Classification and Characterization of Exotic Quantum Systems: From Band Theory to Black Holes
Exotic quantum systems -- those with macroscopic quantum behavior with no classical analogue -- have been a mainstay of condensed matter theory since the discovery of the quantum Hall effect. Since then, several families of related systems have been uncovered. Some, such as topological insulators, were predicted first and found experimentally later. Others are more elusive, like strongly correlated bosonic symmetry protected topological phases in high dimension, of which concrete evidence is still lacking. In this dissertation, we study several examples of exotic quantum systems. For SPT phases, we present a physically motivated classification scheme for interacting bosons and a bulk signature independent of boundary. We then construct a new, beyond Landau-Ginzburg second-order phase transition between two ordered phases of the Heisenberg magnet on the triangular lattice. Finally, we investigate an infinite family of spin liquid states, and conjecture on their connection to black holes.