  We give a characterization of the class of gapped Hamiltonians introduced in
PartI [O]. The Hamiltonians in this class are given as MPS (Matrix product
state) Hamiltonians. In [O], we list up properties of ground state structures
of Hamiltonians in this class. In this Part II, we show the converse. Namely,
if a (not necessarily MPS) Hamiltonian $H$ satisfies five of the listed
properties, there is a Hamiltonian $H'$ from the class in [O], satisfying the
followings: The ground state spaces of the two Hamiltonians on the infinite
intervals coincide. The spectral projections onto the ground state space of $H$
on each finite intervals are approximated by that of $H'$ exponentially well,
with respect to the interval size. The latter property has an application to
the classification problem.
