Line defects in three-dimensional symmetry-protected topological phases

A three-dimensional (3D) symmetry-protected topological (SPT) phase by definition must have symmetry-protected nontrivial boundary states; namely, its two-dimensional (2D) boundary must be either gapless or degenerate. In this work we demonstrate that once we couple a 3D SPT phase to a lattice dynamical Z2 gauge field, in many cases the Z2 vison line excitation (line defect) can be viewed as a â€one-dimensional (1D) boundary†of the 3D SPT phase, and this line defect is guaranteed to have a gapless or degenerate spectrum, which is also protected by the symmetry of the SPT phase.