Layer construction of 3D topological states and string braiding statistics

TL;DR

This paper introduces a formalism for constructing 3D topological states by stacking 2D layers, revealing new states with unique surface and bulk properties, including nontrivial string braiding statistics and a topological field theory framework.

Contribution

It presents a novel layer construction method for 3D topological states, expanding understanding of their properties and braiding statistics, including non-Abelian strings.

Findings

Construction of 3D topological states from layered 2D states.

Identification of states with surface-only or bulk and surface topological order.

Development of a topological field theory capturing string-particle and string-string braiding.

Abstract

While the topological order in two dimensions has been studied extensively since the discover of the integer and fractional quantum Hall systems, topological states in 3 spatial dimensions are much less understood. In this paper, we propose a general formalism for constructing a large class of three-dimensional topological states by stacking layers of 2D topological states and introducing coupling between them. Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface. For both classes of states we study its generic properties and present several explicit examples. As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations. In addition to studying the string-string braiding in the example system, we propose a generic topological field theory description which can capture both string-particle and string-string braiding statistics. Lastly, we provide a proof of a general identity for Abelian string statistics, and discuss an example system with non-Abelian strings.