Three-component topological superfluid in one-dimensional Fermi gases with spin-orbit coupling

We theoretically investigate one-dimensional three-component spin-orbit-coupled Fermi gases in the presence of the Zeeman field. By solving the Bogoliubov-de Gennes equations, we obtain the phase diagram at a given chemical potential and order parameter. We show that, with increasing the intensity of the Zeeman field, in addition to undergoing a phase transition from Bardeen-Cooper-Schrieffer (BCS) superfluid to topological superfluid, similar to the two-component system, the three-component system may exhibit some other interesting topological phase transitions. For example, by appropriately adjusting the chemical potential $\mu$, the system can be in a nontrivial topological superfluid in the whole region of the Zeeman field $h$. It also may initially be a topological superfluid and then translate to a topologically trivial BCS superfluid with increasing the field $h$. Even more exotically, the system may exhibit a re-entrance behavior, being a topological superfluid at small and large fields but a topologically trivial BCS superfluid in between at a mediate Zeeman field. It can therefore have two regions with zero-energy Majorana fermions. As a consequence of these interesting topological phase transitions, the system of the three-component spin-orbit-coupled Fermi gases in a certain parameter range is more optimizing for the experimental realization of the topological phase due to the smaller magnetic field needed. Thus, a promising candidate for the realization of the topological phase is proposed.

Figure 1

The single-particle band structure of a three-component Fermi gas. The (red) dashed lines are three bands in the absence of the Zeeman field $h/E_r=0$, while the (blue) solid lines refer to $h/E_r=1$.

Figure 2

Spectrum in momentum space for the homogeneous system. The parameters are $\mu =0$ and $\Delta =E_r$. As the Zeeman field increases, we see the gap closes at $k_x=0$ when $h_c=1.28E_r$ and then reopens for $h>h_c$.

Figure 3

Profile of energy spectrum for the system of open boundary conditions. The zero mode emerges at the Zeeman field $h=1.4E_r$. The inset shows the Berry phase $e^{i\gamma }$ varies from $-1$ to 1 as the system evolves from the conventional superfluid to the topological superfluid.

Figure 4

Wave functions for different components with the system of length $x=100/k_r$. By plotting the square root of the wave function ${\Psi }_{\eta }\left(x\right)$, we see that all these wave functions are localized at two edges. Here $|u_i\left(x\right)|$ and $|v_i\left(x\right)|$ refer to the particle and hole part, respectively. From Eq. (7), we know that the Bogoliubov transformation is a combination of quasiparticle operators ${\Gamma }_{-1,\eta }$, ${\Gamma }_{0,\eta }$, and ${\Gamma }_{1,\eta }$. As a result, the Majorana modes are manifested in $|u_i\left(x\right)|$ and $|v_i\left(x\right)|$ $(i=-1,0,1)$.

Figure 5

Band structure of the homogeneous gas under fixed parameters $\mu =1.5E_r$ and $\Delta =E_r$. Panels (a)–(c) show the spectrum at $h=0$, $h=0.6E_r$, and $h=E_r$, respectively; panel (d) illustrates that the Berry phase stays the same even when the gap closes at the nonzero $k_x/k_r$.

Figure 6

Phase diagram at $\Delta =E_r$. Compared with the two-component superfluid (red dotted line), the three-component case (blue lines) is more complicated. Four regions are divided by black dashed lines at different chemical potentials $\mu$. The gray area shows the topologically nontrivial phase. The details of the regions I to IV are explained in the main text.

Figure 7

Profile of the Berry phase $e^{i\gamma }$ at the fixed order parameter $\Delta =E_r$ with different $\mu$. Panels (a)–(d) correspond to the regions I–IV shown in Fig. 6, respectively.

Figure 8

Profile of the energy spectrum at four different phase regions. The parameters are the same as those in Fig. 7. (a) $\mu =-E_r$. The system undergoes a “conventional” topological phase transition which exceeds the critical Zeeman field $h_c=2.2E_r$. (b) $\mu =2E_r$. The system is topologically nontrivial at any values of $h$. (c) $\mu =3.5E_r$. It has two critical Zeeman fields with $h_{c1}=E_r$ and $h_{c2}=3.3E_r$. The system is topologically trivial when $h_{c1}<h<h_{c2}$. (d) $\mu =5E_r$. The zero mode disappears when $h>h_c=1.6E_r$.

Figure 9

Phase diagram at (a) $\Delta =\sqrt{2}{E}_r$ and (b) $\Delta =1.5E_r$, respectively. When $\Delta >\sqrt{2}{E}_r$, it has only three regions. The (blue) solid lines are for the three-component superfluid while the (red) dotted line is for the two-component one. The topologically nontrivial phase is highlighted by the gray area.