{"alternative_title":["ISTA Thesis"],"title":"Translation-invariant quantum systems with effectively broken symmetry","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"9005"}]},"status":"public","publisher":"Institute of Science and Technology Austria","oa":1,"oa_version":"Published Version","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_created":"2023-01-26T10:00:42Z","department":[{"_id":"GradSch"},{"_id":"RoSe"}],"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"}],"date_published":"2022-12-15T00:00:00Z","publication_status":"published","article_processing_charge":"No","supervisor":[{"orcid":"0000-0002-6781-0521","last_name":"Seiringer","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Seiringer, Robert"}],"month":"12","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)","image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)"},"date_updated":"2023-08-07T13:32:09Z","file":[{"content_type":"application/pdf","file_name":"Brooks_Thesis.pdf","creator":"cchlebak","access_level":"open_access","relation":"main_file","checksum":"b31460e937f33b557abb40ebef02b567","file_id":"12391","success":1,"date_created":"2023-01-26T10:02:34Z","file_size":3095225,"date_updated":"2023-01-26T10:02:34Z"},{"date_updated":"2023-01-26T10:02:42Z","file_id":"12392","date_created":"2023-01-26T10:02:42Z","file_size":809842,"relation":"source_file","checksum":"9751869fa5e7981588ad4228f4fd4bd6","content_type":"application/octet-stream","file_name":"Brooks_Thesis.tex","access_level":"closed","creator":"cchlebak"}],"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","type":"dissertation","day":"15","degree_awarded":"PhD","ddc":["500"],"project":[{"call_identifier":"H2020","grant_number":"694227","name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"}],"file_date_updated":"2023-01-26T10:02:42Z","year":"2022","language":[{"iso":"eng"}],"_id":"12390","ec_funded":1,"citation":{"ieee":"M. Brooks, “Translation-invariant quantum systems with effectively broken symmetry,” Institute of Science and Technology Austria, 2022.","mla":"Brooks, Morris. <i>Translation-Invariant Quantum Systems with Effectively Broken Symmetry</i>. Institute of Science and Technology Austria, 2022, doi:<a href=\"https://doi.org/10.15479/at:ista:12390\">10.15479/at:ista:12390</a>.","short":"M. Brooks, Translation-Invariant Quantum Systems with Effectively Broken Symmetry, Institute of Science and Technology Austria, 2022.","ama":"Brooks M. Translation-invariant quantum systems with effectively broken symmetry. 2022. doi:<a href=\"https://doi.org/10.15479/at:ista:12390\">10.15479/at:ista:12390</a>","apa":"Brooks, M. (2022). <i>Translation-invariant quantum systems with effectively broken symmetry</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:12390\">https://doi.org/10.15479/at:ista:12390</a>","chicago":"Brooks, Morris. “Translation-Invariant Quantum Systems with Effectively Broken Symmetry.” Institute of Science and Technology Austria, 2022. <a href=\"https://doi.org/10.15479/at:ista:12390\">https://doi.org/10.15479/at:ista:12390</a>.","ista":"Brooks M. 2022. Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria."},"publication_identifier":{"issn":["2663-337X"]},"page":"196","doi":"10.15479/at:ista:12390","has_accepted_license":"1","author":[{"orcid":"0000-0002-6249-0928","id":"B7ECF9FC-AA38-11E9-AC9A-0930E6697425","full_name":"Brooks, Morris","first_name":"Morris","last_name":"Brooks"}]}