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

Abstract

We examine properties of the mean-field wave function of the one-dimensional Kitaev model supporting Majorana Zero Modes (MZMs) \emph{when restricted} to a fixed number of particles. Such wave functions can in fact be realized as exact ground states of interacting number-conserving Hamiltonians and amount to a more realistic description of the finite isolated superconductors. Akin to their mean-field parent, the fixed-number wave functions encode a single electron spectral function at zero energy that decays exponentially away from the edges, with a localization length that agrees with the mean-field value. Based purely on the structure of the number-projected ground states, we construct the fixed particle number generalization of the MZM operators. They can be used to compute the edge tunneling conductance; however, notably the value 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 for the number-projected wave functions and find that it contains a `robust' $\log(2)$ component as well as a logarithmic correction to the mean field result, which depends on the precise partitioning used to compute it. The presence of the logarithmic term in the entanglement entropy indicates the absence of a spectral gap above the ground state; as one introduces fluctuations in the number of particles, the correction vanishes smoothly.