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Majorana zero modes are predicted to exist in p+ip (either inherent oreffective due to proximity effect) superfluids and are proposed to be used forconstructing topological qubits for topologically protected quantum computing.Existing theories on the subject are mostly based on BCS mean-field theorywhich breaks particle number conservation (U(1) symmetry). More specifically,Bogoliubov-de Gennes (BdG) equations are used to derive Majorana zero modes andtheir braiding properties. The broken particle number conservation is, on theother hand, respected in any fermionic condensed matter system and thereforemay be crucial in studying quantum coherence, essential for quantum computing.In this paper, we work towards a particle-number conserving theory by examiningthe role played by particle number conservation in topological properties ofMajorana zero modes. We conclude that the latter may be affected by the formerand new theoretical framework that respects particle number conservation isneeded in establishing (or dismissing) the existence of Majorana zero modes inp+ip superfluids and their application to topological quantum computing.

Towards a Particle-Number Conserving Theory of Majorana Zero Modes in p+ip Superfluids