Some questions concerning Majorana fermions in 2D (p+ip) Fermi superfluids

Abstract

Most of the discussion in the literature of the Majorana fermions (M.F.’s) believed to exist in so-called 2D p+ip Fermi superfluids, and in particular of their possible application in topological quantum computing (TQC) has analyzed the problem using the familiar Bogoliubov–de Gennes (mean-field) formalism, which conserves total electron number only mod 2. We raise the question: Suppose that we require that electron number be conserved, period, at all stages of the calculation, then what are the consequences for the characteristic properties of M.F.’s, in particular for their (physical) braiding statistics? While we do not provide a definitive answer to this question, our considerations suggest that these consequences could be enough to destroy the possibility of using these excitations for TQC, and certainly indicate that the problem deserves further study.