Mutual Chern-Simons theory for Z 2 topological order

TLDR: In this article, the effect of topological order and symmetry in frustrated spin systems has been studied and it was shown that the different topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory.

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.