Emergent topological quantum orbits in the charge density wave phase of kagome metal CsV3Sb5

The recently discovered kagome materials AV3Sb5 (A = K, Rb, Cs) attract intense research interest in intertwined topology, superconductivity, and charge density waves (CDW). Although the in-plane 2 × 2 CDW is well studied, its out-of-plane structural correlation with the Fermi surface properties is less understood. In this work, we advance the theoretical description of quantum oscillations and investigate the Fermi surface properties in the three-dimensional CDW phase of CsV3Sb5. We derived Fermi-energy-resolved and layer-resolved quantum orbits that agree quantitatively with recent experiments in the fundamental frequency, cyclotron mass, and topology. We reveal a complex Dirac nodal network that would lead to a π Berry phase of a quantum orbit in the spinless case. However, the phase shift of topological quantum orbits is contributed by the orbital moment and Zeeman effect besides the Berry phase in the presence of spin-orbital coupling (SOC). Therefore, we can observe topological quantum orbits with a π phase shift in otherwise trivial orbits without SOC, contrary to common perception. Our work reveals the rich topological nature of kagome materials and paves a path to resolve different topological origins of quantum orbits.

It is commonly believed that the measured π phase shift in a quantum oscillation originates from the π Berry phase (φ B ) of Dirac fermion in the system 36,37 .For example, graphene exhibits the π Berry phase in quantum oscillation 38 .However, as pointed out by Alexandradinata et al. 39,40 , the π phase shift in the quantum oscillation has broader origins than the π Berry phase of a Dirac fermion because of the orbital magnetic moment and Zeeman effect.The success of manifesting the Berry phase of graphene lies in the negligible spin-orbital coupling (SOC) 41,42 , resulting in a zero orbital magnetic moment-induced phase φ R and a small Zeeman effect-induced phase φ Z as a spin reduction factor in the quantum oscillation intensity.In spinful systems (the SOC of AV 3 Sb 5 is about dozens of meV), the non-zero φ R and φ Z due to spin-momentum locking sensitive to the Fermi energy cannot be separated from φ B in the total phase shift.As a result, the SOC-driven band anti-crossing can generate quantum orbits with a π phase shift in a proper energy window.Thus, the origin of observed non-trivial Fermi pockets of CsV 3 Sb 5 may deviate from the vaguely argued π-Berry phase mechanism, which remains an open question.Furthermore, the dissimilar frequencies of non-trivial Fermi pockets of CsV 3 Sb 5 remain to be reconciled.
Besides the discrepancy in the topological nature of Fermi pockets, another intriguing puzzle is their structural origin.Previous calculations based on the 2×2×1 CDW model of CsV 3 Sb 5 cannot rationalize their respective quantum oscillation experiments 28,29 .The 2×2×1 CDW, i.e., the star of David (SD) and inverse star of David (ISD) 4 structures, have been commonly adopted to understand various experiments and conceive theoretical models.Recent experiments suggest two-fold or four-fold interlayer modulation with alternative SD and ISD layers 6,28, [43][44][45][46][47] .
The interlayer structural modulation-induced FS reconstruction, which may help to understand the emergence of novel density wave orders and superconductivity at lower temperatures 48 , remains elusive.
In this work, we resolved the origin and topology of the observed quantum orbits in CsV 3 Sb 5 from theory for the first time.We derived Fermi-energy-resolved and layer-resolved quantum orbits that agree quantitatively with experiments in the cyclotron frequency and non-trivial phase shift.
According to the layer contribution, we classified all quantum orbits into three groups, i.e., SD, ISD, and mixed (SD+ISD) group, where most small quantum orbits show a clear 3D nature.According to the topological origin, the non-trivial quantum orbits are classified into four types, with the topology of one type originating from the π Berry phase and the other three originating from the SOC jointly with the Zeeman effect.Most importantly, the SOC and/or Zeeman effect can lead to a trivial quantum orbit (zero phase shift in the quantum oscillation), even though the orbit has a π Berry phase in the spinless case.A hidden Dirac nodal network protected by mirror symmetries in the 3D CDW phase is revealed, which is weakly gapped in the presence of SOC 49,50 .We not only resolve the topology of quantum orbits but also reveal the appreciable 3D nature of Fermi surfaces in CsV 3 Sb 5 .

