Topology of superconductors beyond mean-field theory

The study of topological superconductivity is largely based on the analysis of mean-field Hamiltonians that violate particle number conservation and have only short-range interactions. Although this approach has been very successful, it is not clear that it captures the topological properties of real superconductors, which are described by number-conserving Hamiltonians with long-range interactions. To address this issue, we study topological superconductivity directly in the number-conserving setting. We focus on a diagnostic for topological superconductivity that compares the fermion parity $\mathcal{P}$ of the ground state of a system in a ring geometry and in the presence of zero vs. $\Phi_{\text{sc}}=\frac{h}{2e} \equiv \pi$ flux of an external magnetic field. A version of this diagnostic exists in any dimension and provides a $\mathbb{Z}_2$ invariant $\nu=\mathcal{P}_0\mathcal{P}_{\pi}$ for topological superconductivity. In this paper we prove that the mean-field approximation correctly predicts the value of $\nu$ for a large family of number-conserving models of spinless superconductors. Our result applies directly to the cases of greatest physical interest, including $p$-wave and $p_x+ip_y$ superconductors in one and two dimensions, and gives strong evidence for the validity of the mean-field approximation in the study of (at least some aspects of) topological superconductivity.