Abstract
The interplay between electronic correlations and topological protection may offer a rich avenue for discovering emergent quantum phenomena in condensed matter. However, electronic correlations have so far been little investigated in Weyl semimetals (WSMs) by experiments. Here, we report a combined optical spectroscopy and theoretical calculation study on the strength and effect of electronic correlations in a magnet Co_{3}Sn_{2}S_{2}. The electronic kinetic energy estimated from our optical data is about half of that obtained from singleparticle ab initio calculations in the ferromagnetic ground state, which indicates intermediatestrength electronic correlations in this system. Furthermore, comparing the energy and sideslope ratios between the interbandtransition peaks at high energies in the experimental and singleparticlecalculationderived optical conductivity spectra with the bandwidthrenormalization factors obtained by manybody calculations enables us to estimate the Coulombinteraction strength (U ∼ 4 eV) in Co_{3}Sn_{2}S_{2}. Besides, a sharp experimental optical conductivity peak at low energy, which is absent in the singleparticlecalculationderived spectrum but is consistent with the optical conductivity peaks obtained by manybody calculations with U ∼ 4 eV, indicates that an electronic band connecting the two Weyl cones is flattened by electronic correlations and emerges near the Fermi energy in Co_{3}Sn_{2}S_{2}. Our work paves the way for exploring flatbandgenerated quantum phenomena in WSMs.
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Introduction
Electronic correlations, which is a type of manybody interactions–Coulomb interactions between electrons, lie at the heart of condensed matter physics due to their crucial roles in producing a variety of novel quantum phenomena, such as unconventional superconductivity^{1,2}, heavyfermion behavior^{3,4}, and Mott insulation^{5,6,7,8}. Thus, theoretical predictions and experimental observations of topological quantum states in real materials with significant electronic correlations have generated tremendous interest in the scientific community^{9,10,11,12}. Therein, Weyl semimetals (WSMs) represent a kind of topological quantum states, which host pairs of bulk Weyl cones and surface Fermi arcs connecting pairs of Weyl points with opposite chirality^{12,13,14,15,16,17,18,19,20,21,22}. Recently, theoretical studies indicate that sufficiently strong electronic correlations can gap out bulk Weyl nodes and thus break WSM states^{23,24,25,26,27,28,29,30,31,32,33}. Therefore, if a system which is predicted to exhibit a WSM phase in a noninteracting singleparticle picture has nonnegligible electronic correlations, it will be significant to investigate the influence of electronic correlations on its predicted WSM state^{34,35,36,37,38}. In addition, several correlated electron systems, such as kagomelattice compounds^{39,40} and heavyfermion materials^{41}, have been reported to host flat bands (i.e., dispersionless bands) which can provide a footstone for the emergence of various quantum phenomena, including superconductivity^{42,43}, ferromagnetism^{44,45}, and fractional quantum Hall effect^{46,47,48,49}. Nonetheless, electroniccorrelationinduced flat bands have rarely been observed in WSMs. Lately, singleparticle ab initio predictions of WSM states in 3dtransitionmetal compounds shed light on searching for correlated WSMs with flat bands, owing to the intimate association between the weak spatial extension of 3d orbitals and large Coulomb interactions^{50,51,52,53}.
A cobaltbased shandite compound, Co_{3}Sn_{2}S_{2}, crystallizes in a rhombohedral structure with the cobalt atoms forming a kagome lattice within one quasitwodimensional Co_{3}Sn layer (see Fig. 1a) and exhibits longrange ferromagnetic (FM) order with a magnetic moment of ∼0.3 μ_{B} (μ_{B} denotes the Bohr magneton) per cobalt atom below temperature T ∼177 K^{54,55,56,57}. Singleparticle ab initio calculations show that the electronic bands of FM Co_{3}Sn_{2}S_{2} near the Fermi energy (E_{F}) are dominated by cobalt 3d orbitals (see the electron density of states (DOS) for Co_{3}Sn_{2}S_{2} with the Co 3d, Sn 5p, and S 3p orbital contributions shaded in red, green, and purple colors, respectively in Fig. 1b)^{40}, but the strength of electronic correlations in this FM 3dtransitionmetal compound remains unclear. Furthermore, singleparticle ab initio calculations suggest that FM Co_{3}Sn_{2}S_{2} is a contender for magnetic WSMs^{50,51,52}. Up to now, important progresses in the experimental studies of the predicted WSM state in FM Co_{3}Sn_{2}S_{2}, which involve the measurements of negative magnetoresistance, giant intrinsic anomalous Hall, Nernst effects, bulk Weyl cones, and surface Fermi arcs^{50,52,58,59,60,61,62,63,64,65}, have been achieved. However, the influence of electronic correlations on the singleparticleabinitiocalculationderived WSM state in this WSM candidate, for example, inducing a flat band, remains elusive.
