Topological superconductors and Majorana fermions based on $X$-wave magnets with $X=p,d,f,g,i $

Abstract

We study superconductors coupled with $X$-wave magnets with $X=p,d,f,g,i$. They are essentially different between altermagnets with $X=d,g,i$ and odd-parity magnets $X=p,f$ due to the compatibility of the spin-singlet formation of the $s$-wave superconductor. The altermagnets and odd-parity magnets show rich superconducting phase diagrams depending on the strength of the magnetic order, the chemical potential and the ratio of the on-site and nearest-neighbor interactions. Especially, we find that the altermagnetic order stabilizes the chiral ($p_{x}+ip_{y}$)-wave superconductor, which is a class D topological superconductor hosting Majorana chiral edge states. It is characterized by the Chern number. Especially, there emerge two chiral edge modes in the $i$-wave altermagnet, which are characterized by the Chern number 2. On the other hand, the $p$ ($f$)-wave odd-parity magnetic order stabilizes the $p$ ($f$)-wave topological superconductor, which is a class DIII topological superconductor hosting Majorana flat bands. It is characterized by the winding number. Majorana flat bands are robust in the presence of the mixing with the $s$-wave superconductivity, which is inevitable for odd-parity magnets although the magnitude of the mixing is tiny.