# Automorphic equivalence preserves the split property

## TL;DR

This paper demonstrates that the split property remains invariant under certain automorphisms in spin chain states and applies this to classify gapped phases using the $Z_2$-index, ensuring its stability under fast-decaying interactions.

## Contribution

It establishes the invariance of the split property under specific automorphisms and applies this to the classification of topological phases via the $Z_2$-index.

## Key findings

- Split property is stable under automorphisms generated by power-law decaying interactions.
- The $Z_2$-index is an invariant for gapped ground states with fast-decaying interactions.
- Application to symmetry protected topological phases confirms the robustness of the $Z_2$-classification.

## Abstract

We prove that the split property is a stable feature for spin chain states which are related by composition with *-automorphisms generated by power-law decaying interactions. We apply this to the theory of the $\mathbb{Z}_2$-index for gapped ground states of symmetry protected topological phases to show that the $\mathbb{Z}_2$-index is an invariant of gapped classification of phases containing fast-decaying interactions.