Results
Reduction of the electronic kinetic energy
Optical spectroscopy is a bulksensitive experimental technique for studying charge dynamics and electronic band structure of a material as it probes both itinerant charge carriers and interband transitions from occupied to empty states^{66,67,68,69,70,71}. Here, to investigate electronic correlations and their effects on the previously predicted WSM state in FM Co_{3}Sn_{2}S_{2}, we measured the optical reflectance spectra R(ω) of its single crystals at low temperatures with the electric field (E) of the incident light parallel to the crystalline abplane over a broad photon energy (ω) range (see the details about the reflectance measurements and the sample growth in “Methods” section). Fig. 1c depicts the R(ω) of Co_{3}Sn_{2}S_{2} single crystals measured at different temperatures. The R(ω) at energies lower than 20 meV not only approach to unity, but also increase as the temperature decreases, which exhibits the optical response of a metal. Moreover, the real parts (i.e., σ_{1}(ω)) of the abplane optical conductivity of Co_{3}Sn_{2}S_{2} in Fig. 1d, which were obtained by the Kramers–Kronig transformation of the R(ω) (see “Methods” section), show Drudelike features of metals at energies lower than 20 meV. The Drudelike features in the lowenergy parts of the σ_{1}(ω) and the fastincreasing value of the R(ω) at low energies indicate the existence of itinerant charge carriers in Co_{3}Sn_{2}S_{2}, which provides an opportunity for studying the electronic correlation effect on the motion of the itinerant charge carriers. Furthermore, several peaklike features arising from interband transitions are present in the highenergy parts of the σ_{1}(ω) (please see the possible relation between the decrease in the intensity of the peaklike feature and the absence of WSM phase in the paramagnetic (PM) state of this system in Supplementary Note 1). Comparing the energies and shapes of the experimental interbandtransitioninduced peaklike features with those of the peaklike features calculated without considering electronic correlations enables us to gain insights into the effect of electronic correlations on the bandwidth.
To study the electronic correlation effect on the motion of the itinerant charge carriers, we compare the experimentally measured kinetic energy with the theoretical kinetic energy calculated without taking any manybody interaction into account. Following the definition of the electronic kinetic energy in the optical study of a multiband system LaOFeP^{68}, we can obtain the linear relationship between the electronic kinetic energy (K) and the spectral weight (S) of the Drude component (i.e., the area under the Drude component) of the σ_{1}(ω):
where ω_{c} is a cutoff energy for integrating the Drude component of the σ_{1}(ω), ℏ is Planck’s constant divided by 2π, e is the elementary charge and d_{0} is the interCo_{3}Snlayer distance. Figure 2a displays the Drude components of the real part of the experimental optical conductivity \({\sigma }_{1}^{{\rm{E}}}\)(ω) at T = 8 K and the real part of the theoretical optical conductivity \({\sigma }_{1}^{{\rm{T}}}\)(ω) obtained by singleparticle ab initio calculations of the FM ground state of Co_{3}Sn_{2}S_{2} (see the Drude components over a broader range of the σ_{1}(ω) in Supplementary Fig. 2a). The cutoff energy ω_{c} is usually chosen as the energy position where σ_{1}(ω) reaches its minimum below the interband transition, so the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the \({\sigma }_{1}^{{\rm{T}}}\)(ω) here have the cutoff energies \({\omega }_{{\rm{c}}}^{8{\rm{K}}}\approx 19.9\pm 4\) meV and \({\omega }_{{\rm{c}}}^{{\rm{T}}}\approx 38.9\) meV, respectively. Integrating the Drude components of the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = K) and the \({\sigma }_{1}^{{\rm{T}}}\)(ω) up to the cutoff energies \({\omega }_{{\rm{c}}}^{8{\rm{K}}}\approx 19.9\pm 4\) meV and \({\omega }_{{\rm{c}}}^{{\rm{T}}}\approx 38.9\) meV yields approximately the spectral weights of the experimental and theoretical Drude components: S^{8K} ≈ (8.6 ± 0.6) × 10^{5} Ω^{−1} cm^{−2} and S^{T} ≈ 1.8 × 10^{6} Ω^{−1} cm^{−2}, respectively (see the red and blue points in Fig. 2b, the details about calculating the theoretical Drude spectral weight in the PM state S^{T} ≈ 4.0 × 10^{6} Ω^{−1} cm^{−2} in “Methods” section and the theoretical Drude component in the PM state in Supplementary Fig. 3), which is consistent with the smaller area under the experimental Drude component compared with that under the calculated Drude component (see the red shaded area and the blue area in Supplementary Fig. 2a). Here, the theoretical Drude spectral weight in the FM ground state S^{T} ≈ 1.8 × 10^{6} Ω^{−1} cm^{−2} is not impacted by the choice of the theoretical scattering rate which can significantly influence the cutoff energy (see Supplementary Note 2). Considering the linear relationship between the K and the S, which is shown in Eq. (1), we get the ratio between the experimental kinetic energy at T = 8 K and the theoretical kinetic energy: \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}={S}^{8{\rm{K}}}/{S}^{{\rm{T}}}\approx 0.47\pm 0.04\). To check the ratio \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}\) between the experimental and theoretical kinetic energies, an alternative method based on the linear relationship between the kinetic energy and the square \({\omega }_{{\rm{D}}}^{2}\) of the plasma energy can be employed^{66,67,70}. By fitting the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of Co_{3}Sn_{2}S_{2} based on a standard Drude–Lorentz model, we can obtain the experimental plasma frequency at T = 8 K in its FM state: \({\omega }_{{\rm{D}}}^{{\rm{E}}}\) = 258 ± 4 meV (see the details in “Methods” section). Furthermore, the theoretical plasma energy \({\omega }_{{\rm{D}}}^{{\rm{T}}}\) of FM Co_{3}Sn_{2}S_{2} can be directly calculated from the singleparticleabinitiocalculationderived band structure, i.e., \({\omega }_{{\rm{D}}}^{{\rm{T}}}=379\) meV. Given the linear relationship between the kinetic energy and the square \({\omega }_{{\rm{D}}}^{2}\) of the plasma frequency ω_{D}, we can get the ratio between the experimental kinetic energy at T = 8 K and the theoretical kinetic energy in the FM ground state: \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}={({\omega }_{{\rm{D}}}^{{\rm{E}}}/{\omega }_{{\rm{D}}}^{{\rm{T}}})}^{2}\approx 0.46\pm 0.02\), which is consistent with the kineticenergy ratio \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}={S}^{8{\rm{K}}}/{S}^{{\rm{T}}}\approx 0.47\pm 0.04\) inferred from ratio between the integrations of the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the \({\sigma }_{1}^{{\rm{T}}}\)(ω) in the FM ground state up to the cutoff energies. Therefore, the ratios \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}\) deduced by the above two methods indicate that the experimental kinetic energy of FM Co_{3}Sn_{2}S_{2} at T = 8 K is significantly smaller than the theoretical kinetic energy obtained by singleparticle ab initio calculations of FM Co_{3}Sn_{2}S_{2}.
