[{"date_published":"2023-09-30T00:00:00Z","doi":"10.15479/at:ista:14374","date_created":"2023-09-28T14:23:04Z","page":"206","day":"30","has_accepted_license":"1","year":"2023","publisher":"Institute of Science and Technology Austria","oa":1,"title":"Boundary superconductivity in BCS theory","author":[{"first_name":"Barbara","id":"5DA90512-D80F-11E9-8994-2E2EE6697425","last_name":"Roos","full_name":"Roos, Barbara","orcid":"0000-0002-9071-5880"}],"article_processing_charge":"No","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","citation":{"chicago":"Roos, Barbara. “Boundary Superconductivity in BCS Theory.” Institute of Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14374.","ista":"Roos B. 2023. Boundary superconductivity in BCS theory. Institute of Science and Technology Austria.","mla":"Roos, Barbara. Boundary Superconductivity in BCS Theory. Institute of Science and Technology Austria, 2023, doi:10.15479/at:ista:14374.","ama":"Roos B. Boundary superconductivity in BCS theory. 2023. doi:10.15479/at:ista:14374","apa":"Roos, B. (2023). Boundary superconductivity in BCS theory. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:14374","short":"B. Roos, Boundary Superconductivity in BCS Theory, Institute of Science and Technology Austria, 2023.","ieee":"B. Roos, “Boundary superconductivity in BCS theory,” Institute of Science and Technology Austria, 2023."},"project":[{"grant_number":"694227","name":"Analysis of quantum many-body systems","call_identifier":"H2020","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"},{"_id":"bda63fe5-d553-11ed-ba76-a16e3d2f256b","name":"Mathematical Challenges in BCS Theory of Superconductivity","grant_number":"I06427"}],"related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"13207"},{"relation":"part_of_dissertation","id":"10850","status":"public"}]},"ec_funded":1,"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","file":[{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_id":"14398","checksum":"ef039ffc3de2cb8dee5b14110938e9b6","file_size":2365702,"date_updated":"2023-10-06T11:35:56Z","creator":"broos","file_name":"phd-thesis-draft_pdfa_acrobat.pdf","date_created":"2023-10-06T11:35:56Z"},{"creator":"broos","date_updated":"2023-10-06T11:38:01Z","file_size":4691734,"date_created":"2023-10-06T11:38:01Z","file_name":"Version5.zip","access_level":"closed","relation":"source_file","content_type":"application/x-zip-compressed","file_id":"14399","checksum":"81dcac33daeefaf0111db52f41bb1fd0"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2663 - 337X"]},"publication_status":"published","degree_awarded":"PhD","month":"09","alternative_title":["ISTA Thesis"],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"Superconductivity has many important applications ranging from levitating trains over qubits to MRI scanners. The phenomenon is successfully modeled by Bardeen-Cooper-Schrieffer (BCS) theory. From a mathematical perspective, BCS theory has been studied extensively for systems without boundary. However, little is known in the presence of boundaries. With the help of numerical methods physicists observed that the critical temperature may increase in the presence of a boundary. The goal of this thesis is to understand the influence of boundaries on the critical temperature in BCS theory and to give a first rigorous justification of these observations. On the way, we also study two-body Schrödinger operators on domains with boundaries and prove additional results for superconductors without boundary.\r\n\r\nBCS theory is based on a non-linear functional, where the minimizer indicates whether the system is superconducting or in the normal, non-superconducting state. By considering the Hessian of the BCS functional at the normal state, one can analyze whether the normal state is possibly a minimum of the BCS functional and estimate the critical temperature. The Hessian turns out to be a linear operator resembling a Schrödinger operator for two interacting particles, but with more complicated kinetic energy. As a first step, we study the two-body Schrödinger operator in the presence of boundaries.\r\nFor Neumann boundary conditions, we prove that the addition of a boundary can create new eigenvalues, which correspond to the two particles forming a bound state close to the boundary.\r\n\r\nSecond, we need to understand superconductivity in the translation invariant setting. While in three dimensions this has been extensively studied, there is no mathematical literature for the one and two dimensional cases. In dimensions one and two, we compute the weak coupling asymptotics of the critical temperature and the energy gap in the translation invariant setting. We also prove that their ratio is independent of the microscopic details of the model in the weak coupling limit; this property is referred to as universality.