Gapped boundaries of (3+1) -dimensional topological order

Abstract

Given a gapped boundary of a $(3+1)$-dimensional [$(3+1)\mathrm{D}$] topological order (TO), one can stack on it a decoupled $(2+1)\mathrm{D}$ 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 $(3+1)\mathrm{D}$ TO should be defined modulo the decoupled $(2+1)\mathrm{D}$ TOs. But is this enough? We examine the possibility of coupling the boundary of a $(3+1)\mathrm{D}$ TO to additional $(2+1)\mathrm{D}$ TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk pointlike excitations, when touching the boundary, become excitations in the added $(2+1)\mathrm{D}$ phase, while the stringlike excitations in the bulk may end on the boundary but with end points dressed by some other excitations in the $(2+1)\mathrm{D}$ phase. For a good definition of ``class'' for gapped boundaries of $(3+1)\mathrm{D}$ TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the stringlike excitations that can end on the boundary, whatever their end points are. A concrete example is the $(3+1)\mathrm{D}$ 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 $(3+1)\mathrm{D}$ TO characterized by a finite group $G$, different classes correspond to different normal subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model, and field theory.