Introduction

Kagome lattice systems lie at the forefront of condensed matter physics, providing a fertile platform for discovering emergent quantum phenomena1,2,3. Their unique corner-sharing triangular lattice constrains electron hopping, leading to characteristic electronic structures such as flat bands (FBs), Dirac points (DPs), and saddle points (SPs)4,5,6,7,8,9,10,11,12,13,14,15,16. Among these features, FBs stand out due to their high density of states, making them susceptible to interaction-driven instabilities and giving rise to a variety of strongly correlated phases, including unconventional superconductivity17,18,19,20,21,22, charge density waves23,24, and various magnetically ordered states25,26,27,28,29,30. More recent studies have revealed that FBs are not merely featureless high-density states but can possess nontrivial quantum geometric and topological properties31,32,33,34,35,36,37,38, akin to Landau levels yet emerging without an external magnetic field39,40,41,42,43,44,45. This realization has sparked intense interest, as such nontrivial FBs provide a fertile setting for novel strongly correlated phenomena, including fractional quantum Hall effects at zero magnetic field46,47,48,49,50,51,52,53,54,55,56,57. These advancements not only deepen our understanding of strongly correlated physics but also unlock new opportunities for quantum information science58.

Heavy fermion systems, first discovered in rare-earth and actinide compounds, have long been recognized as a paradigmatic setting for realizing FBs through strong correlations, where Kondo hybridization between localized f-electrons and dispersive conduction bands (f–c hybridization) gives rise to anomalously large effective masses59,60,61,62,63,64. This hybridization generates heavy FBs spanning the entire Brillouin zone (BZ), with their flatness rooted in the localized nature of f-electrons60,61,62,65,66. Despite this well-established mechanism, recent breakthroughs in Moiré superlattices18,67,68,69,70,71,72,73, where geometric and correlation effects intertwine to produce FBs, have shifted the focus toward their intrinsic topological properties and their connection to quantum geometry74,75,76,77. This renewed perspective has invigorated efforts to explore how geometric and topological FBs intersect with Kondo-driven topological phenomena78,79,80,81,82,83,84, collectively referred to as topological heavy fermion systems. However, the interplay between heavy fermion systems and modern geometric FB concepts remains largely uncharted, hindered by the scarcity of material realizations.

Here we report the coexistence of two FBs near the Fermi level (EF) of the kagome compound YbCr6Ge6 (YCG), where geometrically frustrated FBs (KFBs) originating from the 3\({d}_{{z}^{2}}\) orbitals of Cr85,86,87,88 coexist with Kondo resonance states (KRSs) from Yb f-orbitals within the experimental resolution (~20 meV). Angle-resolved photoemission spectroscopy (ARPES) measurements, complemented by density functional theory (DFT)+DMFT calculations, show that the f-electrons of Yb, intercalated between Cr-kagome layers, undergo Kondo hybridization with the kagome bands, forming FBs that span the entire BZ. Transport measurements corroborate the Kondo resonance behavior with the coherent temperature (Tcoh) of 80 K (see Supplementary Note 1). Our theory shows that crystalline symmetry forbids hybridization along certain high-symmetry lines, stabilizing Dirac crossings of heavy-fermion character. The Fu–Kane parity analysis of the Kondo-coherent low-energy band structure distinguishes gapped weak and strong topological Kondo-insulating phases and reveals a Dirac–Kondo semimetal stabilized by symmetry-protected band crossings. Hence, YCG serves as a prototype of a topological heavy-fermion system and a platform to explore symmetry-protected quasiparticles in strongly correlated quantum matter.

