Gapped boundaries of (3+1)d topological orders

Abstract

Given a gapped boundary of a (3+1)d topological order (TO), one can stack on it a decoupled (2+1)d TO to get another boundary theory. Should one view these two boundaries as "different"? A natural choice would be no. Different classes of gapped boundaries of (3+1)d TO should be defined modulo these decoupled (2+1)d TOs. But is this enough? We examine the possibility of coupling the boundary of a (3+1)d TO to additional (2+1)d TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk point-like excitations, when touching the boundary, become excitations in the added (2+1)d phase, while the string-like excitations in the bulk may end on the boundary but with endpoints dressed by some other excitations in the (2+1)d phase. For a good definition of "class" for gapped boundaries of (3+1)d TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the string-like excitations that can end on the boundary, whatever their endpoints are. A concrete example is the (3+1)d bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the $m$-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a (3+1)d TO characterized by a finite group $G$, different classes correspond to different subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model and field theory.

arXiv commentv2 with minor edits; references added

References in corpus (60)

• String-net condensation: A physical mechanism for topological phases
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• Fractons
• 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
• Condensate induced transitions between topologically ordered phases
• Fracton Phases of Matter
• Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story
• Exotic topological order in fractal spin liquids
• Protected edge modes without symmetry
• Topological order in an exactly solvable 3D spin model
• Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory
• Exotic $U(1)$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Ring exchange, the Bose metal, and bosonization in two dimensions
• Theory of defects in Abelian topological states
• Anyon condensation and its applications
• Towards classification of Fracton phases: the multipole algebra
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Field Theories With a Vector Global Symmetry
• Anomalous Symmetry Fractionalization and Surface Topological Order
• Interferometry of non-Abelian Anyons
• Gapless Bosonic Excitation without symmetry breaking: Novel Algebraic Spin liquid with soft Gravitons
• Probing Non-Abelian Statistics with QuasiParticle Interferometry
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Boson Condensation in Topologically Ordered Quantum Liquids
• Full Dyon Excitation Spectrum in Generalized Levin-Wen Models
• Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory
• Boundary Degeneracy of Topological Order
• X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order
• Emergent Gravity at a Lifshitz Point from a Bose Liquid on the Lattice
• Exotic ${\mathbb{Z}}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Boundary-bulk relation in topological orders
• Symmetric Fracton Matter: Twisted and Enriched
• Layer construction of 3D topological states and string braiding statistics
• Bulk-boundary correspondence in (3+1)-dimensional topological phases
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Emergence of helicity +/- 2 modes (gravitons) from qubit models
• Broken quantum symmetry and confinement phases in planar physics
• Anomaly Inflow for Subsystem Symmetries
• Fractonic Chern-Simons and BF theories
• Generalized ADE Classification of Gapped Domain Walls
• More Exotic Field Theories in 3+1 Dimensions
• Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases
• Pauli stabilizer models of twisted quantum doubles
• Braiding and Gapped Boundaries in Fracton Topological Phases
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• $(k,n)$-fractonic Maxwell theory
• Twisted Quantum Double Model of Topological Orders with Boundaries
• The breaking of quantum double symmetries by defect condensation
• Higher-Rank Tensor Non-Abelian Field Theory: Higher-Moment or Subdimensional Polynomial Global Symmetry, Algebraic Variety, Noether's Theorem, and Gauging
• Decoherence of Anyonic Charge in Interferometry Measurements
• Boundary theory of the X-cube model in the continuum
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• No-Go Theorem for Boson Condensation in Topologically Ordered Quantum Liquids
• Ungappable edge theories with finite dimensional Hilbert spaces

Cited by in corpus (11)

• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• Local Modules in Braided Monoidal 2-Categories
• Non-invertible symmetries in finite-group gauge theory
• Generalized Landau Paradigm for quantum phases and phase transitions
• Remote detectability from entanglement bootstrap I: Kirby's torus trick
• Magic Boundaries of 3D Color Codes
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries