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

Abstract

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 nontopological phase at finite temperatures that is relevant for systems of magnetic adatom chains on a superconductor. The 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. Introduction. Topological superconductors have received much attention recently, partly because they host exotic low energy excitations such as Majorana bound states [1-3], whose non-Abelian statistics are attractive for topological quantum computation [4,5]. As a remarkable feature, topological superconductivity can be created artificially by contacting specific materials with a conventional s-wave superconductor. For instance, it arises at the interface between the surface states of a three-dimensional topological insulator and an s-wave superconductor [6], in one-dimensional (1D) semiconducting wires with a strong spin-orbit interaction (SOI) and a Zeeman magnetic field with proximitized superconductivity [7][8][9][10][11], or in arrays of magnetic nanoparticles or magnetic adatoms on top of a superconducting surface [12][13][14][15][16][17][18][19][20][21][22], such as iron adatoms on lead [23][24][25].The systems we consider in this Rapid Communication exhibit a topological phase emerging from self-organization of magnetic moments embedded in 1D conductors with proximity induced superconductivity. This situation may apply to semiconducting wires with extrinsic magnetic impurities or intrinsic moments such as nuclear spins, or a conducting wire made of magnetic adatoms on a superconducting surface. Due to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated through the electrons, the magnetic moments can undergo an ordering transition below a temperature T * and form a spiral with a spatial period characterized by the wave number 2k m (see Fig. 1) such that k m = k F , for k F the Fermi momentum. This ordering mechanism was first demonstrated for normal conductors [26,27], then conjectured [13] and selfconsistently demonstrated [28][29][30] for the superconducting case. These results were corroborated recently by showing that the spiral order persists beyond the RKKY limit, and k m stays close to k F , as long as k F is away from commensurate band fillings and the coupling strength A between magnetic moments and electrons remains smaller than the electron bandwidth [31,32].The locking of k m to k F has important consequences. The magnetic spiral forms a periodic superstructure that causes a part of the electrons to undergo a spin-selective Peierls transition [33] to a nonconducting spiral electron spin density wave, whereas the remaining conducting electron states become helical (spin filtered). The induced supercon...

Topological superconductivity at finite temperatures in proximitized magnetic nanowires

“…Topological features of the systems with self-organized spiral ordering have been so far studied, focusing mainly on the zero temperature limit. Thermal effects have been partly addressed, taking into account magnon excitations (which suppress a magnitude of the spiral order) 34,42 and investigating a contribution of the entropy term to the free energy (which substantially affects the wave vector of the spiral order, so that magnetic order might be preserved but the electronic state could no longer be topological) 47 . Usually, however, any long-range order hardly exists in one-dimensional systems at finite temperatures and therefore it is important -especially for practical applications of such systems -to estimate the maximum temperature up to which the topologically nontrivial states could survive.…”

Missing Shapiro steps and the 8π -periodic Josephson effect in interacting helical electron systems

“…In such one-dimensional (1D) wires, the combined effects of Rashba spin-orbit coupling (RSOC) or helical nuclear order, [15][16][17] proximity-coupling to a superconductor, and an externally applied magnetic field 13,14 open a band gap in the single-particle spectrum of the wire. This gap only closes at the ends of the wire, leading to localized zero-energy states which are believed to behave as Majorana fermions.…”

Topological state engineering by potential impurities on chiral superconductors

“…Magnetic atoms on s-wave superconductors give rise to Yu-Shiba-Rusinov subgap states [7][8][9][10] which have been probed experimentally by scanning tunneling microscopy (STM) [11][12][13][14]. Superstructures fabricated from magnetic atoms are currently under active experimental [15][16][17] and theoretical research [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Intriguing properties of these systems include the possibility for various one-dimensional (1D) topological superconducting phases with Majorana bound states and rich two-dimensional (2D) topological phases [34][35][36][37].…”

“…A topologically nontrivial phase is known to arise in 1D ferromagnetic arrays when the underlying superconductor has a strong Rasha spin-orbit coupling or in arrays with helical magnetic textures. In 1D structures there are theoretical arguments why magnetic self-tuning could result in a nontrivial ground state [20][21][22]28,32,38], though in real systems there are a number of complications. In particular, in 2D structures the nature and tunability of magnetic textures is a delicate and largely unsolved question.…”