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.
Highlights
Some of the most important discoveries in condensedmatter physics over the last few decades have been about topological states of matter
A subclass of topological states of matter is the topologically ordered states, which are stable against any local perturbations and have topologically protected properties, including fractional quasiparticle statistics, ground-state degeneracy determined by topology of the spatial manifold, etc
The first class of states is those with trivial bulk topological order and nontrivial surface twodimensional topological order
Summary
Some of the most important discoveries in condensedmatter physics over the last few decades have been about topological states of matter. The 3D generalization of the toric code model can capture the topological order of 3D lattice discrete gauge theory in which the ground-state degeneracy is associated with the nontrivial one-cycles and the point particles have nontrivial mutual braiding statistics with flux-string excitations. For more general states with nontrivial 3D topological order, our layer construction enables description of the bulk point particles and stringlike excitations. We provide several examples of different topological orders, including a “conventional” 3D topological order that resembles the lattice gauge theory, and more general systems with coexisting bulk and surface topological order.
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