Energy, decay rate, and effective masses for a moving polaron in a Fermi sea: Explicit results in the weakly attractive limit

TL;DR

This paper provides explicit analytical results for the energy, decay rate, and effective masses of a moving impurity in a polarized Fermi gas in the weakly attractive limit, relevant for cold atom experiments.

Contribution

It derives explicit formulas for the impurity's complex energy and effective masses up to second order in the weak coupling limit, including analysis of singularities and non-differentiability points.

Findings

Explicit analytical expressions for energy and effective masses up to second order in g.

Identification of singularities and non-differentiability in effective mass derivatives.

Discussion of physical implications and relevance to cold atom experiments.

Abstract

We study the properties of an impurity of mass $M$ moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass $m$. In the weakly attractive limit, where the effective coupling constant $g\to0^-$ and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy $\Delta E(\KK)$ of the moving impurity up to order two included in $g$. This also gives access to its longitudinal and transverse effective masses $m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion mass ratio $M/m$ we identify four regions separated by singularities in derivatives with respect to $\KK$ of the second-order term of $\Delta E(\KK)$, and we discuss the physical origin of these regions. Remarkably, the second-order term of $m_\parallel^*(\KK)$ presents points of non-differentiability, replaced by a logarithmic divergence for $M=m$, when $\KK$ is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments.