Excitation spectrum and quasiparticles in quantum gases. A rigorous approach

TL;DR

This thesis rigorously analyzes the excitation spectrum and quasiparticles in quantum gases, proposing a spectral approach, connecting approximations like Bogoliubov to a universal scheme, and justifying the Bogoliubov approximation in the large particle limit.

Contribution

It introduces a spectral and Hamiltonian-based framework for excitation spectrum and quasiparticles, unifies Bogoliubov and Hartree-Fock-Bogoliubov approximations, and rigorously justifies the Bogoliubov approximation for large systems.

Findings

Low-lying spectrum described by Bogoliubov approximation in large systems

Unified scheme for quasiparticle approximations via Gaussian states

Rigorous justification of Bogoliubov approximation in mean-field limit

Abstract

This thesis is devoted to a rigorous study of interacting quantum gases. The main objects of interest are the closely related concepts of excitation spectrum and quasiparticles. The immediate motivation of this work is to propose a spectral point of view concerning these two concepts. In the first part of this thesis we discuss the concepts of excitation spectrum and quasiparticles. We provide an overview of physical motivations and based on that we propose a spectral and Hamiltonian-based approach towards these terms. Based on that, we formulate definitions and propositions related to these concepts. In the second part we recall the Bogoliubov and Hartree-Fock-Bogoliubov approximations, which in the physics literature are used to obtain the quasiparticle picture. We show how these two approaches fit into a universal scheme which allows us to arrive at a quasiparticle picture in a more general setup. This scheme is based on the minimization of Hamiltonians over the so-called Gaussian states. Its abstract formulation is the content of Beliaev's Theorem. In the last part we present a rigorous result concerning the justification of the Bogoliubov approximation. This justification employs the concept of the mean-field and infinite-volume limit. We show that for a large number of particles, a large volume and a sufficiently high density, the low-lying energy-momentum spectrum of the homogeneous Bose gas is well described by the Bogoliubov approximation. This result, which is formulated in the form of a theorem, can be seen as the main result of this thesis.