Frustrated Magnets: Topology, Bose-Einstein Condensation, and Order-by-disorder

If a single phrase can be said to embody modern perspectives on many-body theory, it is P.W. Anderson's famously simple line, "More is different". This idea is well understood by practitioners of many-body theory, and field theory more generally, as drawing a distinction between the degrees of freedom which define a macroscopic system and the emergent degrees of freedom which control its low-energy behavior. This dissertation will center this concept by studying a set of models -- frustrated magnetic systems -- which maximize the distinction between effective and definitional degrees of freedom. Each model highlights distinct sets of "exotic" many-body phenomena.We begin by considering an insulating quantum magnet on the geometrically frustrated kagome lattice. A simple approximation to the physics of some materials, such as the iron jarosites, is achieved by focusing on nearest-neighbor antiferromagnetic exchange and spin-orbit effects in the form of Dzyaloshinskii-Moriya interactions. Through series expansions around the strong-field limit, we will provide evidence that this model generically realizes either topological magnon bands or ``semimetals'' built out of charge-neutral magnetic excitations. We will also use the wavefunctions computed in this process to determine transport functions of interest, such as the thermal Hall conductivity. In the topological phase, we compute the system's Chern numbers and discuss the consequences of topological phase transitions for transport properties.Next, we consider intriguing empirical results for the Lanthanide-based compound Ytterbium Silicate. This quantum dimer magnet is related to other well-known materials which exhibit Bose-Einstein condensation transitions of spin excitations when subjected to a magnetic field. We construct an effective spin model which reproduces the observed behavior of Ytterbium Silicate, and use a broad range of computational techniques to argue that this model captures the essential physics of the material. This example highlights the prospect of producing novel interacting phases in materials with strong spin-orbit coupling.Finally, we consider the statistical mechanics of a model that we call the distorted pyrochlore Heisenberg model. This model constitutes a thermodynamic interpolation between the kagome and pyrochlore Heisenberg models, which each have storied histories in the context of spin liquid physics. We will argue that thermal order-by-disorder -- a phenomenon present in the kagome, but absent in the pyrochlore -- appears above a nonzero transition temperature in a quasi-two dimensional limit. We discuss the thermodynamic signatures of this behavior in the low-temperature limit and their relevance for layered kagome systems