Lobato, José I. (2022) Implementación de transformaciones para optimizaciones al grupo de renormalización con matriz densidad / Implementation of transformations for optimizing the density matrix renormalization group. Maestría en Ciencias Físicas, Universidad Nacional de Cuyo, Instituto Balseiro.
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Resumen en español
El grupo de renormalización numérica con matriz densidad (DMRG) es una técnica variacional empleada para resolver sistemas interactuantes de muchos cuerpos. El método se basa en renormalizar el espacio de Hilbert utilizando los estados más relevantes de la matriz densidad pura del estado fundamental. La eficiencia de este método depende esencialmente de la base en la que se esté trabajando. En este trabajo se busca optimizar el método mediante diferentes transformaciones al hamiltoniano como ser la transformación al espacio de Wavelets, orbitales naturales y a la configuración tipo estrella. Se presentó el DMRG en su formulación MPS/MPO y se presentaron las transformaciones de operadores de una y dos partículas. Se analizó la eficiencia del algoritmo en el modelo de Hubbard unidimensional y bidimensional analizando las bases de Wavelets y de orbitales naturales. Se obtuvo que las transformaciones eran ineficientes en cuanto al uso de recursos computacionales ya que involucraban términos de cuatro operadores. Se analizó el modelo de Anderson, donde una impureza se acopla con un baño, transformando solamente los sitios del baño evitando así transformar términos de cuatro operadores. Se transformó a configuración tipo estrella y a la base de orbitales naturales. Se encontró que la transformación de orbitales naturales es la más eficiente de todas. Se presentaron las transformaciones sucesivas de orbitales naturales, las cuales permiten obtener una mayor eficiencia en el método. Finalmente, se aplicó la transformación a orbitales naturales al DMRG en la teoría de campo medio dinámico (DMFT) y se obtuvo una convergencía más rápida en la base de orbitales naturales. Siendo el DMFT uno de los métodos más usados para el cálculo de propiedades cuánticas de sistemas correlacionados, pero que depende fundamentalmente de la posibilidad de la resolución de una impureza efectiva, esta mejora puede ser relevante para el estudio de sistemas cuánticos más complejos que los estudiados hasta el momento.
Resumen en inglés
The density matrix numerical renormalization group (DMRG) is a variational technique used to solve interacting systems of many bodies. The method is based on renormalizing the Hilbert space using the most relevant states of the pure density matrix of the ground state. The efficiency of this method depends essentially on the basis on which you are working. This paper seeks to optimize the method through different transformations to the Hamiltonian such as the transformation to space of Wavelets, natural orbitals and the star-like configuration. The DMRG was presented in its MPS/MPO formulation and the transformations of one- and two-particle operators were presented. The efficiency of the algorithm in the one-dimensional and two-dimensional Hubbard model was analyzed by analyzing the bases of Waveletsand natural orbitals. It was obtained that the transformations were inefficient in terms of the use of computational resources since they involved terms of four operators. Anderson's model was analyzed, where an impurity is coupled with a bathroom, transforming only the sites of the bathroom, thus avoiding transforming terms of four operators. It was transformed into a star-like configuration and the base of natural orbitals. The transformation of natural orbitals was found to be the most efficient of all. The successive transformations of natural orbitals were presented, which allow obtaining greater efficiency in the method. Finally, the transformation to natural orbitals to DMRG was applied in the dynamic mean field theory (DMFT) and a faster convergence was obtained at the base of natural orbitals. Since DMFT is one of the most used methods for calculating the quantum properties of correlated systems, but which depends fundamentally on the possibility of resolving an effective impurity, this improvement may be relevant for the study of quantum systems more complex than those studied so far.
| Tipo de objeto: | Tesis (Maestría en Ciencias Físicas) |
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| Palabras Clave: | Transformations; Transformaciones; [Many body; Muchos cuerpos; Natural orbitals; Orbitales naturales; Wavelets] |
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| Materias: | Física > Materia condensada |
| Divisiones: | Gcia. de área de Investigación y aplicaciones no nucleares > Gcia. de Física > Materia condensada > Teoría de sólidos |
| Código ID: | 1152 |
| Depositado Por: | Tamara Cárcamo |
| Depositado En: | 07 Aug 2023 12:10 |
| Última Modificación: | 07 Aug 2023 12:10 |
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