Results

The kagome compound YCG crystallizes in a family of kagome layered structure10,11,13 (Fig. 1a) with a unit cell composed of four alternating layers of Yb (red), Cr (blue), and Ge (yellow) atoms. At the base is a Cr kagome layer (Fig. 1b), followed by a Ge honeycomb layer in which a Yb atom occupies the hexagon center (Fig. 1c). This is followed by another Cr kagome layer, capped by a Ge honeycomb layer featuring a perpendicular Ge dimer aligned with the hexagon center (Fig. 1d). These four layers repeat periodically along the vertical axis. The alternating sequence of Cr double kagome layers and Ge honeycomb layers preserves the full crystalline symmetry of space group P6/mmm (#191). Notably, the structure is inversion symmetric—with the inversion center located at either the Yb atom or the Ge dimer—and maintains both planar mirror symmetry (mz) and six-fold rotational symmetry (C6z).

Fig. 1: Atomic and electronic structure of Kondo kagome system YbCr6Ge6 (YCG).
Fig. 1: Atomic and electronic structure of Kondo kagome system YbCr6Ge6 (YCG).The alternative text for this image may have been generated using AI.
Full size image

a Atomic structure of the kagome lattice system YCG, delineating the layered arrangement of Yb, Cr, and Ge. b Kagome layer composed of Cr atoms. c Honeycomb Ge layer with Yb atoms situated at the hexagonal centers. d Honeycomb Ge layer with perpendicular Ge dimers at the hexagonal centers. Schematic illustrations of the Cr kagome bands and localized Yb f-orbitals e without hybridization, f in DFT, and g DFT+DMFT, respectively. The Dirac point (DP), van Hove singularities saddle point (SP), and KFB are highlighted. LS and KRS represent localized states and Kondo resonance states of Yb f-orbitals, respectively. h Brillouin zone (BZ) showing the high-symmetry points used in calculations. i–k DFT Band structures projected onto the f orbitals (red), Cr \(3{d}_{{z}^{2}}\) orbitals (blue), and Cr \({d}_{{x}^{2}-{y}^{2}}/{d}_{xy}\) (green), respectively, obtained with U = 0 for both Cr d and Yb f orbitals. The zero-energy level is set at the charge-neutral electron filling.

Figure 1e–g illustrates the coexistence of KFB and KRS in YCG, providing a schematic overview of the band evolution. The Cr 3d-orbitals in the kagome layers give rise to characteristic kagome bands, including an FB, a DP, and a SP. In YCG, the KFB is located at EF, as shown in Fig. 1e. When there is no f–c hybridization, the Yb 4f levels remain far below EF, corresponding to nearly atomic f7/2 states, forming the localized magnetic moments. As f–c hybridization takes place, quasiparticle peaks with heavy effective masses develop near EF. Figure 1f presents a schematic band structure based on DFT, where the 4f states participate in the band formation near EF while maintaining their non-dispersive nature.

When the strong correlation effects of Yb 4f orbitals are properly taken into account, the quasiparticle bands arising from f–c hybridization undergo significant renormalization, which leads to the KRS being energetically aligned with the KFB right near EF (Fig. 1g). Kondo screening develops as the temperature decreases, leading to renormalized coexisting FB states at EF, with the FBs extending across the entire BZ. In addition, overall Cr 3d bands are also renormalized, leading to an overall narrowing of the bandwidth.

DFT calculations reveal that YCG is a KFB system, where the kagome bands derived from Cr \(3{d}_{{z}^{2}}\) orbitals dominate the density of states near EF, with Yb 4f states positioned nearby in energy. Figure 1i–k displays the band structure of YCG using first-principles calculations (see Supplementary Notes 3.4 and 3.6 for detailed analyses of the wave functions and orbital contributions, respectively). The Cr \({d}_{{z}^{2}}\) orbital contributes to the FBs near EF (Fig. 1j), while the Yb 4f-states form FBs located approximately 0.2–0.3 eV below EF (see Fig. 1i). The localized f-orbitals and the KFBs exhibit distinct characteristics: the f-orbitals are flat across the entire momentum space, whereas the KFBs are dispersionless only within a specific kz plane. The significant out-of-plane dispersion highlights the role of interlayer interactions, suggesting possible hybridization between the intercalated Yb 4f-orbitals and the kagome layers (Fig. 1e). DFT alone cannot capture the effects of strong electronic correlations properly, and this limitation is remedied in our DFT+DMFT approach as discussed in the following sections.

