Theorists have envisioned a great variety of exotic states of quantum matter, including topological phases, states with emergent gauge structure, and non-Fermi liquid quantum critical states. But the realization of such states in reality lags far behind, and it is not always clear where to look. In this dissertation I discuss some efforts to find exotica in rocks : a class of minerals known as pyrochlores. In particular, I focus on three problems: order-by-disorder, quantum spin liquids and quantum criticality in fermionic systems. All were formulated theoretically in the 1970s, but their experimental realization has remained elusive, or their theoretical understanding incomplete. First, this dissertation addresses Er2Ti 2O7, a material which can, remarkably, be definitively shown to be governed by quantum order-by-disorder, a phenomenon such that quantum fluctuations lead to ordering in a system which should otherwise be disordered, classically.

Second, a large fraction of the dissertation is devoted to an example of emergent quantum electrodynamics known as Coulombic quantum spin liquid, which may occur in rare earth pyrochlores such as Yb 2Ti2O7. The theory of these systems and this spin liquid are derived and described at zero and nonzero temperature. Quantum spin liquids are insulating states of matter with "long-range entanglement": quantum fluctuations are huge so that the spins typically break no symmetries but yet dramatically differ from simple paramagnets. Finally, the family of pyrochlore iridates with chemical formula A2Ir2O 7 (A is a rare-earth) is also home to exotic behaviors: for example, a new strongly-coupled three-dimensional antiferromagnetic quantum critical point arises in those materials. This presents a novel interplay of Coulomb interactions and magnetic order parameter fluctuations.

The flurry of resulting unusual behaviors predicted in the vicinity of the quantum phase transition are exposed: a nodal non-Fermi liquid phase, a (topological) Weyl semimetal, extreme anisotropy in real space, emergent internal SO(3) symmetry etc..