Magnetic flux induced higher-order topological superconductivity

TL;DR

The paper demonstrates a route to higher-order topological superconductivity in conventional $s$-wave substrates without spin-orbit coupling, by leveraging antiferromagnetic Yu-Shiba-Rusinov states, staggered magnetic flux, and a perpendicular Zeeman field. Through a detailed 2D model and its low-energy edge theory, it achieves a second-order topological superconducting phase with Majorana corner modes and a bulk quadrupole invariant $Q_{xy}=1/2$. It further shows how to realize 3D HOTSCs by stacking 2D layers into hinge or third-order configurations, yielding Majorana hinge modes and eight corner Majorana modes, respectively, with explicit gap-closing conditions at high-symmetry points. The work discusses experimental paths and challenges, including atomic-scale $rac{π}{2}$-flux junctions and anisotropic YSR states, positioning this approach as a spin-orbit-free platform for higher-order topology with potential realizations in STM-engineered nanostructures.

Abstract

Higher-order topological superconductivity typically depends on spin-orbit interaction, and often necessitates well designed sample structures, nodal superconducting pairings or complex magnetic order. In this work, we propose a model that incorporates a Zeeman field, antiferromagnetic order, and $s$-wave superconducting pairing, all without the need for spin-orbit interaction. In a two-dimensional system, we realize a second-order topological superconductor by utilizing a staggered flux, provided that the Zeeman field is oriented perpendicular to the magnetic order moments. In three-dimensional systems, we achieve second- and third-order topological superconductors in theory, through stacking the two-dimensional second-order topological superconductor.

Figures (5)

• Figure 1: (a) The schematic of two-dimensional square arrays of magnetic atoms deposited on an $s$-wave superconductor. The blue and red dots symbolize opposite spins, while the larger blue dots represent different atoms due to their distinct spatial wave functions. The periodically arranged holes contain staggered quantum flux $\Phi_{0}$ through them. (b) The schematic of the effective tight-binding lattice model. The arrows on the bonds indicate the hopping direction with a positive phase factor, while the colors denote different signs of hopping amplitudes. Each unit cell comprises four atoms, as indicated by the green dashed-line box.
• Figure 2: The energy bands and open boundary spectra of Hamiltonian (\ref{['eq2']}). (a) Shows the dispersions along the path depicted in (b), which represents the 2D Brillouin Zone (BZ). (c) The gaps at the four HSPs in the BZ as a function of $\Delta$ with the energies at $X$ and $Y$ exhibiting two-fold degeneracy. (d) and (e) display the spectra of a system with 24$\times$24 sites under OBCs applied in the $x$- or $y$-directions, while (f) presents the case with OBCs imposed in both directions.
• Figure 3: (a) The density distributions of the four MCMs (red dots in inset) in the system comprising 24$\times$24 sites under simultaneous OBCs applied in both $x$- and $y$-directions, with $\Delta=2.0$, $JS=0.3$, $V=1.2$. (b)(c) illustrate the second order topological phase diagrams, where the red regions indicate $Q_{xy}=1/2$. The green solid lines represent the gap-closing conditions calculated by Eq. (\ref{['eq4']}), with $JS=0.3$ in panel (b) and $V=2.0$ in panel (c).
• Figure 4: (a)(d) The two stacking ways that yield 3D SOTSC and third-order TSCs, respectively. The arrows on the bonds indicate the hopping direction with positive phases, while the colors represent the signs of hopping amplitudes. (b) The energy dispersion of the structure in panel (a) with OBCs applied in the $x$- and $y$-directions and PBCs in the $z$-direction. (c) The density distributions of the MHMs with two hinge modes along each hinge. (e) Highlights the HSPs in a 3D cubic BZ. (f) The gaps at the HSPs of the structure depicted in panel (d) as a function of $\Delta$. The energies at $X,Y,Z$ ($U,S,T$) are threefold degenerate. (g) The low energy spectrum with OBCs imposed on all three directions. (h) The density distributions of the eight MCMs (red dots in inset). The parameters are set to $V=2.0, JS=0.6, t_x=t_y=1.0, \Delta=2.0$. $(\phi_z, t_z)$ are $(0.5\pi, 0.2)$ in (b)(c) and $(0.25\pi, 1.0)$ in (f)-(h).
• Figure 5: The schematic depicts the three-dimensional cubic Brillouin zone. We focus on the three red edges to examine the three-dimensional edge theory.