  We study the dynamics of the quantum Ising chain following a zero-temperature
quench of the transverse field strength. Focusing on the behavior of two-point
spin correlation functions, we show that the correlators of the order parameter
display an effective asymptotic thermal behavior, i.e., they decay
exponentially to zero, with a phase coherence rate and a correlation length
dictated by the equilibrium law with an effective temperature set by the energy
of the initial state. On the contrary, the two-point correlation functions of
the transverse magnetization or the density-of-kinks operator decay as a
power-law and do not exhibit thermal behavior. We argue that the different
behavior is linked to the locality of the corresponding operator with respect
to the quasi-particles of the model: non-local operators, such as the order
parameter, behave thermally, while local ones do not. We study which features
of the two-point correlators are a consequence of the integrability of the
model by analizing their robustness with respect to a sufficiently strong
integrability-breaking term.
