A compilation of studies on random systems: measurements of correlation functions and localization properties
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://www.teses.usp.br/teses/disponiveis/76/76134/tde-03092021-113316/ |
Resumo: | We present a collection of studies about several properties of two random systems, namely (i) the random one-dimensional spin-1/2 chain, and (ii) the random two-dimensional Bose-Hubbard model. In study (i), we consider two variants of random spin chains: the usual case with uncorrelated random couplings, and the correlated case, in which the even and odd sublattices are identical to each other. Using the Jordan-Wigner transformation, we map the spin-1/2 chain Hamiltonian into a noninteracting fermionic model. On this new basis, we perform an exact diagonalization routine, which allows us to compute spin- spin correlation functions and related observables. Our results are presented here as a reproduction of two published papers. In the first one, we measure entanglement properties and the violation of Bell inequalities. We show that the correlated case does not violate the Bell inequality up to a small degree of randomness, thus contradicting the prior belief that any amount of disorder should lead to a violation of Bell inequalities and, equivalently, the existence of nonlocal states. In the second paper, we confirm the strong-disorder renormalization group predictions about the scaling of spin-spin correlation functions for the uncorrelated disorder case. We show that results suggesting a possible correction to the scaling function may be consequence of either a lack of numerical precision or a relatively large crossover length. In addition, we show that much of the nonuniversal properties of the spin-spin correlation functions can be understood from a single parameter scaling perspective. In study (ii), we consider the two-dimensional Bose-Hubbard model with disorder introduced either as random site dilution or as onsite interactions generated from a uniform probability distribution. We investigate the localization properties of collective modes by employing a multifractal analysis and a recursive Green´s function method. Using a variational mean-field approach, we obtain noninteracting Hamiltonians for the Goldstone (phase) and Higgs (amplitude) modes. Our results show that only the lowest-excitation Goldstone mode undergoes a localization-delocalization transition close to the superfluid-Mott insulator phase transition; higher-excitation phase modes and all amplitude modes remain localized. This behavior is observed for the two types of disorder investigated. |
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A compilation of studies on random systems: measurements of correlation functions and localization propertiesUma coletânea de estudos em sistemas desordenados: medidas de funções de correlação e propriedades de localizaçãoComportamento críticoCritical behaviorMétodos numéricosNumerical methodsRandom systemsSistemas desordenadosWe present a collection of studies about several properties of two random systems, namely (i) the random one-dimensional spin-1/2 chain, and (ii) the random two-dimensional Bose-Hubbard model. In study (i), we consider two variants of random spin chains: the usual case with uncorrelated random couplings, and the correlated case, in which the even and odd sublattices are identical to each other. Using the Jordan-Wigner transformation, we map the spin-1/2 chain Hamiltonian into a noninteracting fermionic model. On this new basis, we perform an exact diagonalization routine, which allows us to compute spin- spin correlation functions and related observables. Our results are presented here as a reproduction of two published papers. In the first one, we measure entanglement properties and the violation of Bell inequalities. We show that the correlated case does not violate the Bell inequality up to a small degree of randomness, thus contradicting the prior belief that any amount of disorder should lead to a violation of Bell inequalities and, equivalently, the existence of nonlocal states. In the second paper, we confirm the strong-disorder renormalization group predictions about the scaling of spin-spin correlation functions for the uncorrelated disorder case. We show that results suggesting a possible correction to the scaling function may be consequence of either a lack of numerical precision or a relatively large crossover length. In addition, we show that much of the nonuniversal properties of the spin-spin correlation functions can be understood from a single parameter scaling perspective. In study (ii), we consider the two-dimensional Bose-Hubbard model with disorder introduced either as random site dilution or as onsite interactions generated from a uniform probability distribution. We investigate the localization properties of collective modes by employing a multifractal analysis and a recursive Green´s function method. Using a variational mean-field approach, we obtain noninteracting Hamiltonians for the Goldstone (phase) and Higgs (amplitude) modes. Our results show that only the lowest-excitation Goldstone mode undergoes a localization-delocalization transition close to the superfluid-Mott insulator phase transition; higher-excitation phase modes and all amplitude modes remain localized. This behavior is observed for the two types of disorder investigated.Apresentamos uma coletânea de estudos sobre diversas propriedades referentes a dois sistemas desordenados, (i) a cadeia unidimensional de spin-1/2 desordenada, e (ii) o modelo de Bose-Hubbard bidimensional desordenado. No estudo (i), nós consideramos duas variantes da cadeia de spin randômica: o caso usual com acoplamentos randômicos descorrelacionados, e o caso correlacionado, em que as sub-redes par e ímpar são idênticas entre si. Utilizando-se da transformação de Jordan-Wigner, nós mapeamos o Hamiltoniano da cadeia de spin-1/2 em um modelo fermiônico não-interagente. Nesta nova base, realizamos uma rotina de diagonalização exata, que nos permite calcular as funções de correlação spin-spin e observáveis relacionados. Nossos resultados são apresentados aqui na forma de reprodução de dois artigos publicados. No primeiro artigo, medimos as propriedades de emaranhamento e a violação das desigualdades de Bell. Nós demonstramos que o caso correlacionado não viola a desigualdade de Bell até um certo grau de desordem pequeno, contradizendo assim a crença anterior de que qualquer grau de desordem deveria ocasionar na violação de