Symmetry fractional quantization in two dimensions

Abstract

We introduce a solvable spin-rotational and time-reversal invariant spin-1 model in two dimensions. Depending on parameters, the ground state is an equal-weight superposition of all valence loops called "resonating valence loop" (RVL) or an equal-weight superposition of all valence bonds known as "resonating valence bond" (RVB). In RVL, ends of open loops are deconfined spinons of spin-1/2 that cannot be obtained by simple combinations of spin-1 -- a phenomenon of fractionalization; while in RVB, all quasiparticles carry an integer spin, hence no fractionalization. RVL and RVB are spin liquids with identical topological order but different spin-rotational and time-reversal symmetry quantum numbers of quasiparticles. We propose that quantized symmetry quantum number gives a systematic way to (partially) classify phases with identical topological order in dimensions greater than one.

Figures (1)

• Figure 1: (a) The schematic representation of the honeycomb lattice and a typical loop-covering configuration. The black (white) sites are in A (B) sublattices. Thick bond represent a spin singlet created by $B^\dag_{ij}$ (see text) and they forms a loop-covering on the honeycomb lattice. A hexagon plaquette is flippable if it has three thick and three thin bonds. For instance, the loop configuration is flippable on the hexagon plaquette marked by $a$. (b) A typecal spin-1 dimer-covering configuration in which each dimer consists of two singlet bonds.