Tightening the Lieb-Robinson Bound in Locally Interacting Systems
Zhiyuan Wang, Kaden R.A. Hazzard
Abstract
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity. Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications. We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions. Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D . For example, in the Heisenberg model with spin S, we find v const. compared to the previous v S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v N instead of previous v N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model. Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models. Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.