Emergence of flat bands and ferromagnetic fluctuations via orbital-selective electron correlations in Mn-based kagome metal

Abstract

Kagome lattice has been actively studied for the possible realization of frustration-induced two-dimensional flat bands and a number of correlation-induced phases. Currently, the search for kagome systems with a nearly dispersionless flat band close to the Fermi level is ongoing. Here, by combining theoretical and experimental tools, we present Sc₃Mn₃Al₇Si₅ as a novel realization of correlation-induced almost-flat bands in the kagome lattice in the vicinity of the Fermi level. Our magnetic susceptibility, ²⁷Al nuclear magnetic resonance, transport, and optical conductivity measurements provide signatures of a correlated metallic phase with tantalizing ferromagnetic instability. Our dynamical mean-field calculations suggest that such ferromagnetic instability observed originates from the formation of nearly flat dispersions close to the Fermi level, where electron correlations induce strong orbital-selective renormalization and manifestation of the kagome-frustrated bands. In addition, a significant negative magnetoresistance signal is observed, which can be attributed to the suppression of flat-band-induced ferromagnetic fluctuation, which further supports the formation of flat bands in this compound. These findings broaden a new prospect to harness correlated topological phases via multiorbital correlations in 3d-based kagome systems.

Introduction

A flat band system is characterized by the presence of a dispersionless energy band, where the group velocity of electrons vanishes at every crystal momentum^1,2. Because quenching of kinetic energy promotes the effects of electron correlations, flat band systems offer an ideal platform to examine strongly correlated quantum phenomena, encompassing fractional Chern insulator phases and unconventional superconductivity^3,4,5,6. Theoretically, flat bands have been studied in dice, Lieb, kagome, honeycomb, and Tasaki’s decorated square lattices, where destructive quantum interference from two or more hopping channels produces a flat band^7,8,9,10,11. Experimentally, localized flat band state has been reported in photonic Lieb and kagome lattices^12,13,14 as well as in realistic condensed matter systems^15,16,17.

Among the family of flat band systems, the kagome lattice has been one of the most studied. Geometric frustration inherent in the kagome lattice creates destructive interferences between multiple nearest-neighbor electron hopping channels and yields flat bands^17,18. Because of zero kinetic energy within ideal flat bands, the impacts of electron correlations with such bands can be maximized. Experimentally, engineering flat band systems enables a viable route to realize correlation-induced emergent phenomena such as Chern insulators, fractional quantum Hall states, quantum spin liquid, superconductivity, and topological magnon insulators^{19,20,21,22,23,24,25}. Nonetheless, the realization of ideal flat dispersions has remained elusive due to the presence of long-range hopping paths and the difficulties in placing flat bands in the vicinity of the Fermi level (E_F).

Recently, 3d-transition metal-based two-dimensional layered kagome compounds have captured much research attention because of their multiorbital nature. The most studied compounds are Co₃Sn₂S₂^26,27,28, Fe₃Sn₂^29,30,31, FeSn³², CoSn^17,33,34, YCr₆Ge₆³⁵, and YMn₆Sn₆³⁶. Thanks to the active 3d-orbital degree of freedom in all the compounds listed above, multiorbital phenomena such as orbital magnetization or spin-orbit coupling (SOC)-induced topological properties and anomalous (spin-)Hall responses have been actively explored.

At this point, an interesting question arises about the role of electron correlations in these multiorbital kagome systems, where kagome-induced flat bands coexist with other dispersive bands that are less affected by the kagome-induced destructive interference. In non-kagome multiorbital compounds such as Ca_2−xSr_xRuO₄³⁷ and Fe-based superconductors^38,39,40,41, nontrivial orbital-dependent correlation effects induced by the on-site Coulomb repulsions and Hund’s coupling have been reported, such as orbital-dependent Mott transitions^{42,43,44,45,46} and Hund’s metallic phases^47,48,49,50. However, the impact of electron correlations on the electronic structure, especially in the presence of kagome-induced flat bands and SOC in realistic systems, has not been much discussed in previous studies. (It was speculated in ref. ³⁶ that YMn₆Sn₆ can be a Hund’s metal, but without further elaboration.

In this work, we study the electronic and magnetic properties of Mn-based kagome metal Sc₃Mn₃Al₇Si₅ (SMAS). This compound crystallizes in a hexagonal structure with a space group P6₃/mmc. Figure 1a, b present the crystal structure of SMAS and the underlying Mn kagome network (Fig. 1c showing five Mn d-orbitals in the Mn kagome network schematically). Previous experimental reports on SMAS reveal a predominant metallic character with no signature of static magnetic order down to 1.8 K. The specific heat capacity measurement shows a large Sommerfeld coefficient, suggesting a vital role of electronic correlations⁵¹. The absence of long-range magnetic order at very low temperatures indicates a strong magnetic fluctuation in this system, as further probed by inelastic neutron scattering measurements⁵². On the other hand, the previously reported magnitude of magnetic moment (0.5 μ_B/Mn), compared to the one from Hund’s rule applied to the Mn d⁵ charge state (S = 5/2), implies an itinerant character of magnetism.

Fig. 1: Overview of crystal structure and correlation-induced kagome flat bands.

a Crystal structure of SMAS highlights the formation of a Mn kagome network, Sc equilateral triangles, and distorted Al₈ cubes. b Top and side views of the structure show the connectivity between Mn and Si atoms and the formation of MnSi₄ rectangles, constituting a three-dimensional network. Note that the local cartesian axes for the definition of Mn d-orbitals are depicted as red, green, and blue arrows. c Schematic shape and orientation of Mn d-orbitals. d, e Schematic illustrations of Wannier orbital realizing kagome-induced weakly dispersive bands in SMAS and shift of the kagome-induced bands up to the E_F (in addition to the formation of lower- and upper-Hubbard bands) via orbital-selective electron correlations.