II. RESULTS AND DISCUSSIONS
We consider three hexagonal CDW structures with the P 6/mmm space group symmetry, i.e., the 2×2×1 SD and ISD and the 2×2×2 CDW.The energetic stability of interlayer stacking has been theoretically studied 4,7,9,28,51 .We adopt the 2×2×2 CDW alternating SD and ISD layers without interlayer lateral shift as suggested by recent APRES measurements 46,47 , shown in Fig. 1(a).As we will see, the 2×2×2 model captures the essential interlayer interaction which also occurs in more complicated structures like the 2×2×4 CDW 28 .
The band structure calculated by density-functional theory (DFT) is shown for the 2×2×2 CDW in Fig. 1(b).By layer resolution, bands and FSs are well separated into SD, ISD, and mixed SD+ISD types.The band structure can be regarded as overlapping SD and ISD phases with considerable hybridization (see Supplementary Fig. S1).Because the chemical potential in experiments 11,22,23,28,52 is usually lower than the theoretical charge neutral point (energy zero), we will focus on the Fermi energy window of −100 to 0 meV and investigate corresponding FS properties.We show two typical Fermi energies at -40 and -85 meV in Fig. 1(c)-(j) to demonstrate the layer-and energy-dependence of multiple quantum orbits.
We obtain extremal quantum orbits by extracting the maximal and minimal FS area along the k z axis for the 2×2×2 CDW and show their Fermi energy dependence in Fig. 2(a)&(c).Most extremal orbits distribute in high-symmetry planes of k z = 0/0.5 (in unit of 2π/c, c is the out-of-plane lattice parameter) while some special cases appear at a generic k z (see an example at -40 meV in Table I).We will identify the proper Fermi energy by comparing calculated cyclotron frequencies with experimental values.

