Dark Solitons with Majorana Fermions in Spin-Orbit-Coupled Fermi Gases

We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin imbalance, where spin-orbit coupling favors uniform superfluids over nonuniform Fulde-Ferrell-Larkin-Ovchinnikov states, leading to dark soliton excitations in highly imbalanced gases. Above a critical spin imbalance, two topological Majorana fermions without interactions can coexist inside a dark soliton, paving a way for manipulating Majorana fermions through controlling solitons. At the topological transition point, the atom density contrast across the soliton suddenly vanishes, suggesting a signature for identifying topological solitons.

Figure 1

Profiles of the order parameter $\mathrm{\Delta }$ [(a) and (b)] and the total density $n$ [(c) and (d)] with increasing atom interactions. In (b) and (d), the details of dark solitons in a small region are plotted, where $\mathrm{\Delta }$ and $n$ are, respectively, scaled by ${\mathrm{\Delta }}_a$ and $n_a$, which are the asymptotic value to the origin point without the soliton. In (c), lines corresponding to $\mathrm{\Gamma }=1.5\pi$, $\pi$, $0.65\pi$ are moved up $1.5x_s$, $3.0x_s$, $4.5x_s$ with respect to the line for $\mathrm{\Gamma }=2.5\pi$. In the inset of (d), the density profiles ($n_{\sigma }$ for spin $\sigma$ and $n=n_{\uparrow }+n_{\downarrow }$) for a gas with $\mathrm{\Gamma }=\pi$ without SO coupling are plotted. Here $\alpha k_F=E_F$ and $V_z=0.16E_F$.

Figure 2

(a) Phase diagram in the ($\alpha$, $V_z$) plane. PTS, TS, normal superfluid (NS). (b) Quasiparticle excitation energies (all other lines except light green dashed-dotted and magenta dashed lines) and ${\mathrm{\Delta }}_0$ (maximum of the order parameter in whole region), ${\mathrm{\Delta }}_1$ (minimum of the order parameter in region ($2x_s<x<9x_s$), where the topological superfluids exist). $\alpha K_F=E_F$ and $\mathrm{\Gamma }=\pi$.

Figure 3

The left panel shows the wave functions (solid red line for $u_{\uparrow }$ and dashed green one for $u_{\downarrow }$) of four MFs: two inside the dark soliton plotted in (a) and (b) and two at the edges in the insets. (c) and (d) show the local density of states for spin $\downarrow$ and spin $\uparrow$, respectively. Here $V_z=0.768E_F$, $\alpha K_F=E_F$, and $\mathrm{\Gamma }=\pi$.

Figure 4

Density contrast $P_{\sigma }$ as a function of $V_z$ for the spin $\downarrow$ (circle green line), spin $\uparrow$ (square blue line), and total (diamond red line) densities in the soliton region $-1.5x_s<x<1.5x_s$. The insets show the density profiles ($n_{\sigma }$ for spin $\sigma$ and $n=n_{\uparrow }+n_{\downarrow }$) of the soliton in each region with the unit $1/x_s$, corresponding to $V_z=0.3E_F$, $0.7E_F$, $0.768E_F$ respectively. Here $\alpha K_F=E_F$ and $\mathrm{\Gamma }=\pi$.