  When the material phases exhibit topological quantum numbers, they host
defects protected by the nontrivial topology. Magnetic skyrmions are such
``quantized" objects and although many of them are metals they had been most
likely treated in theories as purely classical spin states. Here, we show that
the electrons described by the Hubbard model with strong spin-orbit couplings
can exhibit various nano- or flake-size skyrmions in their ground state.
Importantly, the conducting electrons forming Fermi pockets themselves carry
textured magnetic moments in both real and momentum space and on top of that,
possess Chern numbers in their energy bands. This quantum and conducting
skyrmion is related to small skyrmions observed in atomic-layered compounds. We
clarify how the effective magnetic interactions and magnetic anisotropies are
tied to the spin-orbit coupling, and how they influence the stability of
skyrmions beyond the phenomenology.
