A review of the physical properties of small systems described by various model Hamiltonians is given. Their physical properties are found either exactly or by using Monte-Carlo methods, and are compared with properties of bulk systems. It is shown that (i) clusters differ in an essential way from the bulk, having electronic states strongly dependent on the geometry and cluster size, but, at the same time, (ii) important insights into the nature of electron correlation and low-energy excitations in solids may be gained by cluster studies. In particular, the cluster calculations allow one to identify the tendencies to various kinds of symmetry-broken states in the (extended) Hubbard model and give a simple qualitative understanding of the low temperature properties of the Kondo and the heavy-fermion systems. Furthermore, cluster calculations allow one to study such problems as the development of magnetic moment at chemisorbed atoms for strong correlations and the suppression of frustration by quantum fluctuations. Recently, cluster calculations gave important information about the possible role of charge transfer processes and magnetic interactions in the microscopic mechanism of pairing in high temperature superconducting oxides.
A review of the physical properties of small systems described by various model Hamiltonians is given. Their physical properties are found either exactly or by using Monte-Carlo methods, and are compared with properties of bulk systems. It is shown that (i) clusters differ in an essential way from the bulk, having electronic states strongly dependent on the geometry and cluster size, but, at the same time, (ii) important insights into the nature of electron correlation and low-energy excitations in solids may be gained by cluster studies. In particular, the cluster calculations allow one to identify the tendencies to various kinds of symmetry-broken states in the (extended) Hubbard model and give a simple qualitative understanding of the low temperature properties of the Kondo and the heavy-fermion systems. Furthermore, cluster calculations allow one to study such problems as the development of magnetic moment at chemisorbed atoms for strong correlations and the suppression of frustration by quantum fluctuations. Recently, cluster calculations gave important information about the possible role of charge transfer processes and magnetic interactions in the microscopic mechanism of pairing in high temperature superconducting oxides.
The electronic, magnetic, and structural properties of small clusters are studied in the framework of the single-band Hubbard Hamiltonian. Results for various ground-state and excited-state many-body properties are presented, which were calculated exactly by means of Lanczos's numerical diagonalization method. A full geometry optimization is performed for N\ensuremath{\le}8 atoms by considering all possible nonequivalent cluster structures with fixed nearest-neighbor bond lengths. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb interaction strength U/t and number of electrons \ensuremath{\nu}. The resulting interplay between electron correlations, magnetism, and cluster structure is analyzed and the main trends as a function of N, U/t, and \ensuremath{\nu} are derived. The stability of cluster ferromagnetism is studied from two complementary points of view. First, for N\ensuremath{\le}8 and \ensuremath{\nu}=N+1, we determine exactly the stability of the ferromagnetic ground state with respect to electronic excitations and structural changes. It is shown that in small clusters the structural changes can be as important to the temperature dependence of the magnetization as the purely electronic excitations. Second, we determine the stability of the saturated ferromagnetic state with respect to single spin flips as a function of the band filling \ensuremath{\nu}/N. In this case a few selected larger clusters (7\ensuremath{\le}N\ensuremath{\le}43) in the strongly correlated limit (U/t\ensuremath{\rightarrow}+\ensuremath{\infty}) are considered. It is shown that the \ensuremath{\nu}/N dependence of the spin-flip energy \ensuremath{\Delta}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{sf}}$ shows interesting electronic-shell-like oscillations, which reflect the characteristics of the single-particle energy-level structure and its dependence on the symmetry and size of the cluster. Finally, we conclude by discussing some of the limitations of the model together with relevant extensions. \textcopyright{} 1996 The American Physical Society.
Unusual normal-state properties of strongly correlated electron materials have been discovered in recent years. They can not be explained by traditional theories. Much work has been devoted to exploring the physical insight into these properties. In the dissertation, the study of spin and charge response to magnetic frustration in the Hubbard Model is presented. Using exact diagonalization techniques, the eigenvalues and eigenvectors of a two-dimensional Hubbard model applied to a cluster with periodic boundary conditions have been obtained. Static and dynamic spin and charge correlation functions as well as one-particle excitation spectra have been calculated. The effects of magnetic frustration due to geometrical competition of different electron hopping terms are studied for various doping and short-range Coulomb interaction parameters. The calculated results are discussed in the framework of many-body physics.
