Competing higher order topological superconducting phases in triangular lattice magnet-superconductor hybrid systems

We demonstrate that a plethora of higher order topological phases emerge in magnet-superconductor hybrid (MSH) systems through the interplay of a stacked magnetic structure and an underlying triangular surface lattice; the latter being of great current experimental interest. Such lattices offer the ability to create three main types of edge terminations — called $x$-, $y$- and $y^{\prime }$-edges — of MSH islands that, in turn, give rise to a complex phase diagrams exhibiting various regions of higher order topological superconducting (HOTSC) phases. We identify the single adatom chain, as well as a pair of adjacent adatom chains (called a double-chain), as the basic building blocks for the emergence of HOTSC phases. Of particular interest are those HOTSC phases which arise from a competition between the topology of single- and double-chain blocks, which are absent for square lattices.

Figure 1

(a) Schematic view of the MSH system with double-chains of magnetic adatoms: dark blue circles represent sites of the superconducting substrate lattice and green circles with red arrows denote the magnetic adatoms. (b) Topological phase diagram of the MSH system, with strong topological phases colored according to their Chern number. Regions with potential HOTSC phases are shown in a striped pattern. Edge gap closings are shown in gray. Zero-energy LDOS for a finite-sized island of magnetic adatoms shown in (a), in the (c) strong topological $C=1$ phase reflecting the existence of a chiral edge modes, and (d) in a HOTSC phase, showing the presence of characteristic Majorana corner modes. Parameters used for (c) and (d) are $(\mu ,\lambda ,\mathrm{\Delta },JS)=(2,0.8,1.2,4)t$ and $(\mu ,\lambda ,\mathrm{\Delta },JS)=(3.2,0.8,1.2,2.1)t,$ respectively.

Figure 2

Phase diagrams of (a) an isolated single adatom chain and (b) an isolated double adatom chain. Dark gray and white regions indicate topological and trivial phases, respectively. Insets show the corresponding zero-energy LDOS of single and double chains in the topological regime. Parameters used are $\mathrm{\Delta }=1.2t$ and $\lambda =0.8t$.

Figure 3

(a)–(d) Four different edge terminations of MSH islands with $x$- and $y$-edges and (e)–(h) the corresponding phase diagrams: (a,e) single-chain termination, pointing out (SO) (b,f) single-chain termination, pointing in (SI) (c,g) double-chain termination, pointing out (DO) and (d,h) double-chain termination, pointing in (DI). HOTSC phases are denoted by pink regions, strong topological phases with nonzero Chern numbers by light gray regions, and edge gap closings by dark gray regions and lines.

Figure 4

Topological phase diagram of (a) a single adatom chain [see Fig. 2] and (b) a double adatom chain [see Fig. 2] overlaid with the phase diagram of Fig. 1. Topological phases of the single-chain and double-chains are denoted by purple regions.

Figure 5

Different ways to pair up the MZMs at the ends of adatom chains when they are in the topological phase. (a) Single-chains are topological, and MZMs hybridize via vertical intrapair couplings, (b) Single-chains are topological, and MZMs hybridize via interpair couplings, (c) Double-chains are topological, and their MZMs hybridize when two double-chains are pointing at each other, (d) Single- and double-chains are topological, so an MZM from a double-chain can pair with an MZM from a single-chain.

Figure 6

(a) A parallelogram MSH island shown in light gray, overlaid with the zero-energy LDOS in the HOTSC phase revealing Majorana corner modes at both acute and obtuse angle corners. (b) Zoom-in of (a), showing one acute angle corner. Topological phase diagram for an parallelogram MSH island with (c) a single-chain termination, and (d) a double-chain termination along the $x$-edge. Parameters used in (a) and (b) are $JS=2t$ and $\mu =3.2t$.

Figure 7

Zero-energy LDOS for different MSH island geometries: (a) a trapezoidal island, (b) a triangular island, (c) a hexagonal island formed by stacking of two trapezoids, and (d) a hexagonal island with a single adatom chain inserted in the middle. Parameters used for (a,b) and (c,d) are $(\mu ,\lambda ,\mathrm{\Delta },JS)=(5,0.8,1.2,3.5)t$ and $(\mu ,\lambda ,\mathrm{\Delta },JS)=(4,0.8,1.2,2.5)t,$ respectively.

Figure 8

Phase diagram of an MSH island with $x$- and $y$-edges for $\mathrm{\Delta }=0.9t$ and $\lambda =0.6t$. The regions in which HOTSC phases can occur, depending on the choice of edge termination, are indicated by vertical stripes.