# Tightening the Lieb-Robinson Bound in Locally-Interacting Systems

## TL;DR

This paper introduces a new method that significantly tightens Lieb-Robinson bounds in locally-interacting quantum systems, especially at large local Hilbert space dimensions, improving understanding of information propagation limits.

## Contribution

The authors develop a novel approach that yields much sharper Lieb-Robinson bounds, reducing the growth of the velocity with system parameters compared to previous bounds.

## Key findings

- LR velocity grows slower with local Hilbert space dimension D
- In Heisenberg model, v remains constant at large S
- In multiorbital Hubbard models, v scales as √N instead of N

## 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 an LR velocity that grows much slower than previous bounds with $D$ as $D\to \infty$. For example, in the Heisenberg model with spin $S$, we find $v\leq$ const. compared to the previous $v\propto S$ which diverges at large $S$, and in multiorbital Hubbard models with $N$ orbitals, we find $v\propto \sqrt{N}$ instead of previous $v\propto N$, and similarly in the $N$-state 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.