To check whether the substantial reduction of the experimental kinetic energy (or experimental Drude weight) compared with the theoretical kinetic energy (or theoretical Drude weight) can arise from the change in the Fermi level of FM Co_{3}Sn_{2}S_{2}, we performed singleparticle ab initio calculations of the \({\sigma }_{1}^{{\rm{T}}}\)(ω) with different E_{F}. The above theoretical \({\sigma }_{1}^{{\rm{T}}}\)(ω) in Fig. 2a was obtained with E_{F} = 0 eV. When E_{F} = 0 eV, the corresponding Fermi level is located at ∼ 0.06 eV below the calculated Weyl point^{50,51,52}, while the Fermi level measured by angleresolved photoemission spectroscopy (ARPES) is located at ∼0.05 eV below the Weyl point^{63}. Thus, the Fermi level measured by ARPES is ∼0.01 eV higher than the theoretical one corresponding to E_{F} = 0 eV. Fig. 2c shows the theoretical \({\sigma }_{1}^{{\rm{T}}}\)(ω) calculated with E_{F} = 0.01 eV in the energy range up to 1200 meV. The lowenergy parts of the \({\sigma }_{1}^{{\rm{T}}}\)(ω, E_{F} = 0.01 eV) and \({\sigma }_{1}^{{\rm{T}}}\)(ω, E_{F} = 0 eV) in Fig. 2d indicate that the theoretical Drude weight S^{T}(E_{F} = 0.01 eV) is larger than the theoretical Drude weight S^{T}(E_{F} = 0 eV). As listed in Table 1, the ratio between the calculated S^{T}(E_{F} = 0.01 eV) and S^{T}(E_{F} = 0 eV) is ∼1.03. Thus, if the Fermi level is shifted up by 0.01 eV, the corresponding theoretical Drude weight S^{T}(E_{F} = 0.01 eV) will be larger than the S^{T}(E_{F} = 0 eV), which means that the theoretical kinetic energy at E_{F} = 0.01 eV will be larger than the theoretical one at E_{F} = 0 eV. In addition, we calculated the \({\sigma }_{1}^{{\rm{T}}}\)(ω) with the negative Fermi energies E_{F} = −0.07, −0.12, and −0.26 eV. Fig. 2d, together with Table 1, shows that the theoretical Drude weights S^{T}(E_{F}) corresponding to these negative Fermi energies are also larger than the theoretical Drude weight S^{T}(E_{F} = 0 eV), i.e., the theoretical kinetic energies at these negative Fermi energies are larger than the theoretical one at E_{F} = 0 eV as well. Therefore, the upshifting and lowering of the Fermi energy are unlikely to reduce the electronic kinetic energy of FM Co_{3}Sn_{2}S_{2}. Fig. 2e shows that the deduced ratios \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}\) are distinctly smaller than unity—the kineticenergy ratio in conventional metals (such as Ag and Cu) with quite weak effects of manybody interactions. Here, the substantial reduction in the electronic kinetic energy compared with the K^{T} indicates that manybody interactions which have not been taken into account by singleparticle ab initio calculations in the FM ground state have a pronounced effect of impeding the motion of the itinerant charge carriers in FM Co_{3}Sn_{2}S_{2}. In contrast, ordered spin–spin correlations in itinerant ferromagnets usually correspond to an increase in the kinetic energy of itinerant charge carriers^{72}, because (i) according to the Pauli exclusion principle, a larger kinetic energy is needed for the itinerant charge carriers with parallel spins to meet in the same lattice sites^{73}, and (ii) in the framework of the Stoner model, a phase transition from paramagnetism to itinerant ferromagnetism is accompanied with the increase in the electronic kinetic energy which is outweighed by the lowering of the exchange energy^{74}. Thus, ordered spin–spin correlations in FM Co_{3}Sn_{2}S_{2} are highly likely to be irrelevant with the remarkable reduction of the electronic kinetic energy here. Moreover, note that extremely strong electron–phonon coupling in a polar semiconductor or an ionic crystal can lead to a significant reduction of the electronic kinetic energy owing to the formation of polarons^{75}. Nevertheless, the calculated cobalt3dorbitaldominated bands which cross the E_{F}^{40,50,51,52} and the measured magnetic moment (∼0.3 μ_{B}/Co) which is much smaller than the magnetic moment (3 μ_{B}/Co) of isolated cobalt atoms^{54,55,56,57} strongly suggest that FM Co_{3}Sn_{2}S_{2} should not be a polar semiconductor or an ionic crystal, either of which has been found to host polarons. Generally, electron–phonon coupling in a material with the absence of polarons would not make the ratio between experimental and theoretical kinetic energies much smaller than unity (see the K^{E}/K^{T} in MgB_{2} superconductor with electron–phonon mediated conventional superconductivity in Fig. 2e)^{76,77}, so for FM Co_{3}Sn_{2}S_{2}, electron–phonon coupling is also unlikely to be the main factor causing the substantial decrease in its electronic kinetic energy. Based on the above discussion, electronic correlations, which were previously revealed to result in the remarkable lowerings of the electronic kinetic energies in some transitionmetalbased superconductors, such as the iron pnictides LaOFeP and LaOFeAs (see Fig. 2e)^{68,70}, should play a dominant role in hampering the motion of the itinerant charge carriers in FM Co_{3}Sn_{2}S_{2}. Since the \({K}_{8{\rm{K}}}^{{\rm{E}}}/{K}^{{\rm{T}}}\) in FM Co_{3}Sn_{2}S_{2} is approximately equal to the average of the kineticenergy ratio (∼0) in Mott insulators (like Nd_{2}CuO_{4} and Sr_{2}CuO_{4}) with very strong electronic correlations (see Fig. 2e) and the kineticenergy ratio (∼1) in conventional metals (such as Ag and Cu) with quite weak electronic correlations, the strength of electronic correlations in FM Co_{3}Sn_{2}S_{2} can be regarded to be intermediate.
