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

TL;DR

This paper introduces a measurement method to detect two-dimensional symmetry protected topological order in ground states, linking it to the third cohomology class, and validates it numerically for models with  symmetry.

Contribution

It presents a novel measurement technique for identifying 2D SPT order that correlates with the third cohomology class, validated through numerical simulations.

Findings

Measurement determines the third cohomology class for  symmetry.

Validated the measurement using infinite density matrix renormalization group.

Method distinguishes SPT states from trivial states without local order parameters.

Abstract

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.