Symmetries and topology in quantum baths

In this thesis, I study dissipation in a number of one-dimensional models from condensed matter physics, by coupling them to a quantum bath. The main purpose is to see how the dissipation interacts with other properties, such as topological order and different types of symmetries. The first system studied is the Kitaev chain, a simple non-interacting model with topological order. The presence of a bath drives the system out of equilibrium and into a mixed quantum state, where conventional measures of topological order do not apply. I propose looking for certain dynamical signatures, that can tell us what happens to the topological order of the original system, once the bath is included. In addition, we study the dissipative Kitaev chain with periodic driving, which introduces a host of exotic new topological properties. Finally, I investigate the effect of dissipation on interacting models, such as the XXZ Heisenberg spin chain, and how symmetries might help us analyze this effect. While the dynamics of such a model are very difficult to compute, certain observables like the total magnetization of the spin chain behave in a peculiar way. By studying the symmetry properties of both the system and the bath, I show that the dissipation causes the expectation value of the magnetization to relax in a simple and coherent manner, only adding an overall damping factor to the dynamics of the isolated system.