Rigorous results on topological superconductivity with particle number conservation
Most theoretical studies of topological superconductors and their unpaired Majorana fermions rely on a mean-field (or Bogoliubov-de Gennes) approach to describe superconductivity, which violates particle number conservation (PNC). Recently, A.J. Leggett has suggested that this violation of PNC could pose a serious conceptual problem for the program of Majorana-based quantum computation. In this talk I will present recent work designed to address this concern. I will first introduce a fully number-conserving toy model of a topological superconducting wire in which superconductivity is induced by proximity to a nearby bulk superconductor. I will then present rigorous proofs that this model possesses many of the desired properties of the well-known mean-field models, including a finite energy gap in a sector of fixed total particle number, and the existence of long range “Majorana-like” correlations between the ends of the wire. These results show that many of the remarkable properties of mean-field models of topological superconductivity persist in more realistic models with number-conserving dynamics.