$C^1$-classification of gapped parent Hamiltonians of quantum spin chains

TL;DR

This paper classifies gapped quantum spin chain Hamiltonians based on their edge modes, showing that the number of edge modes fully characterizes their $C^1$-equivalence without breaking translation invariance.

Contribution

It proves that the number of edge modes is the complete invariant for $C^1$-classification of these Hamiltonians, avoiding the need to block the chain.

Findings

Edge mode count equals at both edges and is a complete invariant.

Translation invariance of the bulk state does not need to be broken for $C^1$-equivalence.

The classification applies to Hamiltonians of translation invariant finitely correlated states.

Abstract

We consider the $C^1$-classification of gapped Hamiltonians introduced in [Fannes-Nachtergaele-Werner, Nachtergaele] as parents Hamiltonians of translation invariant finitely correlated states. Within this family, we show that the number of edge modes, which is equal at the left and right edge, is the complete invariant. The construction proves that translation invariance of the `bulk' ground state does not need to be broken to establish $C^1$-equivalence, namely that the spin chain does not need to be blocked.