Implementing Majorana fermions in a cold-atom honeycomb lattice with textured pairings

Recent studies in the realization of Majorana fermion (MF) quasiparticles have focused on engineering topological superconductivity that derives from proximity effects of conventional superconductors and spin textures. We propose an effective model to create unpaired MFs at a honeycomb lattice edge by generalizing a two-dimensional topologically nontrivial Haldane model and introducing textured pairings. The core idea is to add both the spin-singlet and textured spin-triplet pairings to a pseudospin-state-dependent, time-reversal-symmetry (TRS) noninvariant honeycomb lattice, and to satisfy generalized “sweet spot” conditions as in the Kitaev chain model. Our model has a gapped superconducting phase and a gapless phase; either phase may have zero or nonzero topological winding numbers. The discriminant that distinguishes those two phases gives a measure of TRS breaking and may have more general implications. Effective Majorana zero modes arise at edges in distinct phases with different degrees of degeneracy. Our theoretical model motivates concepts, such as “textured pairings” and the “strength” of TRS breaking, that may play important roles in future implementation of MFs with cold atoms in optical lattices.

Figure 1

Physical intuition of generating unpaired MFs at an edge in the case of $\mu =0$. (a) Kitaev's 1D spinless $p$-wave SC quantum wire. The two neighboring MFs constitute a normal fermion. The blue arrows in the upper chain signify the internal pairing of MFs with no unpaired MFs remaining. The red arrows in the lower chain indicate the intercell pairing of MFs with two unpaired MFs at the ends of the chain. (b) MF coupling at a single armchair edge of a 2D honeycomb lattice. The upper two and lower left subfigures are for our model in which there is no threefold rotational symmetry (RS). The solid bonds are the net Majorana couplings contributed by terms related to $(\mathrm{\Delta },t), ({\mathrm{\Delta }}_{\uparrow },t_{\uparrow })$, and $({\mathrm{\Delta }}_{\downarrow },t_{\downarrow })$ in $\widehat{H}$, respectively. The shaded and colored cells denote the dangling MFs in a hexagon at a single armchair edge of the lattice. The lower right subfigure shows an example of unexpected MF couplings in which there is rotational symmetry, just in comparison with that in our model.

Figure 2

Schematic of our physical system. (a) The depictions of the three vectors (${\vec{\delta }}_j$) connecting NN sites and the three vectors (${\vec{r}}_j$) connecting NNN sites for $j=1,2,3$. (b) Angular distribution of the sign in the amplitude and phase of the defined pairing which is similar to the domain wall structure. (c) The distribution of MZMs at a single armchair edge. The heights of the pillars denote the amplitudes of wave function at each site.

Figure 3

Band structure for $\mu =0$ in four cases characterized by different gap conditions and winding numbers $w$. Fixed parameters: $t=1, t_{\uparrow }=0.4, t_{\downarrow }=0.6$. (a) $\vec{p}=(0.9K_x,0)$, (b) $\vec{p}=(0,3K_y)$, (c) $\vec{p}=(0.1K_x,0.2K_y)$, and (d) $\vec{p}=(0.6K_x,3K_y)$, where $K_x=2\pi /3$ and $K_y=K_x/\sqrt{3}$. Each group of blue curves represents a bulk band. The straight dark red line is the MZM and the brown and green curves in (b) and (d) are gapless edge modes. In (a) and (b), the zero modes are fourfold degenerate; in (c) and (d), the zero modes are twofold degenerate. The inset in (c) zooms in on the split zero modes (magenta) and flat MZMs (dark red) separately.

Figure 4

Phase diagram at the generalized sweet spot when $\mu =0$, obtained by numerical simulation. (a) The phase boundary between the gapped SC phase (purple) and gapless phase (light green). (b) The phase boundary between phases with different winding numbers. The green region indicates $w=0$, the blue region $w=1$, and the red region $w=-1$. A combination of these two figures shows the full four-phase diagram.

Figure 5

Regions of the gapped SC phase in the 3D parameter space $(sin{\mathrm{\Phi }}_1,sin{\mathrm{\Phi }}_2,sin{\mathrm{\Phi }}_3)$. The shaded surface near the 12 edges of the cube shows the domain in which at least one inequality $|cos{\mathrm{\Phi }}_j|+|cos{\mathrm{\Phi }}_k|\ge |cos{\mathrm{\Phi }}_l|$ is violated, where $(j,k,l)$ is a permutation of $(1,2,3)$. This surface defines the gapped SC phase. Note that the only points in this parameter space that have physical significance are those for which ${\mathrm{\Phi }}_2={\mathrm{\Phi }}_1+{\mathrm{\Phi }}_3$ (including both the near-edge and kernel regions).