The Landau critical velocity for a particle in a Fermi superfluid

We determine {\`a} la Landau the critical velocity $v\_c^{L}$ of a moving impurity in a Fermi superfluid, that is by restricting to the minimal excitation processes of the superfluid. $v\_c^{L}$ is then the minimal velocity at which these processes are energetically allowed. The Fermi superfluid actually exhibits two excitation branches~: one is the fermionic pair-breaking excitation, as predicted by BCS theory; the other one is bosonic and sets pairs into motion, as predicted by Anderson's RPA. $v\_c^{L}$ is the smallest of the two corresponding critical velocities $v\_{c,f}^{L}$ and $v\_{c,b}^{L}$. In the parameter space (superfluid interaction strength, fermion-to-impurity mass ratio), we identify two transition lines, corresponding to a discontinuity of the first-order and second-order derivatives of $v\_c^{L}$. These two lines meet in a triple point and split the plane in three domains. We briefly extend this analysis to the very recently realized case at ENS, where the moving object in the Fermi superfluid is a weakly interacting Bose superfluid of impurities, rather than a single impurity. For a Bose chemical potential much smaller than the Fermi energy, the topology of the transition lines is unaffected; a key result is that the domain $v\_c^{L}=c$, where $c$ is the sound velocity in the Fermi superfluid, is turned into a domain $v\_c^{L}=c+c\_B$, where $c\_B$ is the sound velocity in the Bose superfluid, with slightly shifted boundaries.