Effective Dynamics of Two Tracer Particles Coupled to a Fermi Gas in the High-Density Limit

Abstract

In a recent paper (Jeblick, Mitrouskas, Petrat and Pickl, Commun. Math. Phys. 356(1), 143–187 (2017) [1]), we studied the dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions. In that paper, closeness of the time evolution to an effective free dynamics for large densities of the gas was proven. The main point of interest of this result was to consider coupling constants that do not scale with the density of the gas. In the present article, we generalize this result to a system of two tracer particles coupled to the ideal Fermi gas. Naively, one might expect this to be a trivial extension of the single particle result. But this is not the case since there is an interesting additional effect: While the dynamics of both tracer particles decouple again from the Fermi gas, there are collision processes which are not present for a single tracer particle and which lead to an effective interaction between the two tracer particles. This effective interaction is mediated through the creation and immediate annihilation of electron–hole pairs, and can thus be interpreted as a Lamb shift-type effect.