Boundary Degeneracy of Topological Order

TL;DR

This work defines boundary degeneracy (GSD on manifolds with gapped boundaries) as a refined probe of topological order, showing that boundary GSD depends on boundary gapping conditions beyond the bulk fusion data. It develops boundary fully gapping rules within a K-matrix Chern-Simons framework, derives a general GSD formula GSD = |L_{qp∩e} / ⊕ Γ^{∂_α}|, and analyzes edge quantization, stability, and the Hilbert-space structure. Through explicit Abelian examples (e.g., Z_k, toric code, double-semion, and U(1)_k × U(1)_{−k}), it demonstrates that boundary GSD can distinguish orders that share the same bulk GSD or fusion algebra, notably separating Z_2 toric code from the Z_2 double-semion model. The results offer a practical diagnostic for intrinsic topological order, provide experimental/testable predictions via flux insertion on cylinders, and point to future extensions to non-Abelian orders and nontrivial symmetry contexts.

Abstract

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. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the $Z_2$ toric code and $Z_2$ double-semion model (more generally, the $Z_k$ gauge theory and the $U(1)_k \times U(1)_{-k}$ non-chiral fractional quantum Hall state at even integer $k$) can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.

Figures (4)

• Figure 1: Topologically ordered states on a 2D manifold with 1D boundaries: (a) Illustration of fusion rules and total neutrality, where anyons are transported from one boundary to another (red arrows), or when they fuse into physical excitations (blue arrows), on a manifold with five boundaries. (b) A higher genus compact surface with boundaries (thus with punctures): a genus-3 manifold with five boundaries.
• Figure 2: (a) The same boundary conditions on two ends of a cylinder allow a pair of cycles $[c_{x}],[c_{z}]$ of a qubit, thus $\mathop{\mathrm{GSD}}=2$. Different boundary conditions do not, thus $\mathop{\mathrm{GSD}}=1$. (b) The same boundary conditions allow z- or x-strings connect two boundaries. Different boundary conditions do not.
• Figure 3: (a) Anyon (qp$_1$) is transported from the bottom to the top of the cylinder. (b) Physical non-fractionalized excitation e$_s$ splits into a pair of anyons (qp$_1$ to the bottom, qp$_2$ to the top).
• Figure 4: Glue the punctured cylinders to form a genus $g$ Riemann surface with $\eta'$ punctures. Start from (a), firstly identify left and right $\vartriangleright$ arrows of each square to form a number $g$ of punctured cylinders. Then glue h$_{j,L}$ and h$_{j,R}$ (red dotted circles) together for $1\leq j \leq g-1$, and glue h$_{i,T}$ and h$_{i,B}$ (blue arrows) together for $1\leq i \leq g$, which yields (b), equivalently as a genus $g$ Riemann surface (c). The extra $\eta'$ punctures are indicated here as a shaded blue puncture in the left most handle.