Correlation driven near-flat band Stoner excitations in a Kagome magnet

Among condensed matter systems, Mott insulators exhibit diverse properties that emerge from electronic correlations. In itinerant metals, correlations are usually weak, but can also be enhanced via geometrical confinement of electrons, that manifest as ‘flat’ dispersionless electronic bands. In the fast developing field of topological materials, which includes Dirac and Weyl semimetals, flat bands are one of the important components that can result in unusual magnetic and transport behaviour. To date, characterisation of flat bands and their magnetism is scarce, hindering the design of novel materials. Here, we investigate the ferromagnetic Kagomé semimetal Co3Sn2S2 using resonant inelastic X-ray scattering. Remarkably, nearly non-dispersive Stoner spin excitation peaks are observed, sharply contrasting with the featureless Stoner continuum expected in conventional ferromagnetic metals. Our band structure and dynamic spin susceptibility calculations, and thermal evolution of the excitations, confirm the nearly non-dispersive Stoner excitations as unique signatures of correlations and spin-polarized electronic flat bands in Co3Sn2S2. These observations serve as a cornerstone for further exploration of band-induced symmetry-breaking orders in topological materials.

1 -Regarding the Stoner excitations from flat bands: I tend to agree with the authors argument that if they are seeing something as well-defined (in the sense that authors used in the previous version) it must be indicating the flat band above the Fermi-level as well. Whether it is a spin-flip transition requires comparison to calculations (as acknowledged by the authors). They provide 4 calculations, 2 of charge and 2 of spin susceptibilities. Only one of them gets close to explaining S1 (the spin susceptibility with correlations). However, believing that this is evidence of spin-flip scattering requires that we believe the calculation of the band structure.
-Question: Is the calculation in agreement with actual reality? Is the structure in the calculations consistent with ARPES, for the occupied side? We need to have some level of confidence in this. The authors should make this important comment or statement in the text.
2 -Correlation effects being required. I generally understand, as an experimentalist, how the vertex corrected calculations and DMFT include correlations. However, I am a bit surprised at how much it changed the calculation of the susceptibilities. Could the authors provide a simple outline for why this is the case? As it is, the manuscript just states that correlations should modify S(q,w) and that DMFT can capture this. But this is impenetrable to a broad audience. A few more sentences trying to provide a physical intuition would be welcome. Fig. 2c: Difficult to see the direction of the arrow. Fig. 2 caption. The letter d in bold is missing to indicate the description of that panel.

Smaller comments
Lines 105-110 took me several readings. The first sentence says there are no similarities between the bare susceptibility and the data. The next sentence says there is a "remarkable similarity". The next sentence refers to a "striking difference". I'm sure the authors can see how it reads in a confusing manner. Please clarify.

Reviewer:
The authors have addressed most of my concerns. Perhaps the paper can now be accepted for publication in Nature Communications. Our response: We thank the referee for their re-reading of our manuscript, and are pleased that they find the resubmitted version of the paper suitable for publication in Nature Communications.

RESPONSE TO REVIEWER #2
Reviewer: I have carefully read the new manuscript by Nag et al, now titled "Correlation driven near-flat band Stoner excitations in a Kagome magnet". The authors have addressed almost all the issues I raised when reviewing the previous version of this manuscript. Still, I have a few important questions/comments that require some clarification. If clarified, I am supportive of publication in Nature Communications. Our response: We thank the referee for their re-reading of our manuscript, and are pleased that we could incorporate and address their comments and suggestions in our resubmitted version.