Figure 2a shows the electronic structure of YCG from ARPES measurements, where the kx–ky constant-energy maps at E = EF and E = EF−0.3 eV reveal two bands crossing EF: a hole-like band centered at \(\overline{\Gamma }\) (the α band) and the other band centered at \(\overline{{\rm{K}}}\) (the β band). The Fermi momentum kF of the β band is located at \(\overline{{\rm{K}}}\) within our experimental resolution, showing that DP of the β band is positioned near EF. The α band is barely visible in the first BZ, likely due to matrix element effects8,89. Both the α and β bands are identified as bulk bands, as they are observed across different surface terminations (see Supplementary Note 2.3).

The band dispersions observed along high-symmetry directions in YCG clearly reveal the Cr d-orbital kagome bands, as predicted by our DFT calculations. Here we annotate high symmetry points using the 2D surface BZ notation as the features are almost kz-independent (Fig. 2b). Figure 2c, d presents the energy-momentum band dispersions along the \(\overline{\Gamma }\)-\(\overline{{\rm{K}}}\)-\(\overline{{\rm{M}}}\) and \(\overline{{\rm{M}}}\)-\(\overline{\Gamma }\)-\(\overline{{\rm{M}}}\) high-symmetry lines (Cut I and Cut II), respectively. Notably, at E = EF, the β band exhibits Dirac-like dispersion at the \(\overline{{\rm{K}}}\) point. For the α band, the FB and SP appear at \(\overline{\Gamma }\) and \(\overline{{\rm{M}}}\) at E = EF and E = EF−0.55 eV, respectively. The distinct visualization of these features underscores the role of kagome-derived states in shaping the electronic properties of YCG, which are also clearly reproduced in our DFT+DMFT calculations that explicitly account for the correlation effects of both Yb 4f and Cr 3d electrons (Fig. 2c) (see Supplementary Note 3.5 for the full DFT+DMFT energy spectrum). Compared to the DFT results shown in Fig. 1, the Yb 4f levels are strongly renormalized, giving rise to a pronounced KRS at EF. In addition, the Cr d bands are also renormalized and become incoherent. We note that these kagome-driven band features are also observed in YCr6Ge685,86,90.

Fig. 2: Momentum-resolved electronic structure of YCG measured by ARPES at 18 K.
Fig. 2: Momentum-resolved electronic structure of YCG measured by ARPES at 18 K.The alternative text for this image may have been generated using AI.
Full size image

a kx-ky constant-energy maps at the Fermi level (EF) and E = EF−0.3 eV, showing the α and β bands crossing EF. b Schematic of the 3D BZ with a projection onto the 2D plane. c Energy–momentum (E–k) cut along \(\overline{\Gamma }\)-\(\overline{{\rm{K}}}-\overline{{\rm{M}}}\) (cut I in a). DP, KRS, and KFB refer to Dirac point, Kondo resonance state, and kagome flat band, respectively. The α and β bands are highlighted with red lines. The corresponding DFT+DMFT-calculated E–k cut is also shown. d E–k cut along \(\overline{{\rm{M}}}\)-\(\overline{\Gamma }-\overline{{\rm{M}}}\) (cut II in a) with the corresponding 2D curvature intensity plot107. Due to the matrix element effect, parts of the α band are not visible and are represented with red dotted lines. e Energy distribution curves (EDCs) at \(\overline{\Gamma },\,\overline{{\rm{K}}}\), and \(\overline{{\rm{M}}}\) points extracted from black rectangular regions in (a). f ky − kz constant-energy map at E = EF−0.3 eV with the corresponding 2D curvature intensity plot. g E–kz cut along Γ−A (cut III in e). h kz-dependent EDCs showing consistently strong peaks at EF.