Here, we combine experimental and first-principles calculation tools to explore potential correlation-induced flat-band physics in SMAS. Our magnetization, magnetic susceptibility, and ²⁷Al nuclear magnetic resonance (NMR) measurements indicate the presence of ferromagnetic fluctuations below T < 30 K, which is attributed to the formation of flat bands and potential negative magnetoresistance in flat band systems^53,54,55. From ab-initio density functional and dynamical mean-field calculations we find that correlation-induced flat bands emerge in the vicinity of the E_F at k_z = 0, which are likely to be strongly linked to the ferromagnetic fluctuations observed in the low-T regime. We propose that the flat bands are induced by i) kagome-induced geometric frustration within a subset of Mn d-orbitals, revealed by constructing electronic Wannier orbitals from the non-correlated electronic structure as depicted in Fig. 1c, and ii) orbital-selective electron correlations which selectively push the kagome-induced weakly dispersive bands up to the E_F and strongly renormalize the bandwidth (see Fig. 1d for a schematic illustration). We further observed a significant negative magnetoresistance in this system, which supports the presence of flat-band-induced ferromagnetic fluctuations in SMAS as suggested in CoSn⁵⁶. Additionally, our dynamical mean-field calculations show a slight upturn in DC resistivity in the low-temperature regime, consistent with our DC resistivity measurement, which can be attributed to the enhanced orbital susceptibility and ferromagnetic fluctuations. These findings make SMAS a promising platform for further exploring correlated and topological phenomena emerging from flat band systems.

Results

Experimental signatures of ferromagnetic instabilities

Figure 2 a presents the temperature and field dependencies of the electrical resistivity ρ(T) of SMAS. On cooling, ρ(T) decreases down to 30 K and reaches a minimum at 25 K, below which it experiences an increase. We note that the essentially same transport behavior was observed in the previous study⁵¹. As evident in the inset of Fig. 2a, the application of an external magnetic field somewhat suppresses ρ(T). This observed upturn below 25 K alludes to the development of an additional scattering channel.

Fig. 2: Electrical and magnetic properties of SMAS.

a Temperature-dependent electrical resistivity ρ(T) of SMAS. The inset plots the field-dependent ρ(T, H) for H//a. b Temperature dependence of the static magnetic susceptibility χ(T) measured at μ₀H = 0.1 T for H//ab and H//c, with an inset showing the ratio of in-plane χ_ab(T) to out-of-plane χ_c(T). c Log-log plot of χ_ab − χ₀ (red circles) and χ_c − χ₀ (cyan triangles) versus temperature. The solid and dashed lines represent fits to a power-law dependence χ(T) ~ T^−α at low temperatures. d Magnetization curves M(H, T) at T = 2 K and 19 K for H//c (open symbols) and H//a (full symbols). e Difference of the magnetization curves M(H, T) − M(H, T = 19 K) at selected temperatures T = 2, 4, 6, 9, and 12 K with solid and dashed lines indicating fits to a modified Brillouin function as described in the text. Note that an additional ${M}_{{{{{{{{\rm{diff}}}}}}}}} \sim {B}_{J}^{2}$ fitting for H//a at T = 2 K, where B_J is the Brillouin function as mentioned in the text, is depicted as a dotted line. f Temperature dependence of the amplitude parameter A(T) for H//c (diamonds) and H//a (spheres) and magnetoresistance at B = 9 T for H//c (open squares) and H//a (full squares) as a function of temperature. The thick line is a guide to the eye.

Figure 2 b shows the temperature dependence of the in-plane and out-of-plane magnetic susceptibilities χ(T) measured in an applied field of 0.1 T. With decreasing temperature, χ(T) increases steeply with no indication of saturation or anomaly, thereby excluding the occurrence of long-range magnetic ordering. Upon closer inspection of χ(T), we observe a notable disparity between the in-plane χ_ab(T) and the out-of-plane χ_c(T). To quantitatively assess the temperature-dependent magnetic anisotropy, we plot the ratio χ_ab(T)/χ_c(T) in the inset of Fig. 2b. Remarkably, a broad maximum is observed with χ_ab/χ_c ≈ 1.16 at approximately T ~ 145 K. The decrease in χ_ab/χ_c below 145 K implies that a weak XY-like magnetism becomes increasingly isotropic as T → 0 K.

To elucidate the anomalous behaviors of the magnetic susceptibility, we first estimate the constant contribution to χ(T), ${\chi }_{0}={\mu }_{0}{N}_{A}{\mu }_{{{{{{{{\rm{B}}}}}}}}}^{2}D({\epsilon }_{F})=7.8\times 1{0}^{-5}$ emu ⋅ Oe⁻¹ ⋅ mol⁻¹ ⋅ Mn⁻¹, where N_A is Avogardro’s number and μ_B is the Bohr magneton. This value is obtained from our DFT calculations, where D(ϵ_F) = 7.24 states/eV/formula unit represents the density of states at the Fermi level. In Fig. 2c, the χ₀-subtracted magnetic susceptibilities χ_ab(T) − χ₀ and χ_c(T) − χ₀ are displayed on a log-log scale. Notable changes in slope and anisotropy are observed around 130 K, where the maximum ratio χ_ab(T)/χ_c(T) occurs, and between 10 K and 25 K, coinciding with the resistivity minimum. The multi-stage evolution of anisotropic magnetic correlations points to the presence of multiple underlying energy scales. Below 8 K, a power-law increase becomes apparent with (χ_ab − χ₀)(T) ~ T ^−0.47(6) and (χ_c − χ₀)(T) ~ T^−0.51(2), signifying the development of critical-like ferromagnetic correlations. Further Curie-Weiss (CW) analysis of 1/(χ_ab(T) − χ₀) above 150 K yields the effective magnetic moment of ${\mu }_{{{{{{{{\rm{eff}}}}}}}}}^{ab}$ = 0.86(3) μ_B/Mn and the CW temperature ${\theta }_{{{{{{{{\rm{CW}}}}}}}}}^{ab}=-421$.(4) K, and from 1/(χ_c(T) − χ₀) (above 150 K) we obtain ${\mu }_{{{{{{{{\rm{eff}}}}}}}}}^{c}$=0.87(2) μ_B/Mn and the CW temperature ${\theta }_{{{{{{{{\rm{CW}}}}}}}}}^{c}=-368$.(9) K. The significantly reduced effective moment, compared to the spin-only value of 4.97 μ_B expected for Mn³⁺ ions, suggests a dominant itinerant character of the magnetism. These CW parameters are, thus, regarded as indicators of correlation-driven magnetism.

