  A self-consistent spectral density approach (SDA) is applied to the Hubbard
model to investigate the possibility of spontaneous ferro- and
antiferromagnetism. 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 (Tc, Tn) 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.
