matrix element depending on the magnetization direction as: $$<d_{yz}|\hat{H}_{SO}^{sc}|d_{yz}> \propto
<Y_1^0|\hat{L}_z|Y_1^0>cos\theta$$ Therefore, when $\theta$=$0^o$ (easy-axis anisotropy), the energy lowering is maximum, and the $d$ states of V with $|m|$=1 interact each other through the $\hat{L}_z$ operator. This is in agreement with the easy-axis anisotropy for V-chain-GY system. As for Cr-chain-GY, HOMO and LUMO state for both AFM and FM coupling are mainly derived from the $|m_l|$=2 states as shown in Fig.\[fig7\](b) and (c). Based on Eq.\[eq7\],\[eq8\], one can see that the operator $\hat{L}_z$ couples the occupied and unoccupied states of $d_{x^2+y^2}$ and $d_{xy}$. Therefore, the coupling lowering the energy of perpendicular magnetization direction, the systems show easy-axis anisotropy. From Fig. \[fig7\](d), one can see that the interaction between the HOMO and LUMO is mainly contributed by the $d_{x^2+y^2}$ and $d_{xy}$ states of minority spin for Mn-chain-GY system. The interaction between $d_{x^2+y^2}$ and $d_{xy}$ through the $\hat{L}_z$ leads to the easy-axis anisotropy. The $|m_l|$-resolved PDOS of Fe-chain-GY system as shown in Fig.\[fig7\](e) indicates that the HOMO is mainly derived from the $m_l$=1 and $|m_l|$=2 states, whereas the LUMO is mainly contributed by $|m_l|$=1 states. The SOC effect induces the interaction between $|m_l|$=1 of HOMO and LUMO by $\hat{L}_z$, which makes system 