2(b)&(d).
Most experimental results can be divided into three groups, i.e., the low-frequency region 0∼250 T, medium-frequency region 550∼950 T, and high-frequency region ≥ 1150 T.Many high frequencies may be induced by the magnetic breakdown by merging neighboring FSs 40 .Thus, we will focus on cyclotron frequencies no larger than 1800 T.There are 4∼6 cyclotron frequencies in the low-frequency region, 2∼6 frequencies in the medium-frequency region, and 1∼6 frequencies in the frequency region of 1150∼1800 T.
Considering both the number and distribution of frequencies in three regions, we find theoretical results at energy ranges of [−50, −35] meV and [−90, −75] meV match well with most experiments.One may also find results in the range of [−20, 0] meV match experiments at medium-and high-frequency regions.However, the agreement in the low-frequency region in this energy range is poor [see Fig. 2(c)-(d)].In the following, we will analyze two representative Fermi energies, −40 and −85 meV, as shown in Fig. 1 and Table I.In detail, the 2×2×2 CDW shows 8 frequencies in the low-frequency region and 7 frequencies in the medium-frequency region at both −40 and −85 meV.In the high-frequency region, it shows 1 frequency at −40 meV and 2 frequencies at −85 meV.In general, theoretical results in low-and medium-frequency regions agree better with experiments than the high-frequency region, as shown in Fig. 2. The highest frequencies at −40 or −85 meV correspond to nearly isotropic FSs centered at Γ (see Fig. 1), which originates in Sb-p z bands.Corresponding FSs are very close to several neighboring pockets.Thus, magnetic breakdown may occur to generate even larger frequencies in experiments, rationalizing the magnetic breakdown observed in Ref. 34.
In contrast, we cannot find a satisfactory agreement between experiments and the 2×2×1 CDW (SD or ISD) simultaneously in all three frequency regions.The quantitative comparison similar to Fig. 2 is shown in Supplementary Fig. S2.Our results are consistent with Refs.28 and 29 where the SD or ISD could not match all frequencies in their respective experiments.
Furthermore, we compare with experiments the cyclotron mass m * (in unit of bare electron mass m 0 ) of quantum orbits in Table I and Fig. 2. In the low-frequency region in Fig. 2(d), reported masses are rather diverse in literature.Several experiments found m * = 0.03∼0.14while Shrestha et al. 31 and Broyles et al. 35 found m * = 0.16∼0.40,[sets F and K in Fig. 2(d)].Our calculations agree better with Shrestha et al. 31 and Broyles et al. 35 (except the large value 2.90 for the observed frequency 221 T) in the low-frequency region.In the medium-and high-frequency regions, experiments 29,31,34,35 found effective masses typical from 0.3 to 1.7.In general, our calculated m * at −40 and −85 meV are within experimental ranges, by considering the uncertainty of Fermi energy in experiments.Therefore, we conclude that our calculations considering the interlayer structural modulation agree with recent quantum oscillation experiments in both fundamental frequency and cyclotron mass.
These quantum orbits exhibit interesting interlayer hybridization, as shown in Fig. 1(c)-(f) and Table I.Approximately half of FSs at -40 and -85 meV are solely contributed by either SD or ISD layer and the rest show a strongly mixed character of two layers, labeled as mixed in Table I.There are two ways to form a mixed SD+ISD FS.In one way, part of the Fermi contour is contributed by the SD layer and the rest by ISD with weak hybridization between two layers.This happens mostly in the k z = 0.5 plane, for example, for orbits of 13 and 30 T at −40 meV.The other way involves strong hybridization between two layers and the Fermi contour has comparable contributions from both layers almost everywhere.This happens mostly in the k z = 0 plane, which is the case for most mixed quantum orbits at k z = 0. Therefore, the interlayer structural modulation is crucial in determining the nature of FSs for the CsV 3 Sb 5 CDW phase.Additionally, one can find by comparing the k z = 0 and 0.5 planes that many large FS pockets are quasi-2D while many small FSs are 3D in geometry.
Next, we discuss the topological properties of these quantum orbits that are debated in experiments.In the following, the sum of the Berry phase, orbital and Zeeman phases under SOC is dubbed the generalized Berry phase λ, i.e., λ = φ B + φ R + φ Z .According to previous discussions 39 , symmetries of quantum orbits introduce additional constraints on λ.For CsV 3 Sb 5 with both inversion and time-reversal symmetries, all bands are doubly degenerate with opposite generalized Berry phases ±λ (reduced to -π ∼ π) in the spinful case.The superposition of the doublets leads to a 2|cos(λ)| scaling of the quantum oscillation intensity, and the λ itself is no longer the total phase shift of the quantum oscillation.Instead, the quantum oscillation always exhibits a quantized phase shift ∆φ (see the method section and Supplementary Note 1-3).Here, ∆φ is 0 or π, depending on whether the reduced |λ| is smaller/larger than 0.5π.According to the origin of λ, non-trivial quantum orbits characterized by ∆φ = π (or |λ| > 0.5π) in CsV 3 Sb 5 can be classified into four types.Type-I refers to the Dirac-point-driven quantum orbit exhibiting a robust φ B = π and small φ R and φ Z .Type-II shows strong SOC-induced large φ B +φ R with a small φ Z .In type-III, the φ Z is dominant over the φ B +φ R .In type-IV, both the SOC-induced φ B +φ R and φ Z are crucial.We point out that strong SOC and/or Zeeman effect can also suppress the λ heavily (|λ| < 0.5π) for some quantum orbits with a robust φ B of π.  ) is in unit of T. kz refers to the kz plane (in unit of 2π/c, c is the lattice constant) the corresponding quantum orbit located in.The wavy-underlined kz indicates a minimal Fermi surface cross-section and the others correspond to maximal ones.The maximal/minimal orbit provides helpful information to determine the sign of the extra π/4 phase shift when analyzing quantum oscillations of a three-dimensional material.The cyclotron mass m * is in unit of bare electron mass m0, where positive and negative values are for electron and hole pockets, respectively.The origin of quantum orbits (i.e., layer-contribution) is indicated by SD, ISD and mixed (SD+ISD), as obtained from Fig. 1(b)-(f).The Berry phase φB of the quantum orbit is calculated without SOC and the generalized Berry phase λ is calculated with SOC (reduced to the range of -π to π).All quantum orbits are doubly degenerate and have λ of the same magnitude but reversed signs (only the positive set is listed).Depending on |λ| being either smaller or larger than 0.5π, the final phase shift (∆φ) in quantum oscillation contributed by two degenerate quantum orbits is either 0 or π.The four types of non-trivial quantum orbits are defined in the main text and classified according to Table S1  We first calculate the accumulated Berry phase along the quantum orbit in the 2×2×2 CDW phase using the Wilson loop method [53][54][55] without including SOC.To demonstrate the type-I quantum orbits, we show the band structure without SOC in Fig. 3 (a).At the K/H point, one can find Dirac points in k z = 0/0.5 planes caused by the kagome structure symmetry, where we mark the higher Dirac point as K1/H1.However, these Dirac points cannot be isolated band crossing points because they locate inside mirror planes in the 3D momentum space.Otherwise, they would exhibit a monopole-like Berry charge which violates the mirror symmetry.Consequently, Dirac points expand inside the mirror plane and present a continual nodal line/ring.Along the K − H line, there indeed exists a Dirac nodal line connecting K1 and H1.As shown in Fig. 3(d), quantum orbits surrounding the nodal line show a π Berry phase, which is the case of two medium-frequency orbits (605 and 617 T) at the Fermi energy -85 meV (Table I).After considering SOC, however, the generalized Berry phases λ of these two quantum orbits are heavily reduced (|λ| < 0.5π) by the large Zeeman phases (roughly proportional to the cyclotron mass, see Supplementary Table S2).Thus such orbits showing no phase shifts in quantum oscillation are trivial.Furthermore, there are additional Dirac points due to band crossings on the M − K line [Di, i = 1 ∼ 8, see Fig. 3(a)] and also on the L − H line (Pi).Because of the mirror symmetry with respect to the M KH plane, Di and Pi are linked by a nodal line inside the M KH plane and related bands have opposite mirror parities.Because k z = 0 and 0.5 planes are mirror planes, some of Di/Pi extend further inside the k z = 0/0.5 planes and form nodal rings.Therefore, intriguing Dirac nodal networks with linked nodal lines and nodal rings emerge in the momentum space.For example, D1 & P1 (also P4 & D4 and D5/6/7 & P5/6/7) are connected to form a cylinder-like network, and P2 & P3 are linked to constitute a 3D network.For the nodal network, the quantum orbit can exhibit a π Berry phase if it surrounds one nodal line and avoids crossing other nodal lines/rings.This is the case for the D5-induced FS (14 T at -85 meV) in Fig. 3(d) and P1-induced FS (30 T at -40 meV) in Fig. 3(b) which are type-I quantum orbits (Table I).In contrast, a Dirac-like crossing cannot generate a non-trivial FS in a plane if there is also a related nodal ring in the same plane.For example, D1/D4 fails to create a non-trivial FS in the k z = 0 plane.In addition, there is another nodal line crossing the X point [Fig.3(b)], leading to a π Berry phase of the 13 T orbit at -40 meV, which belongs to the type-I quantum orbits.These Dirac nodal lines and networks may generate non-trivial quantum orbits in a relatively large energy range besides -40 and -85 meV.We note that Dirac nodal lines were reported in the pristine phase of AV 3 Sb 5 2,56,57 rather than the CDW state.
Other three types of non-trivial quantum orbits originated from either strong SOC and/or strong Zeeman effect are shown in Table I (see details in Supplementary Table S1&S2).Most type-II to type-IV non-trivial quantum orbits are centered at time-reversal invariant points [i.e., Γ, A, M , and L points, see Fig. 3(b)].Thus their Berry phases φ B and orbital phases φ R are strictly zero without SOC due to the time-reversal symmetry constraint (see Supplementary Note 2).Only when SOC is involved the φ B and φ R can appear in quantum orbits.Overall, we found sixteen non-trivial quantum orbits at two Fermi energies.More than half of them originate from a mixture of SD and ISD layers.We list calculated non-trivial quantum orbits and the experimental ones in Table II for convenience of comparison.Except for extremely high-frequency orbits, our calculations agree well with experiments in the fundamental frequency, cyclotron mass, and topology.

