Categories of quantum liquids I

Abstract

We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a $\ast$-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all $n$D quantum liquids form a $\ast$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.

36 pages, a minor revision, including a correction of an typo in Theorem 5.5, an illustrative picture in Hypothesis 5.3 and refinement of a few sentences

References in corpus (7)

• Local stabilizer codes in three dimensions without string logical operators
• Duality and defects in rational conformal field theory
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Cardy algebras and sewing constraints, I
• One dimensional gapped quantum phases and enriched fusion categories
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

Cited by in corpus (26)

• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• String Condensations in 3+1D and Lagrangian Algebras
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Topological Orders in (4+1)-Dimensions
• Quantum Current and Holographic Categorical Symmetry
• Finite Semisimple Module 2-Categories
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Categories of quantum liquids II
• String operators for Cheshire strings in topological phases
• Category of SET orders
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Fusion of one-dimensional gapped phases and their domain walls
• Enriched monoidal categories I: centers
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Generalized Landau Paradigm for quantum phases and phase transitions
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Tube Category, Tensor Renormalization and Topological Holography
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Condensation Completion and Defects in 2+1D Topological Orders
• Categorical computation