\r\n\r\nIn the third part, we study the critical temperature of superconductors in the presence of boundaries. We start by considering the one-dimensional case of a half-line with contact interaction. Then, we generalize the results to generic interactions and half-spaces in one, two and three dimensions. Finally, we compare the critical temperature of a quarter space in two dimensions to the critical temperatures of a half-space and of the full space."}],"department":[{"_id":"GradSch"},{"_id":"RoSe"}],"file_date_updated":"2023-10-06T11:38:01Z","ddc":["515","539"],"supervisor":[{"last_name":"Seiringer","orcid":"0000-0002-6781-0521","full_name":"Seiringer, Robert","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87"}],"date_updated":"2023-10-27T10:37:30Z","status":"public","type":"dissertation","tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)"},"_id":"14374"},{"oa":1,"publisher":"Institute of Science and Technology Austria","date_created":"2023-01-26T10:00:42Z","date_published":"2022-12-15T00:00:00Z","doi":"10.15479/at:ista:12390","page":"196","day":"15","year":"2022","has_accepted_license":"1","project":[{"call_identifier":"H2020","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","name":"Analysis of quantum many-body systems","grant_number":"694227"}],"title":"Translation-invariant quantum systems with effectively broken symmetry","article_processing_charge":"No","author":[{"last_name":"Brooks","full_name":"Brooks, Morris","orcid":"0000-0002-6249-0928","first_name":"Morris","id":"B7ECF9FC-AA38-11E9-AC9A-0930E6697425"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","citation":{"short":"M. Brooks, Translation-Invariant Quantum Systems with Effectively Broken Symmetry, Institute of Science and Technology Austria, 2022.","ieee":"M. Brooks, “Translation-invariant quantum systems with effectively broken symmetry,” Institute of Science and Technology Austria, 2022.","apa":"Brooks, M. (2022). Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:12390","ama":"Brooks M. Translation-invariant quantum systems with effectively broken symmetry. 2022. doi:10.15479/at:ista:12390","mla":"Brooks, Morris. Translation-Invariant Quantum Systems with Effectively Broken Symmetry. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:12390.","ista":"Brooks M. 2022. Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria.","chicago":"Brooks, Morris. “Translation-Invariant Quantum Systems with Effectively Broken Symmetry.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:12390."},"month":"12","alternative_title":["ISTA Thesis"],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"The scope of this thesis is to study quantum systems exhibiting a continuous symmetry that\r\nis broken on the level of the corresponding effective theory. In particular we are going to\r\ninvestigate translation-invariant Bose gases in the mean field limit, effectively described by\r\nthe Hartree functional, and the Fröhlich Polaron in the regime of strong coupling, effectively\r\ndescribed by the Pekar functional. The latter is a model describing the interaction between a\r\ncharged particle and the optical modes of a polar crystal. Regarding the former, we assume in\r\naddition that the particles in the gas are unconfined, and typically we will consider particles\r\nthat are subject to an attractive interaction. In both cases the ground state energy of the\r\nHamiltonian is not a proper eigenvalue due to the underlying translation-invariance, while on\r\nthe contrary there exists a whole invariant orbit of minimizers for the corresponding effective\r\nfunctionals. Both, the absence of proper eigenstates and the broken symmetry of the effective\r\ntheory, make the study significantly more involved and it is the content of this thesis to\r\ndevelop a frameworks which allows for a systematic way to circumvent these issues.\r\nIt is a well-established result that the ground state energy of Bose gases in the mean field limit,\r\nas well as the ground state energy of the Fröhlich Polaron in the regime of strong coupling, is\r\nto leading order given by the minimal energy of the corresponding effective theory. As part\r\nof this thesis we identify the sub-leading term in the expansion of the ground state energy,\r\nwhich can be interpreted as the quantum correction to the classical energy, since the effective\r\ntheories under consideration can be seen as classical counterparts.\r\nWe are further going to establish an asymptotic expression for the energy-momentum relation\r\nof the Fröhlich Polaron in the strong coupling limit. In the regime of suitably small momenta,\r\nthis asymptotic expression agrees with the energy-momentum relation of a free particle having\r\nan effectively increased mass, and we find that this effectively increased mass agrees with the\r\nconjectured value in the physics literature.