Spreading of correlations in exactly solvable quantum models with long-range interactions in arbitrary dimensions

Abstract

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 ^α , 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 interactions, $\alpha \gt D+1$ , 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, α < 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, $D\lt \alpha \lt D+1$ , we find a sub-ballistic, algebraic (bent cone) spreading and determine the corresponding exponent as a function of α . 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. The long-range transverse Ising model in dimensions D = 1, 2, and 3 in the (quadratic) spin-wave approximation is more specifically studied and we also discuss the shape of the correlation front in dimension higher than one. Our results apply to several condensed-matter systems as well as atomic, molecular, and optical systems, and pave the way to the observation of causality and its breaking in diverse experimental realization.

Keywords

• out-of-equilibrium dynamics
• long-range many body quantum systems
• exactly solvable models
• spreading of correlations
• Lieb–Robinson bound