Narrowness of the electronic bandwidth
To investigate the effect of manybody interactions on the electronic bandwidth of FM Co_{3}Sn_{2}S_{2}, we plotted the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the \({\sigma }_{1}^{{\rm{T}}}\)(ω) over a broad energy range up to 1350 meV in Fig. 2f. The overall shape of the \({\sigma }_{1}^{{\rm{E}}}\)(ω) at ω > 20 meV is similar to that of the \({\sigma }_{1}^{{\rm{T}}}\)(ω), but (i) the energy positions of the two peaklike features arising from interband transitions, α_{E} at ∼ 217.4 meV and β_{E} at ∼708.1 meV, in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) are distinctly lower than those of the two corresponding peaklike features in the \({\sigma }_{1}^{{\rm{T}}}\)(ω), α_{T} at ∼ 319.7 meV and β_{T} at ∼ 931.6 meV, respectively; and (ii) the left side of the experimental peaklike feature β_{E} is significantly steeper than that of the theoretical peak β_{T} (the red and blue dashed lines in Fig. 2f are guides for eyes showing the slopes of the left sides of the peaks β_{E} and β_{T}, respectively). Generally, when only the widths of the conduction and valence bands related to interband transitions are reduced, the width of the peaklike feature in σ_{1}(ω) arising from the interband transitions between these related bands decreases due to the reduced widths of these bands (see Supplementary Fig. 5). The reduction in the width of the peaklike feature in σ_{1}(ω), together with the unchanged height of the peaklike feature in σ_{1}(ω), would further lead to the increase in the slope of the sides of the peaklike feature (see the red dashed lines in Supplementary Fig. 5b, d). Therefore, in Fig. 2f, the steeper leftside of the experimental peak β_{E} in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of FM Co_{3}Sn_{2}S_{2}, combined with the comparability between the heights of the experimental and theoretical peaks β_{E} and β_{T}, indicates the manybodyinteractioninduced narrowing of the widths of the electronic bands in FM Co_{3}Sn_{2}S_{2}. In addition, the energy intercept of the red dashed line at the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) = 0, which can approximately represent the energy gap between the conduction and valence bands (see Supplementary Fig. 6) (or the minimal energy difference between the occupied and empty states in the electronic bands displayed in Supplementary Fig. 7), locates at lower energy than the energy intercept of the blue dashed line at the theoretical \({\sigma }_{1}^{{\rm{T}}}\)(ω) = 0. This indicates that in FM Co_{3}Sn_{2}S_{2}, the experimental energy gap between the conduction and valence bands (or the experimental minimal energy difference between the occupied and empty states in the electronic bands) is smaller than the theoretical band gap (or the theoretical minimal energy difference). Besides, it is worth noticing that the change in the energy range (i.e., Δω) from the energy intercept of the dashed lines at σ_{1}(ω) = 0 to the energy position of the opticalconductivitypeakheight maximum can also reflect the renormalization of the widths of the electronic bands. Correspondingly, the ratio (∼0.76) between the energy range Δω_{E} in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the energy range Δω_{T} in the theoretical \({\sigma }_{1}^{{\rm{T}}}\)(ω), which is comparable to the ratio (∼0.74) between the slopes of the left sides of the peaks β_{E} and β_{T}, indicates the narrowing of the widths of the electronic bands in FM Co_{3}Sn_{2}S_{2} as well. Therefore, the redshift of the interbandtransitioninduced peak β_{E} in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) compared with the theoretical peak β_{T} in the calculated \({\sigma }_{1}^{{\rm{T}}}\)(ω) not only indicates the decrease in the energy gap between the occupied and empty band (or the minimal energy difference between the occupied and empty states in the electronic bands), but also reflects the manybodyinteractioninduced narrowing of the widths of the electronic bands in FM Co_{3}Sn_{2}S_{2}.
Given that (i) according to the Pauli exclusion principle and the Stoner model, ferromagnetically ordered spin–spin correlations usually leads to a gain in the electronic kinetic energy^{72,75,74}, (ii) the gain in the electronic kinetic energy mostly corresponds to an extension of the electronic bandwidth^{72,73,74}, and (iii) the cutoff energy (∼50.8 meV) of the phonon spectrum shown in Fig. 2g is much lower than the energies of the interbandtransitioninduced peaks, α_{E}, α_{T}, β_{E}, and β_{T}, ferromagnetically ordered spin–spin correlations and electron–phonon coupling are unlikely to be the leading interactions, which cause the narrowing of the electronic bandwidth here. Therefore, electronic correlations in FM Co_{3}Sn_{2}S_{2} ought to play a major part in narrowing the electronic bandwidth.
