  We study spatial correlation functions of local operators of interacting many
bosons confined in a box of a large, but volume V, for various `ground states'
whose energy densities are almost degenerate. The ground states include the
coherent state of interacting bosons (CSIB), the number state of interacting
bosons (NSIB), and the number-phase squeezed state of interacting bosons, which
interpolates between the CSIB and NSIB. It was shown previously that only the
CSIB is robust (i.e., does not decohere for a macroscopically long time)
against the leakage of bosons into an environment. We show that for the CSIB
the spatial correlation of any local operators A(r) and B(r') (which are
localized around r and r', respectively) vanishes as |r - r' | \sim V^{1/3} \to
\infty, i.e., the CSIB has the `cluster property.' In contrast, the other
ground states do not possess the cluster property. Therefore, we have
successfully shown that the robust state has the cluster property. This ensures
the consistency of the field theory of bosons with macroscopic theories.
