Abstract
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 $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ 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.
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