In addition to the kagome d-orbital bands, the KRS can be identified in the ARPES results, originating from Yb 4f states as anticipated in DFT calculations. As shown in Fig. 2c, this FB is prominently observed near EF and exhibits a defining characteristic: its momentum independence in the kx–ky plane (Fig. 2c, d), along with its coherence. This is evidenced by strong intensity peaks in the energy distribution curves (EDCs) at high-symmetry points (Fig. 2e). Extending this analysis to three dimensions, Fig. 2f presents the ky–kz constant-energy map at E = EF−0.3 eV (see Supplementary Note 2.4 for additional kz-dependent ARPES results). The E–kz cut along the Γ–A high-symmetry line (Cut III in Fig. 2f) further illustrates the presence of the KRS at EF throughout the high-symmetry line (Fig. 2g). Moreover, the kz-dependent EDCs in Fig. 2h demonstrate that peaks from the KRS consistently appear at EF, regardless of the kz momenta. These observations show that the KRS at EF is a highly localized state, extending across the entire BZ and remaining independent of the electron momenta, kx, ky, and kz.

Furthermore, we conduct two additional experiments supporting that the KRS originates from Yb 4f states. First, we perform ARPES on LuCr6Ge6 (LCG), where heavy-fermion behavior is absent (see Supplementary Notes 1.1 and 2.1), and confirm that the KRS does not appear in LCG. Second, we perform Yb-resonant photoemission spectroscopy on YCG and observe that the KRS exhibits a pronounced intensity dependence across the Yb N5 edge (~183 eV, see Supplementary Note 2.2). These results provide strong evidence that the KRS arises from Yb 4f orbitals.

One can expect to observe different temperature-dependent behaviors in KRS and KFB, as they originate from different mechanisms. Figure 3a–c shows E–k cuts along the \(\overline{{\rm{M}}}\)-\(\overline{\Gamma }\)-\(\overline{{\rm{M}}}\) high-symmetry line, measured at temperatures of 220, 80, and 18 K, respectively. Around the first \(\overline{\Gamma }\), where the α band is suppressed due to matrix element effect, the KRS is prominently observed at 18 K but progressively diminishes in intensity as the temperature increases. Eventually, at 220 K, the FB around the first \(\overline{\Gamma }\) becomes nearly undetectable, while the α band remains visible near the second \(\overline{\Gamma }\). This behavior is further illustrated in the EDCs at \(\overline{\Gamma }\) (Fig. 3d), where the FB peak broadens and vanishes with increasing temperature (see Supplementary Note 2.5 for fitting results). The disappearance of the FB peak at higher temperatures suggests possible Kondo behavior, originating from the hybridization between localized Yb 4f orbitals and Cr 3d conduction electrons.

Fig. 3: Temperature-dependent evolution of flat electronic structures.
Fig. 3: Temperature-dependent evolution of flat electronic structures.The alternative text for this image may have been generated using AI.
Full size image

a–c ARPES-measured E–k cuts along the \(\overline{{\rm{M}}}-\overline{\Gamma }-\overline{{\rm{M}}}\) high-symmetry line at temperatures of 220, 80, and 18 K, respectively. d, e Temperature-dependent evolution of EDCs at \(\overline{\Gamma }\) and kx = 1.0 Å−1. f, g DFT+DMFT calculated spectral functions along the L−A–L high-symmetry line at temperatures of 58 and 290 K, respectively. d indicates Cr kagome bands. h Temperature-dependent evolution of the spectral weight, calculated using DFT+DMFT and integrated in momentum space with a Fermi–Dirac distribution.

