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Scientia Plena

Thermodynamic properties of the Hubbard model for a diatomic molecule AB

Marcélio C. B. Sousa, Cláudio Andrade Macedo

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

The Hubbard model has been subject of strong interest in the condensed matter community. It is the  simplest model for strongly correlated electron systems. The text books of physics are based on  applications, under the form of problems, which have exact analytical solutions. Itinerant magnetic  system models, for that reason, are rarely considered in text books of statistical mechanics. 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. In this work, we  have studied, with a pedagogical approach, the thermodynamic properties of the Hubbard model for a  diatomic molecule  AB, that is, we consider a case translationally non-invariant. We apply the grand  canonical ensemble method for allowing the analysis of a general situation of any average electron  number per site. In the applications we consider, in particular, the case in which the average electron  number per site is one. The analysis of the thermodynamic properties makes possible to conclude that the  diatomic molecule AB of Hubbard exhibits ferrimagnetic behavior. Moreover, the studied system presents  a conceptual and mathematical structure accessible to undergraduate students of physics in their last year,  and the results show a rich variety of physics phenomena with details that make feasible a deeper  comprehension of the quantum mechanisms involved in the thermodynamics properties and of the  methods employed to obtain such properties.

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The Hubbard model has been subject of strong interest in the condensed matter community. It is the  simplest model for strongly correlated electron systems. The text books of physics are based on  applications, under the form of problems, which have exact analytical solutions. Itinerant magnetic  system models, for that reason, are rarely considered in text books of statistical mechanics. 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. In this work, we  have studied, with a pedagogical approach, the thermodynamic properties of the Hubbard model for a  diatomic molecule  AB, that is, we consider a case translationally non-invariant. We apply the grand  canonical ensemble method for allowing the analysis of a general situation of any average electron  number per site. In the applications we consider, in particular, the case in which the average electron  number per site is one. The analysis of the thermodynamic properties makes possible to conclude that the  diatomic molecule AB of Hubbard exhibits ferrimagnetic behavior. Moreover, the studied system presents  a conceptual and mathematical structure accessible to undergraduate students of physics in their last year,  and the results show a rich variety of physics phenomena with details that make feasible a deeper  comprehension of the quantum mechanisms involved in the thermodynamics properties and of the  methods employed to obtain such properties.
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.
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.
The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.
By use of the conservation laws a four-site Hubbard model coupled to a particle bath within an external magnetic field in z-direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many-particle states differing in electron number and spin if the electron density is away from half-filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high-temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.
With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.
The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.