  (Abridged) The hyperaccreting neutron star or magnetar disks cooled via
neutrino emission can be a candidate of gamma-ray burst (GRB) central engines.
The strong field $\geq10^{15}-10^{16}$ G of the magnetar can play a significant
role in affecting the disk properties and even lead to the funnel accretion
process. We investigate the effects of strong fields on the disks around
magnetars, and discuss implications of such accreting magnetar systems for GRB
and GRB-like events. We discuss quantum effects of the strong fields on the
disk, and use the MHD conservation equations to describe the behavior of the
disk flow coupled with a large scale field, which is generated by the star-disk
interaction. In general, stronger fields give higher disk densities, pressures,
temperatures and neutrino luminosity, and change the electron fraction and
degeneracy state significantly. A magnetized disk is always viscously stable
outside the Alfv\'{e}n radius, but will be thermally unstable near the
Alfv\'{e}n radius where the magnetic field plays a more important role in
transferring the angular momentum and heating the disk than the viscous stress.
The funnel accretion process will be only important for an extremely strong
field, which creates a magnetosphere inside the Alfv\'{e}n radius and truncates
the plane disk. Because of higher temperature and more concentrated neutrino
emission of the magnetar surface ring-like belt region covered by funnel
accretion, the neutrino annihilation rate from the accreting magnetars can be
much higher than that from accreting neutron stars without fields. Furthermore,
the neutrino annihilation mechanism and the magnetically-driven pulsar wind
from the magnetar surface can work together to generate and feed an
ultra-relativistic jet along the stellar magnetic poles.
