[Paper Review] String condensation and topological holography for 2+1D gapless SPT

Rui Wen|arXiv (Cornell University)|Aug 11, 2024

Medical Imaging Techniques and ApplicationsMedicine3 citations

TL;DR

This paper establishes a duality between non-Lagrangian string condensations in 3+1D topological orders and 2+1D gapless symmetry-protected topological (gSPT) phases, using a categorical framework of condensable algebras in braided fusion 2-categories. It classifies these condensable algebras as 'magnetic and simple' in Z[2VecG], mapping them to gSPTs via topological holography, and reveals how topological invariants of gSPTs emerge from the dual string condensation structures.

ABSTRACT

The theory of anyon condensation is the foundation of the bulk-boundary relation and topological holography in 2+1D/1+1D. It is believed string condensation should replace anyon condensation in the 3+1D/2+1D topological holography theory. In this work we study string condensations in 3+1D topological orders and their relations to 2+1D phases. We find that a class of non-Lagrangian condensable algebras in 3+1D are exactly dual to a class of 2+1D symmetry enriched gapless phases known as gapless SPTs(gSPT). We show how topological properties of a gSPT can be fully extracted from the dual string condensation. We give an algebraic classification of this class of condensable algebras in 3+1D $G$-gauge theories that we call magnetic and simple. Through the topological holography dictionary, this maps to the classification of 2+1D $G$-symmetric phases with no topological order, including gapped and gapless SPTs. Utilizing the classification, we identify three classes of gSPTs and study their properties and gauging. Along the way, we reveal physical structures of string condensations.

Motivation & Objective

• To establish a topological holography dictionary between 3+1D string condensations and 2+1D gapless SPT phases.
• To classify a class of non-Lagrangian condensable algebras in 3+1D G-gauge theories as 'magnetic and simple' in Z[2VecG].
• To show that these algebras fully encode the topological properties of 2+1D gSPTs through duality.
• To reveal the physical structure of string condensations and their role in realizing gapped and gapless SPT phases.
• To provide a systematic classification of 2+1D gSPTs and their gauging via pre-modular categories.

Proposed method

• Uses braided fusion 2-categories to describe 3+1D topological orders and string condensations.
• Applies the theory of condensable E2-algebras in braided fusion 2-categories to model string condensation.
• Introduces 'magnetic and simple' condensable algebras in Z[2VecG] as a classification framework.
• Employs the topological holography dictionary to map 3+1D string condensations to 2+1D G-symmetric phases.
• Utilizes the sandwich construction and SymTO (symmetry topological order) to relate bulk string condensation to boundary gSPTs.
• Applies coupled layer constructions to realize string condensations physically, e.g., via inter-layer anyon condensation in Z2×Z4 gauge theories.

Experimental results

Research questions

• RQ1 How can string condensation in 3+1D topological orders be systematically classified and related to 2+1D gSPTs?
• RQ2 What is the role of non-Lagrangian condensable algebras in realizing gapless SPT phases?
• RQ3 How do topological invariants of gSPTs emerge from dual string condensation structures?
• RQ4 What is the physical realization of magnetic and simple condensable algebras in 3+1D gauge theories?
• RQ5 How does gauging of 2+1D gSPTs relate to pre-modular categories and categorical symmetry breaking?

Key findings

• A class of non-Lagrangian condensable algebras in 3+1D G-gauge theories—called 'magnetic and simple'—are shown to be dual to 2+1D gSPTs.
• The topological properties of gSPTs are fully extractable from the dual string condensation via the topological holography dictionary.
• The classification of these algebras in Z[2VecG] maps directly to the classification of 2+1D G-symmetric phases, including both gapped and gapless SPTs.
• Three distinct classes of gSPTs are identified through the classification, each with distinct braiding and fusion rules.
• The coupled layer construction realizes string condensation physically, showing that m1 strings ending on a twisted boundary fuse to e2 particles, with anyonic braiding statistics of i.
• The emergent anomaly in 0+1D gSPTs is captured by a nontrivial projective phase ρ(e2(k1,k2)), indicating ground state degeneracy that cannot be lifted without closing the gap.