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.
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