  In this paper we systematically study a simple class of translation-symmetry
protected topological orders in quantum spin systems using slave-particle
approach. The spin systems on square lattice are translation invariant, but may
break any other symmetries. We consider topologically ordered ground states
that do not spontaneously break any symmetry. Those states can be described by
Z2A or Z2B projective symmetry group. We find that the Z2A translation
symmetric topological orders can still be divided into 16 sub-classes
corresponding to 16 new translation-symmetry protected topological orders. We
introduced four $Z_2$ topological indices $\zeta_{\v{k}}=0,1$ at $\v
{k}=(0,0)$, $(0,\pi)$, $(\pi, 0)$, $(\pi ,\pi)$ to characterize those 16 new
topological orders. We calculated the topological degeneracies and crystal
momenta for those 16 topological phases on even-by-even, even-by-odd,
odd-by-even, and odd-by-odd lattices, which allows us to physically measure
such topological orders. We predict the appearance of gapless fermionic
excitations at the quantum phase transitions between those symmetry protected
topological orders. Our result can be generalized to any dimensions. We find
256 translation-symmetry protected Z2A topological orders for a system on 3D
lattice.
