Spinodal decomposition in polarized Fermi superfluids

We discuss the dynamics of phase separation through the process of spinodal decomposition in a Fermi superfluid with population imbalance. We discuss this instability first in terms of a phenomenological Landau theory. Working within the mean-field description at zero temperature, we then find the spinodal region in the phase diagram of polarization versus interaction strength and the spectrum of unstable modes in this region. After a quench, the spinodal decomposition starts from the Sarma state, which is a minimum of the free energy with respect to the order parameter at fixed density and polarization and a maximum at fixed chemical potentials. The possibility of observing nontrivial domain structures in current experiments with trapped atomic gases is discussed.

Figure 1

(Color online) Zero temperature mean-field phase diagram for magnetization $m∕n$ versus interaction strength $1∕k_{F}a$. The phase separated (PS) region can be decomposed into two regions—the spinodal or unstable region and the metastable region (darker shade)—divided by the spinodal normal (sp-N) line (dashed) and the spinodal superfluid (sp-SF) line (dot dashed). In addition, the PS spinodal phase is divided in half by the line where the superfluid density $Q$ is zero (thin solid). The tricritical point (orange circle) is at $m∕n=1$ and $1∕k_{F}a\simeq 2.37$. Inset: schematic finite temperature phase diagram of $T_c∕{\epsilon }_F$ versus $m∕n$ for a fixed interaction strength $1∕k_{F}a$. Arrows indicate possible quenches into the spinodal region.

Figure 2

(Color online) Mean-field free energy $\Omega (h∕\mu )$ for the Hamiltonian in Eq. (7) versus $h∕\mu$ at unitarity $1∕k_{F}a=0$ $(\mu >0)$ obtained evaluating the free energy density $f_h(\Delta ∕\mu ,h∕\mu )$, respectively, at the local minimum at $\Delta ∕\mu \simeq 1.15$ (green), at $\Delta =0$ (blue), and at the local maxima (red), i.e., the Sarma state. The value of $h∕\mu$ corresponding to the spinodal normal (sp-N) point and the spinodal superfluid (sp-SF) point and the critical (cr) value for phase separation are indicated. At unitarity, the corresponding values of the magnetization are, respectively, $m_{\mathrm{cr}\text{−}\mathrm{SF}}∕n=m_{\mathrm{sp}\text{−}\mathrm{SF}}∕n=0$, $m_{\mathrm{sp}\text{−}\mathrm{N}}∕n\simeq 0.73$, and $m_{\mathrm{cr}\text{−}\mathrm{N}}∕n\simeq 0.93$ (see Fig. 1). Inset: Plots of the free energy density $f_h(\Delta ∕\mu ,h∕\mu )$ for the values of $h$ corresponding to $h_{\mathrm{sp}\text{−}\mathrm{N}}$, $h_{\mathrm{sp}\text{−}\mathrm{SF}}$, and $h_{\mathrm{cr}}$. Note that the thermodynamic free energy corresponding to the Sarma state has a positive curvature indicating instability. The cusp structure shrinks to a point at the tricritical point.

Figure 3

(Color online) Upper panels: Unstable modes frequency $\omega ∕{\epsilon }_F$ $[\omega \left(\mathbf{q}\right)\equiv -i\epsilon \left(\mathbf{q}\right)]$ versus momentum $q∕q_F$ for different values of the interaction strength $1∕k_{F}a$ across the spinodal region and for fixed polarization $m∕n=0.5$. Note that gaps correspond to complex frequencies. Lower panel: Plot of the most unstable mode frequency $\omega ∕{\epsilon }_F$ (plus, green), momentum $q∕q_F$ (times, blue), and sound velocity $2c_s∕v_F$ (solid, black) across the spinodal region for fixed polarization $m∕n=0.5$.