Isothermal magnetization curves M(H, T) at temperatures T = 2 and 19 K for H//c (open symbols) and H//a (full symbols) are shown in Fig. 2d. At 19 K, M(H) exhibits a linear increase, characteristic of a paramagnetic-like state. As the temperature decreases below 19 K, M(H) progressively develops a convex curvature, indicating the emergence of ferromagnetic correlations. This behavior is consistent with the observed upturn in ρ(T) below 25 K and the power-law increase in χ(T). To further assess the ferromagnetic correlations, we subtract the linear term from M(H, T) and plot the resulting difference in magnetization curves M_diff = M(H, T) − M(H, T = 19 K), as shown in Fig. 2e. We attempted to model M_diff(H, T) using a modified Brillouin function B_J, defined as M_diff = A(T)B_J(gμ_BJ(T)B/k_BT). Here, A(T) is a temperature-dependent amplitude parameter associated with the saturation magnetization of ferromagnetically correlated spins. With lowering the temperature, the spin moment J(T) may be enhanced due to the orbital-selective amplification of ferromagnetic correlations, yet it hardly varies with an external field. We find that M_diff(H, T) follows a linear Brillouin scaling rather than a quadratic relationship, as demonstrated by comparing the solid line for M_diff(H//a, T = 2K) ∝ B_J with the dotted line for ${M}_{{{{{{{{\rm{diff}}}}}}}}}(H//a,T=2K)\propto {B}_{J}^{2}$ in Fig. 2e. Noticeably, a similar linear scaling of M(H) ∝ B_J is observed in the ferromagnetically ordered state of manganites, which exhibit negative magnetoresistance⁵⁷. In this light, the observed linear Brillouin scaling indicates that the studied system is on the verge of ferromagnetic instability. In Fig. 2f, the extracted values of A(T) are plotted together with MR. Here, A(T) essentially conveys the same information as J(T). The similar trend observed between these parameters establishes a direct relationship between the increasing amplitude of ferromagnetic correlations and the negative MR, thereby supporting the proportionality M(H) ∝ B_J ∝ MR.

The presence of ferromagnetic fluctuations in SMAS is also revealed by our ²⁷Al NMR results (see Supplementary Note 4). Further, from the NMR result it is also hinted that the resistivity upturn around 30 K (see Fig. 2a) is related to the onset of the ferromagnetic fluctuations and the resulting enhancement of electron scattering. On the other hand, the formation of Mn d⁵ local moments in SMAS is strongly inhibited due to the strong hybridization between Mn and Si ions. Simple density functional theory calculations (see Fig. 3a–c) or their augmentation with the on-site Coulomb repulsion in a mean-field approximation (i.e. DFT+U methods) do not admit ferromagnetism as well (see Supplementary Note 5). Hence, the observed tendency toward ferromagnetic instability becomes a mystery within the standard band-theoretical and mean-field picture. To explain the observed ferromagnetic tendency, the presence of flat dispersions near the E_F, induced by the dynamical nature of electron correlations, is required, as discussed in the following sections.

Fig. 3: Emergence of correlation-induced flat bands.

a–c Non-spin-polarized band structure of SMAS displays the orbital contribution of Mn B_g (blue), ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ (green), and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ (red) orbitals along the high symmetry paths M-K-Γ-A-H-L, and d PDOS obtained using LDA functional in DFT. e–g Corresponding orbitally-resolved momentum- and frequency-dependent spectra and, h momentum-integrated spectral function from LDA+DMFT calculation for (U, J_H)=(6, 0.8) eV at 210 K with full Coulomb interaction. i Full DFT bands (black) and Wannier interpolated B_g-bands (violet). j Orbital texture with B_g character from constructed Wannier molecular orbitals of non-correlated electronic structure. The inset in a depicts the hexagonal Brillouin zone and spacial k-points therein.

Flat band realized via correlation-induced orbital-dependent band energy renormalization

For a deeper understanding of the ferromagnetic fluctuations in SMAS as revealed through our experimental measurements, we carried out ab initio electronic structure calculations using DMFT methods. We uncover several peculiar features in this compound: (i) Correlation- and frustration-induced nearly flat band in proximity to E_F at the k_z = 0 plane, identified as a correlated metal incipient to an orbital-selective Mott phase. The role of electron correlations shown in Fig. 1d, which selectively promotes kagome bands, is reminiscent of the formation of coherent peak and lower/upper Hubbard bands in the orbital-selective Mott transitions observed in several multiorbital systems such as Ca_2−xSr_xRuO₄ and Fe-based superconductors^38,39,40,41. Hence, the flat bands in SMAS arise from unusual cooperation between kagome-induced kinetic energy quenching and orbital-selective electron correlations, which are likely to be strongly linked to the ferromagnetic fluctuations observed in the low-T regime^58,59,60, (ii) symmetry-protected metallic nodal surface bands at the k_z = π plane, and (iii) strong incoherence driven by Hund’s coupling. In this section, we begin with discussing the electronic structure obtained from nonmagnetic DFT calculations.

