Experimental realization of dice-lattice flat band at the Fermi level in layered electride YCl

Domain: neuroscience
Published: 2026-01-31
Editorial outcome: minor_revision

Abstract
Flat electronic bands, where interactions among electrons overwhelm their kinetic energies, hold the promise for exotic correlation physics. The dice lattice has long been theorized as a host of flat bands with intriguing band topology. However, to date, no material has ever been found to host the characteristic flat bands of a dice lattice. Here, using angle-resolved photoemission spectroscopy (ARPES), we discover a dice-lattice flat band at E_F in the van der Waals (vdW) electride [YCl]2+: 2e-. In this system, excess valence electrons from Y deconfine from the cation framework to form an interstitial anionic electron lattice that constitutes the dice lattice. Our ARPES measurements unambiguously identify two sets of dice-lattice bands in YCl, including a nearly dispersionless band at the Fermi level. The near-E_F electronic structure observed in ARPES, which consists of the flat bands and other dispersive band features, find excellent agreement with first-principles calculations and is well captured by a simple dice-lattice model. Our findings thus end the long quest of a real dice flat band material and establish vdW electride YCl as a prototype of dice metals. Our results further demonstrate the anionic electron lattice as a novel scheme for realizing lattice geometries and electronic structures rare to find in conventional crystalline systems.

Introduction
Flat bands—non-dispersive electronic states with quenched kinetic energy—have emerged as a central concept in quantum materials research1–7. When positioned at the Fermi level (E_F), flat bands exhibit a spiked density of states that enhances electron interactions, driving diverse correlated quantum phases, such as superconductivity8,9, magnetism10, and fractional quantum Hall states11,12. Engineering lattice geometries represents a key strategy for realizing flat bands. Over the past decade, most experimental efforts have focused on the kagome and Moiré lattices, which have been successfully realized in several material systems13–20.

Among lattices that host flat bands, the dice lattice stands out as a prototypical construct that has remained elusive in real materials since its proposal by Sutherland in 198621. Geometrically, the dice lattice consists of two types of lattice sites: the six-fold coordinated C sites (red circles in Fig. 1a) and the three-fold coordinated A/B sites (blue circles in Fig. 1a). In an ideal dice lattice, the C sites equivalently couple with all adjacent A/Bsites, whereas the direct couplings between the neighboring A and B sites are broken21–23. In the electronic states of the dice lattice, one of the eigenfunctions is fully localized on the A/B sites and exhibits zero dispersion, as shown in Fig. 1b, with two dispersive bands that intersect the flat band and form a Dirac cone. Owing to the flat band framework, the dice lattice has attracted considerable theoretical interest, with predicted correlated phases including flat band magnetism24, fractional Chern insulators25, chiral superconductivity26, interaction-driven quantum anomalous Hall effect27, Aharonov–Bohm caging-induced insulators28,29, and so on.

Despite theoretical promise, no material realizes the dice-lattice bands due to its strict geometric and energetic constraints, which require all lattice sites (A, B, and C sites) to host electronic orbitals within an approximate energy window25,30. For instance, 1T-phase transition metal dichalcogenides (1T-TMDs) adopt a dice-like lattice geometry, however, their alternating anion/cation sublattice occupation leads to a large on-site energy difference. The low-energy bands in 1T-TMDs are dominated by transition metal d-orbital states on C-sites only, with chalcogen p-states located at much higher binding energies that are far away from the Fermi level. Thus, such systems assume the minimal tight-binding (TB) model of a triangular lattice rather than a dice lattice31,32. A true dice lattice requires relatively close on-site energies among the two sublattices (A/B and C sites), which requires same-charge ions on all sublattice sites, but the inherent electrostatic repulsion renders such structures unstable and rare to find in nature. Heterostructures and artificial lattices have been proposed33–36, yet to date, no successful experimental attempt has been made along this line.

