| We present results on two kinds of stable quantum spin liquid phases in two and three dimensions. These states lack long-range order, and share the common feature that they are critical, in the sense that they have gapless excitations and power-law correlations. Remarkably, these states are stable zero-temperature phases.; After briefly introducing some of the context and motivation for the work, we consider a model three-dimensional frustrated magnet, and show that it supports a spin liquid phase over a finite extent of its phase diagram. The resulting state has many unusual properties, including fractionalized excitations and an emergent, linearly dispersing photon mode with two transverse polarizations. The low-energy theory is simply scalar quantum electrodynamics with massive electrically and magnetically charged particles.; The final two chapters are concerned with certain algebraic spin liquids---a class of two-dimensional spin liquids of potential interest in the context of the cuprate high temperature superconductors, and also in frustrated quantum magnets. We resolve a longstanding controversy in the literature by showing rather rigorously that these states can indeed exist as stable phases. Finally, we explore some of their potentially observable properties, focusing on the striking consequences of an emergent symmetry present at low energies that is much larger than the symmetry of the models or materials in which these states can arise. Remarkably, this analysis shows that these states contain within them the seeds of many different competing orders. |
