Geometry defects in bosonic symmetry-protected topological phases

Abstract

In this paper we focus on the interplay between geometry defects and\ntopological properties in bosonic symmetry protected topological(SPT) phases.\nWe start from eight copies of 3D time-reversal($\\mathcal{T}$) invariant\ntopological superconductors(TSC) on a crystal lattice. We melt the lattice by\ncondensation of disclinations and therefore restore the rotation symmetry. Such\ndisclination condensation procedure confines the fermion and afterwards turns\nthe system into a 3D boson topological liquid crystal(TCL). The low energy\neffective theory of this crystalline-liquid transition contains a topological\nterm inherited from the geometry axion response in TSC. In addition, we\ninvestigate the interplay between dislocation and superfluid vortex on the\nsurface of TCL. We demonstrate that the $\\mathcal{T}$ and translation invariant\nsurface state is a double $[e\\mathcal{T}m\\mathcal{T}]$ state with intrinsic\nsurface topological order. We also look into the exotic behavior of dislocation\nin 2D boson SPT state described by an $O(4)$ non-linear\n$\\sigma$-model(NL$\\sigma $M) with topological $\\Theta$-term. By dressing the\n$O(4)$ vector with spiral order and gauge the symmetry, the dislocation has\nmutual semion statistics with the gauge flux. Further reduce the $O(4)$\nNL$\\sigma M$ to the Ising limit, we arrive at the Levin-Gu model with stripy\nmodulation whose dislocation has nontrivial braiding statistics.\n