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

TL;DR

This paper analytically computes the second-order self-energy of an impurity in an ideal Fermi gas, revealing singularities and providing insights into the quasiparticle properties of the Fermi polaron.

Contribution

It provides the first explicit analytical expression for the second-order impurity self-energy in a Fermi gas, including singularity analysis and implications for quasiparticle behavior.

Findings

Explicit second-order self-energy expression derived

Singularities identified in derivatives of the self-energy

Regularization approach proposed for divergences in equal mass case

Abstract

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.