  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.
