Nonequilibrium Quantum States of Matter

Among the few lessons that Physics teaches us on a daily basis, one in particular is hard to miss: thermalization processes are ubiquitous. With the same degree of certainty, one can predict an apple to fall onto the ground if its stalk is cut, or a cup of tea to be found cold if left on the table for too long. In this respect, generic classical and quantum mechanical systems seem to behave in the same way. This is also one of the main difficulties in the experimental observation of the bizarre effects predicted from the quantum theory, as several of these are known to disappear at finite temperatures. It has been known for a long time that thermalization is associated with “chaotic” behavior at the microscopic level, but recently systematic efforts, both theoretical and experimental, have allowed us to understand its mechanisms at an unprecedented level of accuracy. Indeed, it has been realized that thermalization generally occurs also in isolated systems, where the absence of interactions with the environment allows for first principle theoretical investigations1. In this case thermalization takes place locally, as the whole system acts as a thermal bath for its own subsystems. Theoretical research has been motivated and paralleled by exciting experimental progress in cold-atom physics, which has provided robust, nearly ideal realizations of several theoretical models2. As an established piece of knowledge, recent research has now confirmed that two outstanding exceptions actually exist in the quantum realm, where thermalization is prevented to occur. These are many-body localized systems3, where disorder plays a crucial role, and integrable ones, which are protected by the existence of higher conservation laws. By their own nature, these systems display exceptional non-equilibrium features, which are not washed out by the onset of thermalization as relaxation occurs. At the core of the present thesis lie the remarkable properties of integrable systems out of equilibrium, with a special attention to physical effects which could be observed in cold-atom realizations. The work collected here is part of a large theoretical effort4 which in the past decade has focused on the study of relatively simple protocols to bring a quantum system out of equilibrium, such as the so-called quantum quench. In these “theoretical laboratories” it has been possible to provide quantitative predictions which helped us to develop an intuition on general questions regarding non-equilibrium and thermalization processes. Furthermore, in some cases these studies have highlighted interesting physical effects which could be directly probed experimentally within cold-atom settings. The contribution of the present thesis is two-fold. On the one hand, we have developed new technical tools for the study of integrable systems out of equilibrium, and for the computation of measurable physical quantities such as correlation functions. The techniques employed are mainly analytical and rooted within the mathematical structures of integrability. On the other hand, we have singled out physically relevant nonequilibrium situations where exotic, non-thermal states of matter emerge after relaxation occurs, and provided quantitative analytical predictions in these cases. A comprehensive discussion of the motivations and results of our work will be presented in Chapter 1, where we provide a complete overview of this thesis and discuss the organization of its content.