Geometry defects in Bosonic symmetry protected topological phases

TL;DR

This paper explores how geometric defects like disclinations and dislocations influence the topological properties of bosonic SPT phases, revealing novel surface states and defect braiding statistics through theoretical models and lattice constructions.

Contribution

It introduces a framework connecting geometry defects with topological phases, including disclination condensation leading to a bosonic topological liquid crystal and analysis of defect braiding in SPT models.

Findings

Disclination condensation transforms TSC into a bosonic topological liquid crystal.

Surface states exhibit double $[e\mathcal{T}m\mathcal{T}]$ topological order.

Dislocation defects exhibit nontrivial braiding statistics with gauge flux.

Abstract

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 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 2D boson SPT state described by an $O(4)$ non-linear $\sigma$-model(NL$\sigma $M) with topological $\Theta$-term. By dressing the $O(4)$ vector with spiral order and gauge the symmetry, the dislocation has mutual semion statistics with the gauge flux. Further reduce 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.