Figure 1a shows the magnetization behavior of high-quality single crystal CsTi3Bi5 under the magnetic field up to 7 T, with the field applied along the c-axis (*θ* = 0°). In the low field regime, a distinct superconducting diamagnetic signal is observed, while pronounced oscillations become noticeable at field strengths above ~ 3 T. The presence of smooth and intricate oscillations at such a low magnetic field reveals the sample's purity and homogeneity, further attesting to the high quality of the samples 29. Figure 1b presents the polynomial background subtracted oscillations at different temperatures. The magnetization exhibits a well-defined periodicity with respect to 1/*B*, which is characteristic of the dHvA oscillation. The amplitude of the oscillation gradually diminishes with increasing temperature and disappears completely around 12 K. The oscillation apparently contains multiple frequencies, implying a complex Fermi surface structure.

Here we use discrete-time Fourier transform to refine the spectrum in order to obtain more precise frequencies (*F*) and amplitude values. As shown in Fig. 1c, the Fourier transform spectrum of the oscillations in Fig. 1b displays six fundamental frequencies and one harmonic frequency, denoted by *F*α = 217 T, *F*β = 281 T, *F*γ = 429 T, *F*δ = 498 T, *F*ζ = 594 T, *F*η = 1013 T and *F*2β = 562 T. Note that there are several small peaks existing in Fig. 1c, such as the lowest frequency (*F* = 12 T) which is most likely caused by extrinsic effects. The frequency of quantum oscillations can be described by the Onsager’s relation27 \(F =(\hslash /2\pi e){A}_{F}\), where *A*F is the extremal cross-sectional area in Fermi surface perpendicular to magnetic field direction. Therefore, the presence of seven frequencies implies that there are seven Fermi wave vectors corresponding to each frequency and that CsTi3Bi5 has complex Fermi surfaces with several pockets. In addition, the magnetization oscillation amplitude could be explained by the Lifshitz-Kosevich (LK) formula34,35 with Berry phase being taken into account:

$$\varDelta M \propto -\sqrt{B}\sum _{i=1}^{n}\sum _{r=1}^{\infty }{r}^{-3/2} {R}_{T}^{i}{R}_{D}^{i}\text{sin}\left[2\pi r\left(\frac{{F}^{i}}{B}-\frac{1}{2}+{\phi }_{B}^{i}\right)\right]$$

The magnetization oscillation is proportional to a linear superposition of sine functions from different orbitals (\(i\)) and their harmonics (*r*). The phase factor \({\phi }_{B}\) stands for Berry phase factor, which is close to 0.5 for a topologically nontrivial band, and 0 for a trivial band36. In general, for 3D electronic structures, the phase term needs to be corrected by adding\(\delta =\pm 1/8\) for maximum and minimum cross area. The decay term \({R}_{T}= \left(\alpha {m}^{*}T/{\mu }_{0}B\right)/\text{sinh}\left(\alpha {m}^{*}T/{\mu }_{0}B\right)\)and \({R}_{D}=\) \(\text{e}\text{x}\text{p}\left(-\alpha {m}^{*}{T}_{D}/{\mu }_{0}B\right)\) come from temperature and scattering. Here,\(\alpha \approx 14.69 T/K\) is a constant. \({m}^{*}\) is the cyclotron effective mass in unit of free electron mass *m**e*. \({T}_{D}= \hslash /2\pi {k}_{B}{\tau }_{q}\), where \({\tau }_{q}\) is quantum scattering time.

Therefore, the effective mass *m** and the quantum scattering time can be obtained by fitting the temperature dependence of the amplitude of each oscillation frequency. Since there are multiple frequencies in the CsTi3Bi5 oscillation data, it is difficult to directly determine the amplitude corresponding to each frequency. To estimate *m**, a common approach is taking the Fourier transform amplitude instead of the original amplitude, and taking the harmonic average magnetic field within Fourier transform window. The best fitting of effective mass for *F*α, *F*β and *F*ζ is shown in Fig. 2a. The dependence of oscillation amplitude on magnetic field is obtained by shifting Fourier transform intervals, and thus \({\tau }_{q}\) is fitted as shown in Fig. 2b. The mean free path \({l}_{q}\)and the quantum mobility \({\mu }_{q}\) are obtained thereafter. The estimated values of Fermi surface parameters for each orbit are presented in Table I. The determined *m** of CsTi3Bi5 are small and comparable with that reported in CsV3Sb522–27, which are usually correlated to Dirac modes. Unlike *m**, the mean free path is obviously larger than that in CsV3Sb522–27, which makes a quite high quantum mobility. \({\mu }_{q}\) is even ten times larger than the classical mobility \({\mu }_{c}\) from Hall effect37. This difference in the two mobilities is common since they represent different scattering processes38.

