  Most theoretical studies of topological superconductors and Majorana-based
quantum computation rely on a mean-field approach to describe
superconductivity. A potential problem with this approach is that real
superconductors are described by number-conserving Hamiltonians with long-range
interactions, so their topological properties may not be correctly captured by
mean-field models that violate number conservation and have short-range
interactions. To resolve this issue, reliable results on number-conserving
models of superconductivity are essential. As a first step in this direction,
we use rigorous methods to study a number-conserving toy model of a topological
superconducting wire. We prove that this model exhibits many of the desired
properties of the mean-field models, including a finite energy gap in a sector
of fixed total particle number, the existence of long range "Majorana-like"
correlations between the ends of an open wire, and a change in the ground state
fermion parity for periodic vs. anti-periodic boundary conditions. These
results show that many of the remarkable properties of mean-field models of
topological superconductivity persist in more realistic models with
number-conserving dynamics.
