Topology of superconductors beyond mean-field theory

Abstract

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 $p_x+i{p}_y$ 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.

arXiv commentv1: 5+1 pages, v2: reformatted with some minor revisions and additional references. Now published in Physical Review Research

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