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

Abstract

The ground state of 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 being always a chiral superconductor, we find that the d + id superconductor undergoes a topological phase transition from a chiral superconductor to a quasi-helical superconductor as the gap amplitude or the chemical potential decreases. The quasi-helical superconducting phase is found to be characterized by a topological invariant in the pseudo-spin 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 new platform for hosting large numbers of Majorana modes at edges.

Non‐Centrosymmetric Superconductors on Honeycomb Lattice

“…We further study the phase diagram for system with Rashba interaction induced triplet pairing (p+ip-wave). It is found the outcomes in the case are similar to that from previous studies 22,45 . When all interactions are included, the phase diagrams show the feature of singlet-triplet mixture.…”

“…The off-diagonal region (blue area) shows Chern number C = 2. This result is similar to that for a chiral superconductor without Zeeman couplings 45 . The grey area indicates fluctuating Chern numbers, which may lie near the phase boundaries.…”

“…It is shown that system with KM and Rashba interaction in the presence of magnetic fields shows the QAH phase with Chern number C = 2 44 . Furthermore, the above model was shown to exhibited equivalent mathematical structure to a spin singlet d+id superconductor via a duality mapping 45 .…”

“…Through a unitary transformation, the Hamiltonian with s-wave pairing and Rashba SO interaction is mapped into the dual p+ip-wave Hamiltonian, which exhibits non-trivial topological superconductivity when Zeeman couplings go beyond a critical value 65 . For honeycomb lattice with singlet d+id-wave pairing and KM interaction, it has been demonstrated that the Hamiltonian is equivalent to a collection of two topological ferromagnetic insulators, as described in Section 2, offers an explanation to why such setup could exhibit chiral edge states 45 .…”

Duality and topology

“…However, it is also known that the Jordan-Wigner transformation may be seen as the result of a local unitary transformation [24] (understood as a product of local transformations across all lattice sites). In this work we will follow a similar procedure and therefore consider a less stringent definition of duality (see also, for instance [35]).…”