The exchange interactions between Ni atoms prevent from the decay of magnetic moments of Ni atoms.

\[Ni\_Chain\] The magnetism of Ni atomic chains
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The initial Ni chains are located in the trough along the A-\[1$\bar{1}$0\] direction with nearest-nearby distance about 2.88$\AA$ the same as the nearest-nearby distance of the missed Au chain \[Fig. \[fig1\]\]. The surface slab includes 4 atomic layers with 56 Au atoms and the size of the super-cell is 23.046$\AA$$\times$8.148$\AA$$\times$15.52$\AA$. We have relaxed the initial structures using the conjugate-gradient Methods. The 2$\times$2$\times$1 K-points mesh is used to sampling Brillouin-zone in the corresponding electronic-structure calculations. Based on the relaxed structures, we calculated their electronic structures more exactly using RMM-DIIS algorithm with the (4$\times$4$\times$1) Monkhorst-Pack grids sampling the first Brillouin zone. The other information is the same as in the calculations for single 3d atoms.

We have also calculated the magnetism of free-standing Ni chains by removing all surface atoms but still preserving in the same box cell. There are mirror atomic chains because of periodic boundary condition along three directions. The interaction of free-standing chains with its mirror atomic chains are small and negligible if the box cell is large enough. We have only chosen short chains 