We investigate the ground-state properties of the spin-1/2 XXZ model with power-law-decaying (1/r α ) interactions, describing spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hardcore bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent α and number of dimensions d: 1) a short-range regime for α > d + σc (where σc = 1 in the gapped Néel antiferromagnetic phase exhibited by the XXZ model, and σc = 2 in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions (α = ∞); 2) a long-range regime α < d, sharing the same properties as those of the infinite-range interactions (α = 0) in the thermodynamic limit; and 3) a most intriguing medium-range regime for d < α < d + σc, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation ω ≈ ∆g + ck z in the gapped phase, and ω ∼ k z in the gapless phase, exhibiting a continuously varying dynamical exponent z = (α − d)/σc. In the gapless phase of the model the z exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal sub-dominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.

The relaxation of uniform quantum systems with finite-range interactions after a quench is generically driven by the ballistic propagation of long-lived quasi-particle excitations triggered by a sufficiently small quench. Here we investigate the case of long-range (1/r α ) interactions for d-dimensional lattice spin models with uniaxial symmetry, and show that, in the regime d < α < d + 2, the entanglement and correlation buildup is radically altered by the existence of a non-linear dispersion relation of quasi-particles, ω ∼ k z<1 , at small wave vectors, leading to a divergence of the quasiparticle group velocity and super -ballistic propagation. This translates in a super-linear growth of correlation fronts with time, and sub-linear growth of relaxation times of subsystem observables with size, when focusing on k = 0 fluctuations. Yet the large dispersion in group velocities leads to an extreme wavelength dependence of relaxation times of finite-k fluctuations, with entanglement being susceptible to the longest of them. Our predictions are directly relevant to current experiments probing the nonequilibrium dynamics of trapped ions, or ultracold magnetic and Rydberg atoms in optical lattices.Introduction. The relaxation of the pure state of a generic quantum many-body system following a quantum quench is a complex phenomenon: there each subsystem sees its reduced state reach thermal equilibrium (or a non-equilibrium steady-state) via the drastic rearrangement of correlations with its exterior, acting as a bath [1, 2]. A particularly dramatic case -yet highly relevant experimentally -concerns the choice of a factorized initial state, in which each subsystem admits a pure-state (zero-entropy) description. The buildup of entropy and fluctuations towards equilibration requires therefore the growth of entanglement and correlations between subsystems, namely the flow of information from each subsystem to its exterior. The post-quench dynamics of entanglement and correlations strongly depends a priori on the peculiar system under investigation. Yet, in generic systems with short-range (SR) interactions and translational invariance, a seemingly universal picture emerges for sufficiently small quenches, as shown both theoretically (see Ref.[3] and refs therein) as well as experimentally [4, 5]. In such a picture (see Fig. 1(a) for a sketch) correlations and entanglement establish because of the ballistic propagation of quasiparticle excitations (QP) triggered by the quench. As a consequence 1) correlations reorganize over distances which grow linearly in time (within a so-called "causal cone" or "light cone") consistent with Lieb-Robinson bounds [6] on the propagation of signals in quantum systems -and, as a consequence, subsystem fluctuations (which are integrals of correlation functions) grow linearly in time as well. The velocity of correlation fronts is twice the maximum group velocity of the QP -this is consistent with correlations among two points being established by counter-propagating QP originating from the m...