Exact Results in Quantum Theory

Stabilizer states provide exactly solvable models of gapped phases in lattice quantum spin systems, including topologically ordered phases that feature anyonic excitations and ground-state degeneracies sensitive to topology. In three and higher dimensions, even more exotic behavior can occur, such as excitations with restricted mobility and subextensive ground-state degeneracy. Underlying these models is a rich mathematical theory based on commutative algebra and algebraic geometry in positive characteristic. I will give a basic introduction to the subject, illustrating the key ideas through examples and outlining the main results.