# The meaning of phase operator in Majorana zero mode

In some article, such as Phys. Rev. B 94, 235446 (2016), they define the Majorana mode operator as follow $$\gamma_j=\int \mathrm{d}r \ [\xi_j(r)e^{-i\theta/2}c^\dagger(r)+\xi_j^*(r)e^{i\theta/2}c(r)]$$

where $$\xi_j(r)$$ is the real space wave function and $$\theta$$ is the phase operator satisfying $$[\theta,N]=2i$$, which gives rise to $$[N,e^{\pm i\theta/2}]=\pm e^{\pm i\theta/2}$$

However, to my best knowledge, MZM comes from diagonalizing the Hamiltonian in Nambu space, by which I could not understand where the phase operator comes from. It would be natural for me to think that $$\theta$$ is just some c-number, where it isn't in the article. Besides, when taking complex conjugate there will exist the commutator $$[e^{i\theta/2},c]$$ which I don't know how to handle.

PS:

1. The author also claims that this is somewhat charge-statistics separation of an electron: "the charge of the electron is spread out over the entire superconductor, while its Fermi statistics are retained by a localized Majorana fermion that is charge neutral.", which I don't quite understand.
2. The other part of the cited article is not relevant to this topic.
• I had that same question some time ago and never found a satisfying answer. Looking forward to see anyone that has one. What I could find, though, is that $\theta$ is definitely not just a c-number. The commutation relation with N is telling us that $\theta$ is reciprocal to N in a similar way momentum is reciprocal with respect to real space. The thing I find most troubling is that the eigenvalues of N are discrete, so I'm not sure it's as simple as $\theta = -2i\partial/\partial N$ Feb 22, 2021 at 5:46
• It's interesting to note, however that although N is discrete and $\theta$ is not, N is not limited, while $\theta$ is. Feb 22, 2021 at 5:55
• @Lucas Baldo I've learnt the phase and particle number uncertainty in superconductor, but I couldn't relate it to this issue. Feb 22, 2021 at 7:03