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, however, A.J. Leggett and others have argued that this violation of PNC may pose a serious conceptual problem for Majorana-based quantum computation. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the 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.