Lieb-Robinson bound and almost-linear light-cone in interacting boson systems

In this work, we investigate how quickly a local perturbation propagates in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, the systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on the specific but experimentally natural setup that the initial state without perturbation is a steady-state where so many bosons do not concentrate on one site. We rigorously prove the existence of the almost linear light cone for the information propagation, where the wavefront grows at most as $t\log^2 (t)$. As practical applications, we give the clustering theorem for gapped ground states and the efficiency guarantee in classically simulating the one-dimensional quench dynamics which is of particular interest in experimental relevance.