Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases

TL;DR

The paper establishes a universal bulk-boundary correspondence for 3D bosonic SPT phases with finite Abelian unitary symmetry by relating bulk loop-braiding invariants $\{\Theta_{i,l}, \Theta_{ij,l}, \Theta_{ijk,l}\}$ to gauged-surface braiding data $\{\Phi_{\mu}, \Phi_{\mu\nu}, \Omega_{i\mu}, \Omega_{ij\mu}, x_{il}^\mu\}$ via three key equations. The approach gauges the symmetry to access braiding statistics of bulk vortex loops and surface excitations, then derives exact, integer-coefficient linear relations that determine bulk invariants from surface data. The work demonstrates the completeness of the bulk and surface data in distinguishing 3D SPTs (for Abelian G) and proves equivalence with Chen–Burnell–Vishwanath–Fidkowsky’s group-cohomology conjecture for Abelian groups, thereby linking topological invariants to surface anomalies. It also provides concrete surface examples, constraints on possible surfaces, and a rigorous bridge to group-cohomology data, suggesting extensions to non-Abelian surfaces and fermionic SPTs as future directions.

Abstract

We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological (SPT) 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 is defined in terms of the statistics of bulk excitations, while the surface data is defined in terms of the statistics of surface excitations. An appealing property of this data is that it is plausibly complete in the sense that the bulk data uniquely distinguishes each 3D SPT phase, while the surface data uniquely distinguishes each gapped, symmetric surface. Our correspondence applies to any 3D bosonic SPT 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.

Figures (14)

• 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 $\Omega_{i\mu}$ [panel (a)] and $\Omega_{ij\mu}$ [panel (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 [panel (a)] in the bulk into a braiding process [panel (d)] of vortex arches near the surface. The deformation splits into three steps, (a)$\rightarrow$(b), (b)$\rightarrow$(c), and (c)$\rightarrow$(d). The gray curves in (a), (b), (c) and (d) are trajectories of some points on $\alpha$; the red curve in (c) is the trajectory of one of the endpoints of $\sigma$.