The top panels of Fig. 3a–c delineate the non-spin-polarized band structure (black lines) calculated using LDA functional without incorporation of additional on-site Coulomb repulsion and spin-orbit coupling. The colored fat bands in Fig. 3a–c highlight the orbital contributions of Mn d orbitals and the corresponding projected density of states (PDOS) (see Fig. 3d). Note that the Mn d orbitals are labeled in terms of irreducible representations defined with respect to local coordinate axes, where the z-axis is set to be parallel to the twofold axis of the C_2h point group of the Mn site, as shown in Fig. 1b. The five d orbitals, as depicted in Fig. 1c, can be divided into three groups: B_g (blue) for {d_xz, d_yz}, ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ (green) for $\{{d}_{{{{{{{{\rm{{x}}}}}}}^{2}-{y}^{2}}}},{d}_{{{{{{{{\rm{xy}}}}}}}}}\}$, and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ (red) for ${d}_{{{{{{{{\rm{{z}}}}}}}^{2}}}}$. In Fig. 3 and for the rest of the paper, we fix this color coding for Mn d orbitals. We comment that despite all the orbitals being nondegenerate, orbitals belonging to the same irreducible representation show almost identical features in the band structure as well as spectral function. In the vicinity of Γ-point, the topmost occupied band shows a strong ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$-character, which is located higher in energy than the blue B_g-bands. Also, note that the bands at the k_z = π plane, especially the ones at the E_F, are degenerate due to nonsymmorphic two-fold screw rotation and time-reversal symmetries in the absence of SOC, consisting of nodal surface bands⁶¹.

The bottom panels of Fig. 3e–g and Fig. 3h show the orbital-resolved spectra and PDOS at 210 K from LDA+DMFT results. In the results presented in Fig. 3e–g, we employed on-site Coulomb parameters (U, J_H) = (6, 0.8) eV. The details on the choice of the parameters and how the result depends on the evolution of parameters will be discussed in the following sections. The quintessential feature of the spectral function is the presence of a nearly flat band, lying just below E_F at the k_z = 0 plane, induced by dynamical correlation effect and mostly consisting of B_g-orbitals as also clearly shown in the PDOS in Fig. 3h. In the DFT bands, the B_g-bands are located around − 0.5 eV (refer to Fig. 3a, d). In the DMFT results, the B_g-bands are pushed up to the E_F with the bandwidth strongly renormalized (see Fig. 3a, e for comparison), while the positions of other bands, especially ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$-bands near Γ and the nodal surface bands at the k_z = π plane, remain only weakly affected. The PDOS of B_g flat bands show an emergence of a sharp peak at E_F (compare Fig. 3d, h), which is identified as a coherent peak emerging from correlated metallic phases in the vicinity of Mott transitions^62,63. This B_g-orbital-selective Mott-like correlations will be discussed later in the next section.

To check whether the inclusion of the U parameter on Mn d orbitals plays a similar role in mean-field treatments of the Coulomb repulsion, we performed DFT+U calculations to compute PDOS (see Supplementary Note 5) and compared them with Fig. 3. Surprisingly, we observe that applying U on Mn d-orbitals i) pushes the B_g-bands downward, contrary to DFT+DMFT, as shown in Supplementary Note 5 and Supplementary Fig. 4 therein, and that ii) DFT+U favors antiferromagnetic order up to U_eff = 4 eV, and beyond that U_eff value large Mn magnetic moments (≥ 3μ_B) set in. Hence, the renormalization of the B_g-bands and the origin of the observed ferromagnetic instabilities with small local moments cannot be captured via the simple DFT or DFT+U description.

As the DMFT spectral function shows Mn B_g-derived almost flat bands close to E_F, an important question arises; to what extent the nature of our flat B_g-bands originates from the kagome lattice physics, namely the frustration-induced destructive interference and the resulting suppression of kinetic energy scale? To answer this, we constructed a set of two electronic Wannier orbitals of B_g-bands from our DFT band structure, where each of two Mn kagome layers in the unit cell hosts one B_g-orbital. Figure 3i, j show our Wannier-projected bands and the real-space Wannier orbital, respectively (see Fig. 1c for a schematic illustration). As discussed in a previous study on CoSn¹⁷, such an orbital shape with alternating sign at neighboring sites of a hexagon suppresses electron hopping between neighboring sites via destructive interferences induced by geometric frustration, which makes the B_g-bands narrower and susceptible to on-site Coulomb correlations.

The role of electron correlations shown in Fig. 3 (also schematically in Fig. 1e), which selectively promotes kagome bands, is reminiscent of the formation of coherent peak and lower/upper Hubbard bands in the orbital-selective Mott transitions observed in several multiorbital systems such as Ca_2−xSr_xRuO₄ and Fe-based superconductors^38,39,40,41. Hence, the flat bands in SMAS arise from an unusual cooperation between kagome-induced kinetic energy quenching and orbital-selective electron correlations.