In stark contrast to the KRS, the KFB from the Cr \(3{d}_{{z}^{2}}\) orbitals remains coherent above the Kondo temperature scale. Fig. 3e presents the temperature-dependent EDC at kx = 1.0 Å−1 as a function of temperature. The peaks near EF, marked with red inverted triangles, persist up to 220 K, indicating that this feature does not share the same origin as the KRS. Thus, we conclude that the dispersionless feature at kx = 1.0 Å−1 (near the second \(\overline{\Gamma }\)) near EF corresponds to the KFB, which originates from the out-of-plane hybridization of Cr \(3{d}_{{z}^{2}}\) orbitals across adjacent kagome layers90.

Figure 3g shows the DFT+DMFT spectral function calculated at 58 K, where the FB is observed near EF, as a direct consequence of Kondo hybridization. Notably, the interaction between the Cr kagome bands (marked as d in Fig. 3f, g) and Yb 4f states gives rise to a characteristic hybridization gap. As the temperature increases, these KRSs become incoherent and suppressed. Eventually, the FBs near EF are no longer clearly defined and the hybridization gap features also diminish at 290 K as shown in Fig. 3f. The temperature-dependent features are also well established in Fig. 3h. Unlike the DFT-only results (Fig. 1i), where the Yb 4f-orbital bands are positioned  ~  0.3 eV below EF with an absence of kagome FBs along the L–A–L direction at EF, the DFT+DMFT calculations properly capture the strong electron correlation effects of the Yb 4f orbitals, yielding strongly renormalized Yb 4f bands that align well with the ARPES observations.

As a consequence of the coexistence of KRS and KFBs near the Fermi energy, we argue that YCG realizes a Dirac–Kondo semimetallic phase, offering a possible manifestation of topological heavy-fermion quasiparticles. In this framework, KFBs—localized within the kagome planes but dispersive along the c-axis due to interlayer coupling via Yb atoms—participate in the Kondo screening of Yb 4f7/2 moments, generating momentum-independent Kondo resonance states at low temperature (T ≪ TK) (see Fig. 4a, b). Crucially, crystalline symmetry forbids hybridization along certain high-symmetry lines, enforcing symmetry-protected Dirac band crossings with heavy-fermion character, while hybridization gaps open elsewhere. These gapless excitations, which retain both strong-correlation coherence and topological protection, are referred to as topological heavy fermions78,79,80,81,91,92,93,94. Moreover, the hybridization-induced gaps that open away from the DPs carry nontrivial \({{\mathbb{Z}}}_{2}\) invariants in the corresponding time-reversal-invariant planes of the BZ, as demonstrated below using DFT calculations and topological analyses.

Fig. 4: Topological classification of hybridization gaps in YbCr6Ge6.
Fig. 4: Topological classification of hybridization gaps in YbCr6Ge6.The alternative text for this image may have been generated using AI.
Full size image

 Schematics of the role of Yb in YCG: a high-temperature state with Cr (dark blue) and Ge (light purple) kagome layers and Yb (cyan); b Low-temperature state where yellow regions denote Dirac-Kondo fermions, localized in-plane but dispersive out-of-plane; red arrows indicate the Kondo coupling JK. c DFT band structure of the high-temperature phase along the high-symmetry lines shown in Fig. 2b. The dashed line denotes the Fermi level EF. d DFT band structure of the low-temperature YCG phase computed with U = 2.6 eV for Cr 3d and U = 0 eV for Yb 4f, with the U values and EF calibrated to reproduce the low-energy dispersion obtained from the preceding DFT+DMFT calculations, consistent with ARPES. Compared with (c), hybridization opens multiple narrow gaps near EF. Colored shading highlights three representative hybridization-gap regions, and arrows indicate their correspondence to the parity-based topological classification in (e–g). Insets (blue boxes) enlarge the representative Dirac crossings. e–g Parity eigenvalues (±) at the time-reversal-invariant momenta for the three colored gaps in (d), used to determine the Fu–Kane Z2 indices. e Weak topological Kondo insulator (TKI) with (ν0; ν1ν2ν3) = (0; 001), implying ν2D(kz = 0) = ν2D(kz = π) = 1. f Strong TKI with ν2D(kz = π) = 1 and ν2D(kz = 0) = 0. g Dirac–Kondo semimetal (DKSM), where the kz = π plane hosts a nontrivial Z2 invariant while symmetry-protected DPs persist along the Γ–A and K–H lines, which prevents the opening of a full bulk gap. The blue Dirac cones indicate the locations of the DPs in the BZ.

