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2012

Boundary Degeneracy of Topological Order

Wang, Juven, Wen, Xiao-Gang

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We introduce the concept of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that the boundary degeneracy provides richer information than the bulk degeneracy.

Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions.
By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries.
By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles.

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2011

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S. B. Bravyi, A. Y. Kitaev, arXiv: quant-ph/9811052

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Later we will show that for the Abelian topological orders, Γ ∂ α \Gamma^{\partial_{\alpha}} and Γ q p ∂ α \Gamma^{\partial_{\alpha}}_{qp} are the charge lattice Hilbert space of the quantized Chern-Simons theory

Cited in the paper.

Explicit results in the appendix of this paper

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For Abelian group, a finite direct product is a finite direct sum, so ⨁ α Γ ∂ α = ∏ α Γ ∂ α \bigoplus_{\alpha}\Gamma^{\partial_{\alpha}}=\prod_{\alpha}\Gamma^{\partial_{\alpha}} , with α \alpha as the finite number of boundaries. The direct sum is indeed the coproduct in the category of abelian groups

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Juven Wang, et al

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