Improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions

Abstract

In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary k-body interactions, so long as they decay with a power law greater than kd, where d is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to R, acting on any one site, decays as a power law 1/R[superscript α], with α>d. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to 1/r[superscript α]. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.