The objective of this work is the theoretical study of the one band Hubbard model for solids and the introduction of an alternative methodology to circumvent its limitations. To this objective, a Hamiltonian composed of two atoms (the Hubbard dimmer) is introduced and solved analitically. From this Hamiltonian, the atomic Green’s functions are obtained for the dimmer, which is used as a seed for a cumulant expansion and to obtain the Green’s functions for the lattice. The density of states for several
values of interest for the parameters of the problem is calculated, as well as the occupation numbers. The analysis of the dimmer Hamiltonian reveals four important competitive interactions: ferromagnetic,
antiferromagnetic and superconducting with positive or negative coupling constants. As a result of using the dimmer solution, the particle-hole simetry for high values of the electronic correlation is recovered, the Mott transition is obtained in a consistent way (the electronic correlation is no longer overestimated) and by the analysis of the ocupation numbers of the model for different values of the chemical potential, one can observe a competition between fundamental states of the model near the energy region Eo + U, as well as a magnetic phase and a Kondo effect region.
The objective of this work is the theoretical study of the one band Hubbard model for solids and the introduction of an alternative methodology to circumvent its limitations. To this objective, a Hamiltonian composed of two atoms (the Hubbard dimmer) is introduced and solved analitically. From this Hamiltonian, the atomic Green’s functions are obtained for the dimmer, which is used as a seed for a cumulant expansion and to obtain the Green’s functions for the lattice. The density of states for several
values of interest for the parameters of the problem is calculated, as well as the occupation numbers. The analysis of the dimmer Hamiltonian reveals four important competitive interactions: ferromagnetic,
antiferromagnetic and superconducting with positive or negative coupling constants. As a result of using the dimmer solution, the particle-hole simetry for high values of the electronic correlation is recovered, the Mott transition is obtained in a consistent way (the electronic correlation is no longer overestimated) and by the analysis of the ocupation numbers of the model for different values of the chemical potential, one can observe a competition between fundamental states of the model near the energy region Eo + U, as well as a magnetic phase and a Kondo effect region.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
The description of a solid at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons.The Hubbard model used in this research describes interacting electrons in narrow energy bands, with application to problems as diverse as high -Tc superconductivity, band magnetism and the metal-insulator transition.This research seeks to solve one of the most challenging problems in Theoretical physics which is to describe electronic correlations.In this research the t -U Hubbard model is used to solve analytically and numerically using the exact diagonalization techniques This research work focused on the spin interaction and magnetic behavior of some systems and it is therefore an investigation into the various phenomena seen in the phase diagram of ferromagnetic systems.We studied the 2 electrons on 2 sites, 2 electrons on 3 sites and 2 electrons on 4 sites all in One-Dimension (1-D), where the onsite coulomb repulsion, U, and the hopping matrix element, t, were varies to determine their magnetic phase diagram.This results obtained shows that, as the values of the onsite coulomb repulsion, U, increases in all the lattice systems studied, the electronic correlation that favours ferromagnetism gets stronger, but there were no clear transition, and at some point the lattices loss its antiferromagnetic properties, hence the instability of ferromagnetism were established.It was also observed that as the hopping matrix element, t, increases the electronic correlation that favours ferromagnetism get stronger and the lattice begins to loss antiferromagnetic properties but beyond the transition point the lattice began to gain a much stronger antiferromagnetism, hence the ferromagnetism is unstable.
The description of a solid at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. The Hubbard model used in this research describes interacting electrons in narrow energy bands, with application to problems as diverse as high – Tc superconductivity, band magnetism and the metal-insulator transition. This research seeks to solve one of the most challenging problems in Theoretical physics which is to describe electronic correlations. In this research the t – U Hubbard model is used to solve analytically and numerically using the exact diagonalization techniquesThis research work focused on the spin interaction and magnetic behavior of some systems and it is therefore an investigation into the various phenomena seen in the phase diagram of ferromagnetic systems. We studied the 2 electrons on 2 sites, 2 electrons on 3 sites and 2 electrons on 4 sites all in One-Dimension (1-D), where the onsite coulomb repulsion, U, and the hopping matrix element, t, were varies to determine their magnetic phase diagram.This results obtained shows that, as the values of the onsite coulomb repulsion, U, increases in all the lattice systems studied, the electronic correlation that favours ferromagnetism gets stronger, but there were no clear transition, and at some point the lattices loss its antiferromagnetic properties, hence the instability of ferromagnetism were established. It was also observed that as the hopping matrix element, t, increases the electronic correlation that favours ferromagnetism get stronger and the lattice begins to loss antiferromagnetic properties but beyond the transition point the lattice began to gain a much stronger antiferromagnetism, hence the ferromagnetism is unstable.
Abstract We use the cumulant Green’s functions method (CGFM) to study the single-band Hubbard model. The starting point of the method is to diagonalize a cluster (‘seed’) containing N correlated sites and employ the cumulants calculated from the cluster solution to obtain the full Green’s functions for the lattice. All calculations are done directly; no variational or self-consistent process is needed. We benchmark the one-dimensional results for the gap, the double occupancy, and the ground-state energy as functions of the electronic correlation at half-filling and the occupation numbers as functions of the chemical potential obtained from the CGFM against the corresponding results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The particle-hole symmetry of the density of states is fulfilled, and the gap, occupation numbers, and ground-state energy tend systematically to the known results as the cluster size increases. We include a straightforward application of the CGFM to simulate the singles occupation of an optical lattice experiment with lithium-6 atoms in an eight-site Fermi-Hubbard chain near half-filling. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and extended to other strongly correlated models, like the Anderson Hamiltonian, the t − J, Kondo, and Coqblin-Schrieffer models.
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.
This thesis discusses mainly two Fermionic lattice systems, first a Kondo lattice with additional Hubbard interaction and second a Hubbard Hamiltonian augmented with additional spin and charge interactions. We first introduce the Quantum Monte Carlo technique, which is then employed to study the two respective systems. We present an innovation that allows to calculate time displaced Greens functions more efficiently. Compared with previously used numerically stable algorithms the new method gains an order of magnitude in speed, but is just as precise, and very simple to implement. In the second chapter we consider the Kondo lattice model in two dimensions at half filling. In addition to the Fermionic hopping integral t and the superexchange coupling J the role of a Coulomb repulsion U in the conduction band is investigated. We find the model to display a magnetic order-disorder transition in the U-J plane with a critical value of Jc which is decreasing as a function of U. The single-particle spectral function A(k,ω) is computed across this transition. We conclude that (i) the local screening of impurity spins determines the low-energy behavior of the spectral function and (ii) one cannot deform continuously the spectral function of the half-filled Hubbard model at J=0 to that of the Kondo insulator at J>Jc. In the third chapter we investigate the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.