Mutual Chern-Simons theory for Z2 topological order
Abstract
We study several different Z2 topological ordered states in frustrated spin systems. The effective theories for those different Z2 topological orders all have the same form -- a Z2 gauge theory which can also be written as a mutual U(1) x U(1) Chern-Simons theory. However, we find that the different Z2 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 quasi-particles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.