Strongly correlated electron systems have the potential to host very exotic

phases of matter. In order to have relevance to real materials, this exotic

physics often must emerge from relatively simple models. The quantum

wavefunctions which describe such phases may bear little resemblance to the

original microscopic models. In these cases a variety of complex analytic

tools often must be supplemented with controlled numerical calculations to

fully understand the essential behavior of these models. In this dissertation,

we study such quantum phases of matter and their relationship to real materials.

We focus on three main problems. First, we explore the relationship between

strong spin-orbit coupling and spin liquid physics by studying a very general

model on the triangular lattice where spin-orbit coupling leads to the presence

of highly anisotropic interactions. We use variational Monte Carlo to study both

U(1) quantum spin liquid states and ordered ones, via the Gutzwiller projected

fermion construction. We thereby obtain the ground state phase diagram in this

phase space. We furthermore consider effects beyond the Gutzwiller wavefunctions

for the spinon Fermi surface quantum spin liquid, which are of particular

importance when spin-orbit coupling is present.

Second we show that the interplay between a high density two-dimensional electron

gas and localized electrons in a neighboring Mott insulator leads to kinetic

magnetism unique to the Mott/band insulator interface. Our study is based upon

a bilayer Hubbard model at $U=\infty$ with a potential difference between the

two layers. We combine analytic results with DMRG simulations to show that

magnetism, and especially kinetic ferromagnetism, is greatly enhanced relative by the

proximity of the two subsystems.

Third we study the effect of interactions on the properties of a model 2D

topological Kondo insulator phase. We introduce a model Hamiltonian which we

believe captures the essential physics of the different competing phases.

Perhaps the most dramatic example of many-body physics in symmetry protected

topological phases is the existence of exotic gapless edge states. We comment on

the potentially dramatic effects that interactions can have on such edge states.