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In this Letter, we first formulate an effective theory, which generallycaptures long-range proximity effects of a surface system weakly coupled to ans-wave superconductor. The long-range proximity effects include both theemergent long-range pairing and hopping interactions in the surface system. Wethen model the Rashba spin-orbit-coupled nanowire in proximity with an s-wavesuperconductor by taking into account the emergent nonlocal effects in theweak-coupling limit. In this limit the induced superconducting pair potentialis found much smaller than that of the host superconductor, which is in goodagreement with recent experiments. Compared with the previously consideredstrong coupling limit with local proximity effects, the long-range interactionscan significantly modify the topological phase diagram, and considerably lowerthe threshold magnetic field for the emergence of Majorana zero modes.

Effective Long-Range Pairing and Hopping in Topological Nanowires Weakly Coupled to $ s $-Wave Superconductors