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Adiabatic theorem states that for sufficiently slow variations of a parameter, a quantum system starting from the ground state will stay in its instantaneous ground state during the evolution; but if we consider a critical system, as the relaxation time diverges as we approach critical point, during a phase transition the evolution will eventually become non-adiabatic and the system will not be able to relax to its instantaneous ground state: this will result in the formation of imperfections in the final state of the system, such as a kink in a ferromagnetic state. The density of such imperfections can be related to the quench velocity in many (classical or quantum, finite or infinite) systems with the help of the Kibble-Zurek mechanism, which essentially divides the transition in adiabatic and impulse-like regimes when the transition velocity exceeds the relaxation time. The Kibble-Zurek mechanism has been successfully applied to superconductors, superfluids and liquid crystals and it has been shown to overlap very well with the more rigorous Landau-Zener transition probability for avoided level crossing. The XXZ model consists of a chain of spin-1/2 interacting via a Heisenberg coupling with anisotropy in the z direction. By changing the intensity and sign of the z coupling relative to the xy coupling one can modify the ground state properties of the system, ranging from the Ising ferromagnet, to the isotropic ferromagnetic or antiferromagnetic Heisenberg model, to the Ising antiferromagnet. In particular for certain values of the z coupling the system shows a critical behaviour. By gradually changing the value of the z coupling we can make a state of the system evolve through this extended critical region. In this thesis we will describe the phase transition of the XXZ model in the framework of the Kibble-Zurek mechanism (and Landau-Zener theory), to see if and in which way these description can be applied to critical regions instead of critical points; our predictions will be compared with numerical findings obtained using the Density Matrix Renormalization Group algorithm, a very powerful tool for the simulation of one dimensional quantum systems, and in particular its recently developed extension which supports time evolution.