On the Effective Dynamics of Interacting Fermionic Many-Body Systems

Abstract

In this thesis, we study the dynamics of quantum many-body systems for $N$ interacting non-relativistic fermions. We establish effective descriptions of the $N$-body systems and construct a rigorous approximation in the limit of $N\to\infty$ . A notable achievement of this work is the treatment of Hamiltonians with coupling parameters of order 1 in physically relevant settings, addressing regimes previously unexplored by existing methods.

In the first part, we will derive the time-dependent Hartree-Fock equations (TDHF) for singular pair interactions $|\cdot|^{-s}$ for $s\in(0,2/3)$ in a newly explored strongly interacting regime. In contrast to other regimes, the Hamiltonian in our setting does not involve $N$-dependent interaction parameters when considered on volumes of order 1 and time scales of order $N^{-2/3}$. Equivalently, on volumes of order $N$ the interaction parameter scales as $g_{N}=N^{-\frac{2-s}{3}}$ for macroscopic time scales. We use and extend the counting functional method to treat this regime by introducing a gauge transformation that extracts the large interaction potential, transforming the Hamiltonian into a magnetic-type structure which involves three-body terms and two-body terms that involve differential operators rather than simple multiplication operators. A central challenge is effectively incorporating these differential operators into the framework. To overcome this, we develop a strategy combining the counting functional with a norm approximation relative to an auxiliary Hamiltonian. This auxiliary Hamiltonian will serve as a simplified reference system, allowing us to rigorously control the bad kinetic energy terms.

In the second part, we study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle in the high density limit. We consider a three-dimensional box with periodic boundary conditions and assume that the initial wave function is a product state between the impurity and a filled Fermi ball. In the limit of large Fermi momentum $k_\text{F}$ , we prove that the effective dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Our method incorporates collective excitations into the effective description by employing almost-bosonic operators recently developed for analyzing the correlation energy. This allows us to describe the formation of collective excitations by approximating the effective dynamics with an explicit coupled coherent state. Our approach covers a broad range of interaction strengths, including couplings of order 1 and time scales of the order $k_\text{F}^{-1}$. As an application, we compute response functions of the interacting system such as the Loschmidt echo and demonstrate that our theoretical predictions align with the universal features observed in recent ultracold atom experiments. These results provide new insights into the formation of polaron quasi-particles in fermionic systems and highlight the effectiveness of our methods for describing the main features of a strongly interacting quantum system in the high density limit.