Detecting two dimensional symmetry protected topological order in a ground state wave function

Understand

Symmetry protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state.

• We demonstrate how to detect whether a two dimensional ground state has symmetry protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons.
• For any finite abelian onsite symmetry, the measurement completely determines the 3rd cohomology class that characterizes the order.
• The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with $\mathbb{Z}_2$ symmetry protected order.