Topological properties of spin block magnetic ladders in proximity of a superconductor: application to BaFe$_{2}$S$_{3}$

Abstract

Monoatomic chains with magnetic order in proximity of a s-wave superconductor can host Majorana edge modes. In this paper, we extend this idea to more complex spin-block chains such as the BaFe$_{2}$S$_{3}$ magnetic material that has a spin-ladder like structure. We investigate the topological phase diagram of such a system as function of the system parameters. We show that the coupling between chains within the ladder leads to the topological phase with a winding number larger than the sum of two single magnetic chains. Furthermore, strong coupling between chains leads to fractal-like substructure in the topological phase diagram. By investigating the real space properties of such a system and particularly its edge modes, we find that the system spectrum contains several in-gap states that we analyze in detail.

Figures (14)

• Figure 1: (a) Crystal structure of BaFe$_{2}$ Ch$_{3}$ ( Ch=S, Se) with $Pnma$ symmetry. (b) The iron ladder can realized canonical ($\pi$,0) AFM-like order in BaFe$_{2}$S$_{3}$. Thus the magnetic unit cell contain two pairs of sites.
• Figure 2: Schematic model of the investigated ladder mode with spin block order. We assume the spin-conserving electron hopping along chains ($-t$) and between chains ($-t_{\perp}$). Similarly the spin--orbit coupling is assumed along chains ($\lambda$) and between chains ($\eta$). Lattice constant is the distance between neighbouring sites along $x$, is taken as $a = 1$. Magnetic sublattices are present by blue and green sites, while the magnetic unit cell is marked by green dashed rectangle.
• Figure 3: The topological phase diagram of the one dimensional AFM chain ($t_{\perp} = \eta = 0$). Results obtained for $m_0 /t = 0.3$, $\Delta / t = 0.2$, and $\lambda / t = 0.15$, correspond to the study presented in Ref. kobialka.sedlmayr.21.
• Figure 4: The topological phase diagram of the ladder with AFM-like order. Results for the same parameters as Fig. \ref{['fig.1dafm_chain']}, while coupling between chains within ladder is set as $t_{\perp} / t = 0.2$ and $\eta / \lambda = 0.2$.
• Figure 5: The same as Fig. \ref{['fig.hmu_ladder']} for $\Delta / t = 0.0015$. Inset show zoom of marked area in the absence and presence of the inter-chain spin--orbit coupling $\eta$ (left and right inset, respectively).