  Scanning tunneling microscopy is a powerful tool to build artificial atomic
structures even not exist in nature but possess exotic properties. We here
constructed Lieb lattices with different lattice constants by real atoms, i.e.,
Fe atoms on Ag(111) and probed their electronic properties. We find a
surprising long-range effective electron wavefunction overlap between Fe
adatoms as it exhibits a 1/r2-dependence with the interatomic distance r
instead of the theoretically predicted exponential one. Combining control
experiments, tight-binding and Green's function calculations, we attribute the
observed long-range overlap to be enabled by the surface state. Our findings
not only enrich the understanding of the electron wavefunction overlap, but
also provide a convenient platform to design and explore the artificial
structures and future devices with real atoms.
