In the last decades, simultaneous progress realized in the field of ultracold atoms permitted to simulate many particle and spin Hamiltonians of condensed-matter physics (Ising, XY, Fermi- and Bose-Hubbard models, ...) with unprecedented control possibilities of the parameters in time allowing to investigate their out-of-equilibrium dynamics. One fundamental issue of the out-of-equilibrium dynamics concerns the spreading of information. It is at the center of many fundamental phenomena including the propagation of correlations and entanglement, relaxation, (pre-)thermalization and the boundary law for the entanglement entropy.
Here, we present the spreading of correlations and entanglement in the paradigmatic, one-dimensional, transverse Ising model with long-range spin exchange of the form 1/Rα . Using a numerically-exact tensor-network approach, we investigate a variety of quenches and observables, and determine the corresponding dynamical scaling laws of their causality cone.
Most importantly, we find that the quasi-local regime (intermediate α) is characterized by fundamentally non-universal behaviours : the dynamical exponent of the entanglement causality edge strongly depends on the entanglement measure and the spin correlation functions are characterized by observable-dependent modulations, whose scaling laws differ from those of the entanglement edge. Nevertheless, some general dynamical behaviours emerge : (i) In the quasi-local regime, the scaling laws for the correlation and entanglement spreading are all found to be algebraic, of the form t ∝ Rβ . The latter contrasts with the linear scaling laws found for the local regime (high α) or Hamiltonians containing short- range couplings [2, 3, 4]. (ii) The causality edge is always sub-ballistic, βCE ≥ 1, and can be bounded by the meanfield scaling law β mf = 3 − α . (iii) Finally, the dynamics of CE the local maxima obey (non-universal) scaling laws that are all ballistic or super-ballistic, βm ≤ 1. Our numerical results are supported by the theoretical predictions of a quasiparticle approach that we discuss.
These results shed new light on the nature of correlation and information spreading in out- of-equilibrium long-range systems and pave the way to future experimental investigations using for instance ultracold trapped ions.