  We discuss how nonlocality originates in long-range quantum systems and how
it affects their dynamics at and out of the equilibrium. We focus in particular
on the Kitaev chains with long-range pairings and on the quantum Ising chain
with long-range antiferromagnetic coupling (both having a power-law decay with
exponent \alpha). By studying the dynamic correlation functions, we find that
for every finite \alpha two different behaviours can be identified, one typical
of short-range systems and the other connected with locality violation. The
latter behaviour is shown related also with the known power-law decay tails
previously observed in the static correlation functions, and originated by
modes, having in general energies far from the minima of the spectrum, where
particular singularities develop as a consequence of the long-rangedness of the
system. We refer to these modes as to "singular" modes, and as to "singular
dynamics" to their dynamics. For the Kitaev model they are manifest, at finite
\alpha, in derivatives of the quasiparticle energy, the order of the derivative
at which the singularity occurs increasing with \alpha. The features of the
singular modes and their physical consequences are clarified by studying an
effective theory for them and by a critical comparison of the results from this
theory with the lattice ones. Moreover, a numerical study of the effects of the
singular modes on the time evolution after various types of global quenches is
performed. We finally present and discuss the presence of singular modes and
their consequences in interacting long-range systems by investigating in the
long-range Ising quantum chain, both in the deep paramagnetic regime and at
criticality, where they also play a central role for the breakdown of conformal
invariance.
