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arXiv:2212.09779 · Physical review. B./Physical review. B

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

Zhu-Xi Luo

Published Mar 31, 2023• 19 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.107.125425

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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 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 aaa around bbb need not yield the same phase as braiding bbb around aaa. 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.
We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold M\mathcal{M}M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as M\mathcal{M}M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M\mathcal{M}M is a disk or a cylinder. The correspondence turns out to be very different from that of the chiral topological orders. When M\mathcal{M}M has the topology of a cylinder, different boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. Charge (smooth) and flux (rough) boundaries of toric code, more general ZN\mathbb{Z}_NZN​ models, and the simplest non-abelian case of doubled Fibonacci are demonstrated.
We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface M\mathcal{M}M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as M\mathcal{M}M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M\mathcal{M}M is a disk or a cylinder. When M\mathcal{M}M has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, ZN{\mathbb{Z}}_{N}ZN​ theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.
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