Diagrammatics for Bose condensation in anyon theories

Abstract

Phase transitions in anyon models in (2+1)-dimensions can be driven by condensation of bosonic particle sectors. We study such condensates in a diagrammatic language and explicitly establish the relation between the states in the fusion spaces of the theory with the condensate, to the states in the parent theory using a new set of mathematical quantities called vertex lifting coefficients (VLCs). These allow one to calculate the full set of topological data ($S$-, $T$-, $R$- and $F$-matrices) in the condensed phase. We provide closed form expressions of the topological data in terms of the VLCs and provide a method by which one can calculate the VLCs for a wide class of bosonic condensates. We furthermore furnish a concrete recipe to lift arbitrary diagrams directly from the condensed phase to the original phase, such that they can be evaluated using the data of the original theory and a limited number of VLCs. Some representative examples are worked out in detail.

20 pages, 1 figure, many diagrams

References in corpus (49)

• Quantum Spin Hall Effect in Graphene
• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Quantum Spin Hall Effect
• String-net condensation: A physical mechanism for topological phases
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Braiding statistics approach to symmetry-protected topological phases
• Models for gapped boundaries and domain walls
• Condensate induced transitions between topologically ordered phases
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Notes on the K3 Surface and the Mathieu group M_24
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Protected edge modes without symmetry
• The Quantum Double Model with Boundary: Condensations and Symmetries
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• A matrix S for all simple current extensions
• Correspondences of ribbon categories
• Quantum groups and nonabelian braiding in quantum Hall systems
• Bicategories for boundary conditions and for surface defects in 3-d TFT
• Explicit tensor network representation for the ground states of string-net models
• A theory of topological edges and domain walls
• Interferometry of non-Abelian Anyons
• Generalizations and limitations of string-net models
• Condensation of achiral simple currents in topological lattice models: a Hamiltonian study of topological symmetry breaking
• Quantum symmetries in discrete gauge theories
• Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory
• Topological Phase Transitions in the Golden String-Net Model
• Measurement-Only Topological Quantum Computation via Anyonic Interferometry
• Phase transitions in topological lattice models via topological symmetry breaking
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Space-Time Geometry of Topological phases
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Non-Abelian anyons: when Ising meets Fibonacci
• Ising anyons with a string tension
• Identifying Non-Abelian Topological Order through Minimal Entangled States
• Decoherence of Anyonic Charge in Interferometry Measurements
• Boundaries, defects and Frobenius algebras
• The modular S-matrix as order parameter for topological phase transitions
• Condensate-induced transitions and critical spin chains
• Clebsch-Gordan and 6j-coefficients for rank two quantum groups
• The classifying algebra for defects
• A hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points
• Tensor-product representations for string-net condensed states

Cited by in corpus (64)

• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Anyon condensation and its applications
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Boson Condensation in Topologically Ordered Quantum Liquids
• Cage-Net Fracton Models
• Entanglement Entropy From Tensor Network States for Stabilizer Codes
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Generalized ADE Classification of Gapped Domain Walls
• Fermion condensation and super pivotal categories
• Gapless edges of 2d topological orders and enriched monoidal categories
• Anisotropic layer construction of anisotropic fracton models
• Ising Anyons in Frustration-Free Majorana-Dimer Models
• Topological holography, quantum criticality, and boundary states
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• A Unified Framework of Topological Phases with Symmetry
• Topological Quantum Computation with Gapped Boundaries
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Simple-current algebra constructions of 2+1D topological orders
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Composing topological domain walls and anyon mobility
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Seeing topological entanglement through the information convex
• Folding approach to topological orders enriched by mirror symmetry
• Evolution of Dynamical Signature in the X-cube Fracton Topological Order
• Entropic Topological Invariants in Three Dimensions
• Topological lattice field theories from intertwiner dynamics
• Constructing topological models by symmetrization: A PEPS study
• Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model
• No-Go Theorem for Boson Condensation in Topologically Ordered Quantum Liquids
• Simple analog of the black-hole information paradox in quantum Hall interfaces
• U(1) spin Chern-Simons theory and Arf invariants in two dimensions
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Signatures of broken parity and time-reversal symmetry in generalized string-net models
• PH-Pfaffian order in a translationally and rotationally invariant system
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Russian doll spectrum in a non-Abelian string-net ladder
• Anyon condensation and confinement transition in a Kitaev spin liquid bilayer
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Domain Walls in 4d N=1 Supersymmetric Yang-Mills
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Ternary Logic Design in Topological Quantum Computing
• Chiral topologically ordered insulating phases in arrays of interacting integer quantum Hall islands
• Mirror anomaly in fermionic topological orders
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase
• Entanglement Entropy of Topological Orders with Boundaries
• Matrix formulation for non-Abelian families
• Edge Theories for Anyon Condensation Phase Transitions
• Ising versus $SU(2{)}_2$ string-net ladder
• Diagrammatics, Pentagon Equations, and Hexagon Equations of Topological Orders with Loop- and Membrane-like Excitations
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Ordering the topological order in the fractional quantum Hall effect
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Anomalies in mirror symmetry enriched topological orders
• Competing automorphisms and disordered Floquet codes
• Proliferation transitions from a topological phase in $2+1$ dimensions
• Studies of braided non-Abelian anyons using anyonic tensor networks
• Note on searching for critical lattice models as entropy critical points from strange correlator