Gapless topological Fulde-Ferrell superfluidity induced by an in-plane Zeeman field

Topological superfluids are recently discovered quantum matter that hosts topologically protected gapless edge states known as Majorana fermions—exotic quantum particles that act as their own antiparticles and obey non-Abelian statistics. Their realizations are believed to lie at the heart of future technologies such as fault-tolerant quantum computation. To date, the most efficient scheme to create topological superfluids and Majorana fermions is based on the Sau-Lutchyn-Tewari-Das Sarma model with a Rashba-type spin-orbit coupling on the x-y plane and a large out-of-plane (perpendicular) Zeeman field along the z direction. Here we propose an alternative setup, where the topological superfluid phase is driven by applying an in-plane Zeeman field. This scheme offers a number of different features, notably Cooper pairings at finite center-of-mass momentum (i.e., Fulde-Ferrell pairing) and gapless excitations in the bulk. As a result, gapless topological quantum matter with an inhomogeneous pairing order parameter appears. It features unidirectional Majorana surface states at boundaries, which propagate in the same direction and connect two Weyl nodes in the bulk. We demonstrate the emergence of such exotic topological matter and the associated Majorana fermions in spin-orbit coupled atomic Fermi gases, and we determine its parameter space. The implementation of our scheme in semiconductor/superconductor heterostructures is briefly discussed.

Figure 1

Formation of an effective p-wave energy band. The energy band of the system splits into two by spin-orbit coupling. A large in-plane Zeeman field along the x direction strongly tilts the energy dispersion. When the Fermi energy lies below the upper band, atoms occupy the lower band only and form an effective “spinless” system, in which the composite particle consists of both spin-up and spin-down ingredients (shown by circles with arrows), and they interact with each other via an effective p-wave interaction.

Figure 2

Zero-temperature phase diagram of the FF superfluid at the interaction parameter $1/\left(k_F{a}_s\right)=-0.5$. With an increasing in-plane Zeeman field, the Fermi cloud changes from a gapped FF superfluid to a gapless FF superfluid, and finally it turns into a gapless topological superfluid. The strength of spin-orbit coupling is in units of $E_F/k_F$.

Figure 3

The evolution of the energy gap and of the topology of the Fermi surfaces at $\lambda =E_F/k_F$ with increasing in-plane Zeeman field. (a) The global energy gap $E_g=min{E}_{2+}\left(\mathbf{k}\right)$ (red dashed line), the energy gap at $\mathbf{k}=\mathbf{0}$ (black solid line), and the minimum energy of the surface states (green solid circles) when an open boundary is imposed on the y-z plane. (b) The energy dispersion $E_{2\pm }\left(k_y\right)$ at $k_x=0$ and $k_z=0$. (c), (d), and (e) The 3D full plot of the energy dispersion $E_{2\pm }(k_x,k_{\perp }=\sqrt{k_y^2+k_z^2})$ at $h_{c1}\simeq 0.3E_F$ (c), $h_{c2}\simeq 0.327E_F$ (d), and $h=0.4E_F$ (e). The wave vector and the energy are in units of $k_F$ and $E_F$, respectively.

Figure 4

Majorana surface states arising from the hard-wall confinement perpendicular to the y-z plane. (a) and (b) The energy spectrum $E_{k_x}^{\left(m\right)}$ as a function of $k_x$ for $m=0$ and 10, respectively. (c) The wave function of the zero-energy Majorana fermions for $m=0$, satisfying the symmetry $u_{\sigma }\left(r\right)=e^{i\vartheta }{v}_{\sigma }^{*}\left(r\right)$, where $\vartheta$ is a constant phase factor and $\sigma =\uparrow ,\downarrow$. (d) The wave function of the zero-energy surface state for $m=10$. In numerical calculations, we have set the radius of the confinement $L=200k_F^{-1}$. Other parameters are $\lambda =E_F/k_F$ and $h=0.4E_F$. The wave function is normalized to unity: ${\int }_0^{L}2\pi rdr{\sum }_{\sigma }[{\left|u_{\sigma }\left(r\right)\right|}^2+{\left|v_{\sigma }\left(r\right)\right|}^2]=1$.

Figure 5

Majorana surface states arising from the hard-wall confinement along the y direction. (a) The surface-state dispersion forms two sheets that cross at the line $k_z=0$. (b) and (c) The full energy spectrum $E_{2\pm }(k_x,k_z)$ along the $k_z$ or $k_x$ axis, respectively. The surface states at the two boundaries are highlighted by red solid circles and blue empty squares, respectively. (d) The wave function of the zero-energy Majorana fermions at $k_x=0$ and $k_z=0$. In numerical calculations, we have set the length of the confinement $L=200k_F^{-1}$. Other parameters are $\lambda =E_F/k_F$ and $h=0.4E_F$ as in Fig. 4. The wave function is normalized to unity: ${\int }_0^{L}dy{\sum }_{\sigma }[{\left|u_{\sigma }\left(y\right)\right|}^2+{\left|v_{\sigma }\left(y\right)\right|}^2]=1$.