an ungapped modes meet, and for $k > k_{c}$, imaginary frequencies appear (Fig. \[fig:-3\](b)). While the exact expression for $k_{c}$ is complicated, and depends on $T$ and density, we estimate the appearance of imaginary wave-vectors when $\Omega_{s} \approx \Omega_{d}$ i.e. $c_{2} \approx c_{0}/3$. The case of $c_{2} = c_{0}$ is special, and there is a single linear mode with frequency $\omega(k) =  \frac{1}{\sqrt{2}}v_{T}k$ (Fig. \[fig:-3\](c)). However as $c_{2} > c_{0}$ (Fig. \[fig:-3\](d)), imaginary frequencies appear once again for a bounded set of $k$ values: $0 < k < k_{max}(c_{2}/c_{0})$.

Summary of Results
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In Fig. \[fig:-4\] we summarize the results of our stability analysis in terms of the interaction parameters $c_{0}$ and $c_{2}$ for zero $q$. The following general features are clear:

(50, 100)(10, 10) (-80, 13)[![\[fig:-4\] Stability phase portrait in terms of the interaction strengths $c_{0}n$ and $c_{2}n$ at zero Zeeman field. The interaction strengths are measured in nK to facilitate comparison with typical experimental values. The four distinct regions are demarcated by solid black lines. I: For $c_{2} < 0$, but $c_{2} > -c_{0}$, the ferromagnetic state is the thermodynamic ground state. II: For $c_{2} > 0$, $c_{2} \le c_{0}/3$, the polar state is the ground state. III: 