Wave Functions of Bosonic Symmetry Protected Topological Phases

TL;DR

This paper explores the ground state wave functions of 3D bosonic SPT insulators, using dual vortex descriptions and vortex ribbon superpositions to reveal their unique topological features.

Contribution

It introduces a simple vortex ribbon wave function representation for 3D bosonic SPT phases, linking effective field theory to ground state structure analysis.

Findings

Wave function expressed as superposition of vortex ribbons with linking factors

Dual vortex description captures differences from conventional insulators

Method reproduces known 2D SPT ground state results

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.