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), 10.1103/PhysRevB.108.L220506; Phys. Rev. B 109, 224501 (2024), 10.1103/PhysRevB.109.224501]. 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.