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Recent high pressure experiments discovered abnormal double-domesuperconductivities in the newly-synthesized kagome materials $A$V$_3$Sb$_5$($A$ = K, Rb, Cs), which also host abundant emergent quantum phenomena such ascharge density wave (CDW), anomalous Hall effect, nontrivial topologicalproperty, etc. In this work, by using first-principles electronic structurecalculations, we have studied the CDW state, superconductivity, and topologicalproperty in CsV$_3$Sb$_5$ under pressures ($<$ 50 GPa). Based on theelectron-phonon coupling theory, our calculated superconducting $T_\text{c}$sare consistent with the observed ones in the second superconducting dome athigh pressure, but are much higher than the measured values at low pressure.The further calculations including the Hubbard U indicate that with modestelectron-electron correlation the magnetism on the V atoms exists at lowpressure and diminishes gradually at high pressure. We thus propose that theexperimentally observed superconductivity in CsV$_3$Sb$_5$ at ambient/lowpressures may still belong to the conventional Bardeen-Cooper-Schrieffer (BCS)type but is partially suppressed by the V magnetism, while thesuperconductivity under high pressure is fully conventional without invokingthe magnetism. We also predict that there are a second weak CDW state andtopological phase transitions in CsV$_3$Sb$_5$ under pressures. Our theoreticalassertion calls for future experimental examination.

First-principles study on the double-dome superconductivities in kagome material CsV$_3$Sb$_5$ under pressure