Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wave Functions of Bosonic Symmetry Protected Topological Phases

Cenke Xu, T. Senthil

TL;DR

The paper provides a bulk, information-rich picture of 3D bosonic SPT phases by deriving ground-state wave functions in dual vortex languages. It shows that a topological phase factor $(-1)^{N_t}$ associated with self-linking of vortex ribbons distinguishes SPTs from trivial Mott insulators, with the structure arising from a bulk $\Theta$-term at $\Theta=2\pi$ or BF-type theories. These bulk wave functions account for boundary phenomena such as fermionic vortex endpoints and mutual statistics in multi-species cases, and they reproduce known 2D SPT results via domain-wall loop gas formalisms. The work links effective field theories to explicit wave-function constructions, offering intuitive, nonperturbative insight and potential routes to lattice realizations and connections to related models like Walker–Wang constructions.

Abstract

We study the structure of the ground state wave functions of bosonic Symmetry Protected Topological (SPT) insulators in 3 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 in the strong coupling limit, and it naturally explains all the phenomena of this SPT phase discussed previously. The ground state structure for other 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.

Wave Functions of Bosonic Symmetry Protected Topological Phases

TL;DR

The paper provides a bulk, information-rich picture of 3D bosonic SPT phases by deriving ground-state wave functions in dual vortex languages. It shows that a topological phase factor associated with self-linking of vortex ribbons distinguishes SPTs from trivial Mott insulators, with the structure arising from a bulk -term at or BF-type theories. These bulk wave functions account for boundary phenomena such as fermionic vortex endpoints and mutual statistics in multi-species cases, and they reproduce known 2D SPT results via domain-wall loop gas formalisms. The work links effective field theories to explicit wave-function constructions, offering intuitive, nonperturbative insight and potential routes to lattice realizations and connections to related models like Walker–Wang constructions.

Abstract

We study the structure of the ground state wave functions of bosonic Symmetry Protected Topological (SPT) insulators in 3 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 in the strong coupling limit, and it naturally explains all the phenomena of this SPT phase discussed previously. The ground state structure for other 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.

Paper Structure

This paper contains 10 sections, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Wave function of trivial 3d Bose Mott insulator
  3. Wave function of 3d Bosonic SPT phases
  4. Ground state wave function of 2d SPT phases
  5. 2d SPT phase with $Z_2$ symmetry
  6. 2d SPT phase with $U(1) \rtimes Z_2^T$ symmetry
  7. Discussion
  8. Appendix: Dynamical $Z_2$ gauge fields in the 2d Ising SPT
  9. Lattice version of effective field theory
  10. Statistics of vortex sources

Figures (4)

  • Figure 1: The wave function of the 3d bosonic SPT discussed in this paper is a superposition of all the configurations of vortex ribbons with factor $(-1)$ associated with each self-linking.
  • Figure 2: $(a)$. When the symmetry is $U(1) \times U(1)$ 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 $Z_2$, 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. $(b) \rightarrow (e)$, 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.