Estudos sobre as equações de Bethe
Autor(a) principal: | |
---|---|
Data de Publicação: | 2015 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFSCAR |
Texto Completo: | https://repositorio.ufscar.br/handle/ufscar/7711 |
Resumo: | In this dissertation we made an analytic study of the Bethe Ansatz equations for the XXZ six vertex model with periodic boundary conditions. We had show that the Bethe Ansatz equations deduced from the algebraic and coordinate Bethe Ansatze are related by a conformal map. This allowed us to reduce the Bethe Ansatz equations to a system of polynomial equations. For the one, two and three magnon sectors, we succeeded in decouple these equations, so that the solutions could be expressed in terms of the roots of some self-inversive polynomials, Pa (z). Through new theorems deduced here about the distribution of the roots of self-inversive polynomials in the complex plane, we did a thorough analysis of the distribution of the Bethe roots for the two-magnon sector. This analysis allowed us to show that the Bethe Ansatz is indeed complete for this sector, except at some critical values of the anisotropy parameter A, in which the polynomials Pa (z) may have multiple roots. Finally, an unexpected connection between the Bethe Ansatz equations and the Salem polynomials was found and a new algorithm for search small Salem numbers was elaborated. |
id |
SCAR_f583cb10fa580ec336d095d55facb91c |
---|---|
oai_identifier_str |
oai:repositorio.ufscar.br:ufscar/7711 |
network_acronym_str |
SCAR |
network_name_str |
Repositório Institucional da UFSCAR |
repository_id_str |
4322 |
spelling |
Vieira, Ricardo SoaresSantos, Antonio Limahttp://lattes.cnpq.br/8463237728503334http://lattes.cnpq.br/82655127397432532a758a3f-78db-437b-858c-c6a00e0690842016-10-07T18:13:48Z2016-10-07T18:13:48Z2015-05-15VIEIRA, Ricardo Soares. Estudos sobre as equações de Bethe. 2015. Tese (Doutorado em Física) – Universidade Federal de São Carlos, São Carlos, 2015. Disponível em: https://repositorio.ufscar.br/handle/ufscar/7711.https://repositorio.ufscar.br/handle/ufscar/7711In this dissertation we made an analytic study of the Bethe Ansatz equations for the XXZ six vertex model with periodic boundary conditions. We had show that the Bethe Ansatz equations deduced from the algebraic and coordinate Bethe Ansatze are related by a conformal map. This allowed us to reduce the Bethe Ansatz equations to a system of polynomial equations. For the one, two and three magnon sectors, we succeeded in decouple these equations, so that the solutions could be expressed in terms of the roots of some self-inversive polynomials, Pa (z). Through new theorems deduced here about the distribution of the roots of self-inversive polynomials in the complex plane, we did a thorough analysis of the distribution of the Bethe roots for the two-magnon sector. This analysis allowed us to show that the Bethe Ansatz is indeed complete for this sector, except at some critical values of the anisotropy parameter A, in which the polynomials Pa (z) may have multiple roots. Finally, an unexpected connection between the Bethe Ansatz equations and the Salem polynomials was found and a new algorithm for search small Salem numbers was elaborated.Nesta tese fizemos um estudo analítico das equações de Bethe para o modelo de seis vértices XXZ com condições de contorno periódicas. Mostramos que as equações de Bethe deduzidas pelo Ansatz algébrico estão relacionadas com as equações de Bethe do Ansatz de coordenadas por uma transformação conforme. Isso nos permitiu reduzir as equações de Bethe a um sistema de equações polinomiais. Para os setores de um, dois e três mágnons, mostramos que essas equações podem ser desacopladas, de modo que as suas soluções podem ser