Self-stabilizing temperature driven crossover between topological and non-topological ordered phases in one-dimensional conductors

We present a self-consistent analysis of the topological superconductivity arising from the interaction between self-ordered localized magnetic moments and electrons in one-dimensional conductors in contact with a superconductor. We show that due to a gain in entropy there exists a magnetically ordered yet non-topological phase at finite temperatures that is relevant for systems of magnetic adatom chains on a superconductor. Spin-orbit interaction is taken into account, and we show that it causes a modification of the magnetic order yet without affecting the topological properties.