Itinerant ferromagnetism in an atomic Fermi gas: Influence of population imbalance

We investigate ferromagnetic ordering in an itinerant ultracold atomic Fermi gas with repulsive interactions and population imbalance. In a spatially uniform system, we show that at zero temperature the transition to the itinerant magnetic phase transforms from first to second order with increasing population imbalance. Drawing on these results, we elucidate the phases present in a trapped geometry, finding three characteristic types of behavior with changing population imbalance. Finally, we outline the potential experimental implications of the findings.

Figure 1

(Color online) (a) shows the magnetization $M$ as a function of population imbalance $P$ and interaction strength $k_{\text{F}}a=\sqrt[3]{3{\pi }^2\left(n_{\uparrow }+n_{\downarrow }\right)}$ in the canonical ensemble at $T=0$ at fixed species populations. The thick line traces system variation at $P=0$, which corresponds to trap profile $\left(P/N=0\right)$ in Fig. 3. (b) shows the phase boundary between “unmagnetized” (UnM) and partially magnetized (PM) region and the line of saturation before the fully magnetized (FM) region. Note that, by unmagnetized, we refer to the not in-plane magnetization. The solid line denotes first-order transitions, the dashed second order and saturation.

Figure 2

(Color online) (a) shows the variation of magnetization $M$ as a function of chemical-potential shift $\Delta \mu$ and interaction strength $k_{\text{F}}a$ in the grand canonical ensemble at $T=0$. The thick lines correspond to trap profiles at $P/N=0$, $P/N=0.4$, and $P/N=0.8$ in Fig. 3. In the region where magnetization is undefined there is phase separation. The lower set of diagrams show the phase boundaries (and saturation line) between “unmagnetized” (UnM), partially magnetized (PM), and fully magnetized (FM) regions, as well as the region of phase separation.

Figure 3

The density of particles at radius $r$ in a trap potential profile at three different values of total population imbalance. The variation in the local particle density $N$ is shown by the solid line, and the variation in the local magnetization $M$ is shown by the dashed line. The plot densities are renormalized by their central density, $N_0$, radii by the outer radius, $r_0$, of noninteracting particles with the same average inner chemical potential, ${\mu }_0$. The upper panel shows the effective $k_{\text{F}}a$ in the $P/N=0$ case.

Figure 4

The re-parameterization of momenta used to ensure momentum conservation. ${\mathbf{k}}_{1,2,3,4}$ represent the momenta appearing in the original integral, whose separate sums are ${\mathbf{q}}_{12}={\mathbf{k}}_1+{\mathbf{k}}_2$ and ${\mathbf{q}}_{34}={\mathbf{k}}_3+{\mathbf{k}}_4$, which the Dirac delta function in Eq. (A1) will ensure that ${\mathbf{q}}_{12}={\mathbf{q}}_{34}$. ${\theta }_{12}$ represents the angle between ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$, ${\mathbf{k}}_{12}^{\perp }$ is the vector perpendicular from ${\mathbf{q}}_{12}$ to ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$, and ${\mathbf{k}}_{34}^{\perp }$ is similarly defined.