Crystalline topological phases as defect networks

Understand

A crystalline topological phase is a topological phase with spatial symmetries.

• In this work, we give a very general physical picture of such phases: a topological phase with spatial symmetry $G$ (with internal symmetry $G_{\mathrm{int}} \leq G$) is described by a *defect network*: a $G$-symmetric network of defects in a topological phase with internal symmetry $G_{\mathrm{int}}$.
• The defect network picture works both for symmetry-protected topological (SPT) and symmetry-enriched topological (SET) phases, in systems of either bosons or fermions.
• We derive this picture both by physical arguments, and by a mathematical derivation from the general framework of [Thorngren and Else, Phys.