Majorana Kramers Pairs in Higher-Order Topological Insulators

We propose a tune-free scheme to realize Kramers pairs of Majorana bound states in recently discovered higher-order topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an $s$-wave superconductor, the competition between local and crossed Andreev pairing leads to the formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electron-electron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions compared with nonhelical nanowires.

Figure 1

Left: In a Bi(111) nanowire (green), the gapless states (blue arrows) propagate along the hinges. The spin-up (-down) hinge states move against (along) the directions of the arrows. The helicities of any two parallel hinge states [along $z\equiv (111)$ axis] on a single lateral facet are opposite. Right: In the proposed setup, a superconducting layer (yellow) covers three parallel hinges (labeled by 1, 2, and 3). The (1,2) pair carries the same helicity, allowing for the crossed Andreev pairing. For other pairs, [(1,3) and (2,3)], such pairing is suppressed. The $x$ axis of the local coordinate is defined along the perimeter of the hexagonal cross section, and the $y$ axis (not shown) is normal to the lateral facets.

Figure 2

Schematics of the parahelical setup in the $xz$ plane of the local coordinate (view from the $+y$ direction) with the hinge coordinate $r$. A superconductor (yellow) covers three long hinges (along the $z$ direction), and several short hinges (along the $x$ direction). In hinges 1 and 2, which are at distance $d$, the spin-up states propagate toward the $-r$ direction (red solid arrows) while the spin-down states toward $+r$ (blue solid arrows). Hinge 3 is decoupled from the others. The local (${\mathrm{\Delta }}_{1,2}$) and crossed Andreev (${\mathrm{\Delta }}_c$) pairing processes are indicated by the dotted and dashed arrows, respectively. Since the short segments along the $x$ direction are not aligned in the laboratory frame, ${\mathrm{\Delta }}_c$ is suppressed if $r\notin [0,L]$, while ${\mathrm{\Delta }}_{1,2}$ remains constant for any $r$, including in the short segments. As a result, the boundaries (black dashed lines) are created at $r=0$ and $r=L$ (the ends of the nanowire), which are assumed to be far apart on the scale of the Majorana localization length. For clarity, only one crossed Andreev pairing process, ${\mathrm{\Delta }}_c{R}_{1,\downarrow }^{†}{L}_{2,\uparrow }^{†}$, is depicted.

Figure 3

RG flow diagram. We take the parameters $K_1(0)=K_2(0)=0.2$, 0.4, 0.6, and 0.8, ${\tilde{\mathrm{\Delta }}}_1(0)={\tilde{\mathrm{\Delta }}}_2(0)=3{\tilde{\mathrm{\Delta }}}_{\mathrm{c}}(0)=0.03$, $a(0)=5\mathrm{nm}$, and $L=1\mu \mathrm{m}$. The crossed Andreev (local) pairing gap ${\mathrm{\Delta }}_c$ (${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$) is plotted in blue solid (red dashed) curves. The blue dots (labeled by I–IV) mark the initial points of ${\mathrm{\Delta }}_c$, and the green arrows and points specify where the RG flows stop. The RG flows labeled by II and III stop at the points at which the renormalized crossed Andreev pairing dominates over the local pairing (${\mathrm{\Delta }}_c>{\mathrm{\Delta }}_1$), indicating the topological superconducting phase.

Figure 4

Phase diagram. The vertical and horizontal axes label the initial values of the gap ratio [${\mathrm{\Delta }}_1(0)/{\mathrm{\Delta }}_c(0)$] and interaction parameter $K_1(0)=K_2(0)$, respectively. The other parameters are the same as those in Fig. 3. The green (yellow) region marks the phase with (without) MKPs. The corresponding RG flows to the blue dots (labeled by I–IV) are shown in Fig. 3. In the red region, both types of the pairing gaps vanish.