On the Effective Dynamics of Impurity Particles in Interacting Fermionic Many-Body Systems

Abstract

Understanding the behaviour of impurity particles in quantum many-body systems is fundamental to both mathematical and theoretical physics. Previous mathematical work established that impurities in a dense, non-interacting Fermi gas effectively decouple from the fermions while developing mediated interactions among themselves. This thesis extends these results to the physically more realistic case of weakly interacting fermions. The central difficulty is that fermion-fermion correlations introduce volume-enhanced terms that scale with positive powers of the system size L, potentially invalidating the effective theory in the thermodynamic limit. We identify the critical scaling \kappa^2 = L^{-2d} for the fermion-fermion coupling that suppresses these contributions. Using a perturbative resolvent method, we prove that under balanced scalings of the coupling constants, the impurities decouple at leading order and evolve according to an effective Hamiltonian with induced pairwise attractions, the same qualitative behavior as in the ideal Fermi gas, now rigorously established for interacting fermions.