Non-equilibrium steady state properties and phase transitions of some one dimensional quantum magnets

The motivation of this thesis was the study of magnetic systems away from equilibrium. For convenience we have chosen to study properties of steady states and also one-dimensional magnets. The main focus has been on some variations of the Transverse Field Ising Model which has served as a fundamental paradigm in condensed matter physics. In particular, firstly we have chosen to drive an energy current through the system and characterize the different phases, reproducing with a novel method available results obtained by other methods, The study was done for both the ferromagnetic and the anti-ferromagnetic case. There is a new phase that emerges upon the application of an energy current, characterized by long-range power law correlations and a finite expectation value of the energy current operator, which we have examined in more detail. We have obtained a better fitting of the correlations while discussing the limits of the method. As a next step we investigated the problem of periodic bond defects in the system and how these affect the phase diagram. The strength of the bond (interaction) was varied as well as the distance between the defected interactions in order to investigate their effect on the phase diagram and the correlations. At the end of this thesis, we derived the energy current operator of a model (called Gu-Wen model) that admits a topological phase (Haldane phase), with the view to continue the study of non-equilibrium physics of richer models. The employed method is a novel one, derived from the Density Matrix Renormalization Group technique. It is known as Time Evolving Block Decimation, using Matrix Product States, and is especially suitable for problems out of equilibrium because it tracks the time dependence. Although it works extremely well for systems with an energy gap in their spectrum, it produces correct results for critical systems as well.