Bell Non-Locality in Many-Body Quantum Systems with Exponential Decay of Correlations

Using Bell inequalities as a tool to explore non-classical physical behaviours, in this paper we analyse what one can expect to find in many-body quantum physics. Concretely, framing the usual correlation scenarios as a concrete spin lattice, we want to know whether or not it is possible to violate a Bell inequality restricted to this scenario. Using clustering theorems, we are able to show that a large family of quantum many-body systems behave almost-locally, violating Bell inequalities (if so) only by a non-significant amount. We also provide examples, explain some of our assumptions via counter-examples and present all the proofs for our results. We hope the paper is self-contained.