[{"ddc":["510"],"quality_controlled":"1","publisher":"Cambridge University Press","article_processing_charge":"No","doi":"10.1017/fms.2020.17","type":"journal_article","_id":"7790","date_updated":"2023-08-21T06:18:49Z","publication":"Forum of Mathematics, Sigma","status":"public","project":[{"_id":"25C6DC12-B435-11E9-9278-68D0E5697425","grant_number":"694227","name":"Analysis of quantum many-body systems","call_identifier":"H2020"}],"date_published":"2020-03-14T00:00:00Z","ec_funded":1,"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"7524"}]},"external_id":{"arxiv":["1910.03372"],"isi":["000527342000001"]},"isi":1,"year":"2020","intvolume":"         8","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","text":"We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 ."}],"has_accepted_license":"1","file_date_updated":"2020-07-14T12:48:03Z","publication_status":"published","publication_identifier":{"eissn":["20505094"]},"title":"The free energy of the two-dimensional dilute Bose gas. I. Lower bound","oa_version":"Published Version","scopus_import":"1","day":"14","author":[{"last_name":"Deuchert","full_name":"Deuchert, Andreas","id":"4DA65CD0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3146-6746","first_name":"Andreas"},{"full_name":"Mayer, Simon","id":"30C4630A-F248-11E8-B48F-1D18A9856A87","last_name":"Mayer","first_name":"Simon"},{"last_name":"Seiringer","full_name":"Seiringer, Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","first_name":"Robert","orcid":"0000-0002-6781-0521"}],"date_created":"2020-05-03T22:00:48Z","article_type":"original","volume":8,"language":[{"iso":"eng"}],"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ista":"Deuchert A, Mayer S, Seiringer R. 2020. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma. 8, e20.","chicago":"Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2020. <a href=\"https://doi.org/10.1017/fms.2020.17\">https://doi.org/10.1017/fms.2020.17</a>.","mla":"Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” <i>Forum of Mathematics, Sigma</i>, vol. 8, e20, Cambridge University Press, 2020, doi:<a href=\"https://doi.org/10.1017/fms.2020.17\">10.1017/fms.2020.17</a>.","apa":"Deuchert, A., Mayer, S., &#38; Seiringer, R. (2020). The free energy of the two-dimensional dilute Bose gas. I. Lower bound. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2020.17\">https://doi.org/10.1017/fms.2020.17</a>","ama":"Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. <i>Forum of Mathematics, Sigma</i>. 2020;8. doi:<a href=\"https://doi.org/10.1017/fms.2020.17\">10.1017/fms.2020.17</a>","short":"A. Deuchert, S. Mayer, R. Seiringer, Forum of Mathematics, Sigma 8 (2020).","ieee":"A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. I. Lower bound,” <i>Forum of Mathematics, Sigma</i>, vol. 8. Cambridge University Press, 2020."},"month":"03","arxiv":1,"file":[{"relation":"main_file","checksum":"8a64da99d107686997876d7cad8cfe1e","file_name":"2020_ForumMath_Deuchert.pdf","access_level":"open_access","content_type":"application/pdf","file_id":"7797","file_size":692530,"date_created":"2020-05-04T12:02:41Z","date_updated":"2020-07-14T12:48:03Z","creator":"dernst"}],"article_number":"e20","department":[{"_id":"RoSe"}]},{"oa_version":"Preprint","title":"The free energy of the two-dimensional dilute Bose gas. II. Upper bound","author":[{"last_name":"Mayer","full_name":"Mayer, Simon","id":"30C4630A-F248-11E8-B48F-1D18A9856A87","first_name":"Simon"},{"first_name":"Robert","orcid":"0000-0002-6781-0521","full_name":"Seiringer, Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","last_name":"Seiringer"}],"day":"22","scopus_import":"1","article_type":"original","date_created":"2020-07-19T22:00:59Z","volume":61,"abstract":[{"lang":"eng","text":"We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]2+) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion."