Single-particle off-diagonal long range order in $|Pf|$ wave function for topological superconductors

We ask what can be glimpsed about a fermion system from a boson wave function obtained as the absolute value of the fermion wave function in the particle number basis. We consider examples of spinless $p$-wave superconductors in one and two dimensions and find that thus constructed wave function shows single-boson off-diagonal long range order when the fermion system is in the weak-coupling topological phase, but only pair-boson order in the strong-coupling non-topological phase. We discuss implications when such fermion states are used as factors in trial wave functions e.g. from slave-fermion construction.