Majorana fermions in one-dimensional spin-orbit-coupled Fermi gases

We theoretically study trapped one-dimensional Fermi gases in the presence of spin-orbit coupling induced by Raman lasers. The gas changes from a conventional (nontopological) superfluid to a topological superfluid as one increases the intensity of the Raman lasers above a critical chemical-potential-dependent value. Solving the Bogoliubov-de Gennes equations self-consistently, we calculate the density of states in real and momentum space at finite temperatures. We study Majorana fermions (MFs) which appear at the boundaries between topologically trivial and topologically nontrivial regions. We linearize the trap near the location of a MF, finding an analytic expression for the localized MF wave function and the gap between the MF state and other edge states.

Figure 1

Band structure of a 1D (pseudo) spin-$1/2$ gas. The red (dashed) curves are the bare bands in the absence of SO coupling. The blue (solid) curves are the upper band $E_{+}$ and the lower band $E_{-}$ in the presence of SO coupling, with the coupling strength $\hslash \Omega /E_r=1$.

Figure 2

Interaction coefficient $V_{kq}$ versus dimensionless momentum $q/k_L$ for $\hslash \Omega /E_r=2$. The blue (solid), green (dashed), and red (dotted) curves correspond to $k=0,0.5k_L$, and $k_L$, respectively.

Figure 3

Band structure of homogeneous gas. From the top to the bottom, the four bands are $E_{+}$, $E_{-}$, $-E_{-}$, and $-E_{+}$, respectively. The parameters are $\mu =E_r$, $\Delta =0.5E_r$, and (a) $\Omega =0$, (b) $\Omega =0.5E_r$, (c) $\Omega =E_r$, and (d) $\Omega =1.5E_r$.

Figure 4

Profiles of order parameter $\Delta \left(x\right)=g_{1\mathrm{D}}{\sum }_n(u_{n\downarrow }\left(x\right)v_{n\uparrow }^{*}\left(x\right)\langle {\xi }_n{\xi }_n^{†}\rangle +v_{n\downarrow }^{*}\left(x\right)u_{n\uparrow }\left(x\right)\langle {\xi }_n^{†}{\xi }_n\rangle )$ (dashed curves) and density $n\left(x\right)={\sum }_{n\sigma }\left(\right|v_{n\sigma }{\left(x\right)|}^2\langle {\xi }_n{\xi }_n^{†}\rangle +|u_{n\sigma }\left(x\right){|}^2\langle {\xi }_n^{†}{\xi }_n\rangle )$ (solid curves) at temperatures $T=0$, $0.1E_r$, $0.2E_r$, and $0.3E_r$. Other parameters are $g_{1\mathrm{D}}=-0.03E_{r}L$, $\hslash \Omega =2E_r$, $\lambda =4$, $k_{L}L=100$, (a) $\mu =E_r$, and (b) $\mu =2.5E_r$.

Figure 5

DOS in real space (left panel) and momentum space (right panel) at $T=0$, $0.1E_r$, $0.2E_r$, and $0.3E_r$ from the top to the bottom, with order parameters identical to those in Fig. 4a. The gray (dashed) curves in the left panels are plotted with $G\left(x\right)=\sqrt{\tilde{\mu }{\left(x\right)}^2+\Delta {\left(x\right)}^2}-\hslash \Omega /2$, where the zero points of $G\left(x\right)$ pinpoint the position of MFs. The brighter color corresponds to the higher spectral weight.

Figure 6

DOS in real space (left panel) and momentum space (right panel) at $T=0$ and $0.3E_r$, with parameters identical to those in Fig. 4b. The brighter color corresponds to the higher spectral weight.

Figure 7

DOS at zero temperature under the LDA. The parameters are identical to those in Fig. 4, except $\Delta \left(x\right)$ is calculated within the LDA. The brighter color corresponds to the higher spectral weight.

Figure 8

The gap between the MF state and excited states as a function of trap stiffness ${\lambda }^{1/4}$: the trapping potential is $V\left(x\right)=\lambda {(x/L)}^2{E}_r$. The black (solid) curve is plotted based on the analytic Eq. (46). The red (dot-dashed), green (dashed), and blue (dotted) curves are the energy levels $E_1/E_r$, $E_2/\sqrt{2}{E}_r$, and $E_3/\sqrt{3}{E}_r$, respectively. They are numerically calculated from Eq. (21) with the parameters identical to those of the solid curve.

Figure 9

Ground-state energy $E_g/E_r$ versus order parameter $\Delta /E_r$. The blue (dashed), black (solid), red (dotted), and green (dot-dashed) curves correspond to $n_{\mathrm{grid}}=400$, 600, 800, and 1000, respectively. Other parameters are ${\tilde{g}}_{1\mathrm{D}}=-0.02E_r$, $\hslash \Omega =2E_r$, $\lambda =0,k_{L}L=100$, and $\mu =E_r$.

Figure 10

Order parameter $\Delta /E_r$ versus ${10}^3/n_{\mathrm{grid}}$. The red dots are calculated from Eq. (A2). The blue (solid) curve is an extrapolation. The parameters here are identical to those in Fig. 4a except for $\lambda =0$.