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

Quantum oscillations can reveal Fermi surfaces and their topology in solids and provide a powerful tool for understanding transport and electronic properties. It is well established that the oscillation frequency maps the Fermi surface area by Onsager's relation. However, the topological phase accumulated along the quantum orbit remains difficult to estimate in calculations, because it includes multiple contributions from the Berry phase, orbital and spin moments, and also becomes gauge-sensitive for degenerate states. In this work, we develop a gauge-independent Wilson loop scheme to evaluate all topological phase contributions and apply it to CsTi$_3$Bi$_5$, an emerging kagome metal. We find that the spin-orbit coupling dramatically alters the topological phase compared to the spinless case. Especially, oscillation phases of representative quantum orbits demonstrate a strong 3D signature despite their cylinder-like Fermi surface geometry. Our work reveals the Fermi surface topology of CsTi$_3$Bi$_5$ and paves the way for the theoretical investigation of quantum oscillations in realistic materials.