Impact of the orbital current order on the superconducting properties of the kagome superconductors

Motivated by recent experimental evidence signaling chiral charge order in vanadium-based kagome superconductors, we investigate theoretically the impact of chiral flux charge order on the experimental outcomes for normal and superconducting (SC) properties. It is revealed that the spectral weight on the Fermi surface (FS) is partially gapped by chiral flux charge order, with the reservation of the spectral weight on the $M$ points and the midpoint between the two adjacent $M$ points, resulting in the momentum-dependent energy gap being consistent with recent experimental observations. More importantly, by considering the influence of chiral flux charge order, we find that a conventional fully gapped SC pairing state evolves into a nodal gap feature for the spectral weight due to the spectral gap modulations on the FS. As a result, the U-shaped density of states (DOS) deforms to the V-shaped DOS along with the residual DOS near the Fermi energy. These results bear some resemblance to the experimental observations, and they may serve as a promising proposal to mediate the divergent or seemingly contradictory experimental outcomes regarding SC pairing symmetry.

Figure 1

(a) The lattice structure of the kagome superconductors, made out of three sublattices $A$ (green dots), $B$ (red dots), and $C$ (blue dots). The arrows on the lattice depict the configuration of the orbital current, and the dashed lines denote the enlarged unit cell. (b) Fermi surface produced by the spectral weight distribution $A^0(\mathbf{k},E)$ at zero energy $E=0$. (c) The spectral weight distribution $A(\mathbf{k},E)$ at zero energy $E=0$ obtained in the $2\times 2$ chiral flux phase with $\lambda =0.1$. The hexagonal area in (c) surrounded by the white dashed lines shows the reduced Brillouin zone in the $2\times 2$ chiral flux phase. (d) The dispersion of the chiral flux phase along high-symmetry cuts in the reduced Brillouin zone for a typical value of $\lambda =0.1$.

Figure 2

The unfolded spectral weight distribution $A(\mathbf{k},E)$ at $E=0$ in the primitive Brillouin zone for $\lambda =0.02$ (a) and $\lambda =0.1$ (b), respectively. The dashed orange arrow in (a) indicates the momentum cut from the $(\pi ,-\sqrt{3}\pi /3)$ to the $(\pi ,\sqrt{3}\pi /3)$ direction. (c) The momentum distribution curves of the spectral weight along the momentum cut shown by the dashed orange arrow in (a) for different $\lambda$. The dashed line in (c) portrays one-third of the intensity of $\lambda =0$. (d) Evolution of the CDW gap along the momentum cut shown by the dashed orange arrow in (a) for different $\lambda$.

Figure 3

(a) The unfolded dispersion of the chiral flux phase along high-symmetry cuts in the primitive Brillouin zone for a typical value of $\lambda =0.1$. (b) The folded energy band dispersion of the chiral flux phase along high-symmetry cuts in the primitive Brillouin zone for a typical value of $\lambda =0.1$. (c) Density of states in a wide energy range for the normal state (the red dotted curve) and the chiral flux state with $\lambda =0.1$ (the black solid curve). The dashed line is the Fermi level corresponding to the van Hove filling. (d) Density of states in the small energy scale for $\lambda =0.05$ and 0.1 in the chiral flux phase.

Figure 4

The intensity plots of the unfolded spectral function as functions of the momentum and energy in the coexistent state for different strength of $\lambda =0$ (a), $\lambda =0.01$ (b), $\lambda =0.02$ (c), $\lambda =0.03$ (d), and $\lambda =0.04$ (e), respectively. The momentum in each panel is along the cut from the $(\pi ,-\sqrt{3}\pi /3)$ to the $(\pi ,\sqrt{3}\pi /3)$ direction, as denoted by the dashed orange arrow in Fig. 2. (f) Evolution of the gap along the Fermi surface as a function of ${\theta }_{k_F}$ for different strength of $\lambda$. The gap is extracted from the peak positions of the spectral function in panels (a)–(e).

Figure 5

(a)–(e) The energy dependence of the DOS for different strength of $\lambda$. (f) Evolution of the SC pairing $\mathrm{\Delta }$ as a function of $\lambda$.

Figure 6

The folded (left column) and unfolded (right column) band structures of the chiral flux charge order phase along high-symmetry cuts in the primitive Brillouin zone for $\lambda =0.05$ (a), (b), $\lambda =0.1$ (c), (d), $\lambda =0.2$ (e), (f), and $\lambda =0.3$ (g), (h), respectively.