Duality in topological superconductors and topological ferromagnetic insulators in a honeycomb lattice

Abstract

The ground state of the large Hubbard U limit of a honeycomb lattice near half filling is known to be a singlet d + id-wave superconductor. It is also known that this d + id superconductor exhibits a chiral p + ip pairing locally at the Dirac cone, characterized by a 2Z topological invariant. By constructing a dual transformation, we demonstrate that this 2Z topological superconductor is equivalent to a collection of two topological ferromagnetic insulators. As a result of the duality, the topology of the electronic structures for a d + id superconductor is controllable via the change of the chemical potential by tuning the gate voltage. In particular, instead of always being a chiral superconductor, we find that the d + id superconductor undergoes a topological phase transition from a chiral superconductor to a quasihelical superconductor as the gap amplitude or the chemical potential decreases. The quasihelical superconducting phase is found to be characterized by a topological invariant in the pseudospin charge sector with vanishing both the Chern number and the spin Chern number. We further elucidate the topological phase transition by analyzing the relationship between the topological invariant and the rotation symmetry. Due to the angular momentum carried by the gap function and spin-orbit interactions, we show that by placing d + id superconductors in proximity to ferromagnets, varieties of chiral superconducting phases characterized by higher Chern numbers can be accessed, providing a platform for hosting large numbers of Majorana modes at edges.