Pressure-tuning topological phase transitions in Kagome superconductor CsTi$_3$Bi$_5$

Recently, the Kagome metal CsTi$_3$Bi$_5$ has exhibited several novel quantum properties similar to CsV$_3$Sb$_5$, such as nontrivial topology, double-dome superconductivity, and flat band features. However, CsTi$_3$Bi$_5$ lacks the charge-density wave (CDW) present in CsV$_3$Sb$_5$, making the study of its emergence of double-dome superconductivity a focus of research. In this work, we have identified an order parameter, the three-band Z$_2$ topological index, that can describe the superconducting phase diagram of CsTi$_3$Bi$_5$ under pressure. Its evolution with pressure follows the expected behavior for superconductivity. Furthermore, the results of the Fermi surface under pressure reveal the potential presence of a Lifshitz transition in the vicinity of the vanishing point of the superconducting temperature change with pressure in CsTi$_3$Bi$_5$. These results indicate that the superconducting behavior of CsTi$_3$Bi$_5$ under pressure is caused by changes in the electronic structure leading to alterations in the topological properties, provide new insights and approaches for understanding the superconducting phenomenon in Kagome metals.