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Scientia Plena

Cálculo de propriedades termodinâmicas do modelo de Hubbard usando teoria de perturbação

Thiago X. R. Souza, Cláudio Andrade Macedo

Published Jun 22, 2012• 0 citations
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Abstract

Magnetic systems of strongly correlated electrons are present in many technologically relevant materials. Models of these systems have been studied using many theoretical approaches. The perturbation theory has been used successfully in one-dimensional systems. In this paper, we extend the perturbation theory developed by Beni and Pincus, originally applied to one-dimensional lattices, for lattices with different geometries. We calculate the specific heat and the magnetic susceptibility of the Hubbard model for the square, ladder and one-dimensional dimerized lattices. Our results, within the limits of validity of the parameters used to perform the perturbative approach, are consistent with those available in the literature, obtained by other techniques. Furthermore, we show that our approach to perturbation theory is feasible to be applied to various other types of lattice, including three-dimensional. We conclude that in the form given in this work, the perturbation theory becomes an important tool for investigating the thermodynamic properties of magnetic systems of strongly correlated electrons.

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Physical review. B, Solid state•Oct 15, 1974•17 citations
Magnetic systems of strongly correlated electrons are present in many technologically relevant materials. Models of these systems have been studied using many theoretical approaches. The perturbation theory has been used successfully in one-dimensional systems. In this paper, we extend the perturbation theory developed by Beni and Pincus, originally applied to one-dimensional lattices, for lattices with different geometries. We calculate the specific heat and the magnetic susceptibility of the Hubbard model for the square, ladder and one-dimensional dimerized lattices. Our results, within the limits of validity of the parameters used to perform the perturbative approach, are consistent with those available in the literature, obtained by other techniques. Furthermore, we show that our approach to perturbation theory is feasible to be applied to various other types of lattice, including three-dimensional. We conclude that in the form given in this work, the perturbation theory becomes an important tool for investigating the thermodynamic properties of magnetic systems of strongly correlated electrons.
In order to contribute to understand well the thermodynamic properties of strongly correlated electron systems, we study numerically, in the grand canonical ensemble, the temperature dependence of the internal energy, specific heat, entropy, magnetization, and magnetic susceptibility of the one-dimensional Hubbard model without transference of moment in the interactions. The results have been obtained for a wide range of values of the on-site Coulombian repulsion U and of the external magnetic field h. Differences between this model and the original Hubbard model have been observed in relation to the number of peaks of the specific heat and to the behavior of the susceptibility in temperatures tending to zero. These differences evidence that the two models possess different magnetic structures of correlation.
In this work we obtain the magnetic and thermodynamic properties of the classical Hubbard model. This one is easier to handle than the original Hubbard model, and it maintains fundamental aspects like charge and particle-hole symmetry. We find the relationship between this and the Ising model. For square and simple cubic lattices, the thermodynamic properties and phase diagrams were calculated by Monte Carlo method. We show the behavior of internal energy, specific heat, and magnetic susceptibility as a function of the temperature. For the considered lattices, the ground state is anti-ferromagnetic. The Néel temperatures were found as a function of the electronic interaction in the case of half-filled band. In the limit of vanishing of diverging interaction, our results agree with the Ising model.
The students interested in the study of strongly correlated electron systems normally feel, in the beginning  of their studies, a great difficulty in understanding the complex correlations of these systems. We consider  that for these students it is very important to have texts that analyze pedagogically the theoretical research  methods of the subject. We show how to determine the thermodynamic properties of the Hubbard model  for the two-dimensional square lattice, utilizing the Hartree-Fock approximation. All the procedures and  peculiarities which lead to the determination of the thermodynamic properties and to the comprehension  of the meaning of the Hartree-Fock approximation applied to models of strongly correlated electron  systems are pedagogically presented in detail. We obtain the magnetization, the magnetic susceptibility,  the internal energy, the specific heat, and also a magnetic phase diagram of the studied system. Our  approach clearly shows that the use of the Hartree-Fock approximation, in this case, is very simple.
O modelo de Hubbard na sua forma original e o mais utilizado no estudo de materiais que apresentam magnetismo itinerante e e amplamente discutido na literatura cientifica. Porem a sua forma com mais de uma banda apresenta uma fonte de informacoes fisicas que ainda e pouco explorada devido ao aumento da complexidade dos calculos envolvidos. Neste trabalho verificamos que a inclusao de mais uma banda no modelo aumenta a precisao com que ele representa a natureza tornando-o ainda mais util no avanco da tecnologia envolvendo materiais com magnetismo itinerante. Palavras-chave: propriedades magneticas, modelo de Hubbard degenerado, sistemas unidimensionais. The Hubbard model in its original form is used in the study of materials that present itinerant magnetism and is widely discussed in the scientific literature. However its form with more than a band presents a source of physical information that has not been more explored due to the increase of the complexity of the calculations. In this work we verify that the inclusion of one more band in the model increases the precision for representing natural phenomena, making it thus more useful in the advance of the technology involving materials with itinerant magnetism.
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.
The dimerized-chain Hubbard model is characterized by a Coulomb energy UUU and two alternating transfer constants, an intradimer constant ttt, and a smaller interdimer constant ${t}^{\ensuremath{'}}$. We have used thermodynamic perturbation theory to treat the half-filled-band case, taking the interdimer transfer as the perturbation term. The unperturbed problem is a lattice of isolated dimers. We give the energy, specific heat, and magnetic susceptibility as functions of temperature for the unperturbed case and for the first nonvanishing order in the perturbation. The results are consistent with the U=0U=0U=0 case, which is solved exactly. The interdimer transfer ${t}^{\ensuremath{'}}$ has the effect of distorting the peaks in the specific heat and reducing the susceptibility; this is shown graphically.