Prediction of double-Weyl points in the iron-based superconductor $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$

Employing a combination of symmetry analysis, low-energy modeling, and ab initio simulations, we predict the presence of magnetic-field-induced Weyl points close to the Fermi level in $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$. Depending on the relative strengths of the magnetic field and of the spin-orbit coupling, the Weyl fermions can carry a topological charge of $\pm 1$ or $\pm 2$, making $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ a rare realization of a double-Weyl semimetal. We further predict experimental manifestations of these Weyl points, both in bulk properties, such as the anomalous Hall effect, and in surface properties, such as the emergence of prominent Fermi arcs. Because $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ displays unconventional fully gapped superconductivity below 30 K, our findings open a route to investigate the interplay between superconductivity and Weyl fermions.

Figure 1

Schematic of the symmetries and band-dispersion degeneracies of a single FeAs layer with $P4/nmm$ space group. Including spin-orbit coupling, the glide-plane symmetry enforces fourfold degenerate bands along the $M-A$ line and twofold degenerate bands along $\mathrm{\Gamma }-M$ (gray curves). Breaking of the Fe-plane inversion symmetry (IS) by making the two As atoms above and below the plane inequivalent (upper inset) lowers the degeneracies of the bands to twofold along $M-A$ and onefold along $\mathrm{\Gamma }-M$ (red curves). Application of a perpendicular magnetic field ($H_z$) in the Fe-plane inversion-symmetry broken (ISB) case makes all bands nondegenerate (green and blue curves). The lower inset shows the Brillouin zone, with $\mathrm{\Gamma }=(0,0,0), M=(\pi ,\pi ,0)$, and $A=(\pi ,\pi ,\pi )$.

Figure 2

(a) Schematic band structure of $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ along $\mathrm{\Gamma }-M$ and $M-A$. Four bands, transforming under the irreps ${\mathrm{\Sigma }}_{1-4}$, become pairwise degenerate at $M$, where they transform as ${\mathrm{M}}_5^{\pm }$. Along $M-A$, they form nodal lines (transforming as ${\mathrm{V}}_5$) which, due to Kramers degeneracy, are fourfold degenerate. (b) In the presence of SOC, the bands transform as the double-valued reducible representations ${\overline{\text{M}}}_5^{\pm }$. The two nodal lines are split into four bands that remain Kramers degenerate. (c) In the presence of a magnetic field (but no SOC), the Kramers degeneracy is split into a spin-$\uparrow$ and a spin-$\downarrow$ branch. The Zeeman splitting $\mathrm{\Delta }$ is proportional to the field $H_z$. (d), (e) In the presence of both SOC and a magnetic field, the bands become nondegenerate and an accidental intersection can occur, resulting in a Weyl point (WP). Its topological charge $\mathcal{C}$ depends on whether the crossing involves bands of the same spin [$\mathcal{C}=\pm 2$, resulting in a double WP, panel (d)] or opposite spins [$\mathcal{C}=\pm 1$, resulting in a single WP, panel (e)].

Figure 3

(a) Upper half of the crystal structure of $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ and (b) top view of four iron plaquettes. (c) DFT band structure in the presence of finite Fe magnetization and SOC calculated with the FPLO basis [50]. A double Weyl-point involving Fe $d_{x^2-y^2}$ orbitals with equal spin emerges. (d) and (e) are zooms of the regions denoted in (c), indicating where the double Weyl point appears.

Figure 4

Berry curvature flux $\mathcal{F}$ on momentum-space spheres of radius $0.002\pi /a$ centered at (a) ${\mathbf{k}}_0^{+}$ and (b) ${\mathbf{k}}_0^{-}$. The vectors have been normalized and their magnitudes are indicated by their color. Upon integration of $\mathcal{F}$, we obtain the topological charges $\mathcal{C}=\pm 2$ characteristic of double-Weyl points. (c) Energy dispersion of the two bands forming the Weyl point near ${\mathbf{k}}_0^{+}$. Although they intersect linearly along $k_z$, a quadratic touching is realized in the $(k_x,k_y)$ plane. (d) Spectral density $A\left(\mathbf{k}\right)$ calculated from DFT across the (100) surface. The Fermi arcs are indicated by white arrows. To enhance the contrast, we performed semislab and bulk calculations and subtracted the latter from the former.