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.
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.
We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a d-dimensional box. The impurities interact with the fermions via a two-body potential λv(x) where λ is a coupling constant and v(x) for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum kF we consider the regime of high density of the fermions, that is, kF large compared to one. For coupling constants that scale like λ2∼kF(2−d) we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.
The object of this thesis is to study the definition and the mathematically rigorous derivation of effective models in many body quantum mechanics. In this field, it is often impossible to compte exactly physically relevant quantities - such as the ground state energy of a system - because of the large number of degrees of freedom. Hence, introducing effective models (depending only on the local density of particles for instance) allows us to compute these quantities. Then, it remains to check that the effective model is indeed an approximation of the exact model in a macroscopic limit. Besides, in some models, some physical quantities (energy, scattering amplitude...), when computed naively, appear to be infinite. One has then to renormalize the theory in order to make these divergences (that are not observed experimentally) disappear. After putting the results of this thesis into context in the introduction, we will focus on the renormalization of a bosonic quadratic Hamiltonian on the Fock space in Chapter 2. This (formal) Hamiltonian is the sum of the second quantization of a one-particle operator and the square of a field operator, and it exhibits an ultraviolet divergence (i.e. coming from local singularities). More accurately, we will try to define a self-adjoint operator, and to show that it is the limit (in the resolvent sense) of the Hamiltonian with ultraviolet cutoff when the cutoff tends to infinity. To do so, we will first diagonalize the Hamiltonian in the regular case, using Bogoliubov transformations. Then, we will prove that the diagonalized Hamiltonian (up to a renormalization of the energy and the charge) still defines a self-adjoint, and we will show the convergence of the Hamiltonian with cut-off. In a second time, we will study the semi-classical limit of a Fermi gas, in two different scalings. In Chapter 3, we will focus on the dilute limite (i.e. with an interaction range very small compared to the typical distance between particles) of an inhomogeneous (trapped with a potential V) Fermi gas, with repulsive interactions and in three dimensions. Due to the Pauli exclusion principle, two fermions of same spin can only rarely be close enough to interact. Thus, we will consider a Fermi gas with two spin components. We will then show that the ground state energy of such a system can be approximated by a Thomas-Fermi energy (in which interactions play the role of a perturbation of the interactionless energy) in the large number of particles limit. Eventually, in Chapter 4, we will study the mean-field limit (with frequent but weak interactions) of an inhomogeneous Fermi gas, with attractive interactions and in one or two dimensions this time. Once more, we will show the convergence of the ground state energy toward a Thomas-Fermi energy. Moreover, we will also prove the convergence of the ground states. More accurately, we will show that the Husimi functions of these ground states converge to a convex combination of minimizers of the Vlasov energy.
We study a one-dimensional system of two-component fermions in the limit of strong attractive particle-particle interactions. First, we analyze scattering in the corresponding few-body problem, which is analytically solvable via Bethe ansatz. This allows us to engineer effective interactions between the system's effective degrees of freedom: fermions and bosonic dimers (tightly bound pairs of fermions). We argue that, although these interactions are strong, the resulting effective problem can be mapped onto a weakly interacting one, paving the way for the use of perturbation theory. This finding simplifies studies of many-fermion systems under confinement that are beyond reach of state-of-the-art numerical methods. We illustrate this statement by considering an impurity atom in a Fermi gas.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding l particles out of their total number N not in the condensate is exponentially small in the number l. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.
The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics. In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.