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We examine properties of the mean-field wave function of the one-dimensionalKitaev model supporting Majorana Zero Modes (MZMs) \emph{when restricted} to afixed number of particles. Such wave functions can in fact be realized as exactground states of interacting number-conserving Hamiltonians and amount to amore realistic description of the finite isolated superconductors. Akin totheir mean-field parent, the fixed-number wave functions encode a singleelectron spectral function at zero energy that decays exponentially away fromthe edges, with a localization length that agrees with the mean-field value.Based purely on the structure of the number-projected ground states, weconstruct the fixed particle number generalization of the MZM operators. Theycan be used to compute the edge tunneling conductance; however, notably thevalue of the zero-bias conductance remains the same as in the mean-field case,quantized to $2e^2/h$. We also compute the topological entanglement entropy forthe number-projected wave functions and find that it contains a `robust'$\log(2)$ component as well as a logarithmic correction to the mean fieldresult, which depends on the precise partitioning used to compute it. Thepresence of the logarithmic term in the entanglement entropy indicates theabsence of a spectral gap above the ground state; as one introducesfluctuations in the number of particles, the correction vanishes smoothly.

Signatures of Majorana Zero-Modes in an isolated one-dimensional superconductor