Lastly, we mention that our DMFT results remain paramagnetic down to T = 116 K when U ≤ 8 eV and J_H = 0.8 eV, contrary to our DFT+U results. This is somewhat consistent with the experimental observation of no long-range order. To check the ferromagnetic instability at the low-temperature regime, which is beyond the power of the quantum Monte Carlo impurity solver, we employed a rotationally-invariant slave boson methodology combined with DFT (DFT+RISB). DFT+RISB method has been known to capture the correlation-induced band renormalization of the so-called coherent peak close to the Mott transition, and has been used to study electronic structures of various correlated metals at the zero-temperature limit^64,65,66. We checked that DFT+RISB reproduces the essential feature of DMFT results, namely the energy renormalization and band flattening of the B_g-bands. A remarkable observation is that, while DFT+RISB results remain paramagnetic most of our choices of U and J_H, ferromagnetism emerges only when the B_g-bands gets very close to the Fermi level (please refer to Supplementary Note 9 for further details). This is an evidence that the presence of the flat B_g-bands in the vicinity of the Fermi level is the origin of the observed ferromagnetic fluctuations in the low-temperature regime.

Dependence of B _g energy renormalization on U, J _H, and double-counting energy

The earlier analysis of DMFT results at T = 210 K and with (U, J_H)= (6, 0.8) eV revealed that the presence of electron correlation shifts the flat band closer to the E_F. In order to understand such nontrivial behavior, we investigated the dependence of band evolution on the change of the on-site Coulomb repulsion U, impurity temperature T, double-counting energy of Mn d-bands, and Hund’s coupling J_H within our DMFT method. In this section, we focus on the effects of U, T, and double-counting shift, while the role of J_H will be discussed in the next section.

Figure 4 summarizes the results by plotting spectral functions and PDOS for three different values of U = 4, 6, 8 eV with a nominal occupancy n = 5.0 in the nominal double-counting scheme⁶⁷ employed at T = 500 and 1500 K. Here, the nominal occupancy describes the position of the correlated Mn d band in energy, which can be shifted via tuning the value of n (see Supplementary Note 7 for results from other choices of n). As shown in Fig. 4a-c, enhancement of U-value from 4 to 8  eV makes the flat B_g bands move towards E_F with the bandwidth more suppressed, while the positions of ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ peaks in the PDOS plots are left almost unaffected. Figure 4g, h depict the mass enhancement and on-site energy renormalization (i.e., ${{{{{{{\rm{Re}}}}}}}}\Sigma (\omega=0)$) induced by U at T = 500 K, respectively, while Fig. 4i, j show the same quantities as a function of temperature at U = 6 eV. These data show that both U and temperature affect the energy renormalization of the B_g-orbitals, and that orbital-dependent mass enhancement of the B_g and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ are stronger than that of ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$.

Fig. 4: U- and T-dependence of orbital-dependent band renormalization.

a–c and d–f Spectral functions plotted at 500 and 1500 K with three sets of (U, J_H)=(4, 0.8), (6, 0.8), and (8, 0.8) eV, respectively. The PDOS are plotted only for four different cases and shown on the left and right side of the spectral functions. g The mass enhancement as a function of on-site Coulomb repulsion U. h On-site energy renormalization: Re(${\Sigma }_{{B}_{{{{{{{{\rm{g}}}}}}}}}}-{\Sigma }_{{A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}}$) (green) and Re(${\Sigma }_{{B}_{{{{{{{{\rm{g}}}}}}}}}}-{\Sigma }_{{A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}}$) (red) versus U at zero frequency. g, h The results are presented at T = 500 K. i Mass enhancement plotted against temperature with error bars for (U, J_H)=(8, 0.8) eV. j On-site energy renormalization: Re(${\Sigma }_{{B}_{{{{{{{{\rm{g}}}}}}}}}}-{\Sigma }_{{A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}}$) (green) and Re(${\Sigma }_{{B}_{{{{{{{{\rm{g}}}}}}}}}}-{\Sigma }_{{A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}}$) (red) versus temperature at zero frequency. All calculations shown in the panels above were obtained using full Coulomb interaction.

In Fig. 4c we observe broad humps of B_g states located around ± 0.5 eV with respect to E_F, which can be identified as the lower and upper Hubbard bands originating from the on-site U. The energy difference between the upper and lower Hubbard bands (~ 1 eV) is a fraction of the magnitude of the on-site Coulomb U (4 ~ 8 eV in Fig. 4) parameter, which can be attributed to the formation of B_g molecular orbitals and the resulting renormalization of U within the molecular orbital sector⁶⁸. Together with the U-induced emergence of a sharp coherence peak at the E_F^62,63, this phase can be considered an incipient orbital-selective Mott phase. Note that similar orbital-selective incipient Mott phase in flat-band systems has been reported in a kagome-induced lattice model^59,60. However the entrance to true orbital-selective Mott-insulating phase is arrested by the presence of other weakly-correlated bands, strong Mn-Si hybridization, and non-negligible inter-orbital hopping channels^44,45,69.

Secondly, in order to see the effect of temperature, in Fig. 4d–f we plot spectral functions at T = 1500 K (U = 4, 6, and 8 eV, respectively). Comparison with the T  = 500 K results (Fig. 4a–c) reveal that an increase in temperature tends to cancel the U-induced renormalization of B_g-bands. The T-induced shifting down of B_g-bands is most significant at U = 8 eV, where the renormalization of the B_g-bands is the strongest, and almost negligible in less-correlated cases with U = 4 eV, aside from a trivial T-induced overall blurring of spectra. This T-induced evolution of electron correlations can also be checked in the T-dependent mass enhancement as shown in Fig. 4i, where the orbital differentiation between B_g, ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$, and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ orbitals become significant below T = 500 K. Note that the mass enhancement of ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ orbital is also comparable to that of B_g, but without a proper kagome-induced destructive interference, it does not exhibit any significant correlation-induced changes in the spectral functions.

