Superconductivity In Doped Quantum Anomalous Hall Insulators

With the rapid development of the topological quantum computing field in recent years,the properties and structure of topological superconductors(TSC)have been widely concerned and discussed as one of the most popular candidates which can realize Majorana fermions(MFs).This quantum state is a novel topological phase with full-gapped bulk energy band and gapless edge states on the open boundary with vacuum,which are topologically protected and tightly connecting with MFs.The Majorana fermions,which are the anti-particle of themselves,are non-Abelian anyons,and their quantum coherence can be hardly influenced by local noise or decoherence,which has huge potential value to realize the hardware of topological computing.In this paper,we mainly study the two-dimensional intrinsic topological superconductors induced by magnetic dopants based on Chern insulators.By forming a reasonable model,we study the topological properties of the doped Chern insulators numerically,plot the topological phase diagram and verify the existence of edge states under different topological phases and Majorana bound state in certain topological phase by means of the bulk-edge energy spectrum and local density of state(LDOS)of the system.Firstly,we build a effective TSC model in two dimensions which is based on a quantum anomalous Hall(QAH)insulator and choose a 48×48 square lattice as the main model,and secondly we solve the Bogoliubov-de Gennes(BdG)equation consistently in the selected parameter range,we can then get the constraint of Zeeman field term m ?band occupied number n and the superconducting order parameter?.Then,by calculating the topological invariant— the Chern number with different parameters through the berry phase of the energy bands,we find that the result divides the whole chosen parameter space into mainly four quantum states which are topologically nonequivalent: metal,theN=0 trivial superconductors,theN=1 TSC phase,and theN=2 TSC phase.Then we compute the bulk-edge energy spectrum of the system in different quantum states,by means of the energy spectrum we can observe the behavior of the edge states.As for topological non-trivial phase with odd chiral edge states,we can calculate the eigenvalues and eigenvectors by diagonalizing the Hamiltonian Matrix to observe the existence of the zero modes and survey sensible zero energy peak by computing the LDOS.According to the methods above,there is a pair of Majorana bound states which is topologically protected in theN=1 TSC state.However,the LDOS of zero energy also contains peaks when the model is under theN=0 orN=2 state,which proposes that the appearance of zero energy peaks in LDOS spectrum is not sufficient evidence of Majorana bound statesWhat's more,we study the topological properties of the system in quasi-one-dimension,and propose that the number of zero energy eigenvalues corresponding to edge states is different from that in two dimensions,because the rotation symmetry of the system is broken,and the continuous position in momentum space lost its correspondence to the discrete lattice in real space.In quasi-one-dimension,when the signal of massive term m changes,the topological phase will change as well,which has a huge difference with that in 2D condition.