  The coupling of spin and orbital degrees of freedom in the trilayer Sr4Ru3O10
sets a long-standing puzzle, due to the peculiar anisotropic coexistence of
out-of-plane ferromagnetism and in-plane metamagnetism. Recently, the induced
magnetic structure by in-plane applied fields has been investigated by means of
spin-polarized neutron diffraction, which allowed to extract a substantial
orbital component of the magnetic densities at Ru sites. It has been argued
that the latter is at the origin of the evident layer dependent magnetic
anisotropy, where the inner layers carry larger magnetic moments than the outer
ones. We present a spin-polarized neutron diffraction study in order to
characterize the nature of the ferromagnetic state of Sr4Ru3O10, in the
presence of a magnetic field applied along the c-axis. The components of the
magnetic densities at the Ru sites reveal a vanishing contribution of the
orbital magnetic moment which is unexpected for a material system where orbital
and spin degeneracy are lifted by spin-orbit coupling and ferromagnetism. We
employ a model that includes the Coulomb interaction and spin-orbit coupling at
the Ru site to address the origin of the suppression of the orbital magnetic
moment. The emerging scenario is that of non-local orbital degrees of freedom
playing a significant role in the ferromagnetic phase, with the Coulomb
interaction that is crucial to make anti-aligned orbital moments at short
distance resulting in a ground state with vanishing local orbital moments.
