The coexistence of unconventional high-temperature superconductivity, topological surface states, and strong electronic correlations makes iron-based superconductors a prominent platform for exploring topological superconductivity and Majorana zero modes. Here, we design a novel intergrowth-structured compound, ${\mathrm{LaNa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4$ (1244-type), by combining uncollapsed tetragonal ${\mathrm{LaFe}}_2{\mathrm{As}}_2$ and NaFeAs. This unique intergrowth structure provides a new route to engineer topological states in iron-based superconductors. Using density functional theory combined with dynamical mean-field theory calculations, we reveal nontrivial band topologies near the Fermi level ($E_F$) in ${\mathrm{LaNa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4$, hosting both topological insulator (TI) and topological Dirac semimetal states, and giving rise to two sets of Dirac-cone-type surface states on the (001) surface. Strong electronic correlations renormalize the band structure, bringing the topological surface states closer to $E_F$. Moreover, slight electron doping at the Fe sites can further tune the TI surface states to $E_F$, enhancing their experimental accessibility. The intrinsic self-doping and multiband Fermi surface of ${\mathrm{LaNa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4$ also suggest potential superconductivity. These findings establish ${\mathrm{LaNa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4$ as a promising candidate for studying topological superconductivity and Majorana zero modes, calling for experimental validation and further exploration of its potential in quantum applications. They also underscore the effectiveness of the intergrowth strategy in optimizing the topological properties of iron-based superconductors.

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• Surface states
• Topological phases of matter
• Iron-based superconductors
• Topological superconductors
• Density functional theory
• Dynamical mean field theory