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 (BdG) formalism, which is ideally suited for an accurate description of the superconducting state in topological superconductors with Majorana fermions. Thanks to medium SNIC grants, we have during the last years investigated a range of systems.
The method traditionally involves diagonalizing large matrices and since we often need a self-consistent solution for superconductivity, this requires finding all eigenvalues. To avoid such costly and badly scalable calculations we have developed our own code that in an efficient way treat these types of systems using a Chebychev polynomial expansion of the Green’s functions. This method takes advantage of recursion relationships for the expansion and can work with just matrix-vector multiplications instead of brute for diagonalization.
Thus 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. In this way we can go to significantly larger lattice sizes and get much more realistic results for a wide variety of systems.
During the last year we have also expanded our capabilities towards ab-initio methods in order to get better access to realistic parameter for our BdG calculations. Here we use density functional theory (DFT) to find accurate energy levels and band structures of materials combinations of relevance for the project.