Gapless edge states of background field theory and translation-symmetric Z2 spin liquids

We study possible gapless edge states of translation-symmetric Z(2) spin liquids. The gapless edge states emerge from dangling Majorana fermions at the boundary. We construct a series of mean-field Hamiltonians of Z(2) spin liquids on the square lattice; these models can be obtained by generalization of Wen's exactly solvable plaquette model. We also study the details of the edge theory of these Z(2) spin liquids and find their effective background field (BF) theory descriptions. The effective BF theories are shown to describe the crystal momenta of the ground states and their degeneracies and to predict the edge theories of these Z(2) spin liquids. As a by-product, we obtained a way to classify the BF theories reflecting the lattice symmetries. We discuss in closing three-dimensional Z(2) spin liquids with gapless surface states on the cubic lattice.