Topological Phases and Strongly Correlated Fermions in Many-Body Systems

In this dissertation we study the properties of quantum many-body systems whose behavior is governed by strong correlation effects between large numbers of degrees of freedom. We begin with a review of various types of condensed matter settings and a general discussion of how interactions can lead to qualitatively new properties. In Chapter 2 we discuss a class of spin models which generate nontrivial realizations of a global symmetry, both in the presence and absence of topological order. In Chapter 3, we examine a model of a non-Fermi liquid. This model is built out of electron-like degrees of freedom and has a Fermi surface, yet has no stable quasiparticle excitations. In Chapter 4 we discuss some aspects of entanglement entropy in gapless multi-component systems, and suggest a way of measuring the topological properties of Kondo lattice wavefunctions