\r\nIn addition we will discuss two unrelated papers written by the author during his stay at ISTA\r\nin the appendix. The first one concerns the realization of anyons, which are quasi-particles\r\nacquiring a non-trivial phase under the exchange of two particles, as molecular impurities.\r\nThe second one provides a classification of those vector fields defined on a given manifold\r\nthat can be written as the gradient of a given functional with respect to a suitable metric,\r\nprovided that some mild smoothness assumptions hold. This classification is subsequently\r\nused to identify those quantum Markov semigroups that can be written as a gradient flow of\r\nthe relative entropy.\r\n"}],"ec_funded":1,"related_material":{"record":[{"relation":"part_of_dissertation","id":"9005","status":"public"}]},"language":[{"iso":"eng"}],"file":[{"file_name":"Brooks_Thesis.pdf","date_created":"2023-01-26T10:02:34Z","file_size":3095225,"date_updated":"2023-01-26T10:02:34Z","creator":"cchlebak","success":1,"file_id":"12391","checksum":"b31460e937f33b557abb40ebef02b567","content_type":"application/pdf","relation":"main_file","access_level":"open_access"},{"file_id":"12392","checksum":"9751869fa5e7981588ad4228f4fd4bd6","relation":"source_file","access_level":"closed","content_type":"application/octet-stream","file_name":"Brooks_Thesis.tex","date_created":"2023-01-26T10:02:42Z","creator":"cchlebak","file_size":809842,"date_updated":"2023-01-26T10:02:42Z"}],"degree_awarded":"PhD","publication_status":"published","publication_identifier":{"issn":["2663-337X"]},"status":"public","tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)"},"type":"dissertation","_id":"12390","file_date_updated":"2023-01-26T10:02:42Z","department":[{"_id":"GradSch"},{"_id":"RoSe"}],"ddc":["500"],"date_updated":"2023-08-07T13:32:09Z","supervisor":[{"last_name":"Seiringer","orcid":"0000-0002-6781-0521","full_name":"Seiringer, Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","first_name":"Robert"}]},{"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"mla":"Mysliwy, Krzysztof. Polarons in Bose Gases and Polar Crystals: Some Rigorous Energy Estimates. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:11473.","ieee":"K. Mysliwy, “Polarons in Bose gases and polar crystals: Some rigorous energy estimates,” Institute of Science and Technology Austria, 2022.","short":"K. Mysliwy, Polarons in Bose Gases and Polar Crystals: Some Rigorous Energy Estimates, Institute of Science and Technology Austria, 2022.","ama":"Mysliwy K. Polarons in Bose gases and polar crystals: Some rigorous energy estimates. 2022. doi:10.15479/at:ista:11473","apa":"Mysliwy, K. (2022). Polarons in Bose gases and polar crystals: Some rigorous energy estimates. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:11473","chicago":"Mysliwy, Krzysztof. “Polarons in Bose Gases and Polar Crystals: Some Rigorous Energy Estimates.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:11473.","ista":"Mysliwy K. 2022. Polarons in Bose gases and polar crystals: Some rigorous energy estimates. Institute of Science and Technology Austria."},"title":"Polarons in Bose gases and polar crystals: Some rigorous energy estimates","article_processing_charge":"No","author":[{"last_name":"Mysliwy","full_name":"Mysliwy, Krzysztof","first_name":"Krzysztof","id":"316457FC-F248-11E8-B48F-1D18A9856A87"}],"project":[{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385"}],"day":"01","year":"2022","has_accepted_license":"1","date_created":"2022-06-30T12:15:03Z","doi":"10.15479/at:ista:11473","date_published":"2022-07-01T00:00:00Z","page":"138","oa":1,"publisher":"Institute of Science and Technology Austria","ddc":["515","539"],"date_updated":"2023-09-07T13:43:52Z","supervisor":[{"last_name":"Seiringer","full_name":"Seiringer, Robert","orcid":"0000-0002-6781-0521","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87"}],"file_date_updated":"2022-07-05T08:17:12Z","department":[{"_id":"GradSch"},{"_id":"RoSe"}],"_id":"11473","status":"public","type":"dissertation","language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"7970714a20a6052f75fb27a6c3e9976e","file_id":"11486","creator":"kmysliwy","file_size":1830973,"date_updated":"2022-07-05T08:12:56Z","file_name":"thes1_no_isbn_2_1b.pdf","date_created":"2022-07-05T08:12:56Z"},{"content_type":"application/zip","relation":"source_file","access_level":"closed","checksum":"647a2011fdf56277096c9350fefe1097","file_id":"11487","file_size":5831060,"date_updated":"2022-07-05T08:17:12Z","creator":"kmysliwy","file_name":"thes_source.zip","date_created":"2022-07-05T08:15:52Z"}],"degree_awarded":"PhD","publication_status":"published","publication_identifier":{"issn":["2663-337X"]},"ec_funded":1,"related_material":{"record":[{"id":"10564","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"8705","status":"public"}]},"oa_version":"Published Version","abstract":[{"text":"The polaron model is a basic model of quantum field theory describing a single particle\r\ninteracting with a bosonic field. It arises in many physical contexts. We are mostly concerned\r\nwith models applicable in the context of an impurity atom in a Bose-Einstein condensate as\r\nwell as the problem of electrons moving in polar crystals.