  We have studied zero temperature metastable states in classical $m$-vector
component spin glasses in the presence of $m$-component random fields (of
strength $h_{r}$) for a variety of models, including the Sherrington
Kirkpatrick (SK) model, the Viana Bray (VB) model and the randomly diluted
one-dimensional models with long-range power law interactions. For the SK model
we have calculated analytically its complexity (the log of the number of
minima) for both the annealed case and the quenched case, both for fields above
and below the de Almeida Thouless (AT) field ($h_{AT} > 0$ for $m>2$). We have
done quenches starting from a random initial state by putting spins parallel to
their local fields until convergence and found that in zero field it always
produces minima which have zero overlap with each other. For the $m=2$ and
$m=3$ cases in the SK model the final energy reached in the quench is very
close to the energy $E_c$ at which the overlap of the states would acquire
replica symmetry breaking features. These minima have marginal stability and
will have long-range correlations between them. In the SK limit we have
analytically studied the density of states $\rho(\lambda)$ of the Hessian
matrix in the annealed approximation. Despite the absence of continuous
symmetries, the spectrum extends down to zero with the usual $\sqrt{\lambda}$
form for the density of states for $h_{r}<h_{AT}$. However, when
$h_{r}>h_{AT}$, there is a gap in the spectrum which closes up as $h_{AT}$ is
approached. For the VB model and the other models our numerical work shows that
there always exist some low-lying eigenvalues and there never seems to be a
gap. There is no sign of the AT transition in the quenched states reached from
infinite temperature for any model but the SK model, which is the only model
which has zero complexity above $h_{AT}$.

Disordered Systems and Neural Networks;Soft Condensed Matter;Statistical Mechanics