expressas em termos das raízes de certos polinómios auto-inversivos, Pa(z). Deduzimos aqui novos teoremas acerca da distribuição das raízes dos polinómios auto-inversivos no plano complexo, o que nos permitiu fazer uma análise minuciosa da distribuição das raízes de Bethe para o setor de dois mágnons. Esta análise nos permitiu mostrar que o Ansatz de Bethe é de fato completo para este setor, exceto para alguns valores críticos do parâmetro de anisotropia A, no qual os polinómios Pa(z) podem apresentar raízes múltiplas. Por fim, uma inesperada conexão entre as equações de Bethe e os polinómios de Salem foi encontrada e um novo algoritmo para se procurar por números de Salem pequenos foi elaborado.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)porUniversidade Federal de São CarlosCâmpus São CarlosPrograma de Pós-Graduação em Física - PPGFUFSCarFísica estatísticaBethe, Equações deModelo de seis vérticesPolinômios auto-inversiveisPolinômios de SalemAnsatz, BetheBethe Ansatz EquationsSix vertex modelSelf-inversive polynomialsSalem polynomialsCIENCIAS EXATAS E DA TERRA::FISICAEstudos sobre as equações de Betheinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisOnline6006004db1ea5a-603f-49ce-8cca-f95ec2fc608dinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFSCARinstname:Universidade Federal de São Carlos (UFSCAR)instacron:UFSCARORIGINALTeseRSV.pdfTeseRSV.pdfapplication/pdf1391601https://repositorio.ufscar.br/bitstream/ufscar/7711/1/TeseRSV.pdffb3e58d9db6c377161785dede432eeeeMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81957https://repositorio.ufscar.br/bitstream/ufscar/7711/2/license.txtae0398b6f8b235e40ad82cba6c50031dMD52TEXTTeseRSV.pdf.txtTeseRSV.pdf.txtExtracted texttext/plain172379https://repositorio.ufscar.br/bitstream/ufscar/7711/3/TeseRSV.pdf.txt3dd0b33674ebf90fcbbc69a9fb1a9bb0MD53THUMBNAILTeseRSV.pdf.jpgTeseRSV.pdf.jpgIM Thumbnailimage/jpeg6729https://repositorio.ufscar.br/bitstream/ufscar/7711/4/TeseRSV.pdf.jpg9c626d95bc47861a7068f34c5718e4a3MD54ufscar/77112023-09-18 18:31:40.077oai:repositorio.ufscar.br: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Repositório InstitucionalPUBhttps://repositorio.ufscar.br/oai/requestopendoar:43222023-09-18T18:31:40Repositório Institucional da UFSCAR - Universidade Federal de São Carlos (UFSCAR)false |
dc.title.por.fl_str_mv |
Estudos sobre as equações de Bethe |
title |
Estudos sobre as equações de Bethe |
spellingShingle |
Estudos sobre as equações de Bethe Vieira, Ricardo Soares Física estatística Bethe, Equações de Modelo de seis vértices Polinômios auto-inversiveis Polinômios de Salem Ansatz, Bethe Bethe Ansatz Equations Six vertex model Self-inversive polynomials Salem polynomials CIENCIAS EXATAS E DA TERRA::FISICA |
title_short |
Estudos sobre as equações de Bethe |
title_full |
Estudos sobre as equações de Bethe |
title_fullStr |
Estudos sobre as equações de Bethe |
title_full_unstemmed |
Estudos sobre as equações de Bethe |
title_sort |
Estudos sobre as equações de Bethe |
author |
Vieira, Ricardo Soares |
author_facet |
Vieira, Ricardo Soares |
author_role |
author |
dc.contributor.authorlattes.por.fl_str_mv |
http://lattes.cnpq.br/8265512739743253 |
dc.contributor.author.fl_str_mv |
Vieira, Ricardo Soares |
dc.contributor.advisor1.fl_str_mv |
Santos, Antonio Lima |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/8463237728503334 |
dc.contributor.authorID.fl_str_mv |
2a758a3f-78db-437b-858c-c6a00e069084 |
contributor_str_mv |
Santos, Antonio Lima |
dc.subject.por.fl_str_mv |
Física estatística Bethe, Equações de Modelo de seis vértices Polinômios auto-inversiveis Polinômios de Salem Ansatz, Bethe |
topic |
