Quite remarkably, the low-energy behavior of most metals is well captured by effectively ignoring the Coulomb repulsion between the constituent electrons. Yet over the past several decades many examples of “strongly correlated systems” have been discovered for which a strong interplay between interactions and quantum mechanics leads rather to spectacular deviations from an independent-electron picture, as exemplified by quantum Hall systems, high-T_c superconductors, heavy fermion compounds, etc. This thesis is concerned with exploring exotic physics that can emerge in a class of correlated systems known as quantum magnets, where electrons “self-localize” around crystal lattice sites due to interactions, leaving residual spin couplings at low energies. Typically the ground states of such systems exhibit spontaneous symmetry breaking, commonly in the form of magnetic order. But sufficiently strong quantum fluctuations can suppress symmetry-breaking order even down to zero temperature, generating exotic “spin liquid” phases that exhibit novel features such as electron fractionalization, topological order, emergent symmetries, etc.

Recent experiments suggest that spin-1/2 triangular antiferromagnets are promising host systems for realizing such states. To explore this possibility, we attack these systems by equivalently reformulating a class of spin models in terms of vortices—topological defects in which spins wind around lattice plaquettes. Both in zero magnetic field and at 1/3 magnetization, by subsequently fermionizing the vortices we are led naturally to novel gapless spin liquids we refer to as “algebraic vortex liquids” (AVLs), whose detailed properties we characterize. The effective theories describing these phases are identical to quantum electrodynamics in 2+1 dimensions, with emergent symmetries that lead to highly nontrivial predictions for the spin structure factor that could be tested via neutron scattering experiments. Potential application to quasi-2D spin-1/2 triangular antiferromagnets Cs₂CuCl₄ and Cs₂CuBr ₄ is discussed. We also explore descendant phases of the AVLs, which include “chiral” spin liquids, a novel gapless “solid” phase, numerous magnetically ordered states, and supersolids. Furthermore, our approach predicts various anomalous “roton” minima in the excitation spectrum for Neel ordered states, explaining recent series expansion results at zero-field. Several interesting future applications are suggested.