Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

TL;DR

We develop a comprehensive tensor-network framework for symmetry-enriched topological order in two dimensions, formulating SET data via $\mathcal{G}$-graded MPO algebras and their Morita extensions. Central to the approach are defect tubes (dubes) and Ocneanu tube algebras, which yield the full spectrum of defect- and domain-wall superselection sectors, modular data, and entanglement signatures. We show how gauging a global symmetry and Rep$(\mathcal{G})$-condensation produce dual SET phases and relate these to Morita-equivalent MPO presentations, enabling systematic exploration of phase transitions between nonchiral topological orders. The theory is illustrated through explicit examples including electromagnetism-enriched toric codes, symmetry-enriched string-nets, SPT gauging, and various anyon-condensation paths, highlighting the practical utility for classifying and engineering SETs on lattices.

Abstract

We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.