On two-dimensional tensor network group symmetries

TL;DR

This paper develops a unified tensor network framework to represent two-dimensional group symmetries with 4-cocycle indices and explores their role in characterizing (2+1)D gapped phases and (3+1)D SPT phases.

Contribution

It introduces a comprehensive tensor network approach for finite group symmetries with cocycle indices, linking (2+1)D and (3+1)D topological phases within a single formalism.

Findings

Unified tensor network representation of 2D group symmetries with 4-cocycle indices.

Characterization of gapped (2+1)D phases with anomalous symmetries.

Construction of tensor network unitaries for (3+1)D SPT phases.

Abstract

We introduce two-dimensional tensor network representations of finite groups carrying a 4-cocycle index. We characterize the associated gapped (2+1)D phases that emerge when these anomalous symmetries act on tensor network ground states. We further develop related tensor network unitaries that generate symmetric states representing (3+1)D symmetry protected topological phases. Although aspects of these constructions have been previously addressed, our contribution unifies them within a single tensor network framework and emphasizes the explicit formulation of local tensor equations encoding global consistency conditions.