# Local order parameters for symmetry fractionalization

## TL;DR

This paper introduces local order parameters capable of detecting the symmetry fractionalization class of anyons in 2D topological phases, providing a more refined and practical tool than previous methods based on dimensional reduction.

## Contribution

The authors develop local, two-dimensional order parameters for symmetry fractionalization, enabling finer phase distinctions and applicability to non-abelian topological orders and anyon permutations.

## Key findings

- Order parameters can be measured locally on 2D geometries.
- They distinguish phases beyond dimensional compactification.
- Effective for non-abelian topological orders and anyon permutation cases.

## Abstract

We propose a family of order parameters to detect the symmetry fractionalization class of anyons in 2D topological phases. This fractionalization class accounts for the projective, as opposed to linear, representations of the symmetry group on the anyons. We focus on quantum double models on a lattice enriched with an internal symmetry in the framework of $G$-isometric projected entangled pair states. Unlike previous schemes based on reductions to effective 1D systems (dimensional compactification), the order parameters presented here can be probed on genuinely two-dimensional geometries, and are local: they rely on operations on few neighbouring particles in the bulk. The power of these order parameters is illustrated with several combinations of topological content and symmetry. We demonstrate that a strictly finer phase distinction than that provided by dimensional compactification can be obtained. As particular examples, the resolution power of these order parameters is illustrated for a case with non-abelian topological order, and for another with symmetries that involves permutation of anyons.