# Symmetry protected phases in inhomogeneous spin chains

## TL;DR

This paper explores how folding inhomogeneous spin chains can induce symmetry-protected topological phases, transforming long-range entanglement into short-range and revealing connections between symmetry, topology, and spin chain properties.

## Contribution

It demonstrates that folding inhomogeneous spin chains can induce topological phases protected by symmetries, linking entanglement structure to topological classifications.

## Key findings

- Folding transforms long-range entanglement into short-range, enabling topological phases.
- Site-centered folding yields non-trivial SPT phases, including SSH and Haldane phases.
- Results extend to higher spin chains, connecting gapped and gapless phase protection.

## Abstract

It has been shown recently that inhomogenous spin chains can exhibit exotic phenomena such as the breaking of the area law of the entanglement entropy. An example is given by the rainbow model where the exchange coupling constants decrease from the center of the chain. Here we show that by folding the chain around its center, the long-range entanglement becomes short-range which can lead to topological phases protected by symmetries (SPT). The phases are trivial for bond-centered foldings, and non trivial for site-centered ones. In the latter case, the folded spin 1/2 chain with $U(1)$ symmetry belongs to the Su-Schrieffer-Heeger class, while the folded chain with $SU(2)$ symmetry is in the Haldane phase. Finally, we extend these results to higher spin chains where we find a correspondence between the symmetry protection of gapped and gapless phases.