  We present the first 3D fully coupled magneto-thermal simulations of neutron
stars (including the most realistic background structure and microphysical
ingredients so far) applied to a very complex initial magnetic field topology
in the crust, similar to what recently obtained by proto-neutron star dynamo
simulations. In such configurations, most of the energy is stored in the
toroidal field, while the dipolar component is a few percent of the mean
magnetic field. This initial feature is maintained during the long-term
evolution (1e6 yr), since the Hall term favours a direct cascade (compensating
for Ohmic dissipation) rather than a strong inverse cascade, for such an
initial field topology. The surface dipolar component, responsible for the
dominant electromagnetic spin-down torque, does not show any increase in time,
when starting from this complex initial topology. This is at contrast with the
timing properties of young pulsars and magnetars which point to higher values
of the surface dipolar fields. A possibility is that the deep-seated magnetic
field (currents in the core) is able to self-organize in large scales (during
the collapse or in the early life of a neutron star). Alternatively, the
dipolar field might be lower than is usually thought, with magnetosphere
substantially contributing to the observed high spin-down, via e.g., strong
winds or strong coronal magnetic loops, which can also provide a natural
explanation to the tiny surface hotspots inferred from X-ray data.
