Gap inversion in quasi-one-dimensional Andreev crystals

Abstract

We study a periodic arrangement of magnetic regions in a 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 a close analogy to the charge fractionalization in Dirac systems with a mass inversion, we find a fractionalization of the interface spin.

Spectral properties of Andreev crystals

“…As it was predicted in previous calculations under the first-neighbours tight-binding approximation (Ref. 23 and Sec. III), these two bands touch each other only at half-integer values of Φ/π closing the gap around the Fermi level (see Fig.…”

“…On the one hand, our solution provides the exact energy and spatial distribution of the density of states and spin polarization of the system. On the other hand, our results demonstrate the validity of the first neighbours' tightbinding approximation regarding the gap closing in infinite antiferromagnetic ACs at half-integer values of Φ/π, the appearance of a pair of states bounded to the interface between two antiferromagnetic ACs with inverted gaps, and the fractionalization of the surface spin polarization in such junctions 23 .…”

“…The dotted lines correspond to energy values for which ω 2 = ω 2 0 sin 2 α and they indicate the maximum possible energy of a bound state [Eq. (23)]. Close to the gap-closing point, cos Φ 1, the interfacial states are present at any value of α, excluding ferromagnetic ordering of the impurities, sin α = 0.…”

“…As discussed in Section III, inverted junctions of antiferromagnetic ACs host a pair of states bounded to the interface. Moreover, inverted of antiferromagnetic ACs may present fractionalization of the surface spin polar-ization per Fermi valley 23 . In this section we show that this result holds beyond the tight-binding approximation used above, by solving the Eilenberger equation in junctions of ACs.…”

“…In Ref. 23 we proposed the analog of such atomic chains in a mesoscopic structure with lateral dimensions smaller than the superconducting coherence length, ξ 0 . The magnetic impurities are replaced by semiclassical magnetic regions realized, for example, by the contact to magnetic materials (see the sketch in Fig.…”