}],"intvolume":"        61","publication_identifier":{"issn":["00222488"]},"publication_status":"published","month":"06","arxiv":1,"article_number":"061901","department":[{"_id":"RoSe"}],"language":[{"iso":"eng"}],"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","issue":"6","citation":{"apa":"Mayer, S., &#38; Seiringer, R. (2020). The free energy of the two-dimensional dilute Bose gas. II. Upper bound. <i>Journal of Mathematical Physics</i>. AIP Publishing. <a href=\"https://doi.org/10.1063/5.0005950\">https://doi.org/10.1063/5.0005950</a>","mla":"Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. II. Upper Bound.” <i>Journal of Mathematical Physics</i>, vol. 61, no. 6, 061901, AIP Publishing, 2020, doi:<a href=\"https://doi.org/10.1063/5.0005950\">10.1063/5.0005950</a>.","ista":"Mayer S, Seiringer R. 2020. The free energy of the two-dimensional dilute Bose gas. II. Upper bound. Journal of Mathematical Physics. 61(6), 061901.","chicago":"Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. II. Upper Bound.” <i>Journal of Mathematical Physics</i>. AIP Publishing, 2020. <a href=\"https://doi.org/10.1063/5.0005950\">https://doi.org/10.1063/5.0005950</a>.","ieee":"S. Mayer and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. II. Upper bound,” <i>Journal of Mathematical Physics</i>, vol. 61, no. 6. AIP Publishing, 2020.","short":"S. Mayer, R. Seiringer, Journal of Mathematical Physics 61 (2020).","ama":"Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. II. Upper bound. <i>Journal of Mathematical Physics</i>. 2020;61(6). doi:<a href=\"https://doi.org/10.1063/5.0005950\">10.1063/5.0005950</a>"},"publisher":"AIP Publishing","doi":"10.1063/5.0005950","article_processing_charge":"No","type":"journal_article","date_updated":"2023-08-22T08:12:40Z","_id":"8134","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2002.08281"}],"external_id":{"isi":["000544595100001"],"arxiv":["2002.08281"]},"year":"2020","isi":1,"project":[{"name":"Analysis of quantum many-body systems","grant_number":"694227","call_identifier":"H2020","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"}],"status":"public","publication":"Journal of Mathematical Physics","date_published":"2020-06-22T00:00:00Z","ec_funded":1},{"month":"02","supervisor":[{"last_name":"Seiringer","full_name":"Seiringer, Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","first_name":"Robert","orcid":"0000-0002-6781-0521"}],"file":[{"checksum":"b4de7579ddc1dbdd44ff3f17c48395f6","relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_name":"thesis.pdf","file_id":"7515","date_updated":"2020-07-14T12:47:59Z","creator":"dernst","date_created":"2020-02-24T09:15:06Z","file_size":1563429},{"creator":"dernst","date_updated":"2020-07-14T12:47:59Z","date_created":"2020-02-24T09:15:16Z","file_size":2028038,"file_id":"7516","content_type":"application/x-zip-compressed","access_level":"closed","file_name":"thesis_source.zip","checksum":"ad7425867b52d7d9e72296e87bc9cb67","relation":"source_file"}],"department":[{"_id":"RoSe"},{"_id":"GradSch"}],"language":[{"iso":"eng"}],"oa":1,"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"short":"S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, Institute of Science and Technology Austria, 2020.","ieee":"S. Mayer, “The free energy of a dilute two-dimensional Bose gas,” Institute of Science and Technology Austria, 2020.","ama":"Mayer S. The free energy of a dilute two-dimensional Bose gas. 2020. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7514\">10.15479/AT:ISTA:7514</a>","mla":"Mayer, Simon. <i>The Free Energy of a Dilute Two-Dimensional Bose Gas</i>. Institute of Science and Technology Austria, 2020, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7514\">10.15479/AT:ISTA:7514</a>.","apa":"Mayer, S. (2020). <i>The free energy of a dilute two-dimensional Bose gas</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:7514\">https://doi.org/10.15479/AT:ISTA:7514</a>","chicago":"Mayer, Simon. “The Free Energy of a Dilute Two-Dimensional Bose Gas.” Institute of Science and Technology Austria, 2020. <a href=\"https://doi.org/10.15479/AT:ISTA:7514\">https://doi.org/10.15479/AT:ISTA:7514</a>.","ista":"Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas. Institute of Science and Technology Austria."