The self-energy of an impurity in an ideal Fermi gas to second order in the interaction strength

• Published in: arXiv.org 2014-12
• Main Authors: Trefzger, Christian, Castin, Yvan
• Format: Article
• Language: English
• Subjects: Damping; Derivatives; Divergence; Fermions; Heuristic methods; Impurities; Mathematical analysis; Momentum; Particle spin; Perturbation theory; Scaling laws; Singularities
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1405.6155

We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass \(m\) perturbed by the presence of a single distinguishable impurity of mass \(M\). The interaction between the impurity and the fermions involves only the partial \(s\)-wave through the scattering length \(a\), and has negligible range \(b\) compared to the inverse Fermi wave number \(1/\kf\) of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasi-particle, which will be here a Fermi polaron (or more precisely a {\sl monomeron}). We consider the general case of an impurity moving with wave vector \(\KK\neq\OO\): Then the quasi-particle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around \(\KK=\mathbf{0}\). We rely thus upon perturbation theory, where the small and negative parameter \(\kf a\to0^-\) excludes any branches other than the monomeronic one in the ground state (as e.g.\ the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy \(\Sigma^{(2)}(\KK,\omega)\) up to second order included in \(a\). Remarkably, we obtain an analytical explicit expression for \(\Sigma^{(2)}(\KK,\omega)\) allowing us to study its derivatives in the plane \((K,\omega)\). These present interesting singularities, which in general appear in the third order derivatives \(\partial^3 \Sigma^{(2)}(\KK,\omega)\). In the special case of equal masses, \(M=m\), singularities appear already in the physically more accessible second order derivatives \(\partial^2 \Sigma^{(2)}(\KK,\omega)\); using a self-consistent heuristic approach based on \(\Sigma^{(2)}\) we then regularise the divergence of the second order derivative \(\partial\_K^2 \Delta E(\KK)\) of the complex energy of the quasi-particle found in reference [C. Trefzger, Y. Castin, Europhys. Lett. {\bf 104}, 50005 (2013)] at \(K=\kf\), and we predict an interesting scaling law in the neighborhood of \(K=\kf\). As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea, at the level of Fermi's golden rule.