Antiferromagnetism in the Hubbard model

The energy spectrum of the two-sublattice Hubbard model is obtained in the static-fluctuation approximation. It is shown how the structure of the energy spectrum is modified as the parameters of the Hubbard model are varied. The ground state of the simple Hubbard model of dimension d=2 is the dielectric antiferromagnetic state. The author derives a consistency equation for the magnetization, which has an antiferromagnetic solution. Fiz. Tverd. Tela (St. Petersburg) 39, 1594–1599 (September 1997)     

Ferromagnetism in the Hubbard model

Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation hasN _d-fold degenerate ground states. The degeneracyN _d is proportional to the total number of sites ||. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to=N _d/(2||), and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.     

Magnetism in the single-band Hubbard model

A self-consistent spectral density approach (SDA) is applied to the Hubbard model to investigate the possibility of spontaneous ferro- and antiferromagnetism. The starting point is a two-pole ansatz for the single-electron spectral density, the free parameter of which can be interpreted as energies and spectral weights of respective quasiparticle excitations. They are determined by fitting exactly calculated spectral moments. The resulting self-energy consists of a local and a non-local part. The higher correlation functions entering the spin-dependent local part can be expressed as functionals of the single-electron spectral density. Under certain conditions for the decisive model parameters (Coulomb interaction U, Bloch bandwidth W, band occupation n, temperature T) the local part of the self-energy gives rise to a spin-dependent band shift, thus allowing for spontaneous band magnetism. As a function of temperature, second-order phase transitions are found away from half-filling, but close to half-filling, the system exhibits a tendency towards first-order transitions. The non-local self-energy part is determined by use of proper two-particle spectral densities. Its main influence concerns a (possibly spin-dependent) narrowing of the quasiparticle bands with the tendency to stabilize magnetic solutions. The non-local self-energy part disappears in the limit of infinite dimensions. We present a full evaluation of the Hubbard model in terms of quasiparticle densities of states, quasiparticle dispersions, magnetic phase diagram, critical temperatures (T_c, T_N) as well as spin and particle correlation functions. Special attention is focused on the non-locality of the electronic self-energy, for which some rigorous limiting cases are worked out.     

Kinetic coefficients in the Hubbard model

The properties of the electron system with the strongest electron-electron coupling are studied. The kinetic equation in the Keldysh-Mills representation has been obtained. A new term has been discovered in the collision integral; this term is due to the relaxation of end multipliers and vanishes in the low-temperature limit as T ²/|t|, where |t| is the quantity on the order of the nearest-neighbor hopping integral and T is the temperature. The speed of sound has been calculated and the temperature behavior of the kinetic coefficients has been estimated.     

Correlation effects in the Hubbard model

We investigate the influence of electron-electron correlation on the zero temperature behavior of the Hubbard model by combining a variational ansatz for the groundstate wave-function with a cluster-expansion for the groundstate energy. In the non-magnetic case our 2-site cluster-approximation predicts, for a half-filled band, a metal-insulator transition as the intrasite Coulomb interaction is increased. The application to a split band antiferromagnetic state shows that the inclusion of correlation effects results in a tendency to suppress the ordered state.     

Pseudogaps in the three-band Hubbard model

Using the strong coupling diagram technique, the energy spectrum of the three-bandHubbard model is investigated. In these calculations, the series in powers of thecopper-oxygen hybridization for the irreducible part is approximated by two lowest-orderterms. For parameters of hole-doped cuprates the calculated energy spectrum consists oflower and upper Hubbard subbands of predominantly copper nature, oxygen bands with someadmixture of copper states and the Zhang-Rice states of mixed nature. The spectrumcontains two pseudogaps, the lower of which separates the Hubbard subband from Zhang-Riceand oxygen bands. The pseudogaps arise due to multiple reabsorption of carriers in stateswith double occupancy of sites by holes or electrons.     

On ferromagnetism in the Hubbard model

We use a simple variational procedure for strictly excluding a totally ferromagnetic groundstate of the Hubbard model for the cubic lattices and a certain range of values of the electron concentrationn _e and the Coulomb repulsionV _o. The curve bounding this domain becomes exact in the limit ofn _e → 1, previously considered by Nagaoka, and leads to qualitatively correct results also in the opposite limiting cases ofn _e → 0, and 2, which were investigated by Kanamori.     

Ferromagnetism in the degenerate Hubbard model

We study the ground state of the doubly degenerate Hubbard model in the strong coupling limit. For this limit, we obtain new exact results: when there are N ? 1 electrons, the ground state is ferromagnetic and there is a ferromagnetic orbital ordering; when the number of electrons is between N and 2N, the ground state is also ferromagnetic due to the intra-atomic Hund's coupling. For this second case, we give an estimate of the Curie temperature.     

Fermi-liquid effects in the Hubbard model

Properties of an electronic system with a very strong electron-electron interaction are studied. The equation of state is obtained, and the temperature and concentration dependences of the hydrodynamic sound and magnetic susceptibility are calculated. A low-temperature domain of instability corresponding to phase segregation is found. This domain is located inside the wide region of Cooper instability. The equation for zero-sound oscillations is obtained with regard to the substantial anisotropy of the energy spectrum.     

Phase stability in the Hubbard model

The Hubbard Model is shown to explain qualitatively the phase diagram of chromium doped V₂O₃ through a study of phase stability.     

Simulations of the Hubbard model

We discuss results of simulations of the Hubbard model of interacting electrons on a lattice. We start with a brief discussion of methodology and point out some of the outstanding problems. We then discuss results of simulations of the model in three, two, and one dimension, particularly in connection with its magnetic and superconducting properties. We conclude with a brief discussion of future directions.     

Spin-charge density waves in the Hubbard model

The ground state of the Hubbard model in a square lattice is examined in the Hartree-Fock mean field approximation at zero temperature. At small finite hole doping, the system has periodically distributed soliton like structures whose modulations are incommensurate. In a self-consistent way, the Fermi energy can always be located in a gap. The incommensurate states have lower energies than the commensurate antiferromagnetic states calculated at the same filling. These soliton structures persist even when a sizeable nearest neighbor repulsive interaction is included.     

Electron correlations in the effective Hubbard model

A new general unitary transformation is obtained, which allows to get in a controllable manner the effective Hamiltonian of the Hubbard model at an arbitrary sign and value of the intraatomic constantU and for any given filling number of electrons per atomn. It is shown that atU<0 the effective Hamiltonian has a multipseudospin exchange form for an arbitrary filling and there exist hidden localSU(2) andU(1) gauge symmetries in the restricted Hilbert space.