Efficient studies of large-scale topological and unconventional superconductors

## Abstract

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 perform calculations and can be exponentially faster than any classical supercomputer. However, quantum systems are generally extremely sensitive to disturbances and we are still not able to construct useful quantum computers. Topological superconductors avoid this extreme sensitivity by using the non-local nature of the Majorana fermions. The long-term goal of this project is to theoretically investigate the currently most promising topological superconductors as well as discover new topological superconductors.
We already have many years of experience studying topological superconductors using a microscopic lattice tight-binding Bogoliubov-de Gennes (BdG) formalism. Thanks to medium SNIC grants, we have successfully investigated a wide range of systems (see webpage for a full publication list) using GPU resources. 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 poorly scalable calculations we have developed our own code, called TBTK (SoftwareX 9, 205 (2019)) that efficiently treat these types of systems using a Chebychev polynomial expansion of the Green’s functions. The method takes advantage of recursion relationships and can work with just matrix-vector multiplications giving a (A+BN)*N scaling, with A and B being constants and N the number of degrees of freedom (lattice sites, spin, orbitals, etc), instead of the brute force N^6 scaling of exact diagonalization. Due to the calculations to a large part consisting of repeated matrix-vector multiplications it is optimally suited for GPUs.
We have the last two years also heavily used quasiclassical theory to be able to study even larger, mesoscopic, topological superconducting systems. In particular, we are using the continuously developed code SuperConga (Appl. Phys. Rev. 10, 011317 (2023)), an open-source framework for simulating physics of mesoscopic superconductors using the quasiclassical Eilenberger theory. This results in a complex ray-tracing problem, which due to it being a quantum field theory, consists of multiple source points, a very high degree of freedoms, and non-linear differential equations. We also require a self-consistent solution, which results in an optimization problem solved using an involved adaptive scheme.
By running our software on high-end GPUs, we are continuously pushing the boundaries of scientific research in our field.