The purpose of this application is to seek supercomputer resources for continuing the research studies on realizing topological superconductors using superconductor-semiconductor hybrids and multiband materials. Topological superconductivity is a new superconducting state of matter that hosts Majorana fermions, charge neutral quasiparticles that hold promise for quantum computation free of decoherence [1]. Despite the advances, it is still unclear whether topological superconductivity has been observed or not [2]. Moreover, the detection of topological superconductivity has mostly been done by zero-energy quantized conductance peaks, which exploits the zero-energy nature of its Majorana states [3]. However, there exist topologically trivial states producing similar conductance signatures [4], thus challenging the advance of the entire field [2]. This application comprises the goals of several projects aiming to realize and detect topological superconductivity in superconductor-semiconductor hybrids and in multiband two-dimensional materials coupled to conventional superconductors. Our research to reach these goals is funded by the Swedish Research Council (grant No.2021- 04121), Carls Trygger Foundation (Grant No. 22: 2093), and Göran Gustafson Foundation (Grant No. 2216). The specific goals of this application can be summarised as follows: 1. Realize topological superconductivity in realistic superconductor-semiconductor hybrids 2. Realize topological superconductivity in realistic multiband two-dimensional materials coupled to conventional superconductors 3. Detect topological superconductivity using spectral observables 4. Detect topological superconductivity by the Josephson effect 5. Characterize the emergent superconductivity in topological matter The first two goals represent the cornerstones of this proposal and will help to understand the conditions where topological superconductivity emerges. Of particular relevance will be to explore realistic geometries, involving large systems with active spins and orbitals so that our systems get closer to the experimental setups [2,3]. We will focus on superconductor-semiconductor hybrids but also on multiband two-dimensional materials coupled to conventional superconductors. Among the multiband materials, we expect to explore Kagome and Weyl materials. The third and fourth goals will add new knowledge by identifying signatures of topological origin. We will explore spectral observables, such as the density of states and conductance, but also the Josephson effect when two coupled superconductors have a finite phase difference between their order parameters. To assess the Majorana origin in said observables, it will relevant to design protocols that sense the inherent Majorana spatial nonlocality, which I showed before to be useful for detecting Majorana states in a simpler setup [5]. In this part we will also investigate the possible realization of topological Josephson diodes [6], which will be a natural effect to obtain from the supercurrents in the Josephson effect. Given the realistic size and properties of the hybrid junctions we aim here, it will require to explore a large parameter space. The last goal will help to understand the fundamental properties of superconductivity in the discovered topological phases. Moreover, by knowing the type of superconductivity, it will be possible to design potential superconducting applications protected by topology. References: [1] Y. Tanaka, S. Tamura, J.Cayao, Prog. Theor. Exp. Phys. ptae065 (2024). [2] Prada el al., Nat. Rev. Phys. 2, 575 (2020) [3] Zhang et al., Nat. Commun. 10, 5128 (2019) [4] J. Cayao, Phys. Rev. B 110, 085414 (2024). [5] Jorge Cayao, Annica M. Black-Schaffer, Phys. Rev. B 104, L020501 (2021). [6] J. Cayao, N. Nagaosa, Y. Tanaka, Phys. Rev. B 109, L081405 (2024).