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We give a characterization of the class of gapped Hamiltonians introduced inPartI [O]. The Hamiltonians in this class are given as MPS (Matrix productstate) Hamiltonians. In [O], we list up properties of ground state structuresof Hamiltonians in this class. In this Part II, we show the converse. Namely,if a (not necessarily MPS) Hamiltonian $H$ satisfies five of the listedproperties, there is a Hamiltonian $H'$ from the class in [O], satisfying thefollowings: The ground state spaces of the two Hamiltonians on the infiniteintervals 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 tothe classification problem.

A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization II