Measuring symmetry fractionalization in topological orders: application to Z2 spin liquids on kagome lattice

Mounting numerical evidence for a gapped $Z_2$ spin liquid in the kagome Heisenberg model urges us to develop methods to measure the global and space group symmetry properties of fractional excitations. We show that the universal symmetry characterization of fractional quasiparticles, the projective symmetry group (PSG), can be measured by a dimensional reduction scheme, which relates two-dimensional (2d) symmetric topological orders to 1d symmetry protected topological phases. This general framework allows us to unify $Z_2$ spin liquids in different slave-particle (parton) constructions on the kagome lattice. It is also directly applicable to numeric results obtained in 2d DMRG studies.