Física estatística Bethe, Equações de Modelo de seis vértices Polinômios auto-inversiveis Polinômios de Salem Ansatz, Bethe Bethe Ansatz Equations Six vertex model Self-inversive polynomials Salem polynomials CIENCIAS EXATAS E DA TERRA::FISICA |
dc.subject.eng.fl_str_mv |
Bethe Ansatz Equations Six vertex model Self-inversive polynomials Salem polynomials |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::FISICA |
description |
In this dissertation we made an analytic study of the Bethe Ansatz equations for the XXZ six vertex model with periodic boundary conditions. We had show that the Bethe Ansatz equations deduced from the algebraic and coordinate Bethe Ansatze are related by a conformal map. This allowed us to reduce the Bethe Ansatz equations to a system of polynomial equations. For the one, two and three magnon sectors, we succeeded in decouple these equations, so that the solutions could be expressed in terms of the roots of some self-inversive polynomials, Pa (z). Through new theorems deduced here about the distribution of the roots of self-inversive polynomials in the complex plane, we did a thorough analysis of the distribution of the Bethe roots for the two-magnon sector. This analysis allowed us to show that the Bethe Ansatz is indeed complete for this sector, except at some critical values of the anisotropy parameter A, in which the polynomials Pa (z) may have multiple roots. Finally, an unexpected connection between the Bethe Ansatz equations and the Salem polynomials was found and a new algorithm for search small Salem numbers was elaborated. |
publishDate |
2015 |
dc.date.issued.fl_str_mv |
2015-05-15 |
dc.date.accessioned.fl_str_mv |
2016-10-07T18:13:48Z |
dc.date.available.fl_str_mv |
2016-10-07T18:13:48Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
VIEIRA, Ricardo Soares. Estudos sobre as equações de Bethe. 2015. Tese (Doutorado em Física) – Universidade Federal de São Carlos, São Carlos, 2015. Disponível em: https://repositorio.ufscar.br/handle/ufscar/7711. |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufscar.br/handle/ufscar/7711 |
identifier_str_mv |
VIEIRA, Ricardo Soares. Estudos sobre as equações de Bethe. 2015. Tese (Doutorado em Física) – Universidade Federal de São Carlos, São Carlos, 2015. Disponível em: https://repositorio.ufscar.br/handle/ufscar/7711. |
url |
https://repositorio.ufscar.br/handle/ufscar/7711 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.relation.confidence.fl_str_mv |
600 600 |
dc.relation.authority.fl_str_mv |
4db1ea5a-603f-49ce-8cca-f95ec2fc608d |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de São Carlos Câmpus São Carlos |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Física - PPGF |
dc.publisher.initials.fl_str_mv |
UFSCar |
publisher.none.fl_str_mv |
Universidade Federal de São Carlos Câmpus São Carlos |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFSCAR instname:Universidade Federal de São Carlos (UFSCAR) instacron:UFSCAR |
instname_str |
Universidade Federal de São Carlos (UFSCAR) |
instacron_str |
UFSCAR |
institution |
UFSCAR |
reponame_str |
Repositório Institucional da UFSCAR |
collection |
Repositório Institucional da UFSCAR |
bitstream.url.fl_str_mv |
https://repositorio.ufscar.br/bitstream/ufscar/7711/1/TeseRSV.pdf https://repositorio.ufscar.br/bitstream/ufscar/7711/2/license.txt https://repositorio.ufscar.br/bitstream/ufscar/7711/3/TeseRSV.pdf.txt https://repositorio.ufscar.br/bitstream/ufscar/7711/4/TeseRSV.pdf.jpg |
bitstream.checksum.fl_str_mv |
fb3e58d9db6c377161785dede432eeee ae0398b6f8b235e40ad82cba6c50031d 3dd0b33674ebf90fcbbc69a9fb1a9bb0 9c626d95bc47861a7068f34c5718e4a3 |
bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 |
repository.name.fl_str_mv |
Repositório Institucional da UFSCAR - Universidade Federal de São Carlos (UFSCAR) |
repository.mail.fl_str_mv |
|
_version_ |
1813715565972291584 |