Superconducting pairing symmetries in charge-ordered kagom\'e metals

We investigate the superconducting state in a kagom\'e lattice, with intertwined charge order and time-reversal symmetry-breaking loop current, using self-consistent Bogoliubov-de Gennes formalism to find the emergent pairing symmetries. Using local and nearest-neighbor attractive interactions, treated within Hartree-Fock mean-field approximation, we obtain all possible pairing symmetries in position space. Our findings indicate that the uniform $s$-wave symmetry, arising in the absence of the charge order and the loop current, modifies to a pair density wave of $s$-wave symmetry of 2$\times$2 lattice periodicity in the presence of the charge order, and a chiral pair density wave of $d_{x^2-y^2}\!+\!id_{xy}$-wave symmetry of the same 2$\times$2 periodicity in the presence of the charge order and loop current order, in both onsite and nearest-neighbor channels. In the absence of inversion symmetry, such as in the thin-film geometry, Rashba spin-orbit coupling appears, inducing an additional nearest-neighbor triplet $p_x\pm ip_y$-wave pairing. The results are relevant to superconductivity found in $A$V$_{3}$Sb$_{5}$ ($A$ = K, Rb, Cs), coexisting with a charge order that breaks time-reversal symmetry. We discuss fingerprints of these different pairing symmetries in scanning tunneling microscopy experiments.