Spin-1 Haldane chains of superconductor-semiconductor hybrids

We theoretically explore the possibility of realizing the symmetry-protected topological Haldane phase of spin-1 chains in a tunable hybrid platform of superconducting islands (SIs) and quantum dots (QDs). Inspired by recent findings suggesting that an appropriately tuned QD-SI-QD block may behave as a robust spin-1 unit, we study the behavior of many such units tunnel-coupled into linear chains. Our efficient and fully microscopic modeling of long chains with several tens of units is enabled by the use of the surrogate model solver V. V. Baran et al., Phys. Rev. B 108, L220506 (2023); Phys. Rev. B 109, 224501 (2024). Our numerical findings indicate that the QD-SI-QD chains exhibit emblematic features of the Haldane phase, such as fractional spin-1/2 edge states and nonvanishing string order parameters, and that these persist over a sizable region of parameter space.