{"day":"23","date_published":"2024-09-23T00:00:00Z","alternative_title":["ISTA Thesis"],"publisher":"Institute of Science and Technology Austria","has_accepted_license":"1","doi":"10.15479/at:ista:18135","year":"2024","page":"353","month":"09","degree_awarded":"PhD","article_processing_charge":"No","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"date_created":"2024-09-24T10:56:25Z","publication_status":"published","citation":{"mla":"Lauritsen, Asbjørn Bækgaard. <i>Energies of Dilute Fermi Gases and Universalities in BCS Theory</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:18135\">10.15479/at:ista:18135</a>.","ista":"Lauritsen AB. 2024. Energies of dilute Fermi gases and universalities in BCS theory. Institute of Science and Technology Austria.","apa":"Lauritsen, A. B. (2024). <i>Energies of dilute Fermi gases and universalities in BCS theory</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18135\">https://doi.org/10.15479/at:ista:18135</a>","ieee":"A. B. Lauritsen, “Energies of dilute Fermi gases and universalities in BCS theory,” Institute of Science and Technology Austria, 2024.","chicago":"Lauritsen, Asbjørn Bækgaard. “Energies of Dilute Fermi Gases and Universalities in BCS Theory.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:18135\">https://doi.org/10.15479/at:ista:18135</a>.","ama":"Lauritsen AB. Energies of dilute Fermi gases and universalities in BCS theory. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:18135\">10.15479/at:ista:18135</a>","short":"A.B. Lauritsen, Energies of Dilute Fermi Gases and Universalities in BCS Theory, Institute of Science and Technology Austria, 2024."},"abstract":[{"text":"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 $d=3$ \r\nand 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. \r\n\r\nThese 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.\r\n\r\nIn 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 $d=1,2,3$.","lang":"eng"}],"author":[{"first_name":"Asbjørn Bækgaard","last_name":"Lauritsen","full_name":"Lauritsen, Asbjørn Bækgaard","orcid":"0000-0003-4476-2288","id":"e1a2682f-dc8d-11ea-abe3-81da9ac728f1"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","corr_author":"1","file_date_updated":"2024-09-26T13:12:55Z","title":"Energies of dilute Fermi gases and universalities in BCS theory","ddc":["515","539"],"publication_identifier":{"isbn":["978-3-99078-042-8"],"issn":["2663-337X"]},"oa":1,"_id":"18135","date_updated":"2026-04-16T08:17:55Z","status":"public","language":[{"iso":"eng"}],"type":"dissertation","oa_version":"Published Version","OA_place":"publisher","file":[{"date_updated":"2024-09-26T13:11:24Z","relation":"main_file","checksum":"c7bc3b31e430d57c65393051ca439575","success":1,"file_name":"Lauritsen-thesis-final.pdf","access_level":"open_access","date_created":"2024-09-26T13:11:24Z","file_size":3648831,"file_id":"18147","content_type":"application/pdf","creator":"alaurits"},{"date_updated":"2024-09-26T13:12:55Z","relation":"source_file","checksum":"39f6b1b7f83e25a3bf9f933f1ea0bc06","date_created":"2024-09-26T13:12:55Z","file_size":1625888,"file_name":"Lauritsen-thesis-source.zip","access_level":"closed","content_type":"application/x-zip-compressed","file_id":"18148","creator":"alaurits"}],"ec_funded":1,"department":[{"_id":"GradSch"},{"_id":"RoSe"}],"supervisor":[{"orcid":"0000-0002-6781-0521","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","last_name":"Seiringer","full_name":"Seiringer, Robert","first_name":"Robert"}],"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"11732"},{"relation":"part_of_dissertation","id":"14542","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18107"},{"id":"17240","relation":"part_of_dissertation","status":"public"},{"relation":"part_of_dissertation","id":"14931","status":"public"}]},"project":[{"grant_number":"I06427","_id":"bda63fe5-d553-11ed-ba76-a16e3d2f256b","name":"Mathematical Challenges in BCS Theory of Superconductivity"},{"name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","grant_number":"694227","call_identifier":"H2020"}]}