A promising route to overcome these limitations lies in the emerging class of two-dimensional (2D) electrides37–39. These materials, such as [Ca2N]+:e−37,40, [Y2C]2+:2e−41,42, and [YCl]2+: 2e−43,44, are characterized by excess electrons that act as anions and are spatially confined within interlayer gaps or channels between positively charged cationic frameworks. These interstitial anionic electrons (IAEs) behave as quasi-metallic species, which circumvent the electrostatic penalties associated with densely packed metallic ions45–49. This electrostatically favorable arrangement provides a promising platform for realizing complex electron lattices50–52, which enables the engineering of lattice geometries that are unlikely to be achieved in conventional ionic lattices41,53,54. Further, being largely detached from the host ions, these weakly bound IAEs typically give rise to low-energy electronic states near E_F37,40,55. Thus, when combined with special lattice geometries, these anionic electron lattices offer a viable route to achieving exotic flat band structures near E_F. The van der Waals (vdW) 2D electride [YCl]2+: 2e− (YCl) exemplifies such an approach43,44,56. In YCl, the anionic electrons are confined within interstitial sites that arrange into an electron lattice, which opens up the possibility of forming a dice lattice structure (Fig. 1c–e).

Here, we report the realization of a dice-lattice flat band at E_F in the vdW 2D electride YCl using angle-resolved photoemission spectroscopy (ARPES). Our ARPES measurements resolve two sets of dice-lattice bands: one featuring a flat band at E_F and another one with the flat band located approximately 600 meV below E_F. The near-E_F electronic structure of YCl, comprising these flat bands and two dispersive band features observed by ARPES (Fig. 1f), agrees well with the density functional theory (DFT) prediction and mirrors the characteristic band structure of a dice lattice model. In YCl, the electronic states near E_F are dominated by the IAEs43,44 (Fig. 1g). These IAEs, located at the A, B, and C sites, form a unique lattice architecture (Fig. 1c–e). Crucially, direct hopping between the A- and B-site IAEs is strongly suppressed, providing the essential geometric constraint that enables the formation of dice-lattice flat bands (Fig. 1a–e). Comparing ARPES results with theoretical calculations, our work demonstrates that the anionic electron lattice (Fig. 1c–e) in YCl gives rise to the dice-lattice electronic structure near E_F, establishing YCl as a dice metal. This material serves as a prototype of a broader class of rare-earth metal halide electrides ReX (Re = rare-earth metal and X = halogen), in which similar exotic flat band structures are theoretically anticipated. Our findings further demonstrate the power of anionic electron engineering in designing exotic lattice geometries and electronic structures, opening new avenues in the search for emergent quantum phenomena.

Results
YCl adopts a rhombohedral phase (space group R-3m, point group D_3d) at ambient conditions (Fig. 1c), featuring a layered Cl–Y–Y–Cl slab structure stacked along the c-axis (see crystal structure information in Supplementary Information S1). This arrangement turns YCl into a vdW material, in which the interlayer Cls are held together by a weak vdW force. In YCl, Y donates one electron to Cl and transfers its remaining two valence electrons to the interstitial voids43,44 (see differential charge density diagram in Supplementary Information Fig. S6f), making the overall oxidation state of Y3+. X-ray photoemission spectroscopy (XPS) measurement confirms the trivalent oxidation state of Y in our sample (see Supplementary Information Fig. S5), which is consistent with a previous study on YCl44. These excess electrons in YCl occupy two spatially distinctive sites (site A/B and site C), as shown by the electron localization function (ELF) maps in Fig. 1d, e. The IAEs located at sites A and B are derived from Y4d_z2 electrons, while the IAEs at site C, located at the center of each hexagon, originate from the Y5s electrons (Fig. 1g, and more details in Supplementary Information S2). This arrangement of IAEs in YCl gives rise to an electron lattice framework that mirrors the architecture of the dice lattice (Fig. 1a, d).

With the IAEs forming a dice lattice, electronic states arising from the IAEs present characteristics of the dice lattice bands. In particular, two nearly dispersionless bands are identified near the Fermi level in the ARPES spectrum (Fig. 1f), consistent with the DFT calculation (Fig. 1g). These two bands of narrow bandwidth correspond to the peaks in the integrated spectral weight near E_F in ARPES and the pronounced peaks in the DFT-derived density of states. Since trivalent Y has a closed-shell configuration, the IAEs dominate the near-E_F electronic structure, as confirmed by the DFT fat-band plot (Fig. 1g). In addition, the near-E_F IAE states are well-isolated in energy, where the localized Cl bands are separated by a substantial energy gap of over 1.5 eV (Fig. 1f), which minimizes hybridization and suppresses coupling between the Cl states and the near-E_F bands.