To demonstrate that YCG realizes a topological heavy-fermion system, we construct an effective DFT band structure that reproduces the DFT+DMFT spectral function and, consistently, the low-temperature ARPES data (see Supplementary Note 3.4 for a direct comparison between the DFT+DMFT and DFT+U band structures). Fig. 4c, d highlights the contrast between the high- and low-temperature electronic structures. Fig. 4c shows the band structure of LCG, taken as a reference for the high-temperature limit of YCG (see Supplementary Note 3.1 for the full energy spectra of YCG and LCG). In this case, only kagome-derived states are present, and no f states appear near the Fermi level. The kagome bands are clearly resolved on the kz = 0 plane, and the nominally FBs disperse upward with kz, demonstrating noticeable interlayer hopping characteristic of the double-kagome stacking. In contrast, Fig. 4d (YCG), obtained in an effective DFT calculation with UCr d = 2.6 eV and UYb f = 0 eV, captures the KRS-KFB hybridization and reproduces the DFT+DMFT/ARPES results; the eight branches of the Yb 4f7/2 manifold (four Kramers-degenerate pairs) overlap the kagome FBs.

The emergence of nontrivial Z2 topology in multiple hybridization gaps demonstrates that topological heavy-fermion quasiparticles are stabilized by the interplay between KRSs and KFBs over a finite range of chemical potential near the Fermi level rather than requiring fine-tuning to a single filling. This conclusion is substantiated by a Fu–Kane parity analysis of the hybridization-induced gaps highlighted in Fig. 4d–g. When the chemical potential is fixed to the experimentally determined Fermi level, the low-energy Kondo-hybridized quasiparticle spectrum encounters three distinct low-energy regions, which are marked by red, green, and blue shading in Fig. 4d. The red-shaded gap realizes a weak topological Kondo insulator (TKI) with (ν0; ν1ν2ν3) = (0; 001) and therefore ν2D(kz = 0) = ν2D(kz = π) = 1, whereas the green-shaded gap corresponds to a strong TKI characterized by (ν0; ν1ν2ν3) = (1; 001). By contrast, the blue-shaded region realizes a topological Dirac–Kondo semimetal (DKSM) because the kz = π plane hosts a nontrivial Z2 invariant while symmetry-protected DPs persist along the high-symmetry lines Γ–A and K–H, which prevents the opening of a fully gapped quasiparticle spectrum. The protection of these DPs follows directly from symmetry since along Γ–A the flat f-derived states transform as \({\overline{\Delta }}_{9}\) whereas the dispersive kagome states transform as \({\overline{\Delta }}_{7}\), which forbids their hybridization and stabilizes the DPs. Analogous crossings arise along K–H from the crossing between \({\overline{P}}_{4}\oplus {\overline{P}}_{5}\) and \({\overline{P}}_{6}\) bands (see Supplementary Notes 3.2 and 3.3 for detailed topological analysis).