To estimate the Coulombinteraction strength U of electronic correlations in FM Co_{3}Sn_{2}S_{2}, we performed manybody calculations, i.e., combination of density functional theory and dynamical meanfield theory (DFT + DMFT) (see the details in “Methods” section)^{78,79}, and then obtained the U dependences of the electronicbandwidth renormalization factor quantifying the effect of electronic correlations on narrowing the electronic bandwidth (see Fig. 2h). The ratio between the slopes of the left sides of the experimental and theoretical peaks, S(β_{T})/S(β_{E}) ≈ 0.74 and the energy ratios between the experimental and theoretical peaks in Fig. 2f, E(α_{E})/E(α_{T}) ≈ 0.68 and E(β_{E})/E(β_{T}) ≈ 0.76, which are comparable to the ratio (∼0.70) between the recently measured bandwidth and the calculated bandwidth^{63}, reflect the electronicbandwidth renormalization effect of the electronic correlations with U ∼ 4 eV, shown in Fig. 2h. Here, the difference in the two energy ratios may arise from the discrepancy between the renormalization factors of the five 3d orbitals (see Fig. 2h) and the difference between the energy distributions of the 3delectron DOS gotten by DFT + DMFT calculations (see Fig. 2i).
Persistence of a Weyl semimetal state
To check whether a WSM state still exist in correlated Co_{3}Sn_{2}S_{2}, we carried out DFT+DMFT calculations with U ∼ 4 eV to obtain its electronic surface and bulk states (see “Methods” section). In Fig. 3a, compared with the abinitiocalculationderived bulk bands (see the red dashed curves) along the highsymmetry lines of the Brillouin zone (see the upleft inset), the bulk momentumresolved electronic spectra (see the bright yellow curves) gotten by DFT + DMFT calculations are indeed renormalized. In Fig. 3b, the Fermiarc structures on the (001) surface, which are based on the quasiparticle bands from DFT + DMFT calculations, connect three pairs of Weyl points, respectively. Fig. 3c depicts a pair of bulk Weyl cones along the direction (i.e., W_{1}–W_{2}) connecting the Weyl points W_{1} (i.e., blue point) and W_{2} (i.e., red point) in Fig. 3b. To study the chirality of these two Weyl points W_{1} and W_{2}, we calculated the Berry curvature around each Weyl point. As displayed in Fig. 3d, e, W_{1} and W_{2} act as a source and a sink of Berry curvature, respectively, so W_{1} and W_{2} have opposite chirality^{12,13,14,15,16,17}. The surface Fermi arc connecting each pair of Weyl points with opposite chirality and the bulk Weyl cones in FM Co_{3}Sn_{2}S_{2}, which were obtained by our DFT + DMFT calculations, indicate the existence of a WSM state in this magnetic system with intermediatestrength electronic correlations.
Flat band connecting the two Weyl cones
To further search for possible effects of intermediatestrength electronic correlations on the WSM state in FM Co_{3}Sn_{2}S_{2}, we derived its quasiparticle band structure along the direction W_{1}–W_{2} connecting the two Weyl points via DFT + DMFT calculations. The left panel of Fig. 4a shows that (i) a band B_{0} obtained by singleparticle ab initio calculations, which not only is a part of the two Weyl cones but also links the two Weyl cones, is turned into a flat band B_{1} near E_{F} in the quasiparticle band structure by electronic correlations, and that (ii) two bands B_{2} and B_{3}, which have dispersionless parts and are absent along W_{1}–W_{2} in the singleparticle band structure, emerges below E_{F} in the quasiparticle band structure. Since (i) the flat band B_{1} and the dispersionless parts of bands B_{2} and B_{3} have divergent DOS (see the right panel of Fig. 4a) and (ii) optical absorptions are determined by the joint DOS of the initial and final state, the four interband transitions related to the flat band B_{1} or the dispersionless part of band B_{2} (or B_{3}), which are illustrated by the four colored arrows in the left panel of Fig. 4a, cause the four obvious peaklike features T_{1}, T_{2}, T_{3}, and T_{4} around 38, 70, 113, and 131 meV in the real part of the optical conductivity \({\sigma }_{1}^{{\rm{QP}}}\)(ω) contributed by the direct optical transitions between the calculated quasiparticle bands, respectively (see the green spectrum Fig. 4b and the details on calculating the \({\sigma }_{1}^{{\rm{QP}}}\)(ω) in “Methods” section). Therein, (i) the strongest peaklike feature T_{1} comes primarily from the optical transitions between the flat bands B_{1} and the top of band B_{2} (see the yellow arrow in Fig. 4a), and (ii) the second strongest peaklike feature T_{2} arises mainly from the optical transitions between the flat band B_{1} and the top of band B_{3} (see the pink arrow in Fig. 4a). Considering that the present of the peaklike features T_{1} and T_{2} are intimately associated with the existence of the flat band B_{1}, the peaklike features T_{1} around 38 meV and T_{2} around 70 meV can be regarded as two spectroscopic signatures of the existence of the flat band B_{1}. It is worth noticing that the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of FM Co_{3}Sn_{2}S_{2} has an asymmetric peaklike feature around 36 meV, which cannot be well reproduced by only one Lorentzian term in a standard DrudeLorentz model or a single TaucLorentzian term in the Drude–Tauc–Lorentz model (see the Drude–Tauc–Lorentz fit to the asymmetric peaklike feature around 36 meV in Fig. 4c and the details about the Drude–Tauc–Lorentz fit in “Methods” section)^{66,67,68,69,70,71,80}. By fitting the lowenergy part of the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) based on the Drude–Lorentz model, we find that this experimental peaklike feature can be decomposed into four components: a Lorentzian peak with the strongest intensity around 36 meV, a Lorentzian peak with the second strongest intensity around 70 meV, a Lorentzian peak around 113 meV and a Lorentzian peak around 131 meV (see “Methods” section, the gray spectrum and the shaded peaks in Fig. 4b), which are consistent with the four peaklike features originating from the optical transitions related to the flat band B_{1} or the dispersionless part of band B_{2} (or B_{3}). Besides, this asymmetric peaklike feature in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) becomes weaker as the temperature increases and disappears completely above the FM transition temperature (i.e., not in the WSM state) (see Fig. 1d), which is in agreement with the absence of the Weyl cones and the flat band connecting the Weyl cones above the FM transition temperature. Therefore, the experimental peaklike feature around 36 meV in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K), which is obvious only in the FM state at low temperatures (i.e., in the WSM state) and includes the two Lorentzian peaks around the energy positions of T_{1} and T_{2}, provides spectroscopic evidence for the existence of the flat band B_{1} near E_{F} in FM Co_{3}Sn_{2}S_{2}.