\r\nThe model has a simple structure in which the interaction of the particle with the field is given\r\nby a term linear in the field’s creation and annihilation operators. In this work, we investigate\r\nthe properties of this model by providing rigorous estimates on various energies relevant to the\r\nproblem. The estimates are obtained, for the most part, by suitable operator techniques which\r\nconstitute the principal mathematical substance of the thesis.\r\nThe first application of these techniques is to derive the polaron model rigorously from first\r\nprinciples, i.e., from a full microscopic quantum-mechanical many-body problem involving an\r\nimpurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas\r\nin the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak\r\ninteractions as a low-energy effective theory for this problem.\r\nIn the second part, we investigate rigorously the ground state of the model at fixed momentum\r\nand for large values of the coupling constant. Qualitatively, the system is expected to display\r\na transition from the quasi-particle behavior at small momenta, where the dispersion relation\r\nis parabolic and the particle moves through the medium dragging along a cloud of phonons, to\r\nthe radiative behavior at larger momenta where the polaron decelerates and emits free phonons.\r\nAt the same time, in the strong coupling regime, the bosonic field is expected to behave purely\r\nclassically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to\r\nbe asymptotically equal to the one obtained from the semiclassical counterpart of the problem,\r\nfirst studied by Landau and Pekar in the 1940s. For polaron models with regularized form\r\nfactors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear\r\nfunction of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove\r\nthat for a large window of momenta below the radiation threshold, the energy-momentum\r\nrelation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the\r\nLandau–Pekar effective mass, as expected.\r\nFor the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is\r\nof the optical type and the form factor is formally UV–singular due to the nature of the point\r\ncharge-dipole interaction, we are able to give the corresponding upper bound. In contrast to\r\nthe regular case, this requires the inclusion of the quantum fluctuations of the phonon field,\r\nwhich makes the problem considerably more difficult.\r\nThe results are supplemented by studies on the absolute ground-state energy at strong coupling,\r\na proof of the divergence of the effective mass with the coupling constant for a wide class of\r\npolaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model\r\nand the application of the techniques used for its removal for the energy estimates.\r\n","lang":"eng"}],"acknowledged_ssus":[{"_id":"SSU"}],"month":"07","alternative_title":["ISTA Thesis"]},{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nd/4.0/legalcode","image":"/image/cc_by_nd.png","name":"Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)","short":"CC BY-ND (4.0)"},"type":"dissertation","status":"public","_id":"9733","file_date_updated":"2022-03-10T12:13:57Z","department":[{"_id":"GradSch"},{"_id":"RoSe"},{"_id":"JaMa"}],"date_updated":"2024-03-06T12:30:44Z","supervisor":[{"full_name":"Seiringer, Robert","orcid":"0000-0002-6781-0521","last_name":"Seiringer","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","first_name":"Robert"},{"first_name":"Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas"}],"ddc":["515","519","539"],"alternative_title":["ISTA Thesis"],"month":"08","abstract":[{"lang":"eng","text":"This thesis is the result of the research carried out by the author during his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich polaron model, specifically to its regime of strong coupling. This model, which is rigorously introduced and discussed in the introduction, has been of great interest in condensed matter physics and field theory for more than eighty years. It is used to describe an electron interacting with the atoms of a solid material (the strength of this interaction is modeled by