Some of these Fermi surfaces have nontrivial topological properties based on theoretical calculations and previous ARPES experiments31–33. To investigate the topological nature, Landau level fan diagrams are constructed for *F*α and *F*β. After using filters to isolate a given frequency, Landau levels are labeled corresponding to half integer levels for maxima, and integer filling at minima. Frequencies obtained from the slope of the Landau level fan diagrams are in good agreement with the frequencies obtained from the Fourier transform, as shown in Fig. 2c, indicating the signals are well preserved when filters are applied. A nonzero intercept of 0.6 is revealed by linear extrapolation to zero in 1/*B* for *F*β, close to the predicted \({\phi }_{B}\) ~ 0.5 for topological orbits, which implies a Berry phase close to π.

The oscillation data was also fitted by the full LK formula in order to double check the non-trivial topological features of the system. It is convenient to set only the frequency and phase factor as fitting arguments based on the parameters obtained from the previous fits. And since the amplitude of the low frequency components is much larger than the high frequency components, only *F*α and *F*β and their harmonic components were considered in the fit in order to reduce the uncertainty caused by more parameters. As shown in Fig. 2d, the fitting results are very close to the experimental data, illustrating the reasonableness of the fitting parameters and the procedure. The phase factor corresponding to the best fit is 0.02 and 0.41 for *F*α and *F*β, which is consistent with the Landau index analysis above (Fig. 2c).

To further investigate the anisotropic electronic structure of CsTi3Bi5, we measured the magnetization in selected angles *θ* and applied Fourier transform analysis, as shown in Fig. 3a and 3b. The amplitude of oscillations is decreased with increasing *θ*, and there is no oscillation observed above *θ* = 60°, which is a character of cylindric Fermi surface. The frequencies move higher along with *θ*. The angle dependence *F*(*θ*) can be well described by a cylindric Fermi surface with *F*0/cos(*θ*), which is consistent with theoretical prediction.

Subsequently, the numeric calculations of the extremal cross area on the Fermi surface based on density functional theory (DFT) were carried out to further understand the Fermiology getting from dHvA oscillation experiments. All Fermi surface pockets display clear 2D features. The Г-centered circular pocket (colored in red) originates from Bi-*p*z orbits, and the remaining ones are mainly contributed by Ti-*d* orbits. Features of the Fermi surface in Fig. 4a have already been experimentally confirmed by quasiparticle interference and ARPES results29,32. The quantum oscillation frequencies are directly proportional to AF as expressed in the equation of Onsager's relation. In Fig. 4c of side views of Fermi surface, the cross-sectional area of these Fermi surface pockets reaches maximum or minimum at *k*c=0 or *k*c = π/c slices, respectively. This means that all extremal orbits are located in either *k*c=0 or π/c slices when *B* is along the c-axis. We concentrate on those closed orbits in *k*c=0 and π/c slices with corresponding oscillation frequencies less than the experimental measurement limit of about 1400 T. Six orbitals with oscillation frequencies in the range from 367–723 T at Fermi level *E*F,exp determined by the ARPES experiment32 are identified and labeled as *F*1-*F*6 in Fig. 4b. A well-defined Fermi level is necessary to determine the orbits accurately. Therefore, an exhaustive search for all oscillation frequencies is performed within a range of *E*-*E*F,exp = ± 0.15eV as plotted in Fig. 4e. Experimental oscillation frequencies *F*α−η show the qualitative consistent with DFT calculations and each of them intersects several orbits (Fig. 4e). When taking the experimental values *E* = *E*F,exp, three frequencies are close to the calculated values. Different from CsV3Sb5 with CDW, both experimental measurements and phonon spectrum calculations show the absence of CDW order and Fermi surface distortion in CsTi3Bi526. Though the number of peaks is less than that in CsV3Sb5, which may come from interlayer coupling23, there is no singular choice of *E*F that can match all experimental frequencies. The quantitatively numerical disagreements between quantum oscillation frequencies and DFT-derived results may imply the complexity of electronic structures of CsTi3Bi5. Moreover, given its high structural symmetry and strong spin-orbit coupling (SOC), this kagome metal is anticipated to exhibit abundant topological properties in its electronic structures. The measured Berry phase of *β* orbit close to π in this work also provides clear evidence experimentally for the non-trivial topology of CsTi3Bi5.

In conclusion, through dHvA oscillation analysis, we have identified multiple frequencies with small effective masses in the CsTi3Bi5 system. Several parameters related to the Fermi surface of CsTi3Bi5 were obtained by fitting the oscillation pattern with the LK formula. The effective masses of CsTi3Bi5 are in the range of 0.27 ~ 0.54 *m*0, which are comparable to that of CsV3Sb5. The angle-dependent oscillations imply a quasi-2D electronic structure. In addition, we find the frequency *F*β is related to a non-zero Berry phase, which implies the nontrivial topological properties in CsTi3Bi5. Hence, the discovery of these properties in CsTi3Bi5 establishes it as a promising avenue for exploring the interplay between different ordered states in topological materials.