Finally, we comment that the change of U and T shows a common feature with respect to the renormalization of the B_g bands. As U is enhanced or T is lowered, the Mn d-orbital occupation reduces and gets closer to 5. For example, at the nominal charge of n = 5.0 and at T = 500 K, increasing the value of U from 4 to 8 eV reduces the d-occupancy from 5.48 to 5.33. On the other hand, at U = 8 eV, cooling the system from T = 1500 to 500 K, reduces the d-occupancy from 5.34 to 5.32. Although the change in the Mn d occupancy in solid is not dramatic, it follows the same trend. This observation is consistent with a previous theoretical result, where the orbital-selective correlation effect is found to become stronger as the system approaches the half-filling regime^40,41. Shifting the entire Mn d-orbitals in energy by tuning the nominal charge n further confirms this observation; pushing the d-orbitals downward makes their d-orbital occupancy enhanced and tends to remove the orbital-dependent correlation effects, and vice versa (See Supplementary Note 7 for further details).

Orbital decoupling and bad metallic phase by Hund’s coupling

So far, the DMFT results with a fixed value of Hund’s coupling J_H = 0.8 eV have been presented. In d-orbital systems like SMAS understanding the role of Hund’s coupling is essential because (i) orbital-selective Mott character, which is essential to the emergence of the B_g kagome bands, has been reported to be strongly enhanced by the Hund’s coupling^44,45, and (ii) in the d⁵-limit, where the effects of Hund’s coupling is the strongest⁴⁹, the most strongest orbital-selective correlation effects are observed both in our results and previous studies^40,41.

We begin with presenting the probability distribution of impurity multiplet states from the Monte Carlo solver with (U, J_H)=(8, 0.8) eV at T = 300 K as depicted in Fig. 5a. The predominance of high-spin configurations in each charge occupation sector is clear, which overall yields the estimate of the size of the Mn moment to be 0.90  μ_B/Mn. Increasing J_H up to 1.0 eV induces strong blurring of band dispersions, which can be attributed to the enhancement of local moment-induced scattering, where the size of the moment also increases up to 1.16 μ_B/Mn at J_H = 1.0 eV. Because the Mn moment 0.86 μ_B/Mn is deduced from our Curie-Weiss fit, we concluded that J_H = 0.8 eV is a reasonable choice and adopted this value for obtaining most of the presented results in this work unless specified.

Fig. 5: Tendency toward local moment formation and DC susceptibilities.

a Histogram of Mn d atomic configurations showing the probability in descending order. For each n, high spin configuration carrying maximum probability is marked. Ising-type Coulomb interactions were employed. b, c DC orbital and spin susceptibilities versus temperature plot, computed for (U, J_H)=(8, 0.8) eV using full Coulomb interaction. The figure legends, within B_g and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ stand for difference in orbital susceptibilities within B_g and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ orbital sectors, respectively.

The nature of the Hund’s-coupling-induced incoherent metallic phase can be further substantiated by the imaginary part of Mn self-energy in the Matsubara frequency axis. By examining the power-law behavior of imaginary part of Mn self-energy $-{{{{{{{\rm{Im}}}}}}}}\Sigma (\beta {\omega }_{n}) \sim \gamma+K{\omega }_{n}^{\alpha }$ at (U, J_H) = (8, 0.8) eV, we investigate the J_H-induced deviation of our system from Fermi-liquid-like behavior. Here, γ ($\equiv -{{{{{{{\rm{Im}}}}}}}}\Sigma ({\omega }_{n}\to 0)$) stands for the low-frequency scattering rate and ω_n ( = (2n + 1)πT) is the Matsubara frequency with α being exponent. From the fitting, the exponent α is found to vary between 0.35–0.39 for a wide temperature range T = 120–2000 K, implying a significant deviation from Fermi-liquid behavior (α = 1)⁷⁰. From low-frequency DMFT self-energy, we computed band renormalization factor (Z⁻¹) and mass enhancement (m^*) for each orbital. The mass enhancement and renormalization factor are connected by the following equation, ${m}^{*}/m={Z}^{-1}=1-\partial {{{{{{{\rm{Im}}}}}}}}\Sigma (i\omega )/\partial \omega {| }_{\omega \to {0}^{+}}$⁷¹. In the studied T = 120–2000 K range, we observe three different m^*/m, viz., 2.07–1.38 for B_g, 1.85–1.39 for ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$, and 2.14–1.35 for ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ orbitals (see Fig. 4i). From this result, we conclude that B_g and ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{in}}}}}}}}}$ show clear orbital-dependent correlations in comparison to ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$, driven by the Hund’s coupling⁷².

