  We investigate interaction effects in three dimensional weak topological
insulators (TI) with an even number of Dirac cones on the surface. We find that
the surface states can be gapped by a surface charge density wave (CDW) order
without breaking the time-reversal symmetry. In this sense, time reversal
symmetry alone can not robustly protect the weak TI state in the presence of
interactions. If the translational symmetry is additionally imposed in the
bulk, a topologically non-trivial weak TI state can be obtained with helical
edge states on the CDW domain walls. In other words, a CDW domain wall on the
surface is topologically equivalent to the edge of a two-dimensional quantum
spin Hall insulator. Therefore, the surface state of a weak topological
insulator with translation symmetry breaking on the surface has a "half quantum
spin Hall effect", in the same way that the surface state of a strong
topological insulator with time-reversal symmetry breaking on the surface has a
"half quantum Hall effect". The on-site and nearest neighbor interactions are
investigated in the mean field level and the phase diagram for the surface
states of weak topological insulators is obtained.