To check whether the experimental peaklike feature around 36 meV in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) could be explained by the exchange splitting of the double exchange model, we investigated the relationship between the relative square \(\Delta {\omega }_{{\rm{scr}}}^{2}\) of the screened plasma frequency normalized to the \({\omega }_{{\rm{scr}}}^{2}\) at T ≈ 200 K and the square χ^{2} of the magnetic susceptibility because a linear relationship between \(\Delta {\omega }_{{\rm{scr}}}^{2}\) and χ^{2} was not only expected in the double exchange model but also observed in the manganites and FM semiconductors Ga_{1−x}Mn_{x}As^{81,82,83,84,85}. Fig. 4d shows the real part ε_{1}(ω) of the dielectric function of Co_{3}Sn_{2}S_{2} at different temperatures, which can be obtained from the imaginary part σ_{2}(ω) of its experimental abplane optical conductivity. Since the screened plasma frequency ω_{scr} is equal to the energy at which the real part of the dielectric function ε_{1}(ω) = 0, we can plot the \({\omega }_{{\rm{scr}}}^{2}\) as a function of temperature in the inset of Fig. 4d. The experimental \({\omega }_{{\rm{scr}}}^{2}\)(T = 8 K) in the FM state is smaller than the experimental \({\omega }_{{\rm{scr}}}^{2}\)(T = 200 K) in the PM state, which is consistent with the singleparticleabinitiocalculationderived result that the theoretical \({\omega }_{{\rm{scr}}}^{2}\) in the FM ground state is lower than the theoretical one in the PM state (see Supplementary Fig. 3 and the details about the singleparticle ab initio calculations of the PM state in Methods section). Fig. 4e displays the \(\Delta {\omega }_{{\rm{scr}}}^{2}\) and the χ of Co_{3}Sn_{2}S_{2} at different temperatures. As shown in the inset of Fig. 4e, the \(\Delta {\omega }_{{\rm{scr}}}^{2}\) in the FM state of Co_{3}Sn_{2}S_{2} does not exhibit a linear dependence on χ^{2}, which is inconsistent with the linear relationship between the \(\Delta {\omega }_{{\rm{scr}}}^{2}\) and the χ^{2} within the double exchange model. Therefore, it seems unlikely that the experimental peaklike feature around 36 meV in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of FM Co_{3}Sn_{2}S_{2} could arise from exchangesplittinginducedinterband transitions.
In addition, singleparticle ab initio calculations indicate that the Weyl points of FM Co_{3}Sn_{2}S_{2} are located at ∼60 meV above the Fermi level^{50,51,52}. According to Pauli’s exclusion principle, the onset energy E_{onset} of the interband transitions between the occupied and empty states of the Weyl cones with the linear dispersions is about double the energy difference between the Weyl point and the Fermi level, i.e., E_{onset} ∼ 120 meV, which is much higher than the energy position (∼36 meV) of the asymmetric peaklike feature in the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K). Thus, the asymmetric peaklike feature around 36 meV cannot originate from the interband transitions between the occupied and empty states of the Weyl cones with linear dispersions.
Moreover, in stark contrast to the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the calculated \({\sigma }_{1}^{{\rm{QP}}}\)(ω), the interbandtransitioncontributed part of the \({\sigma }_{1}^{{\rm{T}}}\)(ω) obtained by singleparticle ab initio calculations in the FM ground state with the different Fermi energies has no distinct peaklike feature around 38 meV (see Figs. 2c and 4b), which further supports that electronic correlations in FM Co_{3}Sn_{2}S_{2} flatten the band linking the two Weyl cones and induce the emergence of the flat band B_{1}.
Discussion
In summary, we have investigated electronic correlations in FM Co_{3}Sn_{2}S_{2}. The electronic kinetic energy extracted from the measured optical data is about half of that deduced by singleparticle ab initio calculations, which indicates that the strength of electronic correlations in Co_{3}Sn_{2}S_{2} is intermediate. The energies of the two interbandtransitioninduced peaks in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of Co_{3}Sn_{2}S_{2} are significantly lower than those in the \({\sigma }_{1}^{{\rm{T}}}\)(ω) obtained by singleparticle ab initio calculations in the FM ground state. In addition, the left side of the experimental peak β_{E} is distinctly steeper than that of the theoretical peak β_{T}. The redshift and the steeperside of the interbandtransitioninduced peak in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) compared with the theoretical peak in the theoretical \({\sigma }_{1}^{{\rm{T}}}\)(ω) indicate that its electronic bandwidth and band gap (or the minimal energy difference between the occupied and empty state in the electronic bands) are narrowed by electronic correlations. Furthermore, by comparing the energy ratios between the interbandtransition peaks in the experimental and singleparticleabinitiocalculationderived real parts of the optical conductivity with the electronicbandwidth renormalization factors gotten by DFT + DMFT calculations, we estimated the Coulombinteraction strength (U ∼ 4 eV) of electronic correlations in this material. Our DFT + DMFT calculations with U ∼ 4 eV show that a WSM state still exists in this correlated system. Besides, the consistence between the asymmetric peaklike feature around 36 meV in the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) and the DFT + DMFTcalculationderived peaklike features in the \({\sigma }_{1}^{{\rm{QP}}}\)(ω) reveals an electronic band connecting the two Weyl cones is flattened by electronic correlations and is present near E_{F} in FM Co_{3}Sn_{2}S_{2}. Our results not only reveal the effects of electronic correlations in FM Co_{3}Sn_{2}S_{2}, but also open an avenue for deeply investigating exotic quantum phenomena dominated by flat bands in WSMs. After submission of this work, we became aware of similar optical data about Co_{3}Sn_{2}S_{2}^{86}.
