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

TL;DR

This paper classifies asymmetric gapped Hamiltonians on quantum spin chains by boundary degeneracies, establishing that these degeneracies serve as complete invariants for the Hamiltonians' classification.

Contribution

It introduces a $C^1$-classification framework considering boundary effects and proves that boundary degeneracies fully classify certain gapped Hamiltonians.

Findings

Left and right degeneracies of edge ground states are complete invariants.

All Hamiltonians with specified properties are equivalent in the bulk classification.

Boundary degeneracies determine the classification of asymmetric gapped Hamiltonians.

Abstract

In this paper, we consider 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 that1.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.