The near-E_F electronic structure of YCl is highlighted in the ARPES spectra as shown in Fig. 2a and further revealed by the dispersions extracted from the peak positions of the energy distribution curves (EDCs) in Fig. 2e. In particular, the two flat bands near E_F are consistent with the narrow bands originating from the IAEs at sites A/B, as evidenced by the band-projected electron wavefunction (Fig. 2b, c). In contrast, the strongly dispersive band at lower energy (with the band minimum at around –2 eV) arises from the IAEs at site C (Fig. 2b, d). These key band features below E_F, which are observed in both ARPES and DFT results, can be described by a dice-lattice TB model (Fig. 2f–h), constructed based on the IAE geometry derived from the ELF map (Fig. 1d).

In an ideal dice lattice, the on-site energies of A, B, and C sites are identical, with neighboring hoppings between A and B sites being broken21–23. Regardless of the coupling between the A/B site and the central C site, there always exists a parity-odd AB-sublattice antibonding state that is fully decoupled from the parity-even state on site C. Combining with the suppression of A–B hoppings, the kinetic energy of this AB-sublattice antibonding state is thus quenched, leading to a dispersionless band and a three-fold band crossing at the K point as shown in Fig. 1b21,25.

The IAE architecture of YCl deviates slightly from an ideal dice lattice, where the on-site energy of site-C electrons is approximately 0.6 eV lower than that of site-A and B electrons. The on-site energy difference gives a TB band structure (Fig. 2g) that is comparable to the ARPES and DFT results (Fig. 2a, b), and reduces the three-fold band crossing at the K point to two-fold27,57 (Figs. 1b and 2g). Nevertheless, the dice flat band remains intact under the sublattice asymmetry between site C and sites A/B (Fig. 2g). This is because the on-site energy difference between A/B and C sites does not break spatial inversion symmetry (with site C defined as the inversion center).

In addition, the IAEs of YCl at sites A and B exhibit electronic wavefunctions resembling the dumb-bell shape of the d_z2 orbital (Fig. 2c). These IAEs at A and B sites are spatially separated at different layers, providing the suppression of the A–B couplings. The alternating-sign wavefunction lobes of the flat-band states corroborate with the A–B antibonding state of the dice lattice, where the kinetic energy is quenched. While a perfect dice lattice requires fully broken A–B couplings, realistically, the hopping between the A and B sites may not be fully quenched. A finite but small hopping parameter t_AB can generate weakly dispersed band features, which can account for the small but finite bandwidth shown in the DFT result (Fig. 2h). Nevertheless, the ARPES data present an even flatter dispersion compared to the DFT bands, revealing that the YCl electride is well in the dice-lattice limit with nearly suppressed couplings between A and B sites.

The ARPES spectra and Fermi surface map in Fig. 3 provide a comprehensive view of the electronic structure. Along high-symmetry cuts (Fig. 3a–k), the four bands arising from the IAEs (α, β, γ, and δ) are observed. EDC stacking plots (Fig. 3c, g, k) highlight the dispersionless nature of the α and γ bands, contrasting with the dispersive δ band. The β band, which cannot be identified from the EDCs due to the highly dispersive nature with large band velocity, is evident in the stacking plot of the momentum distribution curves (MDCs) (Fig. 3d, h). Notably, the β, γ, and δ bands exhibit an accidental degeneracy near the zone corner K point at around -0.55 eV (triangle marks in Fig. 3a, i). Along the M-K cut (Fig. 3j), DFT predicts the α band residing above the Fermi level, while the ARPES reveals substantial spectral weight below E_F, with spectral peaks resolved in the stacking EDC plot (Fig. 3k). This is further supported by the Γ-K and Γ-M cuts (Fig. 3c, g), where spectral peaks right below E_F appear near both M and K points. The ARPES data conclusively show that the α band is even flatter than the DFT prediction, forming a truly dispersionless feature.

