Polaron residue and spatial structure in a Fermi gas

We study the problem of a mobile impurity of mass \(M\) interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass \(m\). Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue \(Z\), and the impurity-to-fermion pair correlation function \(G(x)\). We show that \(G(x)\) deviates from unity at large distances as \(-(A\_4+B\_4 \cos 2 k\_F x)/(k\_F x)^4\), where \(k\_F\) is the Fermi momentum; since \(A\_4>0\) and \(B\_4>0\), the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality catastrophe; at distances between \(\sim 1/k\_F\) and the asymptotic distance where the \(1/x^4\) law applies, they reveal that \(G(x)\) exhibits an intriguing multiscale structure.