Braiding and Gapped Boundaries in Fracton Topological Phases

Abstract

We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it \textit{insufficient} to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is "nonreciprocal," that is, braiding an excitation $a$ around $b$ need not yield the same phase as braiding $b$ around $a$. Instead, we define fractonic "boundary braiding," which resolves these difficulties in the presence of a boundary. We then conjecture that a boundary of an Abelian fracton system is gapped if and only if a "boundary Lagrangian subgroup" of excitations is condensed at the boundary, this is a generalization of the condition for a gapped boundary in two spatial dimensions, but it relies on boundary braiding instead of bulk braiding. We also discuss the distinctness of gapped boundaries and transitions between different topological orders on gapped boundaries.

arXiv comment18 pages, 21 figures. v2: Typo fixes, added references. v3: Published version. Some corrections to figures

References in corpus (58)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Topological quantum memory
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• Models for gapped boundaries and domain walls
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids
• Quantum codes on a lattice with boundary
• Condensate induced transitions between topologically ordered phases
• Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code
• Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story
• Exotic topological order in fractal spin liquids
• Protected edge modes without symmetry
• Topological boundary conditions in abelian Chern-Simons theory
• Topological order in an exactly solvable 3D spin model
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Quantum Field Theory of X-Cube Fracton Topological Order and Robust Degeneracy from Geometry
• Glassy quantum dynamics in translation invariant fracton models
• Fracton Models on General Three-Dimensional Manifolds
• Fracton topological order via coupled layers
• Three-dimensional topological lattice models with surface anyons
• Fractal symmetries: Ungauging the cubic code
• A Time-Reversal Invariant Topological Phase at the Surface of a 3D Topological Insulator
• Foliated fracton order from gauging subsystem symmetries
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• A symmetry-respecting topologically-ordered surface phase of 3d electron topological insulators
• The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories
• Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory
• Classification of Topological Defects in Abelian Topological States
• Topological Quantum Glassiness
• Gapless Bosonic Excitation without symmetry breaking: Novel Algebraic Spin liquid with soft Gravitons
• Emergent Phases of Fractonic Matter
• Fracton topological phases from strongly coupled spin chains
• Topological Entanglement Entropy of Fracton Stabilizer Codes
• X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order
• Quantum mechanical and information theoretic view on classical glass transitions
• Twisted Fracton Models in Three Dimensions
• Boundary-bulk relation in topological orders
• Symmetric Fracton Matter: Twisted and Enriched
• Fractons from Partons
• Layer construction of 3D topological states and string braiding statistics
• Correlation function diagnostics for type-I fracton phases
• Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases
• Stable Gapless Bose Liquid Phases without any Symmetry
• Recoverable Information and Emergent Conservation Laws in Fracton Stabilizer Codes
• Deciphering the nonlocal entanglement entropy of fracton topological orders
• Universal entanglement signatures of foliated fracton phases
• Generalized ADE Classification of Gapped Domain Walls
• Foliated fracton order in the checkerboard model
• A simple anisotropic three-dimensional quantum spin liquid with fracton topological order
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Novel Algebraic Boson Liquid phase with soft Graviton excitations
• Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
• Generalized $U(1)$ Gauge Field Theories and Fractal Dynamics
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Projective plane and planar quantum codes

Cited by in corpus (35)

• Fracton Phases of Matter
• Towards classification of Fracton phases: the multipole algebra
• Foliated Quantum Field Theory of Fracton Order
• Topological Defect Networks for Fractons of all Types
• Fractonic Chern-Simons and BF theories
• Twisted foliated fracton phases
• Fracton fusion and statistics
• Sorting topological stabilizer models in three dimensions
• Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies
• Anisotropic layer construction of anisotropic fracton models
• Gauging permutation symmetries as a route to non-Abelian fractons
• Bifurcating entanglement-renormalization group flows of fracton stabilizer models
• Compactifying fracton stabilizer models
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases
• Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter
• Majorana Quantum Lego, a Route Towards Fracton Matter
• Boundary theory of the X-cube model in the continuum
• Planar p-String Condensation: Chiral Fracton Phases from Fractional Quantum Hall Layers and Beyond
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Gapped boundaries of (3+1)d topological orders
• Fracton Self-Statistics
• Designer non-Abelian fractons from topological layers
• Non-Abelian defects in Fracton phase of matter
• Boundary Modes in the Chamon Model
• Screw dislocations in the X-cube fracton model
• Emergent fermionic gauge theory and foliated fracton order in the Chamon model
• Renormalization of Ising cage-net model and generalized foliation
• No Strings Attached: Boundaries and Defects in the Cubic Code
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• A New Model for Fractons, Fluxons, and Freeons
• Non-invertible symmetries in finite-group gauge theory
• Breakdown of the thermodynamic limit in quantum spin and dimer models
• There and Back Again: A Gauging Nexus between Topological and Fracton Phases