III. EXTENDED DISCUSSIONS
Recently, Zhou and Wang proposed an effective single-orbital model for the 2 × 2 × 1 CDW with time-reversal symmetry breaking loop-currents 48 , where reconstructed small Chern Fermi pockets carrying concentrated Berry curvature of the Chern band emerge at M points.These pockets are connected by one-quarter and three-quarters of the primitive reciprocal lattice vectors and may play a crucial role in the observed pair density wave order in the superconducting state 5 .We verified the existence of such Fermi pockets at M for the realistic 2 × 2 × 2 CDW.They can be seen at -40 meV (20 T) and -85 meV (99 T) in Figs.1(g) and 1(j), respectively.Moreover, we find that under SOC the 20 T pocket does not have a Berry phase without time-reversal symmetry breaking, while the 99 T pocket has a Berry phase close to π. Upon closer examination (Supplementary Fig. S4 for FS unfolding), the 20 T pocket around M at -40 meV originates from the ISD layer due to folding a pocket centered at the crossing point between Γ−K and M −M in the primitive Brillouin zone, which can be captured by a 2 × 2 × 1 real CDW with ISD bond order 58 .In contrast, the 99 T pocket in the k z = 0.5 plane is of mixed ISD and SD character generated by folding a pocket at the k z = 0.25 plane in the primitive zone, which is unique to the alternate stacking of the ISD and SD layers of the 3D CDW.
Additionally, recent Raman experiments 59,60 and time-resolved angle-resolved photoemission spectroscopy 61 detected CDW-driven phonon modes.Measured CDW modes can be described by intralayer vibrations, except that the lowest-energy phonon is attributed to the interlayer Cs vibrations.Compared to phonons, electrons experience a more three-dimensional CDW structure and show stronger interlayer interaction in CsV 3 Sb 5 .The present 2 × 2 × 2 CDW is a minimal model to capture the interlayer coupling and reproduce all quantum orbits measured in experiments (except large orbits due to magnetic breakdown).If we adopt the 2 × 2 × 4 model in calculations, it will bring a much larger number of quantum orbits and complicate the physics understanding.
We point out that Dirac nodal lines and networks may be a general characteristic of kagome materials.For example, the K1 − H1 nodal line comes from the characteristic Dirac points of the kagome lattice.This scenario applies both to nonmagnetic and magnetic kagome materials.
For instance, a similar K1 − H1 nodal line was also found in the noncollinear kagome antiferromagnets Mn 3 Sn/Ge 62 .Because many mirror planes (e.g., the M KH plane and ΓM L) exist, degenerate band crossing points form nodal lines inside mirror planes and nodal lines from intersecting planes may link to each other.In the presence of strong SOC, these nodal lines are usually gapped out 49,50 and generate large Berry curvature 63 or spin Berry curvature 64 .Some nodal lines also lead to drumhead-like topological surface states 65 on a specific facet.

