Majorana corner pairs in a two-dimensional $s$-wave cold atomic superfluid

We propose a method to prepare Majorana pairs at the corners of imprinted defects on a two-dimensional cold-atom optical lattice with $s$-wave superfluid pairing. Different from previous proposals that manipulate the effective Dirac masses, our scheme relies on the sign flip of the spin-orbit coupling at the corners, which can be tuned in experiments by adjusting the angle of incident Raman lasers. The Majorana corner pairs are found to be located at the interface between two regimes with opposite spin-orbit coupling strengths in an anticlockwise direction and are robust against certain symmetry-persevered perturbations. Our work provides a way for implementing and manipulating Majorana pairs with existing cold-atom techniques.

Figure 1

Illustration of system setup on a 2D optical lattice, where two counterpropagating Raman beams are incident with angle $\theta$. A dip potential is applied on a rectangle geometry (indicated by the blue and red curves) through single-site addressing. Under the configuration $\theta =\pi /4$, the sign of effective SOC is positive ($+$)/negative ($-$) on the blue/red lines in edge coordinate along the arrows. Two spheres (encircled by the red dashed oval) at the interface denote the Majorana corner pair (MCP). The inset above shows the level diagram.

Figure 2

(a) SF order parameter ${\mathrm{\Delta }}_s$ vs interaction $U$ and chemical potential $\mu$. A phase transition occurs between metal (M) and superfluid (SF) phases. (b) Similar as panel (a) but plotted with Bogoliubov quasiparticle energy gap $E_g$. In both panels we take $t_x=t_y=1, t_{\mathrm{sox}}=t_{\mathrm{soy}}=2, h_z=1.4$.

Figure 3

(a) Self-consistent $s$-wave pairing order parameters in real space. (b) Energy-level diagrams. (c), (d) The real-space distributions of MZMs. For all panels we choose $t_x=t_y=1,t_{\mathrm{sox}}=t_{\mathrm{soy}}=2, \mu =-7.95, {\mu }_d=-4.0, h=1.4, U=6.3$. The incident angle is $\theta =\pi /4$ in (a)–(c) and $\theta =-\pi /4$ in (d).

Figure 4

Phase diagram for 2D system with the defective outer rectangular region. In region “T,” Majorana bound states exist at corners. In (a), ${\mu }_d=-4.0, h_z=1.4, U=6.3$ are used. In (b), $\mu =-7.8, h_z=1.4, U=6.3$ are used.

Figure 5

(a) Illustration of a ring trap in which the atoms are confined. Two counterpropagating Raman lasers with wave vector $\pm {\vec{k}}_0$ couple the atomic hyperfine states. The notations $+$ and $-$ indicate the sign of SOC. Two spheres encircled by the red dashed circle denote a MCP. (b) The SOC term $\alpha \left(\varphi \right)$ plotted to varying polar angle $\varphi$. (c) The eigenspectrum computed through plane-wave expansion. (d) The particle density ${\rho }_{{}_{\mathrm{MZ}}}$ of Majorana zero modes vs the angle $\varphi$. The parameters are chosen as ${\tilde{\alpha }}_0=0.04, \eta =0.005,{\mu }^{\prime }=0.2,h_z=-1.4,{\mathrm{\Delta }}_s=0.8$.

Figure 6

(a) The amplitude of SF order parameter in real space. (b) The density distribution of the MZM. (c), (d) $z\left(k_x\right)$ in the complex plane for opposite winding number. In all panels we set $t_x=t_y=1, h_z=1.4, \mu =-8.0, {\mu }_d=-4, {\mathrm{\Delta }}_s=0.3$, and ${\mathrm{\Delta }}_a=0.11$. We choose $t_{\mathrm{sox}}=t_{\mathrm{soy}}=2$ in (a–c) and $t_{\mathrm{sox}}=t_{\mathrm{soy}}=-2$ in (d).

Figure 7

The real and imaginary parts of coefficients $\left(a_m,b_m,c_m,d_m\right)$ for the wave functions of four Majorana modes, respectively. The parameters ${\tilde{\alpha }}_0=0.04, \eta =0.005,{\mu }^{\prime }=0.2,h_z=-1.4,\text{and}\mathrm{\Delta }=0.8$ are used for all panels.