We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a υ=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk. Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling.
Abstract We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double Dϕ(Z2) D ϕ ( Z 2 ) .