Two-dimensional topological superconducting phases emerged from d-wave superconductors in proximity to antiferromagnets

Abstract: Motivated by the recent observations of nodeless superconductivity in the monolayer CuO${2}$ grown on the Bi${2}$Sr${2}$CaCu${2}$O${8+\delta }$ substrates, we study the two-dimensional superconducting (SC) phases described by the two-dimensional $t$-$J$ model in proximity to an antiferromagnetic (AF) insulator. We found that (i) the nodal d-wave SC state can be driven via a continuous transition into a nodeless d-wave pairing state by the proximity induced AF field. (ii) The energetically favorable pairing states in the strong field regime have extended s-wave symmetry and can be nodal or nodeless. (iii) Between the pure d-wave and s-wave paired phases, there emerge two topologically distinct SC phases with ($s+$i$d$) symmetry, i.e., the weak and strong pairing phases, and the weak pairing phase is found to be a $Z{2}$ topological superconductor protected by valley symmetry, exhibiting robust gapless non-chiral edge modes. These findings strongly suggest that the high-$T_{c}$ superconductors in proximity to antiferromagnets can realize fully gapped symmetry protected topological SC.