Several remarks on the filling-tunable topological phases in YCG, including the DKSM regime, help clarify the robustness and implications of our prediction. The effective DFT bands, constructed in the zero-temperature quasiparticle limit, qualitatively reproduce the DFT+DMFT spectral function and ARPES data. Importantly, not only the Yb f states but also the Cr d orbitals require on-site interactions to reproduce the ARPES energy spectrum, indicating that strong correlation effects are essential for the kagome FBs as well. Although static DFT does not capture the full renormalization of the KFBs and KRSs, the central conclusions remain robust. Crystalline symmetry forbids hybridization along specific high-symmetry lines and therefore protects the DPs in the correlated regime, while the Fu–Kane parity analysis assigns nontrivial Z2 indices to multiple hybridization gaps over nearby chemical potentials. Nevertheless, a quantitative description of the correlation-driven renormalization and the detailed evolution among the weak and strong TKI regimes and the DKSM as the chemical potential is tuned remain important direction for future study. Equally crucial, experiments such as high-resolution ARPES, quantum oscillations, magnetotransport, and thermodynamic measurements could provide direct evidence, with expected signatures including nontrivial Berry phases, anomalous or topological Hall responses, and enhanced low-temperature specific heat, thereby establishing the phenomenology of topological heavy fermions.

Discussion

Although the KFBs are anticipated to host strong correlation physics, observing such phenomena in most kagome systems has been challenging. This difficulty arises from the presence of many other dispersive conduction bands and the positioning of the FBs far away from EF. Interestingly, recent studies on Ni3In, a d-electron kagome system characterized by an FB precisely located at EF, have demonstrated heavy fermion-like behavior despite the absence of local moments from f electrons95. In this system, both essential components for the manifestation of heavy fermion behavior—localized moments and conduction electrons—are derived entirely from band features. The hopping-frustrated KFB acts analogously to the localized moments observed in conventional heavy fermion systems, providing a distinct mechanism for the emergence of heavy fermion-like phenomena and non-Fermi liquid behavior. In contrast, YCG combines f-electron Kondo physics with kagome FBs, thereby realizing topological heavy fermions rather than band-derived heavy fermion-like states.

YCG is a unique system distinguished by the coexistence of two distinct FBs right near EF: (i) the KRSs and (ii) the KFB induced by hopping frustration (Fig. 4a, b). The coexistence of the KRSs and the KFB not only enriches the correlated phase diagram but also yields filling-tunable weak and strong TKI regimes together with symmetry-protected Dirac crossings, thereby establishing YCG as a platform for topological heavy fermions. Notably, the Cr \({d}_{{z}^{2}}\) orbital, which forms the KFB, exhibits hybridization with the Yb 4f states, opening the possibility of a unique interplay between the two different correlated FBs. When the KFB can be precisely located at EF, strong correlation physics can be induced, as observed in the Ni3In system. As a result, the temperature scale of the system is suppressed, and strange metallic behavior can occur. On the other hand, an increase in the density of states at EF may enhance the Kondo hybridization and temperature scales. Therefore, it would be interesting to investigate experimentally how the two different energy scales arising from these two distinct FBs manifest. For instance, one can tune the KFB position by introducing strain96 or doping to investigate the interplay between two distinct FBs near EF, thereby not only exploring the mutual influence of geometrically frustration-induced KFBs and f-orbital KRSs but also providing experimental access to Dirac–Kondo crossings and to distinct weak and strong TKI regimes across nearby fillings, together with characteristic signatures of topological heavy fermions.

Methods

Crystal growth

Single crystals of YCG and LCG were synthesized using tin as a flux in a sealed quartz tube under a partial argon atmosphere. Yb (grain, 99.9%), Lu (grain, 99.9%), Cr (powder, 99.995%), Ge (lump, 99.999%), and Sn (shot, 99.995%) were placed in an alumina crucible with an atomic ratio of Yb(or Lu):Cr:Ge:Sn = 1:3:6:20. The crucibles were sealed in quartz tubes under 5 Torr of Argon (99.9999%) after purging 3–5 times. The ampules were heated to 1150 °C and held for 24 h, then cooled to 850 °C at a rate of 250 °C/h. Further cooling to 600 °C was performed at a rate of 1 °C/h, at which point the Sn flux was removed by centrifugation. Single crystal X-ray diffraction analysis of YCG single crystals confirmed full stoichiometry with the absence of site disorder across all crystallographic sites.

