  The study of topological superconductivity is largely based on the analysis
of simple mean-field models that do not conserve particle number. A major open
question in the field is whether the remarkable properties of these mean-field
models persist in more realistic models with a conserved total particle number
and long-range interactions. For applications to quantum computation, two key
properties that one would like to verify in more realistic models are (i) the
existence of a set of low-energy states (the qubit states) that are separated
from the rest of the spectrum by a finite energy gap, and (ii) an exponentially
small (in system size) bound on the splitting of the energies of the qubit
states. It is well known that these properties hold for mean-field models, but
so far only property (i) has been verified in a number-conserving model. In
this work we fill this gap by rigorously establishing both properties (i) and
(ii) for a number-conserving toy model of two topological superconducting wires
coupled to a single bulk superconductor. Our result holds in a broad region of
the parameter space of this model, suggesting that properties (i) and (ii) are
robust properties of number-conserving models, and not just artifacts of the
mean-field approximation.
