Cleverson Goulart - Academia.edu (original) (raw)

Papers by Cleverson Goulart

Research paper thumbnail of The double Dyson index β effect in non-Hermitian tridiagonal matrices

Entropy, 2023

The Dyson index β plays an essential role in random matrix theory as it labels the so-called "thr... more The Dyson index β plays an essential role in random matrix theory as it labels the so-called "three fold way" that refers to the symmetries satisfied by the ensembles under unitary transformations. As it is known, its 1, 2 and 4 values denote the Orthogonal, the Unitary and the Symplectic classes whose matrix elements are real, complex and quaternion numbers, respectively. It functions therefore as a measure of the number of independent non-diagonal variables. On the other hand, in the case of the β-ensembles, which are the tridiagonal form of the theory, it can assume any real positive value loosing this way that function. Our purpose, however is to show that, when the Hermitian condition of the real matrices generated with a given value of β, is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value 2β. So, it is as if the β index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the β-Hermite, the β-Laguerre and the β-Jacobi ensemble.

Research paper thumbnail of Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality

We study the localization properties of the eigenvectors, characterized by their information entr... more We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős-Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α, and the losses-and-gain strength γ. Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude iγ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ ≡ ξ(N, α, γ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ < 0.1 (10 < ξ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1 < ξ < 10. Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ, the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

Research paper thumbnail of Proton conductivity of perfluorosulfonate ionomers at high temperature and high relative humidity

Applied Physics Letters, 2014

The proton transport properties of Nafion membranes were studied in a wide range of temperature b... more The proton transport properties of Nafion membranes were studied in a wide range of temperature by using an air-tight sample holder able to maintain the sample hydrated at high relative humidity. The proton conductivity of hydrated Nafion membranes continuously increased in the temperature range of 40–180 °C with relative humidity kept at RH = 100%. In the temperature range of 40–90 °C, the proton conductivity followed the Arrhenius-like thermal dependence. The calculated apparent activation energy Ea values are in good agreement with proton transport via the structural diffusion in absorbed water. However, at higher measuring temperatures an upturn of the electrical conductivity was observed to be dependent on the thermal history of the sample.

Research paper thumbnail of The Double Dyson Index β Effect in Non-Hermitian Tridiagonal Matrices

Entropy

The Dyson index, β, plays an essential role in random matrix theory, as it labels the so-called “... more The Dyson index, β, plays an essential role in random matrix theory, as it labels the so-called “three-fold way” that refers to the symmetries satisfied by ensembles under unitary transformations. As is known, its 1, 2, and 4 values denote the orthogonal, unitary, and symplectic classes, whose matrix elements are real, complex, and quaternion numbers, respectively. It functions, therefore, as a measure of the number of independent non-diagonal variables. On the other hand, in the case of β ensembles, which represent the tridiagonal form of the theory, it can assume any real positive value, thus losing that function. Our purpose, however, is to show that, when the Hermitian condition of the real matrices generated with a given value of β is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value 2β. Therefore, it is as if the β index were, in this way, ...

Research paper thumbnail of Entanglement of Pseudo-Hermitian Random States

Entropy, 2020

In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal... more In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that although the formalism is practically the same, the entanglement is not of Fock states but of Bell states.