Reviewer:
The main observation is made by E-q mapping with RIXS. Two features (S1 and S2) are observed and characterized. The authors use both bare and vertex-corrected calculations of the charge and spin susceptibility to identify the origin of these features. Correctly, they start with the simplest calculation. However, the bare susceptibility calculation does not yield spectral weight or features that would account for the experiments -this is clear. Next they use a vertex correction calculation for the susceptibility. This calculation does a much better job at capturing the most salient features of the experimental data. The authors then reach 2 conclusions: 1) the S1 feature is from Stoner excitations from flat bands and 2) Correlations are necessary to explain the experimental data.
In more detail (I apologize that I am re-writing the authors arguments, but I wanted to be clear regarding the logic I am using to get to my questions).: 1 -Regarding the Stoner excitations from flat bands: I tend to agree with the authors argument that if they are seeing something as well-defined (in the sense that authors used in the previous version) it must be indicating the flat band above the Fermi-level as well. Whether it is a spin-flip transition requires comparison to calculations (as acknowledged by the authors). They provide 4 calculations, 2 of charge and 2 of spin susceptibilities. Only one of them gets close to explaining S1 (the spin susceptibility with correlations). However, believing that this is evidence of spin-flip scattering requires that we believe the calculation of the band structure.
-Question: Is the calculation in agreement with actual reality? Is the structure in the calculations consistent with ARPES, for the occupied side? We need to have some level of confidence in this. The authors should make this important comment or statement in the text. Our response: In Fig. R1, we present a comparison of the available ARPES results and our band structure calculations obtained from a combination of DFT+DMFT. It can be seen from the ARPES intensity plots [ Fig. R1(a, d)] and our calculated band structure [ Fig. R1(b, e)] of Co 3 Sn 2 S 2 that, the overall dispersions and energy scales are in good agreement. It should be noted that while the calculations are at the exact high symmetry points in the Brillouin zone, the experiments probe different values of k z along the projected high symmetry paths. For example, the gray curve in Fig. R1c shows the variation of k z for incident photon energy of 115 eV. Additionally, while DFT based band structure needs to be bandwidth renormalised and shifted to match the ARPES data (see red curves in Fig. R1d), our DFT+DMFT calculations show band structure having similar energy scales as experiment without any renormalisation. We have now included a sentence in the manuscript stating that our band structure calculations are in good agreement with reported ARPES results.
Reviewer: 2 -Correlation effects being required. I generally understand, as an experimentalist, how the vertex corrected calculations and DMFT include correlations. However, I am a bit surprised at how much it changed the calculation of the susceptibilities. Could the authors provide a simple outline for why this is the case? As it is, the manuscript just states that correlations should modify S(q,w) and that DMFT can capture this. But this is impenetrable to a broad audience. A few more sentences trying to provide a physical intuition would be welcome.
As the referee mentions, it is important to include the vertex corrections to obtain the twoparticle dynamic susceptibilities of correlated electronic systems. The dynamic spin susceptibility [S(q, ω)] is related to the bare spin susceptibility [S 0 (q, ω)] as S(q, ω) = S 0 (q, ω)/[1−Γ V (ω)S 0 (q, ω)], where Γ V (ω) is the two-particle vertex function. For weak or uncorrelated metals, the dynamic spin susceptibility resembles the bare susceptibility owing to a small or zero vertex correction. For correlated systems, a significant renormalisation of spectral shape occurs as Γ V (ω)S 0 (q, ω) values approach 1 giving rise to the poles in S(q, ω). Following the referee's suggestion to include this description, we have now included a few sentences in the manuscript. Fig. 2c: Difficult to see the direction of the arrow. Our response: We have changed the position of the arrow heads, so the arrow directions are now clearer. Fig. 2 caption. The letter d in bold is missing to indicate the description of that panel. Our response: We have added now the description of panel d in Fig. 2 caption.

Reviewer:
Lines 105-110 took me several readings. The first sentence says there are no similarities between the bare susceptibility and the data. The next sentence says there is a "remarkable similarity". The next sentence refers to a "striking difference". I'm sure the authors can see how it reads in a confusing manner. Please clarify. Our response: We thank the referee for this suggestion. We have now rewritten this section to clarify the effects of correlations, vertex corrections and comparison of theory and experimental results.

List of changes
All changes in the main text have been highlighted in red in the new version of our manuscript.
1. We have added the following text in line 80: The overall low-energy band dispersions and energies are in good agreement with ARPES data on Co 3 Sn 2 S 2 .
2. We have added in the caption of Fig. 2 of main text, the description of panel d: d, Stoner continuum of excitations corresponding to the electronic bands shown in panel c.
3. We have rewritten the section describing the vertex correction and comparison of experimental results and theory following the suggestions of the referee. The effect of correlations in the two-particle response functions are generally taken into account following a vertex correction [35]. For example, the dynamic spin susceptibility [S(q, ω)] is related to the bare spin susceptibility [S 0 (q, ω)] as S(q, ω) = S 0 (q, ω)/[1 − Γ V (ω)S 0 (q, ω)], where Γ V is the twoparticle vertex function. For weak or uncorrelated itinerant systems, the dynamic spin susceptibility resembles the bare susceptibility owing to a small or zero vertex correction. For correlated systems, a significant renormalisation of spectral shape occurs as Γ V (ω)S 0 (q, ω) values approach 1 giving rise to the poles in S(q, ω). We calculated the bare spin and charge susceptibilities, i.e., the electron-hole pair excitations, based on the DFT+DMFT band structure results with the inclusion of the screened Coulomb interaction and found that both the bare susceptibilities are featureless in the energy range of S1, and have no spectral weight in the energy range of S2 (see Methods and Supplementary Information Fig. S7). Fig.3e and f show the S(q, ω), obtained after vertex correction (see Methods). The remarkable similarity in the energy scales and the dispersions of S1 (S2) peaks between the experiment and the calculated S(q, ω), suggests their origin from the Stoner excitations in Co 3 Sn 2 S 2 , owing to schematically shown U 1 → D 2 or D 1 → U 3 (U 2 → D 2 )-like transitions in Fig 2e. At the same time, the significant difference between S(q, ω) and S 0 (q, ω) (see Supplementary Information Fig. S7), underlines the importance of the correlations effects in both the ground state and the collective excitations. We also note that the experimental distributions are distinct to those of the vertex corrected charge susceptibilities representing non-spin-flip excitations as shown in Fig.3g