Entanglement entropy of gapped phase and topological order in three dimensions

We discuss entanglement entropy of gapped ground states in different dimensions, obtained on
partitioning space into two regions. For trivial phases without topological order, we argue that
the entanglement entropy may be obtained by integrating an ‘entropy density’ over the partition
boundary that admits a gradient expansion in the curvature of the boundary. This constrains
the expansion of entanglement entropy as a function of system size, and points to an even-odd
dependence on dimensionality. For example, in contrast to the familiar result in two dimensions,
a size independent constant contribution to the entanglement entropy can appear for trivial phases
in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE)
that cannot be obtained by adding local contributions. We find that in three dimensions there is
just one type of TEE, as in two dimensions, that depends linearly on the number of connected
components of the boundary (the ‘zeroth Betti number’). In ????3 dimensions, new types of
TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct
generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in ? ?? 3.