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Journal of Physics Conference Series

Spectra of the energy operator of four-electron systems in the triplete state in the Hubbard model

Sadulla Mamarazhabovich Tashpulatov

Published Mar 1, 2016• 14 citations
Publisher

Abstract

We investigate spectral properties of a four-electron system in the Hubbard Model framework in the v — dimensional lattice Z v . We prove that the essential spectrum of the system in a quintet state consists of a single segment and the four-electron bound state or four-electron anti-bound state is absent. We show that the essential spectrum of the system in a triplete states is the union of at most three segments. We also prove that four-electron bound states or a four-electron anti-bound state exists in triplete states. We prove that in the system exists three triplete states and their spectrum are different.

DOI
10.1088/1742-6596/697/1/012025

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We investigate spectral properties of a four-electron system in the Hubbard Model framework in the v — dimensional lattice Z v . We prove that the essential spectrum of the system in a quintet state consists of a single segment and the four-electron bound state or four-electron anti-bound state is absent. We show that the essential spectrum of the system in a triplete states is the union of at most three segments. We also prove that four-electron bound states or a four-electron anti-bound state exists in triplete states. We prove that in the system exists three triplete states and their spectrum are different.
We consider the energy operator of four-electron systems in an impurity Hubbard model and investigated the structure of essential spectra and discrete spectrum of the system in the first triplet state in a one-dimensional lattice. For investigation the structure of essential spectra and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model. The investigations show that there are such cases: 1) the essential spectrum of the system consists of the union of no more than eight segments, and the discrete spectrum of the system consists of no more than three eigenvalues; 2) the essential spectrum of the system consists of the union of no more than sixteen segments, and the discrete spectrum of the system consists of no more than eleven eigenvalues; 3) the essential spectrum of the system consists of the union of no more than three segments, and the discrete spectrum of the system is the empty set. Consequently, the essential spectrum of the system consists of the union of no more than sixteen segments, and the discrete spectrum of the system consists of no more than eleven eigenvalues.
We consider an energy operator of four-electron system in the Impurity Hubbard model with a coupling between nearest-neighbors. The spectrum of the systems in the second triplet state in a ν-dimensional lattice is investigated. For investigation the structure of essential spectra and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model for the second triplet state of the system. The investigations show that the essential spectrum of the system consists of the union of no more than sixteen segments, and the discrete spectrum of the system consists of no more than eleven eigenvalues.
We consider the energy operator of four-electron systems in the Hubbard impurity model and investigate the structure of the essential spectrum and discrete spectra for the second singlet state of the system. It is shown that in the one-dimensional and two-dimensional cases the following situations exist for the essential and discrete spectrum: (a). the essential spectrum of the operator of the second singlet state of four electrons in the Hubbard impurity model consists of a union of eight segments, and the discrete spectrum of the operator consists of six eigenvalues; (b). the essential spectrum of the operator consists of a union of sixteen segments, and the discrete spectrum of the operator consists of fourteen eigenvalues; (c). the essential spectrum of the operator consists of a union of thirteen segments, and the discrete spectrum of the operator consists of nine eigenvalues; (d). the essential spectrum of the operator consists of a union of three segments, and the discrete spectrum of the operator consists of three eigenvalues. In the three-dimensional case the following situations arise: (a). the essential spectrum of an operator consists of unions of eight segments, and the discrete spectrum of the operator consists of six eigenvalues, or the essential spectrum of an operator consists of unions of three segments, and the discrete spectrum of the operator consists of three eigenvalues; (b). the essential spectrum of an operator consists of unions of eight segments, and the discrete spectrum of the operator consists of six eigenvalues; (c). the essential spectrum of an operator consists of unions of sixteen segments, and the discrete spectrum of the operator consists of fourteen eigenvalues; (d). the essential spectrum of an operator consists of unions of three segments, and the discrete spectrum of the operator consists of three eigenvalues.
We consider two-electron systems for the impurity Hubbard Model and investigate the spectrum of the system in a singlet state for the v-dimensional integer valued lattice Zv. We proved the essential spectrum of the system in the singlet state is consists of union of no more then three intervals, and the discrete spectrum of the system in the singlet state is consists of no more then five eigenvalues. We show that the discrete spectrum of the system in the triplet and singlet states differ from each other. In the singlet state the appear additional two eigenvalues. In the triplet state the discrete spectrum of the system can be empty set, or is consists of one-eigenvalue, or is consists of two eigenvalues, or is consists of three eigenvalues. For investigation the structure of essential spectra and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model.