PhD Defense Johannes Agerskov
Title: One-dimensional Dilute Quantum Gases and Their Ground State Energies
In this thesis, we study the ground state energy of one-dimensional dilute quantum systems with repulsive pair potentials. We review part of the general theory of many-body quantum mechanics. We then prove results describing conditions under which, we can associate a unique self-adjoint many-body Hamiltonian to certain repulsive pair-potential.
The point-interacting solvable models in one dimension, i.e. the Lieb- Liniger and Yang-Gaudin models, are reviewed and certain results related to their ground state energy in the dilute limit are proved.
We proceed by proving a ground state energy expansion for the Bose gas. This is done by proving first an upper bound and next a matching lower bound. The ground state energy is found, up to next-to-leading order, to depend on the potential only through the scattering length. Thus the sys- tem exhibits universality similar to that observed for higher dimensional systems. Our result covers the well known results on the ground state energy of the Lieb-Liniger model in the Tonks-Girardeau (dilute) limit. However, our result allows for a very general class of potentials, including potential that differ significantly from the point interacting δ-potentials for example by having positive scattering length. As corollaries, we find similar result for spin polarized Fermi gases and gases with intermediate particle statistics, i.e. anyons.
Finally we study the spin–1/2 Fermi gas. Here we conjecture a ground state energy expansion based on the solvable models at hand. The up- per bound from the bosonic case is generalized by realizing the spins, in a given trial state, to be described by an effective antiferromagnetic Heisenberg chain. Thereby, we prove an upper bound matching our con- jecture. As corollaries, we find similar results for spin-1/2 bosons and for fermions and particles with spatial symmetry with spin-dependent potentials. Furthermore, we generalize parts of the lower bound proof from the bosonic case, and prove in this case for spin–1/2 fermions a lower bound related to the Lieb-Liniger-Heisenberg ground state energy. We notice that for spin-dependent potentials in certain regimes iden- tified with a ferromagnetic phase, the lower bound is reduced to that of the Lieb-Liniger model. Thus a lower bound, matching the previous upper bound, is proved in the ferromagnetic phase for spin-dependent potentials.
Advisor: Jan Philip Solovej,University of Copenhagen, Copenhagen, Denmark
Søren Fournais (Chair)
University of Copenhagen, Copenhagen, Denmark
Aarhus University, Aarhus, Denmark
Simona Rota Nodari, Université, Côte d’Azur, Nice, France
Marcello Porta, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy