One dimensional gapped quantum phases and enriched fusion categories

2021/08/31 by Liang Kong, Xiao-Gang Wen, Hao Zheng · 31 citations
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Categorical variable #Condensed matter physics #Dimension (graph theory) #Fusion #Fusion rules #Gapless playback #Geometry #Mathematics #Physics #Physics of Superconductivity and Magnetism #Pure mathematics #Quantum #Quantum many-body systems #Quantum mechanics #Quantum phase transition #Quantum phases #Statistics #Symmetry (geometry) #Theoretical physics #cond-mat.str-el #hep-th #math.CT

paper · pdf · open access · doi:10.1007/jhep03(2022)022

published in Journal of High Energy Physics 2022(3) (Springer Nature) · 27 pages. We add some remarks and references

openalex publication_date 2022/03/01 · arxiv created 2022/03/10 · arxiv updated 2022/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these models. In the end, we provide a classification of and the categorical descriptions of all 1-dimensional (the spatial dimension) gapped quantum phases with a finite onsite symmetry.

Citations

• Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Symmetry protected topological orders and the group cohomology of their symmetry group
• String-net condensation: A physical mechanism for topological phases
• Local stabilizer codes in three dimensions without string logical operators
• Models for gapped boundaries and domain walls
• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Quantum codes on a lattice with boundary
• Classical and quantum conformal field theory
• Condensations in higher categories
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Orbifolds and minimal modular extensions

Cited by

• von Neumann Subfactors and Non-invertible Symmetries