Kramers pairs of Majorana fermions and parafermions in fractional topological insulators

We propose a scheme based on topological insulators to generate Kramers pairs of Majorana fermions or parafermions in the complete absence of magnetic fields. Our setup consists of two topological insulators whose edge states are brought close to an $s$-wave superconductor. The resulting proximity effect leads to an interplay between a nonlocal crossed Andreev pairing, which is dominant in the strong electron-electron interaction regime, and usual superconducting pairing, which is dominant at large separation between the two topological insulator edges. As a result, there are zero-energy bound states localized at interfaces between spatial regions dominated by the two different types of pairing. Due to the preserved time-reversal symmetry, the bound states come in Kramers pairs. If the topological insulators carry fractional edge states, the zero-energy bound states are parafermions, otherwise, they are Majorana fermions.

Figure 1

Parahelical setup. Sketch of two topological insulators (green rectangles, labeled by $\tau =1,\overline{1}$) that are tunnel coupled to an underlying $s$-wave superconductor (yellow) which induces superconducting proximity effects in the TIs. The chemical potentials, controlled by the gates (light blue), are the same in the two samples ${\mu }_1={\mu }_{\overline{1}}$. The helical edge modes (blue and red lines with arrows) in the TIs circulate in opposite directions due to opposite spin-orbit interaction vectors. The crossed Andreev pairing dominates over the usual pairing along the common boundary of the two TIs (along the $x^{\prime }$ direction). In contrast to that, along the $y^{\prime }$ direction, the dominant pairing is the usual one produced by Cooper pairs tunneling into each of the TIs as a whole. The propagation direction of spin downs defines the $x$ axis. The interfaces between two such regions at $x=0$ and at $x=L$ (shown by dashed lines) host the zero-energy bound states such as Kramers pairs of Majorana fermions or Kramers pairs of parafermions depending on the nature of the TI (see text).

Figure 2

Alternative realization of the parahelical setup. A bilayer consists of two identical topological insulators (dark and light green slabs) tuned at the same chemical potential. The shape of an $s$-wave superconductor (yellow L-shaped slab) is designed such that the crossed Andreev pairing (coupling states from different TIs) dominates along one of the edges (here, along the long edge), whereas the usual superconducting pairing (coupling states from the same TI) dominates at the other edge (here, along the short edge). Again, there are two interfaces separating the two regions, each of them containing Kramers pairs of bound states. Note that while at one interface (located at the corner of the L-shaped slab) the bound states are well localized, they are spread out over the normal region edge states forming the other interface.

Figure 3

The spectrum of two pairs of edge modes in topological insulators labeled by $\tau$ corresponding to the parahelical setups shown in Figs. 1 and 2. The proximity-induced pairing amplitude ${\Delta }_{\tau }$ induces coupling between edge modes from the same TI, resulting in proximity gaps at the Fermi level in each TI separately. If the chemical potentials ${\mu }_{\tau }$ are the same in the two samples, ${\mu }_1={\mu }_{\overline{1}}$, the crossed Andreev pairing with amplitude ${\Delta }_c$ couples a spin-up state at the Fermi level in the upper TI ($\tau =1$, red arrow) with a spin-down state at the Fermi level in the lower TI ($\tau =\overline{1}$, blue arrow), and vice versa. The crossed pairing is dominant in the regime of strong electron-electron interactions since it costs more energy to insert two charges simultaneously into the same edge of a given TI than each of them separately into different TIs.

Figure 4

Alternative realization of the parahelical setup based on a spatial decay of the crossed Andreev pairing. The setup consists of two identical topological insulators (green slabs) tuned at the same chemical potential. The shape of the $s$-wave superconductor (yellow T-shaped slab) is designed such that the crossed Andreev pairing (coupling states from different TIs) dominates around $x=0$, where the distance between the samples is smallest, and decays away from this point, giving way to the usual superconducting pairing (coupling edge states from the same TI). The decay along the $y^{\prime }$ axis is determined by the coherence length and the Fermi wavelength of the superconductor. In contrast, the decay along the $x^{\prime }$ axis is much faster and arises due to a momentum mismatch between the spin-up helical edge states of one TI and the spin-down helical edge states of the other TI. As a result, there are two interfaces between different superconducting pairing regimes at $x=-{\ell }_2$ and ${\ell }_1$.

Figure 5

Orthohelical setup. Sketch of two topological insulators ($\tau =\pm 1$, green rectangles) brought into proximity to a superconductor (yellow). The chemical potentials, controlled by the gates (light blue), are tuned to be opposite in the two samples ${\mu }_1=-{\mu }_{\overline{1}}$. The edge modes circulate in the same direction in both samples as their spin-orbit interaction vectors are codirectional. As a result of strong electron-electron interactions, the crossed Andreev superconducting pairing dominates over the usual superconducting pairing along the common boundary of the two samples ($x^{\prime }$ direction). In contrast to that, in the region where the edge states propagate along the $y^{\prime }$ direction, the only present superconducting pairing is the usual one produced by Cooper pairs tunneling into one of the samples as a whole.

Figure 6

The spectrum of two pairs of edge modes in topological insulators corresponding to the orthohelical setup shown in Fig. 5. The superconducting pairing is similar to the one shown in Fig. 3, with the difference that here the chemical potentials ${\mu }_{\tau }$ are opposite at the two edges ${\mu }_1=-{\mu }_{\overline{1}}$. In this case, the crossed Andreev pairing with amplitude ${\Delta }_c$ couples a spin-up state at the Fermi level in the upper TI ($\tau =1$, blue arrow) with a spin-down state at the Fermi level in the lower TI ($\tau =\overline{1}$, red arrow), and vice versa. Again, the crossed pairing is dominant in the regime of strong electron-electron interactions.

Figure 7

The energy spectrum $E\left(k\right)$ of the system shown in Fig. 5 around the Fermi points $\pm k_F$ [see Eq. (14)]. The gap at $k=0$ closes at ${\Delta }_c=\sqrt{{\Delta }_1{\Delta }_{\overline{1}}}$, indicating the topological transition separating (a) the trivial (characterized by ${\Delta }_{k=0}>0$) and (b) the topological (characterized by ${\Delta }_{k=0}<0$) phases. We note that (a) in the trivial phase the gap is minimal at $k=0$, whereas (b) in the topological phase the gap is minimal at a finite wave vector and is given by $|{\Delta }_1-{\Delta }_{\overline{1}}|$. The parameters are chosen to be (a) ${\Delta }_1/{\Delta }_c=4$ and ${\Delta }_{\overline{1}}/{\Delta }_c=2$; (b) ${\Delta }_1/{\Delta }_c=\frac{2}{3}$ and ${\Delta }_{\overline{1}}/{\Delta }_c=\frac{1}{3}$.

Figure 8

Pinning of bosonic fields in the regime of strong electron-electron interactions. The field ${\varphi }_1$ (blue line) is pinned uniformly in the entire system. The field ${\varphi }_{\overline{1}}$ (${\theta }_{\overline{1}}$) is pinned in the region where the usual pairing term ${\Delta }_{\tau }$ (crossed Andreev pairing term ${\Delta }_c$) is dominant, indicated by green (red) lines. The different regions are separated by interfaces of size $\ell$ located at $x=0$ and $L$ (yellow circles). Within these interfaces, the noncommuting fields ${\varphi }_{\overline{1}}$ and ${\theta }_{\overline{1}}$ overlap and fluctuate, giving rise to zero-energy bound states that can be identified as Kramers pairs of parafermions (one pair at each interface).