In the last ten years the search for Majorana fermions in real materials have generated immense amount of attention. Apart from the academic excitement they generate, the interest is largely driven by proposals to use Majorana fermions for encoding information in topological fault tolerant quantum computers. The reason for this is that Majorana fermions are non-local in nature, and have non-Abelian statistics, which endows them with robustness against local perturbation, and hence, by extension immunity against decoherence when used as qubit in quantum computers.
There have been enormous efforts, theoretically and experimentally, to create Majorana fermions in hybrid superconductor-nonsuperconductor platforms, yet there is no clear identification of Majorana fermions despite several experimental reports of signatures that are similar to theoretical predictions. One major reason for this is due to the discrepancies between experimental data and theoretical predictions. Neither side of the field can be faulted to this mismatch because of the one hand, due to computational cost and dearth of efficient methods, theoretical approach are approximations with some effects ignored while on the other hand experimental data might include unwanted data caused by unintended states in the sample despite the tremendous advancement in material science and material growth in recent times.
Secondly, it seeks to develop an efficient simulation approach for Majorana platforms, surpassing the capabilities of existing methods. Our primary focus within this endeavor lies mainly in two distinct Majorana platforms: the superconductor-semiconductor and superconductor-magnetic impurities systems. We've already amassed significant theoretical expertise in these domains, employing techniques such as the tight-binding Bogoliubov-deGennes (BdG) formalism in conjunction with exact diagonalization and Arnoldi iteration methods. However, achieving accurate and efficient simulations, especially when considering realistic experimental parameters, entails a substantial increase in computational demands. These demands are often beyond the scope of small-scale simulations. To optimize our utilization of SNIC resources, we are currently developing a new approach for studying real materials in experimental scenarios. This approach leverages Green's function methodology, allowing us to extract localized information from any region within the sample. This not only conserves memory but also significantly reduces computational time by eliminating the need to calculate extraneous data, a capability unattainable through traditional diagonalization methods. In tandem with this, we will conduct ab-initio calculations utilizing Density Functional Theory (DFT) to both validate and fine-tune the accuracy of the results obtained through our proposed Green's function method. Additionally, we will employ the Arnoldi iteration method to swiftly extract low-energy states.
Thanks to the medium-scale resources provided by SNIC, we are poised to embark on this research journey, making meaningful contributions toward the goal of realizing topological quantum computers.