Statistical mechanics characterizes systems in or near equilibrium using in terms of a handful of "state" variables, e.g. temperature, rather than infinitely many degrees of freedom. Statistical physics describes the expansion of the early universe, aspects of black holes, and most fruitfully, phases of matter and their properties. Quantum considerations have improved this understanding over time and revealed new phenomena, especially in complicated "strongly correlated" systems. Topological phases of matter, e.g., are of both fundamental and practical interest: these phases cannot be distinguished locally, unlike ice and water, which also allows them to store and process quantum information in a "fault-tolerant" manner, recently proposed for application to quantum computation. However, above zero temperature, thermal effects can overwrite this information.
Recent experiments on isolated systems have raised fundamental questions and revealed new routes to quantum computing. We now know that entanglement, generated dynamically as a quantum state evolves, "hides" local information about the past, producing familiar equilibrium states, described by a temperature. However, many systems do not thermalize: strong disorder can lead to MBL, which supports numerous phenomena forbidden in equilibrium and can protect quantum information at infinite temperature. In particular, both MBL and thermal systems are robust phases of matter, with a novel, athermal phase transition between them.
This thesis begins with an overview of MBL and thermalization, followed by an overview of exactly soluble quantum systems. We then turn to an important result in the field by this author: we introduce the first nontrivial example of an integrable Floquet model and comment on its solution and salient features. We then discuss how integrable models can provide insight into quantum thermalization, e.g. in terms of entanglement growth and demonstrating that conserved charges diffuse. We then investigate thermalization away from the integrable limit, also known as "quantum chaos." We review the standard techniques in this field and, briefly, several important results, before reproducing work by this author establishing definitively the long-conjectured result that the onset of thermalization in the presence of a conserved charge is governed by diffusion of said charge. We then investigate the interplay of conventional and topological order with nonequilibrium phase structure, with applications to quantum computation in mind. We review localization-protected quantum order in several models. We then investigate two models with non-Abelian symmetry, and show that MBL in such models can only realize if the symmetry breaks spontaneously to an Abelian subgroup. Finally, we conclude by examining open quantum systems, where we find several counterintuitive results that show that baths can, in some cases, enhance localization in certain systems, which may have use in realizing quantum computation.
