Efficient studies of large-scale topological superconductors
Topological superconductors are a newly discovered class of materials with features uniquely advantageous for quantum computing. They have lately generated an immense amount of attention due to the possibility of them having effective Majorana fermions at surfaces, vortices, and other defects. Approximately one can say that a Majorana fermion is half an electron, or more accurately, in a system with Majorana fermions the wave function of an electron has split up into two separate parts. This non-local property of two Majorana fermions can be used for exceptionally fault-tolerant quantum computing. A quantum computer uses the quantum nature of matter to represent data and preform calculations and can be exponentially faster than any classical supercomputer. However, quantum systems are generally extremely sensitive to disturbances and we are still far from being able to construct useful quantum computers. Topological superconductors with Majorana fermions avoid this extreme sensitivity by using the non-local nature of the Majorana fermions, which automatically make them very robust. The goals of this project are to theoretically 1) discover new and experimentally viable topological superconductors with Majorana fermions and 2) determine the properties of the Majorana fermions and the conditions necessary for feasible topological quantum computation in real materials. The project will focus both on the currently most promising topological superconductors found in superconducting hybrid structures of well-known spin-orbit coupled materials and on discovering new topological superconductors in graphene and related materials. We already have many years of experience studying these types of systems using a microscopic lattice tight-binding Bogoliubov-de Gennes formalism, which is ideally suited for an accurate description of the superconducting state in topological superconductor with Majorana fermions. Thanks to medium SNIC grants, we have during the last years investigated a range of systems. The method usually involves diagonalizing large matrices and we often need to find a self-consistent solution for superconductivity, which requires us to find all eigenvalues. We have, however, recently developed a much more efficient way of treating these types of systems using a Chebychev expansion of the Green’s functions. The large advantage of this approach is that it avoids full diagonalization of large matrices, which scales at N^6 with N being the number of lattice sites, but instead we get a (A+BN)*N scaling, with A and B being constants. Thus we can go to significantly larger lattice sizes and thus get more realistic results for a wide variety of systems. This scaling is possible thanks to a sparse format and matrix-vector multiplications instead of full diagonalization.