Defects between gapped boundaries in two-dimensional topological phases of matter

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 multifusion 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 $\mathfrak{D}\left(S_3\right)$.

Figure 1

Square lattice for the toric code Hamiltonian. For the example vertex $\mathfrak{v}$ (plaquette $\mathfrak{p}$), the edges in $\text{star}\left(\mathfrak{v}\right)$ $\left[\text{boundary}\right(\mathfrak{p}\left)\right]$ 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 $\left\{\right|0\rangle ,|1\rangle ,\cdots ,|p-1\rangle \}$, 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.

Figure 6

Definition of the Hamiltonian $H_{\text{dft}}$. The line $\mathfrak{L}$ consists of all vertices and edges along the purple line which divides the blue and red regions. The region ${\mathfrak{r}}_1$ consists of all vertices, plaquettes, and edges within the blue shaded rectangle, and all blue vertices on the border of the hole. ${\mathfrak{r}}_2$ is defined similarly. As before, the bulk $\mathfrak{B}$ denotes all other vertices, edges, and plaquettes (i.e., the black and white region). The resulting defects live on the orange cilia.

Figure 7

More general definition of the Hamiltonian $H_{\text{dft}}$. There are $n$ nonintersecting piecewise-linear borders ${\mathfrak{L}}_1,\cdots {\mathfrak{L}}_n$, which separate the hole into $n+1$ regions ${\mathfrak{r}}_1,\cdots {\mathfrak{r}}_{n+1}$. Each ${\mathfrak{L}}_i$ is formed from a chain of edges on the lattice or its dual. A total of $2n$ defects are created, one at each endpoint of each border ${\mathfrak{L}}_i$. Each border ${\mathfrak{L}}_i$ consists of all vertices and edges that the corresponding purple line(s) cross, and includes the vertex on the boundary of the hole (purple dots). Each region ${\mathfrak{r}}_j$ consists of all vertices, plaquettes, and edges within the corresponding blue shaded region, including the vertices along the border of the hole (blue dots), but not along the purple lines. The bulk $\mathfrak{B}$ will denote all other vertices, edges, and plaquettes (i.e., the black and white region).

Figure 8

Ground-state degeneracy of $n$ boundary defects on a hole with $n$ boundary types given by indecomposable modules ${\mathcal{M}}_1$,..., ${\mathcal{M}}_n$.

Figure 9

Crossed condensation. (A) A symmetry defect $Y_{-}$ and its dual $Y_{+}$ in the bulk. (B) After the crossed condensation, the bulk symmetry defect turns in to a boundary defect. Furthermore, the condensation procedure modifies one side of the gapped boundary ${\mathcal{A}}_i$ by the ${\mathbb{Z}}_2$ action to produce the gapped boundary ${\mathcal{A}}_j$.

Figure 10

Six boundary defects on a ring. The mathematical braiding of two defects $X_{12}$ and $X_{21}$ may be implemented physically by introducing ancilla boundary defects and using forced measurements.

Figure 11

Illustration of a defect site absorbing a bulk anyon $a$.

Figure 12

Definition of the Hamiltonian $H_{\text{gn}}$. The region ${\mathfrak{r}}_1$ (respectively, ${\mathfrak{r}}_2$) consists of all blue (red) vertices, plaquettes, and edges. The bulk $\mathfrak{B}$ consists of everything else (black and white). The genons are created on the two orange cilia.

Figure 13

Illustration of the two-step construction for the $A+B+2C/A+B+2F$ boundary defects.