From nodal-ring topological superfluids to spiral Majorana modes in cold atomic systems

In this work, we consider a three-dimensional (3D) cubic optical lattice composed of coupled 1D wires with 1D spin-orbit coupling. When the $s$-wave pairing is induced through Feshbach resonance, the system becomes a topological superfluid with ring nodes, which are the ring nodal degeneracies in the bulk, and supports a large number of surface Majorana zero-energy modes. The large number of surface Majorana modes remain at zero energy even in the presence of disorder due to the protection from a chiral symmetry. When the chiral symmetry is broken, the system becomes a Weyl topological superfluid with Majorana arcs. With 3D spin-orbit coupling, the Weyl superfluid becomes a gapless phase with spiral Majorana modes on the surface. A spatial-resolved radio-frequency spectroscopy is suggested to detect this nodal-ring topological superfluid phase.

Figure 1

Schematic illustration of the gapless topological superfluids evolution: (a) ring nodes with 1D spin-orbit coupling and (d) the localized Majorana zero modes, (b) Weyl nodes in zero energy with 2D spin-orbit coupling and (e) the chiral Majorana zero modes, and (c) Weyl nodes shifted to different energies with Weyl spin-orbit coupling and (f) the spiral Majorana zero modes. The zero-energy spectral function for the surface states is also schematically shown in the surface Brillouin zone.

Figure 2

(a) The Fermi pockets of the spin-orbit-coupled Fermi gas with $\mu =0$ in the $\left(k_x,k_y\right)$ plane. The whole Fermi spheres can be obtained through rotating the Fermi pockets around the $k_x$ axis. It is noted that in the $k_x=0$ plane, all states are spin doubly degenerated. (b) The phase diagram for the spin-orbit-coupled Fermi gas with $\mu =0$ and ${\lambda }_{\mathrm{SOC}}=0.25t$. The nodal-ring superfluid phase is sandwiched between the normal superfluid phase and the normal Fermi gas.

Figure 3

The ring nodes in the superfluid phase with $U=9t$ and $\hslash {\mathrm{\Omega }}_0=0.6t$. The spectral function at zero energy for the superfluid shows the single-ring node (a) and the double-ring nodes (b) in the $k_x=0$ plane. Along the path $\left(k_{y}a,k_{z}a\right)=\left(0,\pi \right)\to \left(0,0\right)\to \left(\pi ,\pi \right)$ of the surface Brillouin zone, the quasienergy edge spectra (c) for the single-ring node and (d) for the double-ring nodes are plotted. The states in the flat band compose the surface Majorana pocket.

Figure 4

The Weyl nodes evolved from the single-ring node in the superfluid phase with $U=9t$ and $\hslash {\mathrm{\Omega }}_0=0.6t$. The spectral function at zero energy in the $k_x=0$ plane is plotted for (a) the Weyl superfluid with the Weyl nodes at the same energy and for (b) the Weyl superfluid with the energy shifted Weyl nodes. The insets in panels (a) and (b) correspond to the bulk energy spectrum crossing the Weyl nodes. In the surface Brillouin zone, the drumheadlike Majorana edge states are reduced to the Majorana arc. The Majorana arc is the flat band in panel (c) for the Weyl nodes with equal energy while it supports the unidirectional Majorana edge states in panel (d) when the Weyl nodes are energy shifted.

Figure 5

Schematic illustration of the spin-orbit coupling induced by a pair of Raman lasers. (a) The cubic optical lattice with a pair of counterpropagating Raman lasers. (b) The Raman lasers couple the magnetic sublevels $\left|F=9/2,m_F=-7/2\right.〉$ and $\left|F=9/2,m_F=-9/2\right.〉$ of ${}^{40}\mathrm{K}$ atoms trapped in the cubic optical lattice.

Figure 6

The winding number $N_{\mathrm{BDI}}$ for the nodal superfluid with $U=9t$, $\hslash {\mathrm{\Omega }}_0=0.6t$, and (a) $\mu =0$ and (b) $\mu =4t$. (c) The zero-energy edge-state wave function with $U=9t$, $\hslash {\mathrm{\Omega }}_0=0.6t$, $\mu =0$, and $k_y=k_z=0$.