Nonsymmorphic symmetry protected spin-orbit semi-Dirac fermions in two dimensions

Semi-Dirac fermions (SDFs) in two-dimensional (2D) systems, which simultaneously exhibit linear and quadratic dispersion around them, are a bridge linking linear and quadratic Dirac cones. However, the so-called SDFs in the reported 2D materials are doubly degenerate (namely semi-Weyl fermions), and are vulnerable against spin-orbital coupling (SOC). Here, we propose a 2D SOC-robust SDF which arises from nonsymmorphic symmetries. Unlike the known 2D SDFs, due to the presence of inversion and time reversal symmetries ($\mathcal{I}\mathcal{T}$), each crossing band is doubly degenerate, making the SDF of fourfold degeneracy. By high throughput screening, we find that 26 candidate 2D materials (such as ${\mathrm{CuCN}}_{2}$, ${\mathrm{CaI}}_{2}$, TlF) can hold SDFs, which belong to layer groups 40 ($pmam$), 43 ($pbaa$), and 45 ($pbma$), respectively. Furthermore, the symmetry protection mechanism, associated Fermi arc edge states, the catalytic performance, and the topological phase transitions under symmetry breaking of SDFs are revealed, providing a way for understanding 2D SOC-robust SDFs. Overall, our work proposes a class of topological phases, and provides a platform to study their fascinating physical effects.