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

TL;DR

This paper establishes a new Lieb-Robinson bound for quantum lattice systems with power-law decaying long-range interactions, refining the understanding of information propagation and defining a more accurate light-cone in such systems.

Contribution

It introduces a novel family of Lieb-Robinson bounds applicable to long-range interactions decaying faster than a critical power-law, and defines a stricter light-cone for these systems.

Findings

Quantum information decay follows approximately 1/r^α at fixed times.

The new light-cone is stricter than the conventional one in long-range systems.

Bounds are valid for arbitrary k-body interactions with decay rate above a threshold.

Abstract

In this work, we prove a new family of Lieb-Robinson bounds for lattice 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 $1/R^\alpha$, with $\alpha > d$. These new bounds allow us to prove that, at any fixed time, the spatial decay of quantum information follows arbitrarily closely to $1/r^{\alpha}$. Moreover, we define a new light-cone 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.