In addition to the incoherent metallicity, another major role of Hund’s coupling is quenching the orbital degree of freedom and decoupling orbitals, thereby enabling orbital-dependent correlation effects^44,45. An additional role is inducing the so-called Hund’s metal phase⁴⁷, where the spin and orbital degrees of freedom decouple and the orbital degree of freedom is more quenched at higher energy than the spin one⁷³. To get insight into these aspects, we computed static spin and orbital susceptibilities as functions of temperature using the following formulae, ${\chi }_{{{{{{{{\rm{sp}}}}}}}}}=\int_{0}^{\beta }\langle {S}_{z}(\tau ){S}_{z}(0)\rangle d\tau \,{{{{{{{\rm{and}}}}}}}}\,{\chi }_{{{{{{{{\rm{orb}}}}}}}}}=\int_{0}^{\beta }\langle \Delta {N}_{{{{{{{{\rm{orb}}}}}}}}}(\tau )\Delta {N}_{{{{{{{{\rm{orb}}}}}}}}}\rangle d\tau -\beta {\langle \Delta {N}_{{{{{{{{\rm{orb}}}}}}}}}\rangle }^{2}$, where S_z is the total spin angular momentum of Mn d orbitals and ΔN_orb = N_a − N_b is defined as the occupation difference between two orbitals (or the difference in average occupations of the two groups of orbitals) within the chosen orbital multiplet⁷⁴. In other words, we are interested in charge fluctuations within an orbital sector at our choice, and contrasting behaviors of χ_orb depending on the choice of orbital sectors can be a signature of orbital differentiation between them. In Fig. 5b, the DC orbital susceptibilities (ω = 0) for ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ and B_g orbitals are plotted as functions of temperature. The χ_orbs for ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ and B_g indeed show stark contrast; while χ_orb for ${A}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{out}}}}}}}}}$ orbitals remains almost constant, that for B_g shows sharp enhancement as T is lowered. This shows that the two orbital sectors are decoupled by J_H, and that the correlation-induced localization of electrons, via orbital-selective correlations and the emergence of correlation-induced B_g kagome bands, induces strong charge fluctuations within the B_g sector. Note that in the presence of spin-orbit coupling, this can lead to enhancement of spin fluctuation within the B_g sector as well.

In Fig. 5c, the DC spin susceptibility is plotted as a function of T. We observe that with decreasing temperature, the spin susceptibility gradually enhances. Because of computational cost issue, we could not lower the temperature below T = 116 K, so the quenching of orbital degree of freedom (i.e. peak of orbital susceptibility) prior to the spin one with decreasing T could not be captured in this study. In another study on the three-orbital Hubbard model, it has been argued that Mott physics is more dominant closer to the half-filling limit⁷³. For a better understanding of the nature of the low-T phase of SMAS, further studies are necessary in the near future.

Signatures of flat bands in magnetoresistance and optical conductivity

Signatures of correlation-induced flat dispersions can be further explored through magnetoresistance (MR) signals. Figure 6a, b present the temperature dependence of a transverse MR ratio measured in the field range of − 9 < B < 9 T for H//a and H//c orientations. Below 11 K, a negative MR is observed with its magnitude rapidly increasing as the temperature is lowered. This negative MR grows without saturation in fields up to 9 T and temperatures down to 2 K. Noteworthy is that the emergence of the negative MR effect is correlated with the observed upturn in resistivity (see Fig. 2a) and the Brillouin scaling, specifically the amplitude parameter A(T) exhibited in Fig. 2f. The similarities between A(T) and MR(T) suggest that the negative MR signal is linked to the formation of flat bands, which promote ferromagnetic fluctuations. Further evidence is seen in the deviation of MR(B) from the conventional B² (or B^1.5) dependence at fields above 4.5 T, as depicted by the dotted and solid lines in Fig. 6a, b. However, the B^−1/3 dependence typically expected for nearly ferromagnetic materials could not be identified within the measured field range up to 9 T, possibly due to the moderate ferromagnetic correlations⁷⁵. Additional high-field measurements are essential to definitively corroborate this dependence.

Fig. 6: Negative magnetoresistance signature.

Transverse magnetoresistance of SMAS measured at selected temperatures with a magnetic field applied to H//a a and H//c b. The dotted lines represent fits to a B² dependence and the solid lines to a B^1.5 dependence of the low-field magnetoresistance. c Angle-dependent magnetoresistance measured at T = 2 K in a magnetic field of B = 9 T.

The optical conductivity shown in Fig. 7a (see Supplementary Note 3 for more details) further provides evidence for enhancement of the electron scattering below 30 K. Below 1500 cm⁻¹, where the intra- and interband transitions are roughly divided, a spectral weight transfer from high- to low-frequency regimes occurs as the temperature is lowered, resulting in the sharpening of the Drude peak. But, there is a slight suppression of the Drude peak below 30 K, as can be seen in the inset of Fig. 7a, where the DC resistivity extracted by the extrapolation of the optical conductivity (σ₁(ω)) to zero frequency is shown; the DC resistivity exhibits a slight upturn below 30 K, which is consistent with the transport measurement (see Fig. 2a). The optical scattering rate (see Supplementary Note 3) presents a similar trend. Such a trend is observed in our computed optical conductivity shown in Fig. 7b, where the optical conductivities were obtained from DFT and DMFT (U = 8 eV and J_H = 0.8 eV) results. The DMFT optical conductivity spectra show similar frequency- and temperature-dependent behaviors as the optically measured ones; both optical conductivity spectra show a broad dip near 1500 cm⁻¹. Note that the suppression of the Drude peak by the inclusion of dynamical correlations is noticeable by comparing DFT (black curve) and DMFT (colored curves), which is consistent with the large values of the effective mass (see Fig. 4g, i, and Supplementary Note 3). DMFT σ₁(ω) also shows the enhancement of electron scattering as the temperature is lowered from 210 to 120 K; a slight suppression of σ₁(0) between T = 210 and 120 K is in qualitative agreement with experimental observations (note that the temperature scale is overemphasized in DMFT, where only the electronic temperature contributions at impurity sites are incorporated). The suppression of the Drude contribution in the low temperature regime can be attributed to the enhancement of orbital fluctuations, as shown in Fig. 5b, which may lead to the enhanced magnetic fluctuations as the spin-orbit coupling is included.

Fig. 7: Comparison between experimental and computed optical conductivities.

a Experimental optical conductivity of SMAS at various temperatures. In the inset, the DC resistivity obtained from the optical conductivity and measured DC resistivity are compared. b Computed optical conductivity from DMFT calculations for (U, J_H)= (8, 0.8) eV. In both cases normal incidence is considered.

