This thesis is devoted to the study of one-dimensional quantum spin chains using matrix product state-based techniques known. We deal mostly with the transverse field Ising model and perturbations. Firstly, we consider an out of equilibrium steady state by applying an energy current. We confirm the phase diagram, the correlations and the entanglement scaling. Subsequently, we introduce two kinds of perturbations. Firstly, we add an interaction that takes the system away from integrability and we study the correlations and the entanglement scaling, determining its phase diagram. Secondly, we weaken the strength of the interaction every nth pair with focus on the simplest case n=2. In every case we compute correlations and central charge (entanglement scaling) and we show that in the presence of an energy current, there is no conservation of energy. Finally, we motivate briefly work on a system that exhibits the Haldane phase.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.