Kagome superconductors AV₃Sb₅ (A = K, Rb, Cs) have garnered significant attention due to their intertwined topology, superconductivity, and charge density waves (CDWs). While the in-plane 2 × 2 CDW is well-understood, the out-of-plane structural correlation with Fermi surface properties remains less clear. Numerous quantum oscillation experiments on CsV₃Sb₅ have revealed complex oscillation frequencies, varying with Fermi energy across different samples. However, inconsistencies exist in reports regarding the nature of nontrivial Fermi pockets exhibiting a π phase shift in fundamental quantum oscillations. The prevailing belief attributes this phase shift to the π Berry phase of Dirac fermions; however, studies have shown that orbital magnetic moment and Zeeman effects also contribute. In systems with significant spin-orbital coupling (SOC), like AV₃Sb₅ (with SOC of tens of meV), separating these contributions from the Berry phase becomes challenging. Furthermore, previous calculations based on the 2 × 2 × 1 CDW model have failed to fully explain experimental observations, highlighting the need for a more comprehensive theoretical understanding. This study aims to resolve the origin and topology of observed quantum orbits in CsV₃Sb₅ by utilizing a more realistic 3D CDW model, clarifying the role of SOC and reconciling discrepancies in experimental findings.

The literature review extensively cites prior research on AV₃Sb₅ materials, highlighting the intriguing interplay of charge density waves (CDWs), potential symmetry breaking, topological band structures, and exotic superconductivity. Numerous experimental studies using quantum oscillations are reviewed, emphasizing the inconsistencies in reported nontrivial Fermi pockets and their associated π phase shifts. Existing theoretical models based on a 2 × 2 × 1 CDW structure are shown to be inadequate in fully explaining the observed experimental data. The limitations of solely attributing the π phase shift to the Berry phase of Dirac fermions are discussed, considering the contributions of orbital magnetic moment and Zeeman effect, particularly in the presence of spin-orbit coupling. The need for a more accurate 3D CDW model and a more nuanced understanding of the topological origins of the observed quantum orbits is emphasized, setting the stage for the current research.

The study employs density-functional theory (DFT) calculations using the Vienna ab initio Simulation Package (VASP) to investigate the electronic structure of CsV₃Sb₅. Three different hexagonal CDW structures (2 × 2 × 1 SD, 2 × 2 × 1 ISD, and 2 × 2 × 2 CDW) are considered, with the 2 × 2 × 2 CDW model, incorporating alternating SD and ISD layers, ultimately chosen for its better agreement with experimental data. Spin-orbit coupling (SOC) is included in the electronic structure calculations. High-resolution Fermi surfaces are calculated using Wannier functions. Extremal quantum orbits are determined by analyzing the changes in Fermi surface slices along the k_z direction. The cyclotron frequency is calculated using Onsager's relation, and the effective mass is calculated from the derivative of the extremal orbit area with respect to energy. The topological properties of the quantum orbits are investigated using the Wilson loop method to calculate the Berry phase, both with and without SOC. The generalized Berry phase, incorporating Berry, orbital, and Zeeman phases, is calculated and used to classify the quantum orbits into four types based on their topological origins. The analysis accounts for the double degeneracy of quantum orbits due to inversion and time-reversal symmetries. Detailed calculations of the various phases contributing to the quantum oscillation, including the de Broglie phase, Aharonov-Bohm phase, Maslov correction, Berry phase, orbital phase, and Zeeman phase are provided in the supplementary materials.

The 2 × 2 × 2 CDW model accurately reproduces the experimentally observed quantum oscillation frequencies and cyclotron masses. The study reveals a hidden Dirac nodal network protected by mirror symmetries in the 3D CDW phase, which is weakly gapped by SOC. The nontrivial quantum orbits are classified into four types based on their topological origins: Type-I orbits driven by Dirac points; Type-II, III, and IV orbits significantly influenced by SOC and/or Zeeman effects. The calculations show that SOC and/or the Zeeman effect can induce a π phase shift in orbits that would otherwise be trivial in the absence of SOC. The analysis distinguishes between quantum orbits originating from SD, ISD, or mixed SD+ISD layers, revealing a significant three-dimensional character for many of the smaller orbits. The study identifies sixteen nontrivial quantum orbits at two representative Fermi energies (-40 meV and -85 meV), with over half of these exhibiting a mixed SD and ISD layer contribution. The calculated frequencies, cyclotron masses, and topological properties of these orbits show good agreement with experimental observations, except for extremely high-frequency orbits potentially resulting from magnetic breakdown. The existence of Chern Fermi pockets at the M points, as predicted by a previous single-orbital model, is also confirmed. The 2 × 2 × 2 CDW model emerges as a minimal model to capture the interlayer coupling and reproduce the experimental results, suggesting stronger interlayer interaction than previously considered.

The findings address the long-standing puzzle surrounding the origin and topological properties of quantum orbits in CsV₃Sb₅. The successful reproduction of experimental data using the 2 × 2 × 2 CDW model, considering the interlayer structural modulation and the impact of SOC, provides strong evidence for the three-dimensional nature of the CDW and its influence on the Fermi surface. The classification of nontrivial quantum orbits into four types based on their topological origins offers a refined understanding of the underlying physics. The discovery of a hidden Dirac nodal network reveals a richer topological landscape than previously anticipated and highlights the critical role of SOC in shaping the quantum oscillation behavior. The agreement between calculated and experimentally observed frequencies, cyclotron masses, and topological properties validates the model and provides a framework for understanding similar systems. The study's focus on disentangling the contributions of Berry phase, orbital moment, and Zeeman effect advances the understanding of quantum oscillations in layered materials.

This research provides a comprehensive theoretical explanation for the experimentally observed quantum oscillations in CsV₃Sb₅. The 2 × 2 × 2 CDW model accurately predicts the frequencies, masses, and topological nature of the observed quantum orbits. The identification of a hidden Dirac nodal network and the classification of nontrivial orbits based on their topological origins contribute significantly to the understanding of topological phenomena in kagome materials. Future research could explore more complex CDW models to further refine the understanding and investigate the role of these topological features in the superconducting state. The methodology developed in this study can be applied to analyze similar systems.

The study primarily focuses on the 2 × 2 × 2 CDW model, and while it provides excellent agreement with a wide range of experimental data, more complex CDW structures might provide even more accurate descriptions. The analysis primarily focuses on the magnetic field parallel to the c-axis, and further studies could extend the analysis to other field orientations. While the study accurately models many high-frequency oscillations, some of the highest frequencies might be caused by magnetic breakdown, which is not explicitly modeled. Finally, the experimental uncertainties in Fermi energy could influence the quantitative comparison between theory and experiments.