Research paper thumbnail of Ensemble Beta-Laguerre pseudo-hermitiano

Aos meus pais, familiares e amigos pelo apoio e compreensão. Um agradecimento em especial para mi... more Aos meus pais, familiares e amigos pelo apoio e compreensão. Um agradecimento em especial para minhas amigas Flávia e Gisele, ichis, que sempre incentivaram e estiveram por perto durante todo o meu processo acadêmico, vocês são incríveis! À minha Tia Isabel pela ajuda na revisão ortográfica do texto, suas dicas foram, sem dúvida, úteis. Ao Gabriel Marinello, pelas conversas e dicas que ajudaram a deixar o texto mais claro e preciso. Aproveito e agradeço, também:  Ao meu orientador Prof. Dr. Mauricio Porto Pato, pela confiança e aprendizado na realização do trabalho.  À Universidade de São Paulo e em especial a comunidade do Instituto de Física, professores e colegas, pelo valioso aprendizado nessa bela ciência.  Ao conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), pelo apoio financeiro (Processo 133394/2019-0). Resumo Na década de 30, as matrizes aleatórias (RM) foram introduzidas pelo matemático John Wishart em um estudo estatístico de sistemas multivariados. Algumas décadas depois a primeira aplicação em física aconteceu no estudo dos níveis energéticos de núcleos atômicos pesados com Eugene P. Wigner. Com o desenvolvimento de trabalhos recentes, construindo matrizes aleatórias tridiagonais conhecidas como beta-ensembles, investigou-se o efeito do índice beta no contínuo, ou seja, fora do threefold way de Dyson, a saber 1, 2, 4  . Motivados por buscar no contexto das matrizes aleatórias elementos com a mesma propriedade de hamiltonianas não hermitianas, mas invariantes, simultaneamente, por simetrias de paridade (P) e inversão temporal (T) estudamos, neste trabalho, a estatística de sistemas pseudohermitianos no ensemble Beta-Laguerre. Essa classe de elementos compartilham o espectro com o seu respectivo adjunto via transformação de similaridade. Os resultados obtidos mostram a concordância entre as expressões analíticas, revisadas ao longo do estudo desenvolvido, com o perfil de histogramas produzidos empregando cálculo numérico para diferentes parâmetros em matrizes no ensemble utilizado. Destaca-se o resultado observado na estatística de espaçamento de autovalores, que difere entre os casos hermitianos e pseudo-hermitianos. Esse fenômeno, que também é observado no ensemble Beta-Hermite, indica que embora a distribuição dos elementos pseudo-hermitanos tenda, no limite assintótico, a distribuição dos elementos hermitianos, o parâmetro  que ajusta as expressões difere, concluímos que a relação é 2 pseudo hermitiano hermitiano     . Palavras-chave: Mecânica quântica pseudo-hermitiana; Teoria de matrizes aleatórias; β-Laguerre ensemble; Estatística de níveis;

Research paper thumbnail of The double Dyson index β effect in non-Hermitian tridiagonal matrices

Entropy, 2023

The Dyson index β plays an essential role in random matrix theory as it labels the so-called "thr... more The Dyson index β plays an essential role in random matrix theory as it labels the so-called "three fold way" that refers to the symmetries satisfied by the ensembles under unitary transformations. As it is known, its 1, 2 and 4 values denote the Orthogonal, the Unitary and the Symplectic classes whose matrix elements are real, complex and quaternion numbers, respectively. It functions therefore as a measure of the number of independent non-diagonal variables. On the other hand, in the case of the β-ensembles, which are the tridiagonal form of the theory, it can assume any real positive value loosing this way that function. Our purpose, however is to show that, when the Hermitian condition of the real matrices generated with a given value of β, is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value 2β. So, it is as if the β index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the β-Hermite, the β-Laguerre and the β-Jacobi ensemble.

Research paper thumbnail of Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality

We study the localization properties of the eigenvectors, characterized by their information entr... more We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős-Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α, and the losses-and-gain strength γ. Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude iγ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ ≡ ξ(N, α, γ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ < 0.1 (10 < ξ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1 < ξ < 10. Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ, the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

Research paper thumbnail of Proton conductivity of perfluorosulfonate ionomers at high temperature and high relative humidity

Applied Physics Letters, 2014

The proton transport properties of Nafion membranes were studied in a wide range of temperature b... more The proton transport properties of Nafion membranes were studied in a wide range of temperature by using an air-tight sample holder able to maintain the sample hydrated at high relative humidity. The proton conductivity of hydrated Nafion membranes continuously increased in the temperature range of 40–180 °C with relative humidity kept at RH = 100%. In the temperature range of 40–90 °C, the proton conductivity followed the Arrhenius-like thermal dependence. The calculated apparent activation energy Ea values are in good agreement with proton transport via the structural diffusion in absorbed water. However, at higher measuring temperatures an upturn of the electrical conductivity was observed to be dependent on the thermal history of the sample.

