A K matrix Construction of Symmetry Enriched Phases of Matter

TL;DR

<3-5 sentence high-level summary> The paper develops a K-matrix framework for constructing and classifying symmetry enriched topological (SET) phases in 2+1 dimensions, focusing on Abelian, non-chiral orders and exploring how global symmetries act on anyons via group extensions of the fusion algebra. By combining K-matrix Chern-Simons theory with Higgs (condensation) terms, it derives conditions under which symmetry-preserving edges can be fully gapped and reveals a quasi-group structure SEP(\mathfrak{F},G_s) that organizes SETs with the same fusion algebra and symmetry. It provides explicit constructions for SEP phases in Z2×Z2 and Z4×Z4 gauge theories, including non-local symmetry actions that exchange anyons, and discusses how stacking and extensions yield a rich landscape beyond central extensions. The work connects to and contrasts with existing SPT/SET frameworks, offering a practical path toward systematic classification and highlighting avenues for extending to non-Abelian orders and higher dimensions.

Abstract

We construct in the K matrix formalism concrete examples of symmetry enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and non-chiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)-group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic non-local anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.