Discussion

From our experimental results, we observed several interesting phenomena at low-temperature regimes, such as an upturn in resistivity below 30 K, deviation from Curie-Weiss behavior below 100 K, the power-law behavior of the magnetic susceptibility and internal fields suggesting ferromagnetic fluctuation, and non-saturating negative magnetoresistance down to T = 2 K. On the other hand, studying low-temperature phenomena below 100 K is computationally limited to our case due to the computational costs of CTQMC solver in the presence of significant hybridizations.

Nevertheless, our DMFT calculations reveal an unexpected emergence of kagome-induced flat band physics via electron correlations, which seems the only viable way to understand the observed ferromagnetic fluctuations. We speculate that as the temperature is lowered below 100 K, the flat band may shift even closer to E_F and may lead to various electronic instabilities including ferromagnetic ones as suggested in another kagome magnet FeSn^76,77. Indeed, our DFT+RISB^64,65 result shows that the presence of the flat bands close to the Fermi level can create the ferromagnetic instability, supporting our speculation above. Further investigation is necessary to explore the nature of the low-T ground state of this system.

There are several heavy-fermion compounds such as CeRh₆Ge₄, which have both the flat and dispersive bands at the Fermi level that host ferromagnetism^78,79,80. Despite differences in chemical compositions and correlated subspaces involved (d vs. f), there has been a growing interest in the universality between f-orbital-based Kondo systems and d- or p-orbital-based kagome flat band systems, where the heavy electron bands in f-based Kondo systems are analogous to the flat bands in d-based kagome lattices or even p-orbital-based twisted bilayers^81,82. Given that our system shows a large Sommerfeld coefficient⁵², and power-law behavior of magnetic susceptibility and specific heat in the low-temperature regime (below 10 K), we believe that our Mn-kagome Sc₃Mn₃Al₇Si₅ may share an interesting universality with a broader class of correlated materials.

Finally, we comment that a recent theoretical study suggests that the orbital-selectivity of electron correlations found in our system can be a general phenomenon in transition-metal-based kagome metal systems, where nearly flat kagome bands coexist with wide dispersive bands (such as ligand-originated bands or ones irrelevant to kagome-induced kinetic energy quenching)^60,83. Additionally, it has been also suggested that many kagome metals may host universal long-range Coulomb interactions. In combination with the SOC-induced gap opening and the wider spread of Berry curvature over momentum space induced via flat dispersion, one may ask about the possibility of realizing interesting correlated phenomena such as fractional Chern insulators^3,5,21 and Weyl-Kondo semimetal phase⁸⁴ on top of the on-site-correlations-induced flat bands in SMAS.

In summary, we have investigated the nature of electronic correlations and magnetic properties of Mn-based kagome metal Sc₃Mn₃Al₇Si₅, combining a multitude of experimental and theoretical techniques. The temperature and field-dependent magnetization measurement signifies the presence of ferromagnetic fluctuations at very low temperatures. The upturn in the resistivity alludes to the development of electron correlations below 30 K. The dynamical mean-field calculations reveal correlation-induced flat bands close to E_F at k_z = 0 with an additional nodal surface band at k_z = π guaranteed by nonsymmorphic twofold screw rotation and time-reversal symmetries. With the inclusion of spin-orbit coupling, a gap opens up at the Dirac points and the flat bands are likely to become topologically nontrivial. Therefore, SMAS constitutes a potentially promising platform to explore the interplay between electron correlations and SOC in kagome flat band systems.

Methods

Sample synthesis, magnetic and transport properties

High-quality SMAS single crystals were prepared using the self-flux method. The magnetic measurements were performed using a superconducting quantum interference device vibrating sample magnetometer (SQUID VSM). The NMR measurements were done by employing a home-made NMR spectrometer and an Oxford Teslatron PT superconducting magnet. The magnetoresistance (MR = [ρ(B) − ρ(0)]/ρ(0)) measurements were performed at ambient pressure using the electrical transport option of the Quantum Design Physical Properties Measurement System with a four-point contact configuration. To measure reflectance at various temperatures, a commercially available spectrometer Vertex 80v and a continuous liquid helium flow cryostat were used.

Electronic structure calculations

For an accurate and appropriate treatment of dynamic electron correlations in the electronic structure of SMAS, an ab-initio density functional theory (DFT) and dynamical mean-field theory (DMFT) methods were employed. The DFT calculations were carried out within the framework of local density approximation (LDA)⁸⁵, using a full potential linearized augmented plane wave plus local orbital (LAPW+lo) method. For investigation of dynamical correlation effect, a fully charge-self-consistent DMFT method, as implemented in the embedded DMFT functional code^67,86, was employed in combination with the wien2k package⁸⁷. Throughout the entire manuscript, rotationally-invariant full Coulomb interactions were used for the impurity problem unless otherwise specified. In some cases Ising-type density-density interactions were adopted. We checked that the choice of the Coulomb interactions does not affect our core results. A nominal double-counting scheme⁶⁷ with the Mn nominal charge n = 5.0 was adopted, where the validity of the double-counting parameter was justified by comparison to results from an exact double-counting scheme⁸⁸ (see Supplementary Note 7 and Supplementary Fig. 9 therein for more details). For our DFT+RISB calculations, we employed cygutz (https://cygutz.readthedocs.io/) package in combination with wien2k^64,65. RK_max = 9.0 was employed, and for a better convergence, a non-shifted k-grid of up to 17 × 17 × 14 was used. Mn d-orbital was set to be the correlated active subspace. More details on our experimental methods and computational parameters are listed in Supplementary Note 1 and 2, respectively.