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Quantum oscillations can reveal Fermi surfaces and their topology in solidsand provide a powerful tool for understanding transport and electronicproperties. It is well established that the oscillation frequency maps theFermi surface area by Onsager's relation. However, the topological phaseaccumulated along the quantum orbit remains difficult to estimate incalculations, because it includes multiple contributions from the Berry phase,orbital and spin moments, and also becomes gauge-sensitive for degeneratestates. In this work, we develop a gauge-independent Wilson loop scheme toevaluate all topological phase contributions and apply it to CsTi$_3$Bi$_5$, anemerging kagome metal. We find that the spin-orbit coupling dramatically altersthe topological phase compared to the spinless case. Especially, oscillationphases of representative quantum orbits demonstrate a strong 3D signaturedespite their cylinder-like Fermi surface geometry. Our work reveals the Fermisurface topology of CsTi$_3$Bi$_5$ and paves the way for the theoreticalinvestigation of quantum oscillations in realistic materials.

Quantum oscillations with topological phases in a kagome metal CsTi$_3$Bi$_5$