Rich nature of Van Hove singularities in Kagome superconductor CsV3Sb5

The recently discovered layered kagome metals AV3Sb5 (A = K, Rb, Cs) exhibit diverse correlated phenomena, which are intertwined with a topological electronic structure with multiple van Hove singularities (VHSs) in the vicinity of the Fermi level. As the VHSs with their large density of states enhance correlation effects, it is of crucial importance to determine their nature and properties. Here, we combine polarization-dependent angle-resolved photoemission spectroscopy with density functional theory to directly reveal the sublattice properties of 3d-orbital VHSs in CsV3Sb5. Four VHSs are identified around the M point and three of them are close to the Fermi level, with two having sublattice-pure and one sublattice-mixed nature. Remarkably, the VHS just below the Fermi level displays an extremely flat dispersion along MK, establishing the experimental discovery of higher-order VHS. The characteristic intensity modulation of Dirac cones around K further demonstrates the sublattice interference embedded in the kagome Fermiology. The crucial insights into the electronic structure, revealed by our work, provide a solid starting point for the understanding of the intriguing correlation phenomena in the kagome metals AV3Sb5.

In the reply letter, the authors made serious efforts to address my comments in previous report. In particular, the authors include the new photon energy dependence data at 200K with higher energy photons in the supplementary materials. However, the added photon energy dependence data are only focused at the zone center. While they are useful/sufficient for determining the periodicity in the kz direction and estimating the inner potential, the critical information on whether there is substantial kz dispersion of the VHS at M is still missing. In this regard, the unpublished data I mentioned in previous report are now posted on the web (arXiv:2105.01689v2). In Fig. S5 of this paper, the photon energy dependence data taken along K-M-K direction clearly show that the VHS discussed in current manuscript exhibit substantial kz-dependence: it is located above Ef at 97 eV and below Ef at 106 eV. Note that all data in this figure were taken in LH polarization. At 106 eV, the VHS related band shows a band top below Ef with no suppression of intensity as expected; whereas at 97 eV, the corresponding band loses its intensity around M simply because the VHS shifts above Ef, without resorting to complex matrix element effect of dx2-y2 and dz2 orbitals. For this reason, I cannot recommend the current manuscript for publication in its present format without a satisfactory resolution to the issue of kz dispersion of VHS at M.

Point-to-Point Response to Reviewers' Reports
For clarity, the reviewers' original comments are shown by blue italic characters.
The authors' responses are shown by black normal characters.

Reviewer #2 (Remarks to the Author):
In the reply letter, the authors made serious efforts to address my comments in previous report. In We thank Reviewer #2 for reviewing our paper again and for providing further comments. We are glad to read that the Reviewer is satisfied with our kz (thus inner potential) determination: "In the reply letter, the authors made serious efforts to address my comments in previous report. … they are useful/sufficient for determining the periodicity in the kz direction and estimating the inner potential". The remaining concern is related to the kz dispersion of VHS1 band and the flat-top dispersion of VHS1. In the following, we give our response to Reviewer #2's comments.
Following the Reviewer's suggestion, we display the photon energy-dependent data taken along the K -M -K direction with linear horizontal (LH) polarization at 200 K in Figs. R1. The VHS bands generally exhibit weaker kz dispersion than the DFT calculations, and the evolution of the VHS1 band, as it crosses from below to above EF, is clearly observed (see the black arrow in Fig. R1a), consistent with version 2 of the arXiv preprint 2105.01689v2. Moreover, the flat dispersion of VHS1 band is consistently seen in different kz = 0 planes (as indicated by the orange dashed curve and black arrow in Fig. R1a), despite minor differences in spectral intensity. Notably, the kz evolution of the VHS1 band is consistent with the kz periodicity determined from the Sb pz band around G, i.e., 54 eV, 78 eV and 108 eV all correspond to the kz = 0 plane, unambiguously confirming our previous estimate of the inner potential.
Importantly, we note that the flat feature of VHS1 in 78 eV (also presented in our main text) is relatively weaker compared to the 54 eV and 108 eV data (marked as the vertical arrow in Fig. R1a). On the other hand, the intensity of the VHS1 band below EF in 78 eV is much stronger than the corresponding band in the 54 eV and 108 eV data (highlighted by the red horizontal arrows in Fig.  R1a). We attribute this difference to matrix element effects, that depends on electron momentum, and on the energy and polarization of the incoming photon [1]. This interpretation is supported by our data collected with 78 eV LH polarization along a different momentum path (G1 -K -M, G2 -K -M direction, indicated by the red arrow in Fig. R2a). This data shows that the VHS1 band has a clear nondispersive flat-top dispersion without any suppression of intensity, in contrast to the data along the G -K -M direction (marked as the black arrow in Fig. R2a, compare Fig R1a with Fig R2b).
In conclusion, in order to reveal the true dispersion of the VHS1 band one needs to take into consideration the kz = 0 spectra measured at 54, 108 and 78 eV and also look at different momentum cuts. Hence, we conclude that the flat feature of the VHS1 band in the 78 eV spectra measured along the G1 -K -M direction, with LH polarization, is affected by matrix elements. This is in agreement with our previous theoretical calculations, where we included the matrix element effects to quantitatively explain the diminished intensity of the flat band top in 78 eV data.   Regarding the kz dispersion of VHS2, we did not observe noticeable kz dispersion of the VHS2 bands (Fig. R3a), suggesting the kz dispersion is weaker than the one of the DFT calculations. Actually, the kz dispersion of VHS2 bands shown in the preprint [arXiv:2105.01689v2] is also much weaker than the DFT calculations. Figure R3b shows the measured kz dispersion of the VHS bands in the preprint. Compared to the calculations (Fig. R3c), the experimental kz dispersion of the VHS2 band is very weak (Fig. R3b). For instance, the VHS2 band (see the dashed line) shows a negligible difference in the kz = 0 and kz = p planes (Fig. R3d), in sharp contrast to the DFT calculations (Fig. R3c). These results are fully consistent with our measurements. In summary, we appreciate the constructive comments and suggestions from the Reviewer. Regarding the Reviewer's concern, we have followed the Reviewer's suggestion and modified our paper accordingly: (i) In the second round, as suggested by the Reviewer, we have invested considerable efforts to determine the kz (thus inner potential), which has been recognized by the Reviewer.
(ii) Following the Reviewer's suggestions in the second round, we displayed the spectrum divided by the Fermi-Dirac function and demonstrated that there is no additional band crossing EF along the M-K direction (Fig. R4). In Fig. R2, we have shown new data taken with LH polarization along a different path (namely, along the G2 -K -M direction). This clearly shows that there is no additional band crossing EF and that the VHS1 band below EF has a flat band top without any suppression in intensity.
(iii) Furthermore, we provided additional photon energy-dependent measurements (Fig. R1) to further clarify the kz dispersion of the VHS bands. In agreement with the preprint arXiv:2105.01689v2 (mentioned by the Reviewer), we find that the VHS1 has indeed a nonzero dispersion and crosses from below EF to above EF.