reciprocal exchange coupling [EPL 60, 418 (2002)] which preserves rotational symmetry. The resultant dynamics conserves neither magnetization nor energy. We uncover other local conservation laws in their place in the extreme case of a strictly antisymmetric coupling. We show numerically that the model undergoes an analogue of thermalization. We present results on the presence of conserved quantities, their diffusive spreading and a hydrodynamic picture, and the nature of the decorrelation front upon adding an initial perturbation to the system.
Quantum chaos is a major subject of interest in condensed matter theory, and has recently motivated new questions in the study of classical chaos. In particular, recent studies have uncovered interesting physics in the relationship between chaos and conserved quantities in models of quantum chaos. In this paper, we investigate this relationship in two simple models of classical chaos: the infinite-temperature Heisenberg spin chain, and the directed polymer in a random medium. We relate these models by drawing analogies between the energy landscape over which the directed polymer moves and the magnetization of the spin chain. We find that the coupling of the chaos to these conserved quantities results in, among other things, a marked transition from the rough perturbation profiles predicted by analogy to the KPZ equation to smooth, triangular profiles with reduced wandering exponents. These results suggest that diffusive conserved quantities can, in some cases, be the dominant forces shaping the development of chaos in classical systems.
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.
Abstract The emergence of diffusion is one of the deepest physical phenomena observed in many-body interacting, chaotic systems. But establishing rigorously that correlation functions, say of the spin, expand diffusively, remains one of the most important problems of mathematical physics. We establish for the first time, with Lindbladian evolution, a lower bound on spin diffusion in chaotic, translation-invariant, nearest-neighbor open quantum spin-1/2 chain satisfying a local unitality condition and strong conservation of magnetization. The bound is strictly positive if and only if the local quantum jumps transport spin. Physically, the bound comes from the spreading effects of initial-state macroscopic fluctuations, a mechanism which occurs whenever spin is an interacting ballistic mode. Chaoticity means that the Hilbert space of extensive charges is spanned by magnetization; we expect this to be generic. Our main tool is the Green-Kubo formula, the mathematical technique of projection over quadratically extensive charges, and appropriate correlation decay bounds recently established. Because Lindbladian dynamics is not reversible, the Green-Kubo spin diffusion strength includes a contribution due to irreversibility, which we interpret as encoding the hydrodynamic entropy production that may occur in the forgotten environment. This, we show, vanishes for certain choices of interaction parameters, for which the Lindbladian dynamics becomes reversible. Our methods can be extended to finite or short ranges, higher spins, and other non-Hamiltonian systems such as quantum circuits. As we argue, according to the theory of nonlinear fluctuating hydrodynamics, we further expect these systems to display superdiffusion, and thus have infinite diffusivity; however this appears to still be beyond the reach of mathematical rigor.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
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 vb, 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.