  In this PhD thesis we investigate some properties of one-dimensional quantum
systems, focusing on two important aspects of integrable models: Their
entanglement properties at equilibrium and their dynamical correlators after a
quantum quench. The first part of the thesis will therefore be devoted to the
study of the entanglement entropy in one-dimensional integrable systems, with a
special focus on the XYZ spin-1/2 chain, which, in addition to being
integrable, is also an interacting theory with non perturbative solutions. We
derive its bipartite Renyi entropies in the thermodynamic limit and its
behaviour in different phases and for different values of the mass-gap is
analysed, both analytically and numerically. In particular it is worth
mentioning that the numerical analysis of the entropies of the XYZ model
presented in this dissertation has never been published in literature. In the
second part of the thesis we study the dynamics of correlators after a quantum
quench, preparing the system in a squeezed coherent initial state. The emphasis
will be on the Transverse Field Ising Chain and the O(3) non-linear sigma
model, the latter studied by means of the semi-classical theory, the former by
a form-factor approach. Moreover in the last chapter we outline a general
result about the dynamics of correlation functions of local observables after a
quantum quench. In particular we show that if there are not long-range
interactions in the final Hamiltonian, then the dynamics of the model at long
times (non equal- time correlations) is described by the same statistical
ensemble that describes its statical properties (equal-time correlations). For
the Transverse Field Ising Chain this result means that its dynamics is
described by a Generalized Gibbs Ensemble.
