Models for $d$ Wave Topological Superconductors and Quantum Anomalous Hall Effect with Arbitrary Large Chern Numbers

Abstract

We construct theoretical models for two dimensional(2d) chiral d x 2 −y 2 ±id xy topological superconductors and for three dimensional(3d) d wave topological superconductors. Moreover we build models for any 2d class C and 3d class CI topological superconductors (with any even topological invariants). We also construct concrete models that can realize quantum anomalous Hall effect with arbitrary large Chern numbers. We study the chiral edge modes or gapless surface states of our 2d or 3d models in details. In all the cases, we find novel mechanisms that make the numbers of boundary states always agree with the nontrivial bulk topology, just as required by the bulk boundary correspondence.

Simple Models for All Topological Phases

“…If we compare our present models (37) with the models (for class CI topological superconductors) that we introduced in [21], after an appropriate reordering of the basis, we will find that, although the specific details are different, but the function d(p) in our present models can be naturally corresponded to the function ∆ z p 2l+1 z in the corresponding models of [21], these two functions are all odd pairing functions. Moreover, the pairing function ∆(p) in the present paper is corresponding to the pairing function w(p x , p y ) in [21]. Both these two functions are even, and in the situations with topological invariants 2n, both of them are degree 2n polynomials of the momentum.…”

“…Both these two functions are even, and in the situations with topological invariants 2n, both of them are degree 2n polynomials of the momentum. The pairing function w(p x , p y ) is crucial for the models of [21]. Indeed, as [21] has demonstrated, w(p x , p y ) determines the gapless surface states.…”

“…The pairing function w(p x , p y ) is crucial for the models of [21]. Indeed, as [21] has demonstrated, w(p x , p y ) determines the gapless surface states. In the topologically nontrivial phase, the degree (2n) of w(p x , p y ) determines that the numbers of gapless surface states should be 2n.…”

“…As an illustration, we would like to give a theoretical derivation for the model for two dimensional quantum anomalous Hall insulator with the first Chern number c 1 = 1 [34][35], since this has been realized in magnetic topological insulator of Cr-doped (Bi, Sb) 2 Te 3 [36]. For the quantum anomalous Hall insulators with large Chern numbers we refer the readers to [21,37,38,39,40,41].…”