Electron correlation plays an important role in determining the properties of physical systems, from atoms and molecules to condensed phases. Recent theoretical progress in the field has proven possible using both analytical methods and numerical many-body treatments, for realistic systems as well as for simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model, while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key ingredient in the treatment of electrons in condensed phases within density-functional formalism. The Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situations in which very strong correlations tend to bring about site localization of electrons. The character of the interactions in these lattice models allows for a local treatment of correlations. This is achieved by the use of projection techniques that were first proposed by Gutzwiller for the multicenter problem, being the natural extension of the Coulson-Fischer treatment of the H2 molecule. Much work in this area is analytic or semianalytic and requires approximations. However, a full many-body treatment of both realistic and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum Monte Carlo method. This method, which can be implemented in a number of ways, has been applied to atoms, molecules, and solids. In spite of continuing progress, technical problems still remain. Thus one may mention the fermion sign problem and the increase in computational time with the nuclear charge in atomic and related situations. Still, this method provides, to date, one of the most accurate ways to calculate correlation energies, both in atomic and in multicenter problems.
Electron correlation plays an important role in determining the properties of physical systems, from atoms and molecules to condensed phases. Recent theoretical progress in the field has proven possible using both analytical methods and numerical many-body treatments, for realistic systems as well as for simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model, while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key ingredient in the treatment of electrons in condensed phases within density-functional formalism. The Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situations in which very strong correlations tend to bring about site localization of electrons. The character of the interactions in these lattice models allows for a local treatment of correlations. This is achieved by the use of projection techniques that were first proposed by Gutzwiller for the multicenter problem, being the natural extension of the Coulson-Fischer treatment of the H2 molecule. Much work in this area is analytic or semianalytic and requires approximations. However, a full many-body treatment of both realistic and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum Monte Carlo method. This method, which can be implemented in a number of ways, has been applied to atoms, molecules, and solids. In spite of continuing progress, technical problems still remain. Thus one may mention the fermion sign problem and the increase in computational time with the nuclear charge in atomic and related situations. Still, this method provides, to date, one of the most accurate ways to calculate correlation energies, both in atomic and in multicenter problems.
We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
In order to understand various local phenomena related to electronic phase separation discovered recently in strongly correlated materials, quantum cluster methods are applied to a theoretical model, i.e., Hubbard model, which has successfully predicted basic properties of strongly correlated systems. Our calculations are performed at two levels: exact calculations based on the isolated small clusters via exact diagonalization and approximate calculations applied to large lattices using the variational cluster approximation (VCA).\nWe first study phase separation instabilities with the formation of a pairing gap in the two-dimensional Hubbard model for 8-site Betts clusters. The exact diagonalization method is applied to extract exact ground and excited states of the Hubbard model. The results show that the electronic states with one hole off half filling are unstable and they energetically prefer creating the spatial phase separation. The effect of next nearest hopping is also discussed in this part.\nWe studied the variation of the pairing when the out-of-plane correlation is present by adding an interacting apical site above the two-dimensional cluster plane. The calculations indicate that the out-of-plane correlation can be detrimental to the in-plane pairing effect. When it is not too strong to destroy electron pairs on the plane, the modulation of the apical site can drive the pairing gap to show a sinusoidal variation which is consistent with recent experimental discoveries.\nVariational cluster approximation is introduced to explore local properties of larger lattices. The local nematic state is studied in the square lattice using this method, which indicates that it is possible that the electronic states can locally break the C4 symmetry (90o$ rotational symmetry) to form a pattern where the electronic properties, such as charge and spin correlations, are different in x and y directions.\nThe spatial phase separation in the square and honeycomb lattices is also investigated by the variational cluster approximation. The study shows that Coulomb interaction and lattice geometry are both crucial for electronic phase separation. We propose that electronic phase separation is the result of an instability of the Fermi surface at the boundary of the first Brillouin zone of antiferromagnetic states.