Pinning Effect Of Vortex In Topological Superconductor And The Braiding Of Majorana Fermions

In recent years,topological superconductors have attracted wide attention.This is because topological superconductor has a superconducting gap in the bulk and topologically protected gapless states.There is close relationship between the gapless states and Majorana fermions.Each Majorana fermion is its own antiparticle.Majorana fermion is still a non-Abelian anyon,obeying non-Abelian statistics,which has potential applications in the realization of topological quantum computation.In this paper,by establishing a reasonable theoretical model and sloving the Bogoliubov-de Gennes equation,the vortex pinning effect in topological superconductor is numerically calculated and the braiding of the Majorana fermions is achieved by dragging impurities.Firstly,we give a reasonable Hamiltonian and illustrate the rationality of the model simply.Next,we numerically calculate the vortex pinning effect of topological trivial states and topological non-trivial states,and the value of critical magnetic field by energy spectrum on the N?(28)2448 lattice.The result shows that there are two vortices in the topological superconductors,located at(12,12)and(36,12)respectively.After the addition of the impurity,the vortex is dragged to the location of the impurity.However,there is a critical distance Rc of vortex pinning by the point potential.We suppose the position of the vortex center without impurity is the origin,and when the distance of the impurity to the origin exceeds the corresponding critical distance,the vortex will return to the original position without impurities.We know that superconducting vortices in topological superconductors are most likely associated with the Majorana bound states.By calculating eigenvalues and eigenvectors numerically,we can visually see the existence of two zero-energy eigenvalues,showing the close relationship between vortices and Majorana bound states.By calculating the density of states after vortex pinned,we can observe a significant zero-energy peak,whose appearance indicates that vortex pinning by the point potential does not destroy the Majorana bound states.In summary,impurities in topological superconductors have a pinning effect on vortices,this pinning effect does not destroy the Majorana bound state.Therefore,we can try to use this pinning effect to realize the braiding of Majorana bound states.However,when the two vortices are close,the vortices will repel each other.This will to a certain extent reduce the critical distance of pinning.In addition,the existence of the pinning critical distance makes it impossible to directly exchange the two vortices in the horizontal direction,which is not conducive to the exchanging and braiding of the Majorana bound states.Therefore,we find that the zero-energy eigenvalues and the Majorana bound states still exist on the 4848? lattice,namely there is no essential difference between 2448? lattice and4848? lattice.We achieve the exchange of the two vortices by dragging the vortices slowly on the4848? lattice.Then,we show that the vortices in topological superconductors contact the Majorana bound states by numerically calculating and theoretically analysing.We succeed in exchanging the Majorana bound states by pinning slowly,which is of great significance to the realization of the braiding of Majorana bound states and the topological quantum computer.