Phase and group velocities for correlation spreading in the Mott phase of the Bose-Hubbard model in dimensions greater than one

Lieb-Robinson and related bounds set upper limits on the rate at which information propagates in non-relativistic quantum systems. Experimentally, they have been observed in the spreading of correlations in the Bose-Hubbard model (BHM) after a quantum quench. Using a two particle irreducible (2PI) strong coupling approach to out-of-equilibrium dynamics in the BHM we calculate both the group and phase velocities for the spreading of single-particle correlations in 1, 2 and 3 dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the velocities for the spreading of single particle correlations in both the 1 and 2 dimensional BHM realized with ultra-cold atoms. They also agree with the claim that the phase velocity rather than the group velocity was observed in recent experiments in 2 dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and explore the variation of the anisotropy in the velocity at which correlations spread across the phase diagram of the BHM. Our results establish the 2PI strong coupling approach as a powerful tool to study out-of-equilibrium dynamics in the BHM in dimensions greater than 1.