TY  - MANSCPT
AU  - Lauritsen, Asbjorn Baekgaard
TI  - Energies of dilute Fermi gases and universalities in BCS theory
PY  - 2024///
PB  - Institute of Science and Technology Austria
N1  - Thesis; Abstract; Acknowledgements; About the Author; List of Publications; Table of Contents; List of Figures; Preface; I Energies of Dilute Fermi Gases; 1 Introduction to the theory of dilute quantum gases; 2 Pair of particles in an; 3 Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion; 4 Ground state energy of the dilute spin-polarized Fermi gas: Lower bound; 5 Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion; 6 Pressure of a dilute spin-polarized Fermi gas: Lower bound; 7 Pressure of a dilute spin-polarized Fermi gas: Upper bound; II Universalities in BCS Theory; 8 Brief introduction to the BCS theory of superconductivity; 9 Universality in low-dimensional BCS theory; 10 The BCS energy gap at high density; 11 Universal behaviour of the BCS energy gap; Bibliography
N2  - This thesis consists of two separate parts. In the first part we consider a dilute Fermi gas interacting through a repulsive interaction in dimensions $d=1,2,3$. Our focus is mostly on the physically most relevant dimension $d=3$ and the setting of a spin-polarized (equivalently spinless) gas, where the Pauli exclusion principle plays a key role. We show that, at zero temperature, the ground state energy density of the interacting spin-polarized gas differs (to leading order) from that of the free (i.e. non-interacting) gas by a term of order $a_p^d\rho^{2+2/d}$ with $a_p$ the $p$-wave scattering length of the repulsive interaction and $\rho$ the density. Further, we extend this to positive temperature and show that the pressure of an interacting spin-polarized gas differs from that of the free gas by a now temperature dependent term, again of order $a_p^d\rho^{2+2/d}$. Lastly, we consider the setting of a spin-$\frac{1}{2}$ Fermi gas in $d=3$ dimensions and show that here, as an upper bound, the ground state energy density differs from that of the free system by a term of order $a_s \rho^2$ with an error smaller than $a_s \rho^2 (a_s\rho^{1/3})^{1-\eps}$ for any $\eps > 0$, where $a_s$ is the $s$-wave scattering length of the repulsive interaction. These asymptotic formulas complement the similar formulas in the literature for the dilute Bose and spin-$\frac{1}{2}$ Fermi gas, where the ground state energies or pressures differ from that of the corresponding free systems by a term of order $a_s \rho^2$ in dimension $d=3$. In the spin-polarized setting, the corrections, of order $a_p^3\rho^{8/3}$ in dimension $d=3$, are thus much smaller and requires a more delicate analysis. In the second part of the thesis we consider the Bardeen--Cooper--Schrieffer (BCS) theory of superconductivity and in particular its associated critical temperature and energy gap. We prove that the ratio of the zero-temperature energy gap and critical temperature $\Xi(T=0)/T_c$ approaches a universal constant $\pi e^{-\gamma}\approx 1.76$ in both the limit of high density in dimension $d=3$ and in the limit of weak coupling in dimensions $d=1,2$. This complements the proofs in the literature of this universal behaviour in the limit of weak coupling or low density in dimension $d=3$. Secondly, we prove that the ratio of the energy gap at positive temperature and critical temperature $\Xi(T)/T_c$ approaches a universal function of the relative temperature $T/T_c$ in the limit of weak coupling in dimensions $d=1,2,3$
UR  - https://doi.org/10.15479/at:ista:18135
ER  -