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.

v2 with minor edits; references added

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• 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