Quantum Orders in an Exact Soluble Model

We find all the exact eigenstates and eigenvalues of a spin-$1/2$ model on square lattice: $H=16g\sum _{\mathit{i}}{S}_{\mathit{i}}^y{S}_{\mathit{i}+\widehat{\mathit{x}}}^x{S}_{\mathit{i}+\widehat{\mathit{x}}+\widehat{\mathit{y}}}^y{S}_{\mathit{i}+\widehat{\mathit{y}}}^x$. We show that the ground states for $g<0$ and $g>0$ have different quantum orders described by $Z2A$ and $Z2B$ projective symmetry groups. The phase transition at $g=0$ represents a new kind of phase transition that changes quantum orders but not symmetry. Both the $Z2A$ and $Z2B$ states contain $Z_2$ lattice gauge theories at low energies. They have robust topologically degenerate ground states and gapless edge excitations.