Anisotropic Weyl Fermions from the Quasiparticle Excitation Spectrum of a 3D Fulde-Ferrell Superfluid

Weyl fermions, first proposed for describing massless chiral Dirac fermions in particle physics, have not been observed yet in experiments. Recently, much effort has been devoted to explore Weyl fermions around band touching points of single-particle energy dispersions in certain solid state materials (named Weyl semimetals), similar as graphene for Dirac fermions. Here we show that such Weyl semimetals also exist in the quasiparticle excitation spectrum of a three-dimensional spin-orbit-coupled Fulde-Ferrell superfluid. By varying Zeeman fields, the properties of Weyl fermions, such as their creation and annihilation, number and position, as well as anisotropic linear dispersions around band touching points, can be tuned. We study the manifestation of anisotropic Weyl fermions in sound speeds of Fulde-Ferrell fermionic superfluids, which are detectable in experiments.

Figure 1

Plot of ${\mathrm{\Delta }}_0$ in (a), $\mu$ in (b), and $Q_y$ in (c) as a function of $1/K_F{a}_s$ for different parameters ($h_x$, $h_z$). $\alpha K_F=E_F$ and the temperature $T=0$.

Figure 2

(a) Mean-field phase diagrams of 3D spin-orbit-coupled Fermi gases. The area in the small black box is enlarged in the inset, which shows the gapless FF phase. NG: Normal gas. (b) Contours of the zero energy quasiparticle spectrum in the plane ($k_y$, $k_z$) with $k_x=0$. Here, $h_x=0.5E_F$, $h_z=0$ (blue line), $h_z=0.2E_F$ (red line), $h_z=E_F$ (green line), and $h_x=0.2E_F$, $h_z=E_F$ (black points) correspond to blue, red, green, and black square points in (a). The brown point in the gapped FF phase has no zero energy excitations. In both figures, $\alpha K_F=E_F$ and $1/a_s{K}_F=-0.1$.

Figure 3

(a) The change of the quasiparticle excitation gap $E_g$, ${\mathrm{\Delta }}_0$, $Q_y$, $\mu$ as a function of $h_z$ with $h_x=0.2E_F$. (b) Gap closing points ($E_g=0$) in the ($h_z$, $k_z$) plane for $h_x=0$ (solid blue line), $h_x=0.1$ (dashed red line), and $h_x=0.2$ (dashed-dotted green line). (c) Chern number for a fixed $k_z$ plane. (d) Display of the density of states, which is calculated via the iterative Green function method [49], $k_y=0$. The light blue region and the red line (Fermi arch) represent bulk and surface excitations, respectively. In (c) and (d), $h_x=0.2E_F$, $h_z=0.55E_F$. In all four figures $\alpha K_F=E_F$, $1/a_s{K}_F=-0.1$.

Figure 4

(a) Quasiparticle excitations around the Weyl node ${\mathbf{k}}_W=(0,0,0.77K_F)$ with $h_z=0.55E_F$ and $h_x=0.2E_F$. The inset gives the contours of energy with $E=0.1E_F$ ( red line) and $E=-0.1E_F$ (blue line). (b) Quasiparticle excitation spectrum in the gapless topological FF phase as a function of $k_y$ with fixed $k_z=0.8K_F$ and confinement in the $x$ direction. Black lines are bulk states while the red lines correspond to the surface state. $h_x=0.5E_F$ and $h_z=0.2E_F$. $\alpha K_F=E_F$, $1/a_s{K}_F=-0.1$ for both (a) and (b).

Figure 5

Sound speeds as a function of $h_z$. Dashed red and dotted black lines represent the speed along the $x$ and $z$ directions. Solid blue and dashed-dot green lines represent sound speeds along the negative $y$ and positive $y$ directions, respectively. The unit of sound speed is $v_F=\hslash K_F/m$. Here, $h_x=0.2E_F$, $\alpha K_F=E_F$, and $1/a_s{K}_F=-0.1$.