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 $(\mathcal{T})$ 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 $\mathcal{T}$ and translation invariant surface state is a double $[e\mathcal{T}m\mathcal{T}]$ 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 $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ with topological $\mathrm{\ensuremath{\Theta}}$ 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 $O(4)\phantom{\rule{4pt}{0ex}}\mathrm{NL}\ensuremath{\sigma}M$ to the Ising limit, we arrive at the Levin-Gu model with stripy modulation whose dislocation has nontrivial braiding statistics.