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Известия высших учебных заведений. Поволжский регион. Физико-математические науки

Исследование наносистем в модели Хаббарда в приближении статических флуктуаций

Силантьев Анатолий Владимирович

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Background. The Hubbard model is widely used for theoretical description of strongly correlated electronic systems. Various approximate methods are used for investigation of these systems. At the present time there are four approaches to calculation of the Green’s function in the approximation of static fluctuations. The purpose of this paper is to develop two more approaches to calculation of the Green’s function in the approximation of static fluctuations. Materials and methods. The basis for the approximation of static fluctuations is the method of the motion equations for creation operators. The approximation of static fluctuations allows to obtain a closed system of differential equations for finding creation operators. The Green’s functions, correlation functions and energy spectrum can be calculated using creation operators. Results. The developed approaches for finding the Green’s functions in the Hubbard model by the approximation of static fluctuations were used to obtain the Green’s functions of dimer, hexagon, pentagon and fullerene С 20. Conclusions. This work demonstrates that the Green’s functions of dimer, hexagon, pentagon and fullerene С 20 obtained by the new approaches coincide with the Green’s functions of these systems obtained by the earlier approaches.

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Background. The Hubbard model is widely used for theoretical description of strongly correlated electronic systems. Various approximate methods are used for investigation of these systems. At the present time there are four approaches to calculation of the Green’s function in the approximation of static fluctuations. The purpose of this paper is to develop two more approaches to calculation of the Green’s function in the approximation of static fluctuations. Materials and methods. The basis for the approximation of static fluctuations is the method of the motion equations for creation operators. The approximation of static fluctuations allows to obtain a closed system of differential equations for finding creation operators. The Green’s functions, correlation functions and energy spectrum can be calculated using creation operators. Results. The developed approaches for finding the Green’s functions in the Hubbard model by the approximation of static fluctuations were used to obtain the Green’s functions of dimer, hexagon, pentagon and fullerene С 20. Conclusions. This work demonstrates that the Green’s functions of dimer, hexagon, pentagon and fullerene С 20 obtained by the new approaches coincide with the Green’s functions of these systems obtained by the earlier approaches.
Background. The Hubbard model is widely used for description of strongly correlated electronic systems. Various approximate methods are used for investigation of these systems. The purpose of this paper is to investigate nanosystems within the Hubbard model by the method of motion equations for creation operators. Materials and methods. The mean field approximation allows to obtain a closed system of differential equations for finding creation operators. The Green’s functions, correlation functions and an energy spectrum of nanosystem can be calculated using creation operators. Results. The developed approaches for finding the Green’s functions within the Hubbard model by the mean field approximation were used to obtain the Green’s functions of dimer, hexagon, pentagon and fullerene С20. Conclusions. This work demonstrates that the energy spectrum of a nanosystem, obtained within the Hubbard model by the mean field approximation, is similar to the energy spectrum of this system, obtained within the Huckel model. This work also demonstrates that the Huckel model can be regarded as the Hubbard model within the mean field approximation.
Background. Exactly solvable models are of great importance in quantum physics, as most models in quantum physics are not exactly solvable. The detailed analysis of exactly solvable model’s behavior can be made by exact solutions. The solutions of exactly solvable models, obtained by the approximation method, can be compared with exact solutions, used as standards. A dimer in the Hubbard model is an exactly solvable model. The purpose of this paper is to compare exact solutions obtained for a dimer in the Hubbard model, with solutions obtained for a dimer by approximation of static fluctuations. Materials and methods. In this study, to construct exact solutions and solutions in approximation of static fluctuations the authors used the method of motion equations for creation operators, the method of Green’s functions and the spectral theorem. The energy spectrum of a quantum system can be calculated by Green’s functions, as the poles of Green’s functions correspond to the energy spectrum of a quantum system. Knowing the Green’s functions, the spectral theorem allows to calculate the correlation functions of the quantum system. Results. Resulting from the usage of the methods of the quantum field theory there were obtained anticommutator Green’s functions for a dimer, that were both exact and in approximation of static fluctuations. Using the anticommutator Green’s functions the authors obtained correlation functions and the energy spectrum for a dimer. Conclusions. This work demonstrates that the energy spectrum of a dimer is split by a strong Coulomb repulsion of electrons on two Hubbard subbands and the “metal-insulator” transition occurs in a dimer. This work also demonstrates that several energy states of a dimer, obtained by the approximation of static fluctuations, coincide with energy states of a dimer, obtained by an exact solution.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.