Stability of topological superconducting qubits with number conservation

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 is whether the properties of these mean-field models persist in more realistic models with a conserved particle number and long-range interactions. For quantum computing applications, 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. 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 rigorously prove that properties (i) and (ii) hold in a number-conserving toy model of two topological superconducting wires coupled to the same bulk superconductor. Our result holds in a broad region of the model's parameter space and therefore proves that (i) and (ii) are robust properties of number-conserving models, and not just artifacts of the mean-field approximation.