B1 Norm Approximation for Interacting Many-Body Fermi Gases
Fermi gases at low temperature show many interesting physical effects. Unoccupied states (holes) in the Fermi sea can be understood as quasi-particles of opposite charge. Not only the motion of these holes reminds of a particle with opposite charge, also the influence on other charges does. Exciting a particle in the Fermi sea so that its energy increases above the Fermi level can thus be seen as the creation of a particle-hole pair. In certain regimes these pairs behave like bosons. This bosonization of fermionic degrees of freedom is an important step in understanding superconductivity.
In this project we are going to expand the mathematical understanding of fundamental properties of Fermi gases at low temperature. First of all, we aim at deriving norm approximations for the timeevolution of weakly interacting Fermi gases, starting from a first-principle description based on the manybody Schrödinger equation. In particular, we will consider initially trapped Fermi gases in a mean-field regime, with the interaction varying on the length scale of the trap. As usual for fermionic systems, the mean-field regime will be coupled with a semi-classical limit, to reach a nontrivial competition between kinetic and potential energies. In order to obtain norm approximation, we will need to go beyond the simple self-consistent mean-field approximation and take into account correlations among particles. We will do so by means of a rigorous version of Bogolubov theory.
Another important goal is the understanding of the dynamics of large Coulomb systems at low temperature. In this case, we will first try to establish convergence towards Hartree–Fock theory (describing to leading order the evolution of initial Slater determinants). Afterwards, we plan to go beyond the mean-field description, applying again Bogolubov theory to describe the evolution of the excitations of the Slater determinant. Many important physical phenomena depend on the behavior of these corrections. Finally, we will consider the interaction of an additional charge (“tracer particle”) with the fermions. Due to the rigidity of the Fermi sea the interaction will be significantly suppressed. Excitations of fermions due to the interaction with the charge are only possible close to the Fermi surface. Unoccupied states in the sea will behave like anti-particles. We will also study the influence of the correlated excitations on the dynamics of the tracer.