IV. CONCLUSION
In summary, we studied the Fermi surface properties of the 3D CDW in CsV 3 Sb 5 .The comprehensive analysis of quantum orbits indicates that the 2 × 2 × 2 CDW with interlayer structural modulation agrees broadly with recent quantum oscillation experiments in fundamental frequencies, cyclotron mass, and non-trivial phase shifts.The non-trivial Berry phases reveal a hidden Dirac nodal network in the momentum space due to mirror symmetry protection in the spinless case.We advanced the study of quantum oscillations in layered materials by identifying the different structural and topological origins of all quantum orbits.Our work not only resolves the Fermi surface puzzle in CsV 3 Sb 5 but also serves as the first realistic example to show the crucial role of SOC in determining the topology of quantum orbits in quantum oscillation experiments.

METHODS
We have performed Density functional theory (DFT) calculations with the Vienna ab-initio Simulation Package (vasp) 66,67 and fully relaxed all structures.Except for structural relaxation, spin-orbital coupling (SOC) is considered in electronic structure calculations if otherwise stated.More details about DFT calculations are referred to Ref. 4. High-resolution Fermi surfaces are calculated via Wannier functions 68 (V-d and Sb-p orbitals) extracted from DFT. k-meshes of at least 100×100×100 and 100×100×50 are used to calculate band structures in the full BZs of 2×2×1 and 2×2×2 CDWs, respectively.The in-plane extremal (maximal and minimal) orbits of the closed FS at different energies, which corresponds to the magnetic field parallel to the c axis in quantum oscillation experiments, are determined by tracking the FS slice changes along the k z direction.The cyclotron frequency of an electron on a closed quantum orbit is calculated according to the Onsager's relation 69 , F = c 2πe A e , where A e is the area of an extremal orbit at Fermi energy ε F , e is the electron charge and is the reduced Planck constant.The effective mass of a cyclotron electron is calculated with m * = 2 2πm0 ∂Ae ∂ε F , where m 0 is the bare electron mass.Quantum oscillation phase calculation.Comprehensive explanations of different phases in quantum oscillations are found in Supplementary Notes 1-3.Here we provide a brief introduction on how to calculate them.More theoretical backgrounds can be found in Refs.39 and 40.
In the quantum oscillation experiment, electrons undergo cyclotron motion whose phase interference condition gives rise to Landau level.There are mainly six contributions to the total phase and the quantization rule for the total phase is, where l B = c/eB is the magnetic length.The first term is the sum of the de Broglie phase and Aharonov-Bohm (A-B) phase.
φ M is the Maslov correction and equals to π for a simple closed smooth curve (all the cyclotron orbits considered in this paper belong to this category).The generalized Berry phase λ a (a = 1, 2 the degenerate band indices) has three contributions (Berry phase φ B , orbital phase φ R and spin Zeeman phase φ Z ) and can be calculated from the eigenvalue e iλa of the propagator U[C] of loop C (exp means path-ordered exponential), where the degenerate band group being considered.) is the non-Abelian Berry connection, and the first term is the Berry phase for the multiband case.The second phase which comes from the orbital magnetic moment is with Π ln = u lk |v|u nk being the interband matrix elements of the group velocity operator v.Here we emphasize the importance of the orbital phase which is usually ignored in experiments.Since CsV 3 Sb 5 has inversion and time-reversal symmetries, the orbital phase is identically zero for all quantum orbits without SOC.But in the presence of SOC, it may contribute non-negligibly to the total phase λ a .For example, it can compensate the loss of the π Berry phase in a Dirac model with a Semenoff mass 70 (see also Supplementary Note 3).The last term comes from the spin Zeeman effect and is usually written as a reduction factor 71,72 in quantum oscillation.When SOC is not included, the three terms can be calculated separately.Under SOC, the three contributions to the propagator must be calculated as a whole, so separating the spin reduction factor is no longer possible.In other words, the orbital and Zeeman phases must be calculated together with the Berry phase to get observable results.
The Berry phase with or without the inclusion of the orbital phase and the generalized Berry phase λ a calculated for quantum orbits of the 2×2×2 CDW are listed in Table S1&S2 in the Supplementary.Due to the double degeneracy of quantum orbits ensured by both inversion and time-reversal symmetries, their generalized Berry phases are λ 1 = +λ and λ 2 = −λ.Thus the sum of two oscillations is where the final phase shift ∆φ is 0 or π, depending on whether cos(λ) is positive (|λ| < 0.5π) or negative (|λ| > 0.5π).If |λ| = 0.5π, the oscillation may not observable because cos(λ) = 0. Notice that the ± π 4 phase correction comes from the k z integral in a three-dimensional case (it does not exist in a two-dimensional system).

DATA AVAILABILITY
The data supporting the findings of this study are available within the paper and in the supplementary information.

Table of content for figures, notes, and tables:
• Supplementary Fig. S1 displays band structures of the 2×2×1 CDW structures SD and ISD.
• Supplementary Fig. S3 displays the layer resolved band structure of the 2×2×2 CDW without spin-orbital coupling.
• Supplementary Note 1 provides a general derivation of the Lifshitz-Kosevich formula with discussions on the phase shift.
• Supplementary Note 2 discusses the Berry phase in a time reversal and inversion symmetric system.
• Supplementary Note 3 shows the π total phase of a Dirac Hamiltonian with Semenoff mass.
• Supplementary Table S1 and  For a formal justification of formula (S14) and also the determination of the amplitude c r , we provide a direct calculation of the grand canonical potential taking the finite T and τ into account at the beginning.Here the method is from Champel and Mineev 10 .First, we focus on one band taking possible spin-orbit coupling into consideration (so the wave function is a two-component spinor).In the presence of disorder scattering, if one uses the Born approximation, then the self-energy is where the real part of self-energy is ignored whose effect is only a constant shift of energy in this approximation.τ is the relaxation time used in (S12) and gives the Dingle factor as can be seen later.The retarded Green's function is now modified by the self-energy Σ R (ε, k z ) as where the Landau level ε n (k z ) is determined by (S4).The spectral density is the imaginary part of Green's function The spectral density function can be used to define a generalized density of states D(ε) whose integral is the particle number.In the present case, each Landau level has a degeneracy D = eB/2π c, and D(ε) is given by To calculate the sum over n, we first define a continuous variable x via i.e. n + γ a in (S4) is replaced by x.Then with the help of Poisson summation formula 8 (0 ≤ γ a ≤ 1) D(ε) can be written as The first part of the above formula represents the DOS in the absence of a magnetic field, and the second part is the oscillation part which we shall focus on.Below we calculate the r'th Fourier components in the oscillation part D r (ε).Changing the variable from x to ε(x, k z ) via (S19) and use the definition of effective mass m(ε) (S10), the integral is written as The lower integral limit ε 0 is the energy for the cross section area to vanish, i.e., A (ε 0 , k z ) = 0.In general, ε 0 depends on k z , but if we assume Γ << ε, then the contribution of the integral over ε is mainly from the vicinity of ε, hence the lower integral limit can be extended to −∞.Then interchange the order of integral and do the integral over k z first, we need to calculate This integral can be done by using the stationary phase approximation.Physically, the integral is a superposition of oscillation of DOS from all different k z .If the cross-section area A(ε , k z ) changes rapidly, then these contributions cancel out.The only exception is the extremal orbit where dA(ε , k z )/dk z vanishes and the oscillation from the orbits around it adds coherently.Hence, after integrating over k z , only the extremal orbit has an appreciable contribution.The result is simply where A e (ε ) is the area of extremal orbit and there is an additional phase shift ± π 4 (+ for minimum area and − for maximum area).Substitute this result into (S22), and complete the integral, the r'th component is found as

