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Time-Reversal Symmetry Breaking Superconducting State and Collective Modes in Kagome Superconductors

We comprehensively study the unconventional pairing and collective modes in the multiband kagome superconductors AV$_3$Sb$_5$ (A=$\mathrm{K},\mathrm{Cs},\mathrm{Rb}$). By solving gap equations at zero temperature, we identify a transition from normal $s++/s\pm$-wave pairing to time-reversal symmetry (TRS) breaking pairing with a variation of inter-pocket interactions or density of states. This TRS breaking pairing originates from the superconducting phase frustration of different Fermi pockets and can account for experimental TRS breaking signal in kagome superconductors. Moreover, we investigate collective modes, including the Higgs, Leggett, and Bogoloubov-Anderson-Goldstone modes, arising from fluctuations of the amplitude, relative phase, and overall phase of the superconducting order parameters, respectively. Remarkably, due to the presence of multibands, one branch of the Leggett modes becomes nearly massless near the TRS breaking transition, providing a compelling smoking-gun signature of TRS-breaking superconductivity, in clear contrast to TRS-breaking charge orders. Our results elucidate the rich superconducting physics and its associated collective modes in kagome metals, and suggest feasible experimental detection of TRS breaking pairing.