  It is old folklore that the violation of Leibniz rule on a lattice is an
obstruction for constructing a lattice supersymmetric model. While it is still
true for full supersymmetry, we show that a slightly modified form of the
Leibniz rule, which we call cyclic Leibniz rule (CLR), is actually a criterion
for the existence of partial lattice supersymmetry. In one dimension, we find
sets of lattice difference operator and field multiplication smeared over
lattice which satisfy the CLR under some natural assumptions such as
translational invariance and locality. Thereby we construct a model of
supersymmetric lattice quantum mechanics without spoiling locality. The CLR
relation is coincident with the condition that the vanishing of the so-called
surface term in the construction by lattice Nicolai map. We can construct
superfield formalism with arbitrary superpotential. This also enables us to
apply safely a localization technique to our model, because the kinetic term
and the interaction terms of our model are independently invariant under the
supersymmetry transformation. A preliminary attempt in finding a solution for
the higher dimensional case is also discussed.
