  The variational cluster approach (VCA) is applied to study spontaneous
ferromagnetism in the Hubbard model at zero temperature. We discuss several
technical improvements of the numerical implementation of the VCA which become
necessary for studies of a ferromagnetically ordered phase, e.g. more accurate
techniques to evaluate the variational ground-state energy, improved local as
well as global algorithms to find stationary points, and different methods to
locate the magnetic phase transition. Using the single-site VCA, i.e. the
dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the
model in infinite dimensions is worked out. The results are compared with
previous dynamical mean-field studies for benchmarking purposes. The DIA
results provide a unified picture of ferromagnetism in the infinite-dimensional
model by interlinking different parameter regimes that are governed by
different mechanisms for ferromagnetic order. Using the DIA and the VCA, we
then study ferromagnetism in one-dimensional Hubbard chains with nearest and
next-nearest-neighbor hopping t2. In comparison with previous results from the
density-matrix renormalization group, the phase diagram is mapped out as a
function of the Hubbard-U, the electron filling and t2. The stability of the
ferromagnetic ground state against local and short-range non-local quantum
fluctuations is discussed.