the presence of a coupling constant α in the Hamiltonian of the system). The particular regime examined here, which is mathematically described by considering the limit α →∞, displays many interesting features related to the emergence of classical behavior, which allows for a simplified effective description of the system under analysis. The properties, the range of validity and a quantitative analysis of the precision of such classical approximations are the main object of the present work. We specify our investigation to the study of the ground state energy of the system, its dynamics and its effective mass. For each of these problems, we provide in the introduction an overview of the previously known results and a detailed account of the original contributions by the author."}],"oa_version":"Published Version","ec_funded":1,"license":"https://creativecommons.org/licenses/by-nd/4.0/","related_material":{"record":[{"id":"9787","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"9792","status":"public"},{"id":"9225","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"9781","relation":"part_of_dissertation"},{"status":"public","id":"9791","relation":"part_of_dissertation"}]},"publication_status":"published","degree_awarded":"PhD","publication_identifier":{"issn":["2663-337X"]},"language":[{"iso":"eng"}],"file":[{"file_name":"Thesis_FeliciangeliA.pdf","date_created":"2021-08-19T14:03:48Z","creator":"dfelicia","file_size":1958710,"date_updated":"2021-09-06T09:28:56Z","checksum":"e88bb8ca43948abe060eb2d2fa719881","file_id":"9944","relation":"main_file","access_level":"open_access","content_type":"application/pdf"},{"content_type":"application/octet-stream","access_level":"closed","relation":"source_file","file_id":"9945","checksum":"72810843abee83705853505b3f8348aa","date_updated":"2022-03-10T12:13:57Z","file_size":3771669,"creator":"dfelicia","date_created":"2021-08-19T14:06:35Z","file_name":"thesis.7z"}],"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"},{"call_identifier":"H2020","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","name":"Analysis of quantum many-body systems","grant_number":"694227"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"}],"article_processing_charge":"No","author":[{"full_name":"Feliciangeli, Dario","orcid":"0000-0003-0754-8530","last_name":"Feliciangeli","id":"41A639AA-F248-11E8-B48F-1D18A9856A87","first_name":"Dario"}],"title":"The polaron at strong coupling","citation":{"chicago":"Feliciangeli, Dario. “The Polaron at Strong Coupling.” Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/at:ista:9733.","ista":"Feliciangeli D. 2021. The polaron at strong coupling. Institute of Science and Technology Austria.","mla":"Feliciangeli, Dario. The Polaron at Strong Coupling. Institute of Science and Technology Austria, 2021, doi:10.15479/at:ista:9733.","ieee":"D. Feliciangeli, “The polaron at strong coupling,” Institute of Science and Technology Austria, 2021.","short":"D. Feliciangeli, The Polaron at Strong Coupling, Institute of Science and Technology Austria, 2021.","ama":"Feliciangeli D. The polaron at strong coupling. 2021. doi:10.15479/at:ista:9733","apa":"Feliciangeli, D. (2021). The polaron at strong coupling. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:9733"},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa":1,"publisher":"Institute of Science and Technology Austria","page":"180","date_created":"2021-07-27T15:48:30Z","date_published":"2021-08-20T00:00:00Z","doi":"10.15479/at:ista:9733","year":"2021","has_accepted_license":"1","day":"20"},{"alternative_title":["ISTA Thesis"],"month":"02","abstract":[{"text":"We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case.\r\nWe motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.","lang":"eng"}],"oa_version":"Published Version","ec_funded":1,"related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"7524"}]},"publication_status":"published","degree_awarded":"PhD","publication_identifier":{"issn":["2663-337X"]},"language":[{"iso":"eng"}],"file":[{"file_size":1563429,"date_updated":"2020-07-14T12:47:59Z","creator":"dernst","file_name":"thesis.pdf","date_created":"2020-02-24T09:15:06Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_id":"7515","checksum":"b4de7579ddc1dbdd44ff3f17c48395f6"},{"content_type":"application/x-zip-compressed","relation":"source_file","access_level":"closed","checksum":"ad7425867b52d7d9e72296e87bc9cb67","file_id":"7516","file_size":2028038,"date_updated":"2020-07-14T12:47:59Z","creator":"dernst","file_name":"thesis_source.zip","date_created":"2020-02-24T09:15:16Z"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"type":"dissertation","status":"public","_id":"7514","department":[{"_id":"RoSe"},{"_id":"GradSch"}],"file_date_updated":"2020-07-14T12:47:59Z","date_updated":"2023-09-07T13:12:42Z","supervisor":[{"first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6781-0521","full_name":"Seiringer, Robert","last_name":"Seiringer"}],"ddc":["510"],"oa":1,"publisher":"Institute