Comptes Rendus. Physique (Jan 2021)
Marche au hasard d’une quasi-particule massive dans le gaz de phonons d’un superfluide à très basse température
Abstract
We consider in dimension 3 a homogeneous superfluid at very low temperature $T$ having two types of excitations, (i) gapless acoustic phonons with a linear dispersion relation at low wave number, and (ii) gapped $\gamma $ quasiparticles with a quadratic (massive) dispersion relation in the vicinity of its extrema. Recent works [Nicolis and Penco (2018), Castin, Sinatra and Kurkjian (2017, 2019)], extending the historical study by Landau and Khalatnikov on the phonon–roton interaction in liquid helium 4, have explicitly determined the scattering amplitude of a thermal phonon on a $\gamma $ quasiparticle at rest to leading order in temperature. We generalize this calculation to the case of a $\gamma $ quasiparticle of arbitrary subsonic group velocity, with a rigorous construction of the $S$ matrix between exact asymptotic states, taking into account the unceasing phonon–phonon and phonon–$\gamma $ interaction, which dresses the incoming and emerging phonon and $\gamma $ quasiparticle by virtual phonons; this sheds new light on the Feynman diagrams of phonon–$\gamma $ scattering. In the whole domain of the parameter space (wave number $k$, interaction strength, etc.) where the $\gamma $ quasiparticle is energetically stable with respect to the emission of phonons of arbitrary wavevector, we can therefore characterize the erratic motion it performs in the superfluid due to its unceasing collisions with thermal phonons, through (a) the mean force $F(k)$ and (b) longitudinal and transverse momentum diffusion coefficients $D_{\sslash }(k)$ and $D_{\perp }(k)$ coming into play in a Fokker–Planck equation, then, at long times when the quasiparticle has thermalized, (c) the spatial diffusion coefficient $\mathcal{D}^{\mathrm{spa}}$, independent of $k$. At the location $k_0$ of an extremum of the dispersion relation, where the group velocity of the quasiparticle vanishes, $F(k)$ varies linearly with velocity with an isotropic viscous friction coefficient $\alpha $ that we calculate; if $k_0=0$, the momentum diffusion is also isotropic and $F(k_0)=0$; if $k_0>0$, it is not ($D_{\sslash }(k_0)\ne D_{\perp }(k_0)$), and $F(k_0)$ is non-zero but subleading with respect to $\alpha $ by one order in temperature. The velocity time correlation function, whose integral gives $\mathcal{D}^{\mathrm{spa}}$, also distinguishes between these two cases ($k_0$ is now the location of the minimum): if $k_0=0$, it decreases exponentially, with the expected viscous damping rate of the mean velocity; if $k_0>0$, it is bimodal and has a second component, with an amplitude lower by a factor ${\propto } T$, but with a lower damping rate in the same ratio (it is the thermalization rate of the velocity direction); this balances that. We also characterize analytically the behavior of the force and of the momentum diffusion in the vicinity of any sonic edge of the stability domain where the quasiparticle speed tends to the speed of sound in the superfluid. The general expressions given in this work are supposedly exact to leading order in temperature (order $T^8$ for $F(k)$, order $T^9$ for $D_{\sslash }(k)$, $D_{\perp }(k)$ and $F(k_0)$, order $T^{-7}$ for $\mathcal{D}^{\mathrm{spa}}$). They however require an exact knowledge of the dispersion relation of the $\gamma $ quasiparticle and of the equation of state of the superfluid at zero temperature. We therefore illustrate them in the BCS approximation, after calculating the stability domain, for a fermionic $\gamma $ quasiparticle (an unpaired fermion) in a superfluid of unpolarized spin $1/2$ fermions, a system that can be realised with cold atoms in flat bottom traps; this domain also exhibits an interesting, unobserved first order subsonic instability line where the quasi-particle is destabilized by emission of phonons of finite wave vectors, in addition to the expected sonic instability line resulting from Landau’s criterion. By the way, we refute the thesis of Lerch, Bartosch and Kopietz (2008), stating that there would be no fermionic quasiparticle in such a superfluid.
Keywords
