Bulk-Boundary Correspondence for Three-Dimensional Symmetry-Protected Topological Phases

We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological phases with unitary symmetries. The correspondence consists of three equations that relate bulk properties of these phases to properties of their gapped, symmetry-preserving surfaces. Both the bulk and surface data appearing in our correspondence are defined via a procedure in which we gauge the symmetries of the system of interest and then study the braiding statistics of excitations of the resulting gauge theory. The bulk data are defined in terms of the statistics of bulk excitations, while the surface data are defined in terms of the statistics of surface excitations. An appealing property of these data is that it is plausibly complete in the sense that the bulk data uniquely distinguish each 3D symmetry-protected topological phase, while the surface data uniquely distinguish each gapped, symmetric surface. Our correspondence applies to any 3D bosonic symmetry-protected topological phase with finite Abelian unitary symmetry group. It applies to any surface that (1) supports only Abelian anyons and (2) has the property that the anyons are not permuted by the symmetries.

Popular Summary

For certain classes of insulating materials, it is possible to derive a very precise connection between the properties of the bulk and the properties of the surface. A connection of this kind is known as a bulk-boundary correspondence. Such correspondences are useful both theoretically and experimentally, but they are often only derivable for noninteracting or low-dimensional systems. Here, we take a step toward addressing this deficiency by deriving a bulk-boundary correspondence for a large class of strongly interacting, three-dimensional insulators.

Our theoretical analysis focuses on connecting bulk data (consisting of three tensors) and surface data (consisting of five tensors). The insulators that we study can be thought of as interacting analogs of the recently discovered “topological insulators.” However, they differ from these systems in two respects: (i) They are built of bosons instead of fermions, and (ii) they are invariant under an arbitrary Abelian unitary symmetry group instead of the time-reversal and charge-conservation symmetry that characterize topological insulators. For these bosonic insulators, we can define a set of measurable quantities that characterize the bulk and the boundary of our systems. Our bulk-boundary correspondence consists of a set of three algebraic equations that express the bulk data in terms of the surface data.

While the problem of how to incorporate antiunitary symmetries like time reversal is more challenging and represents an important direction for future work, we expect that our results can be easily extended to fermionic insulators.

Figure 1

(a) Three-loop braiding process. (b) Cross section of the braiding process in the plane containing the base loop $\sigma$.

Figure 2

Sketch of excitations on and near the surface. Above the surface is the gauged SPT model and below is the vacuum.

Figure 3

Braiding processes that define the surface data ${\mathrm{\Omega }}_{i\mu }$ (a) and ${\mathrm{\Omega }}_{ij\mu }$ (b).

Figure 4

Interpreting $x_{il}^{\mu }$ through a 3D thought experiment. See the main text for details.

Figure 5

Deforming a three-loop braiding process (a) in the bulk into a braiding process (d) of vortex arches near the surface. The deformation splits into three steps, $(\mathrm{a})\to (\mathrm{b})$, $(\mathrm{b})\to (\mathrm{c})$, and $(\mathrm{c})\to (\mathrm{d})$. The gray curves in (a)–(d) are trajectories of some points on $\alpha$; the red curve in (c) is the trajectory of one of the end points of $\sigma$.

Figure 6

Splitting $\alpha$ into $\tilde{\alpha }$ and $\mathcal{X}$, and splitting $\beta$ into $\tilde{\beta }$ and $\mathcal{Y}$.

Figure 7

Folding the surface and straightening the vortex lines in Fig. 6. In between the top and bottom parts of the surface is the gauged SPT model, and outside is the vacuum.

Figure 8

A 2D braiding process that shows that ${\mathrm{\Theta }}_{ij,l}$ vanishes for strictly 2D systems. See the main text for details.

Figure 9

Deforming a process in which $\alpha$ is braided around $\beta$ and simultaneously $\overline{\alpha }$ is braided around $\overline{\beta }$ in the opposite direction. In the initial state $|{\mathrm{\Psi }}^{\prime }⟩$ before the braiding starts, the two pairs $\alpha$, $\overline{\alpha }$ and $\beta$, $\overline{\beta }$ are both in the vacuum fusion channel. See the main text for details.

Figure 10

The equivalence between Eq. (61) and the bulk-boundary formulas, Eqs. (30)–(32). The arrow (a) represents the one-to-one mapping between the surface data used in Eq. (61) and in Eqs. (30)–(32). The arrow (b) represents the one-to-one mapping between the bulk data used in Eq. (61) and in Eqs. (30)–(32). Arrows (c) and (d) represent Eq. (61) and Eqs. (30)–(32), respectively. We show that the whole diagram commutes in Sec. 7d.

Figure 11

Definition of $\omega (a,b)$. Panel (a) shows two defect lines labeled by $a$ and $b$. In (b), we fuse the two defect lines and get a defect line corresponding to $a+b$. This defect line differs from the representative $a+b$ defect line shown in (c) by an anyon $\omega (a,b)$.

Figure 12

Diagrammatic proof that ${\mathrm{\Omega }}_{i\mu }$ is well defined.

Figure 13

Diagrammatic proof that ${\mathrm{\Omega }}_{ij\mu }$ is well defined.

Figure 14

Fusing vortex $\alpha$ and its antivortex $\overline{\alpha }$. The resulting doublet $(\mathcal{X},\mathcal{Y})$ is viewed as a single anyon from the 2D slab point of view.