Measuring space-group symmetry fractionalization in ${\mathbb{Z}}_2$ spin liquids

The interplay of symmetry and topological order leads to a variety of distinct phases of matter, the symmetry enriched topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of ${\mathbb{Z}}_2$ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground-state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally, these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular, we show that the kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the kagome in a few different ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to connect bosonic and fermionic mean-field theories of spin liquids. When fusing quasiparticles, it correctly predicts nontrivial phase factors for combining their space group quantum numbers.