Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae

Abstract

We investigate the composite systems consisting of topological orders separated by gapped domain walls. We derive a pair of domain-wall Verlinde formulae, that elucidate the connection between the braiding of interdomain excitations labeled by pairs of anyons in different domains and quasiparticles in the gapped domain wall with their respective fusion rules. Through explicit non-Abelian examples, we showcase the calculation of such braiding and fusion, revealing that the fusion rules for interdomain excitations are generally fractional or irrational. By investigating the correspondence between composite systems and anyon condensation, we unveil the reason for designating these fusion rules as symmetry fractionalized (irrationalized) fusion rules. Our findings hold promise for applications across various fields, such as topological quantum computation, topological field theory, and conformal field theory.

14 pages, 7 figures

References in corpus (21)

• Non-Abelian Anyons and Topological Quantum Computation
• Generalized Global Symmetries
• Condensate induced transitions between topologically ordered phases
• Orbifold groupoids
• Characterizing topological order by studying the ground states of an infinite cylinder
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• A theory of topological edges and domain walls
• Topological Quantum Compiling
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Universal Quantum Computation with Gapped Boundaries
• Categories of quantum liquids I
• Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via Entanglement Entropy and Modular S-matrix
• The fusion algebra of bimodule categories
• Entanglement bootstrap approach for gapped domain walls
• Boundary Hamiltonian theory for gapped topological orders
• Domain wall topological entanglement entropy
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• Boundary and domain wall theories of 2d generalized quantum double model
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Bulk from boundary in finite CFT by means of pivotal module categories

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• Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization
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• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization