Spreading of correlations in exactly-solvable quantum models with   long-range interactions in arbitrary dimensions
  We study the out-of-equilibrium dynamics induced by quantum quenches in quadratic Hamiltonians featuring both short- and long-range interactions. The spreading of correlations in the presence of algebraic decaying interactions, $1/R^\alpha$, is studied for lattice Bose models in arbitrary dimension $D$. These models are exactly solvable and provide useful insight in the universal description of more complex systems as well as comparisons to the known universal upper bounds for the spreading of correlations. Using analytical calculations of the dominant terms and full numerical integration of all quasi-particle contributions, we identify three distinct dynamical regimes. For strong decay of nteractions, $\alpha>D+1$, we find a causal regime, qualitatively similar to what previously found for short-range interactions. This regime is characterized by ballistic (linear cone) spreading of the correlations with a cone velocity equal to twice the maximum group velocity of the quasi-particles. For weak decay of interactions, $\alpha<D$, we find instantaneous activation of correlations at arbitrary distance. This signals the breaking of causality, which can be associated with the divergence of the quasi-particle energy spectrum. Finite-size scaling of the activation time precisely confirms this interpretation. For intermediate decay of interactions, $D<\alpha<D+1$, we find a sub-ballistic, algebraic (bent cone) spreading and determine the corresponding exponent as a function of $\alpha$. These outcomes generalize existing results for one-dimensional systems to arbitrary dimension. We precisely relate the three regimes to the first- and second-order divergences of the quasi-particle energy spectrum for any dimension. 
Lorenzo Cevolani, Giuseppe Carleo and Laurent Sanchez-Palencia
cond-mat.stat-mech