We prove an index theorem for the existence of Majorana zero modes in a semiconducting thin film with a sizable spin{\textendash}orbit coupling when it is adjacent to an s-wave superconductor. The theorem, which is analogous to the Jackiw{\textendash}Rebbi index theorem for the zero modes in mass domain walls in one-dimensional Dirac theory, applies to vortices with odd flux-quantum in a semiconducting film in which s-wave superconductivity and a Zeeman splitting are induced by proximity effect. The momentum space construction of the zero-mode solution presented here is complementary to the approximate real space solution of the Bogoliubov-de Gennes equations at a vortex core (Sau et al., arXiv:0907.2239 [17]), proving the existence of non-degenerate zero-energy Majorana excitations and the resultant non-Abelian topological order in the semiconductor heterostructure. With increasing magnitude of the proximity-induced pairing potential, the non-Abelian superconducting state makes a topological quantum phase transition to an ordinary s-wave superconducting state which no topological order.

}, keywords = {2010, majorana fermions, non-abelian statistics, semiconductor, superconductor, topological quantum computation}, issn = {0003-4916}, url = {http://dx.doi.org/10.1016/j.aop.2009.11.003}, author = {Tewari, Sumanta and Jay D Sau and S. Das Sarma} }