Spontaneous mass current and textures of $p$-wave superfluids of trapped fermionic atom gases at rest and under rotation

Based on the Ginzburg-Landau framework, we find that $p$-wave superfluids of neutral atom gases in three-dimensional harmonic traps exhibit spontaneous mass current at rest with direction dependent on trap geometry. Under rotation, various types of order parameter textures are stabilized including Mermin-Ho and Anderson-Toulouse-Chechetkin vortices. In a cigar shape trap, spontaneous current flows parallel to the rotation axis and thus perpendicular to the ordinary rotational current. The spontaneous mass current at rest and texture formation can be used to investigate $p$-wave superfluidity.

Figure 1

(Color online) Phase diagram of the $p$-wave pairing state in an infinite system showing temperature $\left(t_x=t_y\right)$ versus anisotropy $\alpha \equiv T_{cx}/T_{cz}$. $N$: normal state, $B$: polar state $A_z{p}_z$, $A$: chiral state $A_z\left(p_z+i\gamma p_x\right)$.

Figure 2

(Color online) Stable texture at rest for cigar trap $\lambda =0.2$. (a) Distribution of $\left|l\right|$ in $z\text{−}x$ plane. (b)–(d) Three cross sections indicated in (a), showing $l_z$, and $l_x$ and $l_y$ components (arrows). (e) Spontaneous circulating current flows in $z\text{−}y$ plane along the $z$ direction. Values of the current are denoted in units of $\frac{m}{\hslash }\sqrt{{\alpha }_0^{3}K/{\left({\beta }_{24}\right)}^2}$.

Figure 3

(Color online) Texture change with rotation speed $\Omega$ relative to the in-plane trap frequency ${\omega }_{\perp }$. Central cross sections are displayed showing $l_z$, and $l_x$ and $l_y$ components (arrows). $\Omega =0$ corresponds to (c) in Fig. 1. As $\Omega$ increases, $\vec{l}$ vectors acquire a $z$ component (seen as a color change). At $\Omega =0.4{\omega }_{\perp }$, two MH-like vortices enter from the $y$ direction, seen as yellow objects. At $\Omega =0.5{\omega }_{\perp }$, four MH-like vortices are visible and the condensate expands and distorts.

Figure 4

(Color online) $z\text{−}x$ cross sections corresponding to Fig. 3. The $\vec{l}$ vectors lie almost entirely in the $x\text{−}y$ plane at $\Omega =0$. Under rotation, two polar core vortices appear shown as asterisks. At $\Omega =0.5{\omega }_{\perp }$, two of the four MH-like vortices can be seen indicated by red arrows.

Figure 5

(Color online) Stable texture in the pancake trap with $\lambda =3.0$ at rest. (a) $\vec{l}$-vector pattern in the $z\text{−}r$ plane. The pattern is axis symmetric around $z$. (b) Spontaneous mass current in the $x\text{−}y$ plane at $z=0$.