Lieb-Robinson-type bounds are reported for a large class of classical Hamiltonian lattice models. By a suitable rescaling of energy or time, such bounds can be constructed for interactions of arbitrarily long range. The bound quantifies the dependence of the system's dynamics on a perturbation of the initial state. The effect of the perturbation is found to be effectively restricted to the interior of a causal region of logarithmic shape, with only small, algebraically decaying effects in the exterior. A refined bound, sharper than conventional Lieb-Robinson bounds, is required to correctly capture the shape of the causal region, as confirmed by numerical results for classical long-range XY chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.In many nonrelativistic lattice systems, and despite the absence of Lorentz covariance, physical effects are mostly restricted to a causal region, often in the shape of an effective "light cone," with only tiny effects leaking out to the exterior. The technical tool, known as the LiebRobinson bound [1,2], to quantify this statement in a quantum mechanical context is an upper bound on the norm of the commutator ½O A ðtÞ; O B ð0Þ, where O A ð0Þ and O B ð0Þ are operators supported on the subspaces of the Hilbert space corresponding to nonoverlapping regions A and B of the lattice. The importance of such a bound lies in the fact that a multitude of physically relevant results can be derived from it. Examples are bounds on the creation of equal-time correlations [3], on the transmission of information [4], and on the growth of entanglement [5], the exponential spatial decay of correlations in the ground state of a gapped system [6], or a Lieb-Schultz-Mattis theorem in higher dimensions [7]. Experimental observations related to Lieb-Robinson bounds have also been reported [8].The original proof by Lieb and Robinson [1] requires interactions of finite range. An extension to power-lawdecaying long-range interactions has been reported in Refs. [3,6]. In this case the effective causal region is no longer cone shaped, and the spatial propagation of physical effects is not limited by a finite group velocity [9]. For "strong long-range interactions," i.e., when the interaction potential decays proportionally to 1=r α with an exponent α smaller than the lattice dimension d, the theorems in [3,6] do not apply and no Lieb-Robinson-type results are known. This fact nicely fits into the larger picture that, for α ≤ d, the behavior of long-range interacting systems often differs substantially from that of short-range interacting systems. Examples of such differences include nonequivalent equilibrium statistical ensembles and negative response functions [10], or the occurrence of quasistationary states whose lifetimes diverge with the system size [11,12]. The latter is a dynamical phenomenon, and it has been conjectured in [13] that some of its properties are universal and in some way connected to Lieb-Robinson bounds.In mos...

We prove that any non zero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous, or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of J.D. Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.Understanding synchronization in large populations of coupled oscillators is a question which arises in many different fields, from physics to neuroscience, chemistry and biology [1]. Describing the oscillators with their phases only, Winfree [2], and Kuramoto [3] have introduced simple models for this phenomenon. The latter model, which features a sinusoidal coupling, and an all-to-all interaction between oscillators, has become a paradigmatic model for synchronization, and its very rich behavior prompted an enormous number of studies. Kuramoto model displays a transition between an incoherent state, where each oscillator rotates at its own intrinsic frequency, and a state where at least some oscillators are phase-locked. The degree of coherence is measured by an order parameter r, which bifurcates -continuously for symmetric unimodal frequency distributions-from 0 when the coupling is increased, or the dispersion in intrinsic frequencies decreases. In order to better fit modeling needs, it has been necessary to consider refined models, including for instance, citing just a few contributions: more general coupling [4], noise [5], phase shifts bringing frustration [6], delays [7,8], or a more realistic interaction topology [9,10]. In particular, inertia has been introduced to describe the synchronization of a certain firefly [11], and proved later useful to model coupled Josephson junctions [12,13] and power grids [14,15]; recently, an inertial model on a complex network was shown to display a new type of "explosive synchronization" [16]. It has been quickly recognized [17,18] that inertia could turn the continuous Kuramoto transition into a discontinuous one with hysteresis. At first sight, a natural adaptation of the original clever self-consistent mean-field approach by Kuramoto [3] seems to explain satisfactorily this observation [17,19]: a sufficiently large inertia induces a bistable dynamical behavior of some oscillators, that translates into a hysteretic dynamics at the collective level. However, Fig. 1 makes clear that even a small inertia is enough to trigger a discontinuous transition: this cannot be accounted for by the bistability picture.In this letter, we explain why any non zero inertia, however small, can have a dramatic effect on the transition: it can turn discontinuous an otherwise continuous transition, and the other way round. These results are obtained through a careful unstable manifold expansion in the spirit of [20][21][22] (see also [23] for a very readable ...