We investigate the electronic structure and topological properties of iron-based superconductors ${\mathrm{LaFe}}_2{\mathrm{As}}_2$ using density functional theory plus dynamical mean-field theory. We find that the uncollapsed tetragonal ${\mathrm{LaFe}}_2{\mathrm{As}}_2$ is in a nontrivial $Z_2$ topological phase and has topological Dirac surface states near the Fermi energy which suggests there could be Majorana zero modes in the superconducting ${\mathrm{LaFe}}_2{\mathrm{As}}_2$. In light of the nontrivial topological properties and superconductivity of ${\mathrm{LaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$, we predict a new iron-based compound ${\mathrm{LaBaFe}}_4{\mathrm{As}}_4$ and find it possesses two sets of topological Dirac surface states near the Fermi energy despite of a trivial $Z_2$ topological index. These topological surface states are induced by a nontrivial high-order topological index $Z_8$, a new mechanism that is distinct from all-known iron-based superconductors. Our study not only demonstrates that both ${\mathrm{LaBaFe}}_4{\mathrm{As}}_4$ and uncollapsed tetragonal ${\mathrm{LaFe}}_2{\mathrm{As}}_2$ can be good platforms for exploring topological superconductivity but also paves a new way to realize it with a nontrivial high-order topological index.

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• First-principles calculations
• Topological materials
• Iron-based superconductors
• Strongly correlated systems
• Topological superconductors
• Density functional theory
• Dynamical mean field theory
• Electronic structure
• Wannier function methods