},"oa_version":"Published Version","title":"The free energy of a dilute two-dimensional Bose gas","day":"24","author":[{"full_name":"Mayer, Simon","id":"30C4630A-F248-11E8-B48F-1D18A9856A87","last_name":"Mayer","first_name":"Simon"}],"date_created":"2020-02-24T09:17:27Z","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"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"has_accepted_license":"1","file_date_updated":"2020-07-14T12:47:59Z","publication_status":"published","publication_identifier":{"issn":["2663-337X"]},"related_material":{"record":[{"id":"7524","relation":"part_of_dissertation","status":"public"}]},"year":"2020","status":"public","project":[{"_id":"25C6DC12-B435-11E9-9278-68D0E5697425","name":"Analysis of quantum many-body systems","grant_number":"694227","call_identifier":"H2020"}],"degree_awarded":"PhD","date_published":"2020-02-24T00:00:00Z","ec_funded":1,"publisher":"Institute of Science and Technology Austria","alternative_title":["ISTA Thesis"],"article_processing_charge":"No","doi":"10.15479/AT:ISTA:7514","type":"dissertation","_id":"7514","date_updated":"2023-09-07T13:12:42Z","ddc":["510"],"page":"148"},{"abstract":[{"lang":"eng","text":"We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\\rho$ and inverse temperature $\\beta$ differs from the one of the non-interacting system by the correction term $4 \\pi \\rho^2 |\\ln a^2 \\rho|^{-1} (2 - [1 - \\beta_{\\mathrm{c}}/\\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\\cdot]_+ = \\max\\{ 0, \\cdot \\}$ and $\\beta_{\\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit\r\n$a^2\\rho \\ll 1$ and if $\\beta \\rho \\gtrsim 1$."}],"page":"61","publication_status":"draft","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.03372"}],"oa_version":"Preprint","publisher":"ArXiv","title":"The free energy of the two-dimensional dilute Bose gas. I. Lower bound","day":"08","scopus_import":1,"article_processing_charge":"No","author":[{"first_name":"Andreas","orcid":"0000-0003-3146-6746","last_name":"Deuchert","id":"4DA65CD0-F248-11E8-B48F-1D18A9856A87","full_name":"Deuchert, Andreas"},{"last_name":"Mayer","id":"30C4630A-F248-11E8-B48F-1D18A9856A87","full_name":"Mayer, Simon","first_name":"Simon"},{"orcid":"0000-0002-6781-0521","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Seiringer, Robert","last_name":"Seiringer"}],"date_created":"2020-02-26T08:46:40Z","type":"preprint","_id":"7524","date_updated":"2023-09-07T13:12:41Z","status":"public","publication":"arXiv:1910.03372","language":[{"iso":"eng"}],"project":[{"call_identifier":"H2020","name":"Analysis of quantum many-body systems","grant_number":"694227","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2019-10-08T00:00:00Z","citation":{"short":"A. Deuchert, S. Mayer, R. Seiringer, ArXiv:1910.03372 (n.d.).","ieee":"A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. I. Lower bound,” <i>arXiv:1910.03372</i>. ArXiv.","ama":"Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. <i>arXiv:191003372</i>.","mla":"Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” <i>ArXiv:1910.03372</i>, ArXiv.","apa":"Deuchert, A., Mayer, S., &#38; Seiringer, R. (n.d.). The free energy of the two-dimensional dilute Bose gas. I. Lower bound. <i>arXiv:1910.03372</i>. ArXiv.","ista":"Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. arXiv:1910.03372, .","chicago":"Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” <i>ArXiv:1910.03372</i>. ArXiv, n.d."},"ec_funded":1,"related_material":{"record":[{"id":"7790","status":"public","relation":"later_version"},{"relation":"dissertation_contains","status":"public","id":"7514"}]},"month":"10","year":"2019","department":[{"_id":"RoSe"}]}]