Wave functions of bosonic symmetry protected topological phases

We study the structure of the ground-state wave functions of bosonic symmetry protected topological (SPT) insulators in three space dimensions. We demonstrate that the differences with conventional insulators are captured simply in a dual vortex description. As an example, we show that a previously studied bosonic topological insulator with both global U(1) and time-reversal symmetry can be described by a rather simple wave function written in terms of dual “vortex ribbons.” The wave function is a superposition of all the vortex-ribbon configurations of the boson, and a factor $(-1)$ is associated with each self-linking of the vortex ribbons. This wave function can be conveniently derived using an effective field theory of the SPT phase in the strong-coupling limit, and it naturally explains all the phenomena of this SPT phase discussed previously. The ground-state structure for other three-dimensional (3D) bosonic SPT phases are also discussed similarly in terms of vortex loop gas wave functions. We show that our methods reproduce known results on the ground-state structure of some 2D SPT phases.

Figure 1

The wave function of the 3D bosonic SPT discussed in this paper is a superposition of all of the configurations of vortex ribbons with factor $(-1)$ associated with each self-linking.

Figure 2

(a) When the symmetry is U$\left(1\right)\times \mathrm{U}\left(1\right),$ the bulk wave function is a superposition of two flavors of vortex loops with factor $(-1)$ attached to each linking. (b)–(f) Braiding between two flavors of vortices at the boundary effectively creates one extra linking to the bulk vortex loops, which according to the bulk wave function would contributes factor $(-1)$. This implies that the two flavors of vortices at the boundary have mutual semion statistics.

Figure 3

(a) Skyrmion of O(3) vector $\vec{n}$ in 2D space. (b) If the SO(3) symmetry is broken down to $Z_2$, then the Skyrmion becomes a domain wall of $Z_2$ order parameter $n^z$.

Figure 4

(a) A vortex source is the end point of a bulk vortex ribbon. (b)–(e) Sequence of vortex-ribbon deformation, starting with interchanging two vortex sources in (b). (b) is homotopically equivalent to self-twisting one of the two ribbons by $2\pi$ in (e), which according to the bulk wave function should acquire factor $-1$. (f) Interchanging two vortex sources is also homotopically equivalent to creating one extra vortex ribbon with $2\pi$ self-twist in the bulk.