Explorations in two dimensional strongly correlated quantum matter: from exactly solvable models to conformal bootstrap

Abstract

This dissertation presents two projects that touch upon the role of quantum mechanics in classifying phases of matter and their transitions. In the first project, we set out to answer: is it possible to find a lattice model in the Ising universality class that realizes the Kramers Wannier symmetry in such a way that it squares to 1, rather than a lattice translation as in the usual Ising model? Using insights from symmetry-protected topological phases of matter, we answer in the affirmative, with the caveat that the symmetry, beyond being non-onsite, actually acts on a Hilbert space that is not a local tensor product. The second concerns the nature of the Neel-VBS deconfined quantum critical point. This is thought to be described by the noncompact CP¹ model, which we argue to be continuously connected to the theory accessed by the 2 + ε expansion for the O(3) NLSM. To shed light on the nature of the DQCP, we perform conformal bootstrap studies of the O(3) model in 2 < d < 3.