of Science and Technology Austria","page":"148","date_created":"2020-02-24T09:17:27Z","date_published":"2020-02-24T00:00:00Z","doi":"10.15479/AT:ISTA:7514","year":"2020","has_accepted_license":"1","day":"24","project":[{"grant_number":"694227","name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"article_processing_charge":"No","author":[{"id":"30C4630A-F248-11E8-B48F-1D18A9856A87","first_name":"Simon","full_name":"Mayer, Simon","last_name":"Mayer"}],"title":"The free energy of a dilute two-dimensional Bose gas","citation":{"chicago":"Mayer, Simon. “The Free Energy of a Dilute Two-Dimensional Bose Gas.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7514.","ista":"Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas. Institute of Science and Technology Austria.","mla":"Mayer, Simon. The Free Energy of a Dilute Two-Dimensional Bose Gas. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7514.","ama":"Mayer S. The free energy of a dilute two-dimensional Bose gas. 2020. doi:10.15479/AT:ISTA:7514","apa":"Mayer, S. (2020). The free energy of a dilute two-dimensional Bose gas. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7514","ieee":"S. Mayer, “The free energy of a dilute two-dimensional Bose gas,” Institute of Science and Technology Austria, 2020.","short":"S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, Institute of Science and Technology Austria, 2020."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1"},{"related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"5856"},{"relation":"part_of_dissertation","status":"public","id":"154"},{"id":"1198","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"741","relation":"part_of_dissertation"}]},"publication_identifier":{"issn":["2663-337X"]},"degree_awarded":"PhD","publication_status":"published","file":[{"file_name":"2018_Thesis_Moser.pdf","date_created":"2019-04-09T07:45:38Z","creator":"dernst","file_size":851164,"date_updated":"2020-07-14T12:46:37Z","file_id":"6256","checksum":"fbd8c747d148b468a21213b7cf175225","relation":"main_file","access_level":"open_access","content_type":"application/pdf"},{"relation":"source_file","access_level":"closed","content_type":"application/zip","file_id":"6257","checksum":"c28e16ecfc1126d3ce324ec96493c01e","creator":"dernst","file_size":1531516,"date_updated":"2020-07-14T12:46:37Z","file_name":"2018_Thesis_Moser_Source.zip","date_created":"2019-04-09T07:45:38Z"}],"language":[{"iso":"eng"}],"alternative_title":["ISTA Thesis"],"month":"09","abstract":[{"text":"In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.","lang":"eng"}],"oa_version":"Published Version","department":[{"_id":"RoSe"}],"file_date_updated":"2020-07-14T12:46:37Z","supervisor":[{"id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","first_name":"Robert","last_name":"Seiringer","full_name":"Seiringer, Robert","orcid":"0000-0002-6781-0521"}],"date_updated":"2023-09-27T12:34:14Z","ddc":["515","530","519"],"type":"dissertation","status":"public","pubrep_id":"1043","_id":"52","page":"115","doi":"10.15479/AT:ISTA:th_1043","date_published":"2018-09-04T00:00:00Z","date_created":"2018-12-11T11:44:22Z","has_accepted_license":"1","year":"2018","day":"04","publisher":"Institute of Science and Technology Austria","oa":1,"publist_id":"8002","author":[{"first_name":"Thomas","id":"2B5FC9A4-F248-11E8-B48F-1D18A9856A87","last_name":"Moser","full_name":"Moser, Thomas"}],"article_processing_charge":"No","title":"Point interactions in systems of fermions","citation":{"chicago":"Moser, Thomas. “Point Interactions in Systems of Fermions.” Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:th_1043.","ista":"Moser T. 2018. Point interactions in systems of fermions. Institute of Science and Technology Austria.","mla":"Moser, Thomas. Point Interactions in Systems of Fermions. Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:th_1043.","ieee":"T. Moser, “Point interactions in systems of fermions,” Institute of Science and Technology Austria, 2018.","short":"T. Moser, Point Interactions in Systems of Fermions, Institute of Science and Technology Austria, 2018.","ama":"Moser T. Point interactions in systems of fermions. 2018. doi:10.15479/AT:ISTA:th_1043","apa":"Moser, T. (2018). Point interactions in systems of fermions. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_1043"},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","project":[{"_id":"25C878CE-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Structure of the Excitation Spectrum for Many-Body Quantum Systems","grant_number":"P27533_N27"}]}]