Research paper thumbnail of The Double Dyson Index β Effect in Non-Hermitian Tridiagonal Matrices

Entropy

The Dyson index, β, plays an essential role in random matrix theory, as it labels the so-called “... more The Dyson index, β, plays an essential role in random matrix theory, as it labels the so-called “three-fold way” that refers to the symmetries satisfied by ensembles under unitary transformations. As is known, its 1, 2, and 4 values denote the orthogonal, unitary, and symplectic classes, whose matrix elements are real, complex, and quaternion numbers, respectively. It functions, therefore, as a measure of the number of independent non-diagonal variables. On the other hand, in the case of β ensembles, which represent the tridiagonal form of the theory, it can assume any real positive value, thus losing that function. Our purpose, however, is to show that, when the Hermitian condition of the real matrices generated with a given value of β is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value 2β. Therefore, it is as if the β index were, in this way, ...

Research paper thumbnail of Entanglement of Pseudo-Hermitian Random States

Entropy, 2020

In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal... more In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that although the formalism is practically the same, the entanglement is not of Fock states but of Bell states.

Research paper thumbnail of Ensemble Beta-Laguerre pseudo-hermitiano

Aos meus pais, familiares e amigos pelo apoio e compreensão. Um agradecimento em especial para mi... more Aos meus pais, familiares e amigos pelo apoio e compreensão. Um agradecimento em especial para minhas amigas Flávia e Gisele, ichis, que sempre incentivaram e estiveram por perto durante todo o meu processo acadêmico, vocês são incríveis! À minha Tia Isabel pela ajuda na revisão ortográfica do texto, suas dicas foram, sem dúvida, úteis. Ao Gabriel Marinello, pelas conversas e dicas que ajudaram a deixar o texto mais claro e preciso. Aproveito e agradeço, também:  Ao meu orientador Prof. Dr. Mauricio Porto Pato, pela confiança e aprendizado na realização do trabalho.  À Universidade de São Paulo e em especial a comunidade do Instituto de Física, professores e colegas, pelo valioso aprendizado nessa bela ciência.  Ao conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), pelo apoio financeiro (Processo 133394/2019-0). Resumo Na década de 30, as matrizes aleatórias (RM) foram introduzidas pelo matemático John Wishart em um estudo estatístico de sistemas multivariados. Algumas décadas depois a primeira aplicação em física aconteceu no estudo dos níveis energéticos de núcleos atômicos pesados com Eugene P. Wigner. Com o desenvolvimento de trabalhos recentes, construindo matrizes aleatórias tridiagonais conhecidas como beta-ensembles, investigou-se o efeito do índice beta no contínuo, ou seja, fora do threefold way de Dyson, a saber 1, 2, 4  . Motivados por buscar no contexto das matrizes aleatórias elementos com a mesma propriedade de hamiltonianas não hermitianas, mas invariantes, simultaneamente, por simetrias de paridade (P) e inversão temporal (T) estudamos, neste trabalho, a estatística de sistemas pseudohermitianos no ensemble Beta-Laguerre. Essa classe de elementos compartilham o espectro com o seu respectivo adjunto via transformação de similaridade. Os resultados obtidos mostram a concordância entre as expressões analíticas, revisadas ao longo do estudo desenvolvido, com o perfil de histogramas produzidos empregando cálculo numérico para diferentes parâmetros em matrizes no ensemble utilizado. Destaca-se o resultado observado na estatística de espaçamento de autovalores, que difere entre os casos hermitianos e pseudo-hermitianos. Esse fenômeno, que também é observado no ensemble Beta-Hermite, indica que embora a distribuição dos elementos pseudo-hermitanos tenda, no limite assintótico, a distribuição dos elementos hermitianos, o parâmetro  que ajusta as expressões difere, concluímos que a relação é 2 pseudo hermitiano hermitiano     . Palavras-chave: Mecânica quântica pseudo-hermitiana; Teoria de matrizes aleatórias; β-Laguerre ensemble; Estatística de níveis;