We study several different $Z_2$ topological ordered states in frustrated spin systems. The effective theories for those different $Z_2$ topological orders all have the same form—a $Z_2$ gauge theory which can also be written as a mutual $U\left(1\right)\times U\left(1\right)$ Chern-Simons theory. However, we find that the different $Z_2$ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.

• Received 3 May 2008

DOI:https://doi.org/10.1103/PhysRevB.78.155134