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.
Highlights
A gapped quantum many-body system is said to belong to a nontrivial symmetry-protected topological (SPT) phase if it satisfies three conditions
We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological phases with unitary symmetries
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 study the braiding statistics of excitations of the resulting gauge theory
Summary
A gapped quantum many-body system is said to belong to a nontrivial symmetry-protected topological (SPT) phase if it satisfies three conditions. (2) and (3) are measurable in the sense that they can be extracted from a microscopic model by following a well-defined procedure These quantities are topological invariants; that is, they remain fixed under continuous, symmetry-preserving deformations of the (ungauged) Hamiltonian that do not close the bulk or surface gap, respectively [42]. [17,44,45,46] for other related discussions of anomalies.) If this anomaly is nonzero, the corresponding anyon system cannot be realized in a strictly 2D lattice model Given these two facts, Chen et al conjectured that gapped symmetric surfaces of the group cohomology model always have an anomaly ν that matches the ν ∈ H4ðG; Uð1ÞÞ defining the bulk cohomology model.
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