A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization III

In this paper, we consider the classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in [ O1 ]. We consider the C 1 -classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that (1) are given by translation invariant finite range interactions, (2) are gapped in the bulk, (3) are frustration-free, (4) have uniformly bounded ground state degeneracy on finite intervals, and (5) have a unique bulk ground state. We show that for the bulk-classification, any such Hamiltonians are equivalent.