Lieb-Robinson bounds imply locality of interactions

TL;DR

This paper establishes a precise equivalence between Lieb-Robinson bounds and the exponential decay of interactions in quantum lattice models, linking locality of interactions to finite propagation speed of correlations.

Contribution

It proves that Lieb-Robinson bounds hold if and only if the underlying interactions decay exponentially, generalizing previous results and applying to fermionic models.

Findings

Lieb-Robinson bounds are equivalent to exponential decay of interactions.

Bounds for single-site observables imply bounds for all bounded observables.

The results extend to various decay behaviors and fermionic models.

Abstract

Discrete lattice models are a cornerstone of quantum many-body physics. They arise as effective descriptions of condensed matter systems and lattice-regularized quantum field theories. Lieb-Robinson bounds imply that if the degrees of freedom at each lattice site only interact locally with each other, correlations can only propagate with a finite group velocity through the lattice, similarly to a light cone in relativistic systems. Here we show that Lieb-Robinson bounds are equivalent to the locality of the interactions: a system with k-body interactions fulfills Lieb-Robinson bounds in exponential form if and only if the underlying interactions decay exponentially in space. In particular, our result already follows from the behavior of two-point correlation functions for single-site observables and generalizes to different decay behaviours as well as fermionic lattice models. As a side-result, we thus find that Lieb-Robinson bounds for single-site observables imply Lieb-Robinson bounds for bounded observables with arbitrary support.