# Energies of dilute Fermi gases and universalities in BCS theory

Lauritsen AB. 2024. Energies of dilute Fermi gases and universalities in BCS theory. Institute of Science and Technology Austria.

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*Thesis*|

*PhD*|

*Published*|

*English*

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Abstract

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 $d=1,2,3$.

Publishing Year

Date Published

2024-09-23

Publisher

Institute of Science and Technology Austria

Page

353

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IST-REx-ID

### Cite this

Lauritsen AB. Energies of dilute Fermi gases and universalities in BCS theory. 2024. doi:10.15479/at:ista:18135

Lauritsen, A. B. (2024).

*Energies of dilute Fermi gases and universalities in BCS theory*. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18135Lauritsen, Asbjørn Bækgaard. “Energies of Dilute Fermi Gases and Universalities in BCS Theory.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18135.

A. B. Lauritsen, “Energies of dilute Fermi gases and universalities in BCS theory,” Institute of Science and Technology Austria, 2024.

Lauritsen, Asbjørn Bækgaard.

*Energies of Dilute Fermi Gases and Universalities in BCS Theory*. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18135.**All files available under the following license(s):**

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