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Spatially modulated, orbital selective ferromagnetism in La$_5$Co$_2$Ge$_3$

We present density functional theory calculations for low-$T_c$ metallic ferromagnet La$_5$Co$_2$Ge$_3$ at ambient and applied pressures. Our investigations reveal that the system is a quasi-one-dimensional ferromagnet with a peculiar coexistence of two different orbital-selective magnetic moments at two crystallographically inequivalent cobalt atoms, Co1 and Co2. Namely, due to different crystal-field splitting, the magnetic moment of Co1 atoms predominantly derives from $d_{xz}$ orbital whereas of Co2 atoms from $d_{xy}$ orbital. Consequently, Co1 and Co2 atoms develop unequal net magnetic moments, a feature that gives rise to a periodic, spatial modulation of magnetization along crystallographic $c$-direction. The amplitude of the spatial modulation, small at ambient pressure, drastically increases with applied pressure, until Co2 atoms become nonmagnetic. With the help of a toy model mimicking found orbital-selective ferromagnetic order, we demonstrate that the increasing amplitude of spatial modulation provides a consistent interpretation to the recently observed resistivity anomaly emerging at applied pressure identified as the appearance of the {\it new state}. Although, proposed here structural origin of the spatial modulation of magnetic moments in La$_5$Co$_2$Ge$_3$ is an alternative one to the advocated for this material ferromagnetic quantum criticality avoidance, the effects of quantum fluctuations can still play an important role at pressure larger than up-to-date measured 5GPa.