Methods
Optical reflectance measurements
The optical reflectance measurements in the energy range from 8 to 6000 meV were performed on a Bruker Vertex 80v Fouriertransform spectrometer. The singlecrystal sample was mounted on an optically black cone locating at the cold finger of a helium flow cryostat. A freshlycleaved abplane of the Co_{3}Sn_{2}S_{2} single crystal was obtained just before pumping the cryostat. An in situ gold and aluminum overcoating technique was employed to get the reflectance spectra R(ω). The optical reflectance data are highly reproducible. Moreover, J.A. Woollam RC2 spectroscopic ellipsometer was used to get the optical constants of the Co_{3}Sn_{2}S_{2} single crystals in the energy range from 500 to 6000 meV, which are consistent with the optical constants extracted from the measured reflectance spectra in this energy range.
Singlecrystal growth
The Co_{3}Sn_{2}S_{2} single crystals were grown by a selfflux method. Highpurity elemental Co, Sn, and S with a molar ratio of 3:2:2 were put into an alumina crucible and then sealed in a quartz tube under high vacuum. The quartz tube was slowly heated to 637 K and maintained for two days due to the high vapor pressure of sulfur. Afterward, the quartz tube was heated to 1273 K within 10 h and then slowly cooled down to 973 K before switching off the furnace. Shining crystal faces can be obtained by cleaving the Co_{3}Sn_{2}S_{2} single crystals.
Kramers–Kronig transformation
The σ_{1}(ω) were obtained by the Kramers–Kronig transformation of the R(ω). A Hagen–Rubens relation was used for lowenergy extrapolation, and a ω^{−0.15} dependence was used for highenergy extrapolation up to 80,000 meV, above which a ω^{−4} dependence was employed. The reciprocal value of the obtained σ_{1}(ω = 0) at each temperature coincides with the direct current resistivity obtained by the transport measurements (see Supplementary Fig. 2b), which indicates that the Kramers–Kronig transformation of the R(ω) here is reliable.
Singleparticle ab initio calculations
Our singleparticle ab initio optical conductivity calculations were performed at T = 0 K in the FM and PM ground states of Co_{3}Sn_{2}S_{2} by using the full potential linearized augmented plane wave method implemented in the WIEN2k package (see the spinpolarized bands in Supplementary Fig. 1b)^{87}. The kpoint mesh for the Brillouin zone integration is 36 × 36 × 36, and the plane wave cutoff \({K}_{\max }\) is given by \({R}_{{\rm{mt}}}* {K}_{\max}=8.0\). The spin–orbit coupling effects are included in our calculations. The phonon dispersions were calculated by using the open source code PHONOPY^{88}. The phonon force constants in real space were calculated based on the densityfunctional perturbation theory method using Vienna ab initio simulation package (VASP)^{89} with a 2 ×2 × 2 supercell. The plane wave energy cutoff was chosen as 400 eV, and a Γcentered kpoint grid with 3 × 3 × 3 discretization was used.
Furthermore, we employed two different methods HSE06 hybrid functional method in VASP package and mBJ method in WIEN2k package to calculate the electronic band structure of FM Co_{3}Sn_{2}S_{2}. The electronic band structures calculated by HSE06 hybrid functional method and mBJ method do not exhibit band inversions near the Fermi energy (please see the detailed results in Supplementary Note 3).
Fitting based on the Drude–Lorentz model
We fit the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of Co_{3}Sn_{2}S_{2} using a standard Drude–Lorentz model^{66,67,68,69,70,71}:
where Z_{0} ≈ 377 Ω is the impedance of free space, ω_{D} is the plasma frequency, and Γ_{D} is the relaxation rate of itinerant charge carriers, while ω_{j}, Γ_{j}, and S_{j} are the resonance frequency, the damping, and the mode strength of each Lorentz term, respectively. The first term in Eq. (2) denotes the optical response of free carriers, i.e., Drude response. The Lorentzian terms can describe the contributions from interband transitions. The parameters of the four Lorentzian terms and the Drude term for fitting the lowenergy part of the \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) are listed in Table 2.
We further fit the lowenergy part of the experimental \({\sigma }_{1}^{{\rm{E}}}\)(ω, T = 8 K) of Co_{3}Sn_{2}S_{2} using a Drude–Tauc–Lorentz model. The Tauc–Lorentzian term for fitting can be expressed as^{80}:
Here, E_{g} is the band gap, E_{0} is the peaktransition energy, and C is the peak broadening term. The parameters of the Tauc–Lorentzian term for fitting are listed in Table 3.
