Geometry defects in bosonic symmetry-protected topological phases

In this paper we focus on the interplay between geometry defects and topological properties in bosonic symmetry-protected topological (SPT) phases. We start from eight copies of 3D time-reversal (T) invariant topological superconductors (TSC) on a crystal lattice. We melt the lattice by condensation of disclinations and therefore restore the rotation symmetry. Such a disclination condensation procedure confines the fermion and afterwards turns the system into a 3D boson topological liquid crystal (TCL). The low energy effective theory of this crystalline-liquid transition contains a topological term inherited from the geometry axion response in TSC. In addition, we investigate the interplay between dislocation and superfluid vortex on the surface of TCL. We demonstrate that the T and translation invariant surface state is a double [eT mT] state with intrinsic surface topological order. We also look into the exotic behavior of dislocation in the 2D boson SPT state described by an O(4) nonlinear sigma model (NL sigma M) with topological Theta term. By dressing the O(4) vector with spiral order and gauging the symmetry, the dislocation has mutual semion statistics with the gauge flux. Further reducing the O(4) NL sigma M to the Ising limit, we arrive at the Levin-Gu model with stripy modulation whose dislocation has nontrivial braiding statistics.