MARC details
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04207ntm a22004217a 4500 |
| 003 - CONTROL NUMBER IDENTIFIER |
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AT-ISTA |
| 005 - DATE AND TIME OF LATEST TRANSACTION |
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20250911132645.0 |
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250911s2024 au ||||| m||| 00| 0 eng d |
| 040 ## - CATALOGING SOURCE |
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ISTA |
| 100 ## - MAIN ENTRY--PERSONAL NAME |
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| Personal name |
Lauritsen, Asbjorn Baekgaard |
| 9 (RLIN) |
1084211 |
| 245 ## - TITLE STATEMENT |
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| Title |
Energies of dilute Fermi gases and universalities in BCS theory |
| 260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) |
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| Name of publisher, distributor, etc. |
Institute of Science and Technology Austria |
| Date of publication, distribution, etc. |
2024 |
| 500 ## - GENERAL NOTE |
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| General note |
Thesis |
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Abstract |
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Acknowledgements |
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About the Author |
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List of Publications |
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Table of Contents |
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List of Figures |
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Preface |
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I Energies of Dilute Fermi Gases |
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1 Introduction to the theory of dilute quantum gases |
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2 Pair of particles in an |
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3 Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion |
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4 Ground state energy of the dilute spin-polarized Fermi gas: Lower bound |
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5 Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion |
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6 Pressure of a dilute spin-polarized Fermi gas: Lower bound |
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7 Pressure of a dilute spin-polarized Fermi gas: Upper bound |
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II Universalities in BCS Theory |
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8 Brief introduction to the BCS theory of superconductivity |
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9 Universality in low-dimensional BCS theory |
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10 The BCS energy gap at high density |
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11 Universal behaviour of the BCS energy gap |
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Bibliography |
| 520 ## - SUMMARY, ETC. |
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| Summary, etc. |
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 $d=1,2,3$. |
| 856 ## - ELECTRONIC LOCATION AND ACCESS |
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| Uniform Resource Identifier |
<a href="https://doi.org/10.15479/at:ista:18135">https://doi.org/10.15479/at:ista:18135</a> |
| 942 ## - ADDED ENTRY ELEMENTS (KOHA) |
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| Source of classification or shelving scheme |
Dewey Decimal Classification |