Symmetry Fractionalization in Two Dimensions

Topologically ordered states are often characterized by topological properties, such as braiding statistics and fusion rules, of their excitations. However, excitations also carry symmetry quantum number, namely a representation of a symmetry group, when a topologically ordered state respects the symmetry. If an excitation's symmetry quantum number cannot be obtained from a finite integer number of fundamental constituents of the system, we propose to call such phenomena ``symmetry fractionalization.'' We introduce a solvable SO(3) spin-rotational and time reversal invariant spin-1 model on the honeycomb and decorated honeycomb lattices. We show that the ground state is the equal-weight superposition of all valence loops, which we call ``resonating valence loop'' (RVL) state and which is a quantum spin liquid respecting all the symmetries of the model. Ends of broken loops are excitations with spin-1/2, which are deconfined spinons. Since spin-1/2 cannot be obtained from an integer numbers of spin-1, the system exhibits symmetry fractionalization (specifically the ``SO(3) symmetry fractionalization''). Moreover, for time-reversal $T$, a spinon has $T^{2}$ = -1, while integer spins have $T^{2}$ = +1. Consequently, the system also has ``time-reversal fractionalization.''