FIG. 2 .
FIG. 2. Calculated cyclotron frequencies in comparison with experiments.(a) Cyclotron frequencies of extremal quantum orbits as a function of Fermi energy for the 2×2×2 CDW.The energy of the charge neutral point is set to energy zero.Symbols with the same color and style connected by dashed lines represent the evolution of the same quantum orbit.(b) Comparison of calculated cyclotron frequencies at −40 and −85 meV (on the left side of the red dashed line) with experimental values in literature.Experimental set A is from Ref. 30, B from Ref. 27, C from Ref. 28, D from Ref. 29, E and F from Ref. 31 for the angle between magnetic filed and c axis being 0 • and 20 • , respectively, G from Ref. 32, H and I from Ref. 33 for bulk and flake, respectively, J from Ref. 34, and K from Ref. 35.Numbers marked near the symbols are corresponding cyclotron masses in unit of m0.The experimental values are divided into three regions, i.e., low-frequency region 0∼250 T, medium-frequency region 550∼950 T, and high-frequency region ≥1150 T. (c) and (d) are a zoom-in of the low-frequency region in (a) and (b), respectively, which can be further divided into three sub-regions as indicated by the three grey backgrounds.

FIG. 3 .
FIG. 3. Dirac nodal lines and nodal networks in the band structure.(a) Band structures on kz = 0 (left part) and 0.5 (right part) planes without SOC.Eight Dirac points on the M K (LH) line near the charge neutral point are labeled from D1 to D8 (P1-P8) respectively.Selected Dirac points at K and H are labeled as K1 and H1, respectively.The two dash-dotted horizontal lines indicate the energy positions of −40 and −85 meV, respectively, where non-trivial Fermi surfaces (φB = π) are indicated by blue ellipses.Dirac nodal lines/rings connecting D1 to D8 (P1-P8) Dirac points are divided into four groups, as sketched in (b)-(e).Blue circles in (b) and (d) indicate Fermi surfaces surrounding a nodal line at −40 and −85 meV in (a)&(b), respectively.For simplicity, we show those Dirac nodal lines/rings and Fermi surfaces partly near one M/L and one K/H point.The full Dirac nodal lines/rings and non-trivial Fermi surfaces can be obtained by crystal symmetries (P 6/mmm).We note that nodal lines except the K1−H1 line are not necessarily straight along kz but locate inside the mirror plane.
S7. (a) Band energy and (b) Berry phase and the phase sum of gapped Dirac model (S45).The parameters are v = 1 and Semenoff mass ∆ = 1.(b) Band energy in k z = 0 mirror plane and (c) Berry phase and the phase sum of BHZ model (S49).The parameters are A = B = 3 and M 0 = Ã = 1 which is in Z 2 topological insulator region.
and S2 in Supplementary.

TABLE II .
Frequencies of calculated and experimentally observed non-trivial quantum orbits.

TABLE S1 .
Berry phase φ B , orbital phase φ R and Zeeman phase φ Z of quantum orbits of 2×2×2 CDW at −40 meV calculated by using the expression (S1).Under SOC, the orbital and spin magnetic moment-related phases can only be calculated together with the Berry phase, and (φ B +φ R +φ Z ) is the generalized Berry phase (λ) in the main text.Frequency (Freq.) is unit of T. All phases are in unit of π.For the 912 T, we could not get converged results for φ B +φ R and φ B +φ R +φ Z .Notice that the numerical convergence of λ a for the quantum orbit of 912 T at -40 meV is not as good as others.

TABLE S2 .
Similar to Table S1 but at -85 meV.