Exactly Solvable Models: Quantum Criticality, Dynamics, and Topology

We present a collection of three short stories on the applications of simple toy models indifferent topics in condensed matter theory. These stories describe examples of new toy models and methods for solving them exactly, as well as limitations of certain kinds of toy models. The models we construct provide to a refined understanding of complex physical phenomena, by distilling the phenomena down to their minimal ingredients. The first story describes an exactly solvable model for an unusual kind of phase tran- sition called a deconfined quantum critical point (DQCP). While these kinds of quantum critical points have been hypothesized and intensely studied from field theory and numerical perspectives, their exact nature is still disputed. Our model provides the first example of a DQCP that can be solved exactly on the lattice, and gives a clear picture of the physi- cal mechanism behind the transition. The second story presents a simple, exactly solvable model for a transition in the entanglement dynamics of a quantum system. Like DQCPs, entanglement transitions have been explored mainly through numerical work and field the- ory approximations, with very limited exact results. We present both the model and a novel means of solving it – by a mapping to Möbius transformations. The third story is about limitations of exactly solvable models. We show that, although these models can be con- structed for broad classes of topological phases of matter, they cannot be constructed for certain phases – namely those with a nonzero Hall conductance.