On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter

TL;DR

The paper develops a general, exactly solvable framework for boundary defects between gapped boundaries in two-dimensional topological phases, realized via commuting projector Hamiltonians in untwisted Dijkgraaf-Witten theories and modeled algebraically by a multi-fusion category. It establishes a bulk–edge correspondence that connects boundary defects to bulk symmetry defects, and shows that the projective braiding of boundary defects is governed by the bulk $G$-crossed braiding, enabling analysis of defect-dense networks and their quantum Information properties. Three principal classes—Majorana/parafermion boundary defects, genons, and boundary defects in $rak D(S_3)$—are treated in detail, with explicit simple defect labels $(T,R)$, quantum-dimension formulas, fusion rules, and exact degeneracies, illustrating their potential for topological quantum computation. The work unifies physical realizations with a robust algebraic framework, offering pathways to universal gates and measurement-based braiding through crossed condensation and boundary defect fusion, and suggesting extensions to twisted DW theories and Levin-Wen models.

Abstract

Defects between gapped boundaries provide a possible physical realization of projective non-abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Witten theory, and use these to describe their topological properties such as their quantum dimension. By modeling the algebraic structure of boundary defects through multi-fusion categories, we establish a bulk-edge correspondence between certain boundary defects and symmetry defects in the bulk. Even though it is not clear how to physically braid the defects, this correspondence elucidates the projective braid statistics for many classes of boundary defects, both amongst themselves and with bulk anyons. Specifically, three such classes of importance to condensed matter physics/topological quantum computation are studied in detail: (1) A boundary defect version of Majorana and parafermion zero modes, (2) a similar version of genons in bilayer theories, and (3) boundary defects in $\mathfrak{D}(S_3)$.

Figures (13)

• Figure 1: Square lattice for the toric code Hamiltonian. For the example vertex $\mathfrak{v}$ (plaquette $\mathfrak{p}$), the edges in $\text{star}(\mathfrak{v})$ ($\text{boundary}(\mathfrak{p})$) are highlighted in red (blue).
• Figure 2: Definition of the Hamiltonian for parafermion zero modes. Note the hole $\mathfrak{h}_1$ includes all direct and dual vertices on its border (the red/blue boldfaced lines on the lattice) but NOT the edges on the border. The red lines are rough boundaries, and the blue lines are smooth boundaries. Parafermion zero modes are created at the four corners of $\mathfrak{h}_1$ (purple crosses).
• Figure 3: Ground state degeneracy of the parafermion Hamiltonian. The ground state of $H_{\text{parafermion}}$ is $p$-fold degenerate, with a basis given by $\{ |0\rangle, |1\rangle, ... |p-1\rangle \}$, where $|0\rangle$ is the ground state of the bulk Hamiltonian $H_{\text{bulk}}$, and $|i\rangle$ is obtained from $|0\rangle$ by applying the operator $X_p^i$ to all qudits along the dark red line.
• Figure 4: Example: defining $H_{\text{G.B.}}$ in the case of two holes on an infinite lattice.
• Figure 5: Definition of the Hamiltonian $H_{\text{G.B.}}$, in cases where some (e.g. $\mathfrak{h}_2$) or all (e.g. $\mathfrak{h}_1$) of the hole's sides lie on the dual lattice. For $\mathfrak{h}_2$, four boundary defects are created, one at each corner.

Theorems & Definitions (8)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Definition 5.1
• Definition 5.2
• Definition 5.3
• Corollary 6.2
• Conjecture 7.1