ARPES measurements

ARPES measurements are conducted at beamline 21-ID-1 at the National Synchrotron Light Source II (NSLS-II) of Brookhaven National Laboratory (BNL) using a Scienta DA30 analyzer. Single-crystalline YCG samples are cleaved in an ultra-high vacuum with pressure below 10−11 Torr. kx–ky constant-energy maps are measured at a temperature of 18 K using linearly horizontally (LH) polarized 93 eV light. To obtain a kz-dependent constant-energy map, LH-polarized beams with photon energies ranging from 63 to 135 eV are used. All ARPES map data are acquired using the deflection mode of the Scienta DA30 analyzer. Temperature-dependent ARPES measurements are performed with an LH-polarized 97 eV beam, covering temperatures from 18 to 220 K.

Single-crystalline LuCr6Ge6 samples are cleaved in an ultra-high vacuum with pressure below 10−11 Torr. kx–ky constant-energy maps (see Supplementary Note 2.1) are measured at a temperature of 18 K using LH polarized 200 eV light.

Resonant photoemission spectroscopy measurements

Resonant photoemission spectroscopy measurements are conducted at the 4A1-μ ARPES beamline of the Pohang Accelerator Laboratory using LH-polarized light with photon energies ranging from 180 to 200 eV at a temperature of 80 K. Single-crystalline YCG samples are cleaved in an ultra-high vacuum with a pressure below 10−10 Torr.

DFT calculations

We performed DFT calculations as implemented in the Vienna Ab initio Simulation Package (VASP)97 with pseudopotentials generated by the projector-augmented wave method98. The generalized gradient approximation of Perdew-Burke-Ernzerhof99 was employed for the exchange-correlation energy functional. The kinetic energy cutoff for the plane-wave basis was set to 395 eV. In the case of DFT+U calculations, we employed the rotationally invariant formalism100. We used atomic parameters obtained from single-crystal X-ray analysis on our single-crystalline samples. Self-consistent field calculations were performed with a Γ-centered sampling of k-points on a 13 × 13 × 7 uniform grid, using the tetrahedron method with Blóchl corrections101. The total energy was successfully converged below a threshold of 10−7 eV. To compute the irreducible representations of electronic states, we utilized IRVSP102. The VASPKIT103 code was used to plot the charge density of the wave functions. All band structure calculations included spin-orbit coupling.

DFT+DMFT calculations

We employed a fully charge self-consistent DFT+DMFT method implemented in DFT+Embedded DMFT (eDMFT) functional code104 to describe the f−c hybridization properly. DFT calculations were performed using the WIEN2k code105, and the DMFT loop treats the correlation effect of the Yb 4f orbitals and Cr 3d orbitals. The number of plane wave as set to RKmax = 9, and a 2000 K points set was used. The hybridization energy window from −10 to 10 eV was chosen with respect to the Fermi level. We used U = 7 eV and JH = 0.7 eV for the Yb 4f orbitals, and U = 5 eV and JH = 0.7 eV for the Cr 3d orbitals. The local basis of the Yb atom for the DMFT part is projected to f5/2 and f7/2 states, so that the crystal field splittings are only considered on the level of the lattice (DFT part) but are not normalized by the impurity solver. The continuous-time quantum Monte Carlo solver was adopted to solve the impurity problem106. The nominal double-counting method was used, where the nominal occupancy of the Yb atom was set to 13, and that of the Cr atoms was set to 3.

Transport measurements

The AC resistance was measured using a Lakeshore Cryotronics 370 AC resistance bridge with a standard four-probe configuration, while temperature control was achieved using a Teslatron PT (Oxford Instruments). DC heat capacity measurements were performed using the relaxation method with a Physical Property Measurement System (PPMS, Quantum Design) and an SR850 Lock-In Amplifier (Stanford Research Instruments).