Dynamical instability of a spin spiral in an interacting Fermi gas as a probe of the Stoner transition

We propose an experiment to probe ferromagnetic phenomena in an ultracold Fermi gas, while alleviating the sensitivity to three-body loss and competing many-body instabilities. The system is initialized in a small pitch spin spiral, which becomes unstable in the presence of repulsive interactions. To linear order the exponentially growing collective modes exhibit critical slowing down close to the Stoner transition point. Also, to this order, the dynamics are identical on the paramagnetic and ferromagnetic sides of the transition. However, we show that scattering off the exponentially growing modes qualitatively alters the collective mode structure. The critical slowing down is eliminated and in its place a new unstable branch develops at large wave vectors. Furthermore, long-wavelength instabilities are quenched on the paramagnetic side of the transition. We study the experimental observation of the instabilities, specifically addressing the trapping geometry and how phase-contrast imaging will reveal the emerging domain structure. These probes of the dynamical phenomena could allow experiments to detect the transition point and distinguish between the paramagnetic and ferromagnetic regimes.

Figure 1

(a) The gas is started in a fully polarized state and (b) a normal magnetic field $B$ (purple arrows) is applied to form a spin spiral of pitch $1/Q$. When the interaction strength is ramped upward (c) the spins rotate into the $y$ direction forming a fully polarized state (which we later show has a long wavelength modulation). The thickness of the green plane indicates the growing magnetization. In (b) and (c) the axis basis set co-rotates with the spin spiral.

Figure 2

The variation of the exponential growth rate of the collective mode at wave vector $\mathbf{q}$ with interaction strength $k_{\mathrm{F}}a$. Calculated for an initial seed field of size (a) $\Phi =0$, (b) $\Phi =0.1Q^2/g$, and (c) $\Phi =0.2Q^2/g$. The fields $\{\theta ,\phi \}$ are evaluated at wave vector $\mathbf{q}$, and $\eta$ at wave vector $\mathbf{q}\pm \mathbf{Q}$; we focus on the peak contribution to the growth with $\mathbf{q}\parallel \mathbf{Q}$. The black trajectory highlights the maximum growth rate at a given interaction strength.

Figure 3

The variation of growth rate (red solid, primary $y$ axis) and domain size (blue dashed, secondary $y$ axis) with radius within a harmonic well. (a) corresponds to linear response, and when (b) feedback corrections with a seed field of size $0.2Q^2/g$ are taken into account. The vertical cyan line denotes the radius corresponding to critical slowing and $r_{\max }$ is the outer radius of the atom distribution.

Figure 4

The Fourier transform of the domain distribution within a harmonic well. Without magnetization feedback is shown by the red solid line, and the blue dashed curve is in the presence of magnetization feedback. The peak growth rate at $Q/\sqrt{2}$ is labeled by the vertical cyan (left) line, whereas the maximum growth wave vector $Q$ is shown by the vertical green (right) line.