Pair Density Waves and Vortices in an Elongated Spin-$1/2$ Fermi Gas

We study the vortex structures of a (pseudo)spin-$1/2$ Fermi gas experiencing a uniform effective magnetic field in an anisotropic trap that interpolates between quasi-one dimensional (1D) and quasi-two dimensional (2D). At a fixed chemical potential, reducing the anisotropy (or equivalently increasing the attractive interactions or increasing the magnetic field) leads to instabilities towards pair density waves and vortex lattices. Reducing the chemical potential stabilizes the system. We calculate the phase diagram and explore the density and pair density. The structures are similar to those predicted for superfluid Bose gases. We further calculate the paired fraction, showing how it depends on the chemical potential and anisotropy.

Figure 1

(a) Phase diagram as a function of $\alpha$ and $\mu$. The value of $\xi$ (the number of nonzero ${\Delta }_{|n|K_0}$ ) is denoted in each region. The two black solid curves are analytical calculations for the boundaries of two continuous transitions: $\xi =0\leftrightarrow \xi =1$ and $\xi =1\leftrightarrow \xi =3$. They show a good agreement with numerics. (b) The structures of pair density $|⟨{\Psi }_{\uparrow }{\Psi }_{\downarrow }⟩{|}^2$ and density $⟨{\Psi }_{\uparrow }^{†}{\Psi }_{\uparrow }⟩$ in the corresponding regions. The color key is shown in Fig. 3

Figure 2

The pair fraction $P=2N_{\mathrm{pair}}/N$ versus $\alpha$ with $\mu =-1$, 0, 1. The exponential small $P$ for $\mu =1$ at $\alpha \to 0$ is reminiscent of the BCS limit, and the large value of $P$ for $\mu =-1$ at $\alpha \approx 1.5$ is analogous to the BEC limit. The kink on each curve corresponds to the $\xi =3\leftrightarrow \xi =2$ phase transition.

Figure 3

The profile of density (left panel) and pair density (right panel) at $\alpha =65$, $\mu =2$, where the dimensionless coordinates are $X=x/\sqrt{2}\tilde{\ell }$, $Y=y/\sqrt{2}\tilde{\ell }$. The color key is shown on the top.