Chiral pair density wave as a precursor of the pseudogap in kagome superconductors

Motivated by scanning tunneling microscopy experiments on ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ $(A=\mathrm{Cs}, \mathrm{Rb}, \mathrm{K})$ that revealed periodic real-space modulation of electronic states at low energies, I show using model calculations that a triple-Q chiral pair density wave (CPDW) is generated in the superconducting state by a charge order of $2a\times 2a$ superlattice periodicity, intertwined with a time-reversal symmetry breaking orbital loop current. In the presence of such a charge order and orbital loop current, the superconducting critical field is enhanced beyond the Chandrasekhar-Clogston limit. The CPDW correlation survives even when the long-range superconducting phase coherence is diminished by a magnetic field or temperature, stabilizing an exotic granular superconducting state above and in the vicinity of the superconducting transition. The presented results suggest that the CPDW can be regarded as the origin of the pseudogap observed near the superconducting transition.

Figure 1

Charge order configuration with intertwined orbital loop current on the kagome lattice, analogous to the trihexagonal pattern observed in experiments. The yellow and cyan colors represent the modulation in the chemical potential ${\mu }_{\mathrm{co}}$, while the arrows represent the loop current propagation direction and the associated flux $\phi$.

Figure 2

(a) Profile of the pairing gap $\mathrm{\Delta }\left({\mathbf{r}}_i\right)={\mathrm{\Delta }}_m{e}^{i{\theta }_i}$ solution on the considered $10a\times 10a$ lattice with periodic boundary conditions—the color scale shows the real part; the arrows show the phase ${\theta }_i$. (b) Fourier transform of the pair-pair correlation function $C\left(\mathbf{Q}\right)$ obtained on a $20a\times 20a$ lattice with periodic boundary conditions, showing the three characteristic momenta (six-peak structure), indicative of the CPDW state. The hexagon plotted with dashed lines depicts the Brillouin zone. Parameters used are ${\mu }_{\mathrm{co}}=0.5$ and $t_{\mathrm{lc}}=1$.

Figure 3

(a), (b) Variation of the average gap magnitude $\left|\mathrm{\Delta }\right|$ (without and with the charge order and the loop current) and superfluid density $n_s$ with magnetic field $B_z$ and temperature $T$. (c), (d) Density of states $\rho \left(E\right)$ for different $B_z$ and $T$ near the critical values $B_c$ and $T_c$, determined by vanishing $n_s$. Insets in (c) and (d) show the density of states at zero energy $\rho \left(0\right)$ as a function of $B_z$ and $T$, respectively. The results were obtained on a $20a\times 20a$ lattice with periodic boundary conditions. All other parameters are the same as in Fig. 2. A constant offset has been added to the vertical axis for each curve in (c) and (d) for clarity.

Figure 4

Profile of the absolute value of the pairing gap $\left|\mathrm{\Delta }\right({\mathbf{r}}_i\left)\right|$, self-consistently obtained on a $8a\times 8a$ lattice with periodic boundary conditions, at different magnetic fields across the superconducting transition: (a) $B_z=0$, (b) $B_z=1.1B_c$, and (c) $B_z=1.5B_c$. In the vicinity of the superconducting transition [the case of plot (b)], the pairing gap is locally suppressed in the charge-ordered clusters, creating a granular phase having superconducting patches separated by nonsuperconducting regions. All other parameters are the same as in Fig. 2.

Figure 5

Fourier transformed pair-pair correlation function $C\left(\mathbf{Q}\right)$ at (a) magnetic field $B_z=1.1B_c$, temperature $T=0$, and (b) $B_z=0$, $T=1.03T_c$, indicating the presence of the CPDW correlation above the superconducting transition. (c), (d) Fourier-transformed local density of states $\rho \left({\mathbf{Q}}_p,E\right)$ at a characteristic CPDW momentum ${\mathbf{Q}}_p$, shown in (a), at different values of $B$ and $T$ across the superconducting transition, revealing particle-hole symmetric quasiparticle states with a nonzero weight at $E=0$. The results were obtained on a $20a\times 20a$ lattice with periodic boundary conditions. All other parameters are the same as in Fig. 2. A constant offset has been added to the vertical axis for each curve in (c) and (d) for clarity.