Classification of symmetry fractionalization in gapped ${\mathbb{Z}}_2$ spin liquids

In quantum spin liquids, fractional spinon excitations carry half-integer spins and other fractional quantum numbers of lattice and time-reversal symmetries. Different patterns of symmetry fractionalization distinguish different spin liquid phases. In this work, we derive a general constraint on the symmetry fractionalization of spinons in a gapped spin liquid, realized in a system with an odd number of spin-$\frac{1}{2}$ per unit cell. In particular, when applied to kagome/triangular lattices, we obtain a complete classification of symmetric gapped ${\mathbb{Z}}_2$ spin liquids.

Figure 1

Illustrations of the effect of anyon SOC.

Figure 2

Illustrations of the effect of anyon SOC.

Figure 3

Illustration of the physical process determining the anyonic flux creation $b_y\left(P_x\right)$.

Figure 4

Generators of the space group of the kagome/triangular lattice, $G_s=p6m$.