Particle-Number Conserving Considerations of Majorana Zero Modes and Topological Quantum Computing - beyond BCS Mean-Field Theory and Bogoliubov-de Gennes (BdG) Equations
Majorana zero modes are predicted to exist in p+ip (either inherent or effective due to proximity effect) superfluids and can be used to construct topological qubit as building block for topologically protected quantum computing. Existing theories for the subject are mostly based on BCS mean-field theory which breaks particle number conservation (U(1) symmetry) (except for some (quasi-)1D systems). More specifically, Bogoliubov-de Gennes (BdG) equations are used to derive Majorana zero modes and their braiding properties. The present work is initially motivated by the simple observation that Majorana zero modes in existing theories break particle number conservation which is, on the other hand, respected in any fermionic condensed matter system and which therefore may be important for studying quantum coherence, essential for quantum computing. In this talk, I will examine the role played by particle number conservation in topological properties of Majorana zero modes and conclude that the latter may be affected by the former and new theoretical framework that respects particle number conservation is much needed in establishing (or dismissing) the existence of Majorana zero modes in p+ip superfluids and their application to topological quantum computing. The talk is largely based on joint work with Tony Leggett.