Manybody calculations
The method of density functional theory plus dynamical mean field theory (DFT + DMFT) can capture dynamic quantum fluctuation effects and thus is suitable for investigating the quasiparticles in correlated metals, while DFT+U method is a static HatreeFock approach (see the band structures of Co_{3}Sn_{2}S_{2} obtained by DFT+U calculations in Supplementary Fig. 9 and Supplementary Note 4). The correlated electronic structure of Co_{3}Sn_{2}S_{2} was obtained by DFT + DMFT calculations. A Wannier tight binding Hamiltonian consisting of 3d orbitals of the three Co atoms, and p orbitals of the two Sn atoms and the two S atoms was constructed using the Wannier90 package^{90}. The hybridization between the d orbitals and the p orbitals, together with the spin–orbit coupling effect, is included in our model. Only 3d electrons in Co are treated as correlated ones in DFT + DMFT calculations. We chose the fully localized form \({\Sigma }_{{\rm{D}}C}=U({n}_{d}^{0}\frac{1}{2}) \frac{1}{2}J({n}_{d}^{0}1)\), where \({n}_{d}^{0}\) is nominal occupation of 3d orbitals, as the “doublecounting” scheme.
We used the hybridization expansion version of the continuoustime quantum Monte Carlo (HYBCTQMC) method implemented in the iQIST package^{91,92} as the impurity solver. The local onsite Coulomb interactions are parameterized by the Slater integrals F^{0}, F^{2}, and F^{4}. Hubbard U and Hund’s coupling J amount to U = F^{0}, J = (F^{2} + F^{4})/14. The constrained DFT calculations suggest U = 5.1 eV and J = 0.9 eV for Co^{2+} in CoO^{93}. Besides, the experimental optical absorption data indicate J_{H} ≈ 0.8 eV^{94}. Thus, in order to check the effective U and J related to the renormalization factor \({\mathcal{Z}}\), we fixed the ratio of J/U = 0.2 to change U, which was also used in d^{7} cobalt compounds study^{95}. We only keep the density–density terms of the Coulomb interactions for computational efficiency. The inverse temperature is β = 1/(K_{B}T) = 40 eV^{−1}. The standard deviation of the selfenergy is <0.03 in the last selfconsistent loop. We used the analytical continuation method introduced by K. Haule^{96} to extract the selfenergy Σ(ω) on real axis from the Matsubara selfenergy Σ(iω) obtained from CTQMC.
In order to study the topological electronic structure of Co_{3}Sn_{2}S_{2}, we calculated the momentumresolved spectra, which is defined as
where, H_{0}(k) is the noninteraction Hamiltonian at each kpoint from DFT calculation, \(\tilde{\Sigma }(k,\omega )={\hat{P}}_{k}(\Sigma (\omega ){\Sigma }_{dc})\), \({\hat{P}}_{k}\) are the projection operators.
The lowenergy quasiparticle (QP) behavior is described by the following QP Hamiltonian,
Our (001) surface electronic structure, Fermi arcs and Berry curvature were calculated based on the lowenergy QP Hamiltonian (See Supplementary Note 5). The surface spectra (i.e., Fermi arcs) were calculated by using the iterative Green’s function method^{97} as implemented in the WannierTools package^{98}. The QP band structure of FM Co_{3}Sn_{2}S_{2} obtained from the QP Hamiltonian here cannot totally capture the effect of electronic correlations—the reduction of its Drude spectral weight^{79}.
The real part of the optical conductivity \({\sigma }_{1}^{{\rm{QP}}}\)(ω) contributed by the direct optical transitions between the calculated quasiparticle bands in the main text was calculated by the Kubo–Greenwood formula as implemented in the Wannier90 package^{66,71,90}.
Data availability
Data measured or analyzed during this study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Xi Dai, Hongming Weng, Rui Yu, Quansheng Wu, and Xiaoyu Deng for very helpful discussions. The authors acknowledge support from the National Key Research and Development Program of China (Projects Nos. 2017YFA0304700, 2016YFA0300600, 2017YFA0302901, 2016YFA0300504, 2017YFA0303800, 2016YFA0302400, and 2018YFA0307000), the strategic Priority Research Program of Chinese Academy of Sciences (Project No. XDB33000000), the Pioneer Hundred Talents Program of the Chinese Academy of Sciences, the National Natural Science Foundation of China (Projects Nos. 11604273, 11774399, 11574394, 11774423, 11822412, 11721404, and 11874022), Longshan Academic Talent Research Supporting Program of SWUST (Project No. 17LZX527) and Beijing Natural Science Foundation (Project No. Z180008). A.S. thanks the support of SNSF NCCR MARVEL and QSIT grants and of Microsoft Research and SNSF Professorship. J.Z. acknowledges the Pauli Center for funding his visit in UZH. This work is in memory of Alexey Soluyanov, who passed away during the peer review of this paper. As a friend, his support continue to live on through everyone who knew him. Y.W. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences as a part of the Computational Materials Science Program through the Center for Computational Design of Functional Strongly Correlated Materials and Theoretical Spectroscopy.
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ZG.C. conceived and supervised this project. Y.X. and X.H. carried out the optical experiments. J.Z. did firstprinciple and manybody calculations. C.Y., Q.W., Q.Y., H.L., and Y.S. grew the single crystals. ZG.C., J.Z., Y.W., E.L., L.W., G.X., L.L., A.S., and J.L. analyzed the data. ZG.C. wrote the paper.
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Xu, Y., Zhao, J., Yi, C. et al. Electronic correlations and flattened band in magnetic Weyl semimetal candidate Co_{3}Sn_{2}S_{2}. Nat Commun 11, 3985 (2020). https://doi.org/10.1038/s41467020172340
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DOI: https://doi.org/10.1038/s41467020172340
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