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Understanding the magnetic properties of atomic chains on superconductors isan essential cornerstone on the road towards controlling and constructingtopological electronic matter. Yet, even in simple models, the magnetic groundstates remain debated. Ferromagnetic (FM), antiferromagnetic (AFM), and spinspiral configurations have been suggested and experimentally detected, whilenon-coplanar and complex collinear phases have been additionally conjectured.Here, we resolve parts of the controversy by determining the magnetic groundstates of chains of magnetic atoms in proximity to a superconductor withMonte-Carlo methods. We confirm the existence of FM, AFM and spin spiral groundstates, exclude non-coplanar phases in the model and clarify the parametricregion of a $\uparrow\uparrow\downarrow\downarrow$-phase. We further identify anumber of novel complex collinear spin configurations, including the periodicspin configurations $\uparrow\uparrow\uparrow\downarrow$, and $\uparrow\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \downarrow \uparrow\downarrow$, which are in some cases combined with harmonic and anharmonicspirals to form the ground state. We topologically classify the electronicstructures, investigate their stability against increasing the superconductingorder parameter, and explain the complex collinear order by an effectiveHeisenberg model with dominant four-spin interactions.

Complex magnetic ground states and topological electronic phases of atomic spin chains on superconductors