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We study a class of three dimensional exactly solvable models of topologicalmatter first put forward by Walker and Wang [arXiv:1104.2632v2]. While theseare not models of interacting fermions, they may well capture the topologicalbehavior of some strongly correlated systems. In this work we give a fullpedagogical treatment of a special simple case of these models, which we callthe 3D semion model: We calculate its ground state degeneracies for a varietyof boundary conditions, and classify its low-lying excitations. While pointdefects in the bulk are confined in pairs connected by energetic strings, thesurface excitations are more interesting: the model has deconfined pointdefects pinned to the boundary of the lattice, and these exhibit semionicbraiding statistics. The surface physics is reminiscent of a $\nu=1/2$ bosonicfractional quantum Hall effect in its topological limit, and theseconsiderations help motivate an effective field theoretic description for thelattice models as variants of $bF$ theories. Our special example of the 3Dsemion model captures much of the behavior of more general `confinedWalker-Wang models'. We contrast the 3D semion model with the closely related3D version of the toric code (a lattice gauge theory) which has deconfinedpoint excitations in the bulk and we discuss how more general models may havesome confined and some deconfined excitations. Having seen that there existlattice models whose surfaces have the same topological order as a bosonicfractional quantum Hall effect on a confining bulk, we construct a latticemodel whose surface has similar topological order to a fermionic quantum halleffect. We find that in these models a fermion is always deconfined in thethree dimensional bulk.

Three-dimensional topological lattice models with surface anyons