Estudos sobre as equações de Bethe

Detalhes bibliográficos
Autor(a) principal: Vieira, Ricardo Soares
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.
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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
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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
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dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
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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
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institution UFSCAR
reponame_str Repositório Institucional da UFSCAR
collection Repositório Institucional da UFSCAR
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