Chiral pair density wave as a precursor of the pseudogap in kagom\'e superconductors

Motivated by scanning tunneling microscopy experiments on $A$V$_3$Sb$_5$ ($A$ = Cs, Rb, K) that revealed periodic real-space modulation of electronic states at low energies, I show using model calculations that a triple-{\bf 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.