desigualdades de Bell e, equivalentemente, a existência de estados não-locais. No segundo artigo, nós confirmamos as previsões do método do grupo de renormalização de desordem forte acerca do escalonamento das funções de correlação spin-spin para o caso com desordem descorrelacionada. Mostramos que resultados sugerindo uma possível correção na função de escala podem ser consequência tanto de uma falta de precisão numérica, como de um comprimento de transição relativamente longo. Além disso, demonstramos que grande parte das propriedades não-universais das funções de correlação spin-spin podem ser compreendidas a partir de um escalonamento de parâmetro único. No estudo (ii), consideramos o modelo de Bose-Hubbard bidimensional com desordem introduzida tanto na forma de diluição aleatória de sítios, como na forma de interações no sítio geradas a partir de uma distribuição de probabilidade uniforme. Investigamos as propriedades de localização de modos coletivos através de uma análise multifractal e pelo método da função de Green recursiva. Utilizando-se de um método de campo médio variacional, nós obtemos Hamiltonianos não-interagentes para os modos de Goldstone (fase) e de Higgs (amplitude). Nossos resultados mostram que apenas o modo de Goldstone de energia mais baixa sofre uma transição localização-deslocalização nas proximidades da transição de fase superfluida-isolante de Mott; modos de fase de energia mais alta e todos os modos de amplitude permanecem localizados. Este comportamento é observado para os dois tipos de desordem investigados.Biblioteca Digitais de Teses e Dissertações da USPHoyos Neto, José AbelGetelina, João Carlos de Andrade2021-03-15info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/76/76134/tde-03092021-113316/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2024-08-22T20:57:03Zoai:teses.usp.br:tde-03092021-113316Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212024-08-22T20:57:03Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.none.fl_str_mv |
A compilation of studies on random systems: measurements of correlation functions and localization properties Uma coletânea de estudos em sistemas desordenados: medidas de funções de correlação e propriedades de localização |
title |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
spellingShingle |
A compilation of studies on random systems: measurements of correlation functions and localization properties Getelina, João Carlos de Andrade Comportamento crítico Critical behavior Métodos numéricos Numerical methods Random systems Sistemas desordenados |
title_short |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
title_full |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
title_fullStr |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
title_full_unstemmed |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
title_sort |
A compilation of studies on random systems: measurements of correlation functions and localization properties |
author |
Getelina, João Carlos de Andrade |
author_facet |
Getelina, João Carlos de Andrade |
author_role |
author |
dc.contributor.none.fl_str_mv |
Hoyos Neto, José Abel |
dc.contributor.author.fl_str_mv |
Getelina, João Carlos de Andrade |
dc.subject.por.fl_str_mv |
Comportamento crítico Critical behavior Métodos numéricos Numerical methods Random systems Sistemas desordenados |
topic |
Comportamento crítico Critical behavior Métodos numéricos Numerical methods Random systems Sistemas desordenados |
description |
We present a collection of studies about several properties of two random systems, namely (i) the random one-dimensional spin-1/2 chain, and (ii) the random two-dimensional Bose-Hubbard model. In study (i), we consider two variants of random spin chains: the usual case with uncorrelated random couplings, and the correlated case, in which the even and odd sublattices are identical to each other. Using the Jordan-Wigner transformation, we map the spin-1/2 chain Hamiltonian into a noninteracting fermionic model. On this new basis, we perform an exact diagonalization routine, which allows us to compute spin- spin correlation functions and related observables. Our results are presented here as a reproduction of two published papers. In the first one, we measure entanglement properties and the violation of Bell inequalities. We show that the correlated case does not violate the Bell inequality up to a small degree of randomness, thus contradicting the prior belief that any amount of disorder should lead to a violation of Bell inequalities and, equivalently, the existence of nonlocal states. In the second paper, we confirm the strong-disorder renormalization group predictions about the scaling of spin-spin correlation functions for the uncorrelated disorder case. We show that results suggesting a possible correction to the scaling function may be consequence of either a lack of numerical precision or a relatively large crossover length. In addition, we show that much of the nonuniversal properties of the spin-spin correlation functions can be understood from a single parameter scaling perspective. In study (ii), we consider the two-dimensional Bose-Hubbard model with disorder introduced either as random site dilution or as onsite interactions generated from a uniform probability distribution. We investigate the localization properties of collective modes by employing a multifractal analysis and a recursive Green´s function method. Using a variational mean-field approach, we obtain noninteracting Hamiltonians for the Goldstone (phase) and Higgs (amplitude) modes. Our results show that only the lowest-excitation Goldstone mode undergoes a localization-delocalization transition close to the superfluid-Mott insulator phase transition; higher-excitation phase modes and all amplitude modes remain localized. This behavior is observed for the two types of disorder investigated. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-03-15 |
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.uri.fl_str_mv |
https://www.teses.usp.br/teses/disponiveis/76/76134/tde-03092021-113316/ |
url |
https://www.teses.usp.br/teses/disponiveis/76/76134/tde-03092021-113316/ |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
|
dc.rights.driver.fl_str_mv |
Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Liberar o conteúdo para acesso público. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.coverage.none.fl_str_mv |
|
dc.publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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1809090347703205888 |