Gap inversion in quasi-one-dimensional Andreev crystals

We study a periodic arrangement of magnetic regions in a quasi-one-dimensional superconducting wire. Due to the local exchange field, each region supports Andreev bound states that hybridize, forming Bloch bands in the subgap spectrum of what we call the Andreev crystal (AC). As an illustration, ACs with ferromagnetic and antiferromagnetic alignment of the magnetic regions are considered. We relate the spectral asymmetry index of a spin-resolved Hamiltonian to the spin polarization and identify it as the observable that quantifies the closing and reopening of the excitation gap. In particular, antiferromagnetic ACs exhibit a sequence of gapped phases separated by gapless Dirac phase boundaries. Heterojunctions between antiferromagnetic ACs in neighboring phases support spin-polarized bound states at the interface. In close analogy to the charge fractionalization in Dirac systems with a mass inversion, we find a fractionalization of the interface spin.