The Fermi surface map shown in Fig. 3l reflects the strong spectral contributions from the nearly flat α band at the Fermi level, resulting in broad and intense spectral features. Two prominent regions of spectral weight can be resolved. One of these features centers around the Γ point, arising from the enhanced spectral intensity of the dispersionless α band near the Γ point, consistent with the ARPES results shown in Fig. 3a, e. The other feature exhibits spectral intensity near the zone boundary (Fig. 3l, m), arising from the intersection of the dispersive β band and the α flat band at the E_F (Fig. 3a, e), where the β band forms a large hole pocket centered around the Γ point as indicated in the schematic Fermi surface plot in Fig. 3m. Both features on the Fermi surface present intensity modulation of the hexagon shape (Fig. 3l), consistent with the symmetry of the YCl crystal. Fermi surface from DFT, on the other hand, presents a complicated structure due to the wiggling of the α flat band around the Fermi level (see Supplementary Information Fig. S8).

To further examine the flat band structure of YCl, we present ARPES spectra that cut across different Brillouin zone (BZ) regions in Fig. 4a–c. Two sets of ARPES cuts—parallel cuts to Γ-M direction and to Γ-K direction (with the cut position indicated in Fig. 4a)—reveal the two flat band features that span the entire BZ. The α band and the γ band exhibit dispersionless features throughout the BZ, indicating strong quenching of the electron kinetic energy. In addition to the dispersionless bands, the parallel cuts shown in Fig. 4b, c present well distinguishable dispersions along the path, showing a continuous evolution of the dispersive dice bands across the BZ. Overall, the ARPES data present an even flatter dispersion compared to the DFT bands, revealing that the YCl electride is well in the dice-lattice limit with nearly suppressed couplings between A and B sites. The ARPES spectra of YCl in Figs. 1–4 exhibit large broadening across all bands. This spectral broadening is potentially induced by the two dispersionless bands. With the diminishing band velocity, even modest electron self-energy could introduce a significant broadening effect in the spectral function58. As all the near-E_F bands in YCl arise from the IAE states, the presence of the electronic screening effect59 from the IAEs could potentially provide substantial electron self-energy. The observed broadening could also stem from charge inhomogeneities. Previous study indicates that the IAE states are susceptible to the crystal environment60. Thus, defects and impurities could readily perturb the IAEs and cause spectral broadening on the IAE bands.

Unlike the two well-separated flat bands, the dispersive δ band displays a single prominent dispersion. We assign the higher-energy split component in DFT as the δ band, given that it merges with γ flat band at the K point, consistent with ARPES observation. The lower-energy split component predicted by DFT (δ0 band), however, is absent in the ARPES spectra. To further investigate the absence of the δ band splitting, we explore the matrix element effect with different photon energies (see Supplementary Information Fig. S12). Though there are quite strong modulations of the ARPES intensity with different photon energies, the single δ-band dispersion remains unchanged. It is worth noting that the splitting between the two sets of dice bands in DFT is largely due to the spin-splitting within the monolayer (see discussion of the spin structure in Supplementary Information S3). The splitting of the flat band due to Stoner instability has been predicted by the Hubbard model, where the splitting of dispersive bands is less affected compared with that of the flat band (see Supplementary Information S5.1 and Fig. S15)61. In our case, DFT calculations predict that the splitting of the δ band is approximately 0.25 eV, about half that of the flat bands. In the real material, the δ band splitting could be even smaller than the DFT prediction and thus, not resolvable in the ARPES spectra.

Comparing the ARPES data to the DFT result and the TB model simulation shown in Fig. 2, YCl exhibits a remarkably simple electronic structure with two sets of dice-lattice bands near the Fermi level, free from complications or contamination by other electronic states. This is in stark contrast with most kagome intermetallic compounds5,62,63, where multiple metallic ions contribute to the low-energy bands. This energetic separation of the Y-derived valence states near E_F makes YCl an ideal, simplified flat-band system for study.

Our analysis shows that the formation of dice flat bands in electrides requires two key ingredients: (i) IAEs at site C that couple with those at sites A and B, and (ii) sufficiently weak direct A–B hopping. Both conditions emerge naturally from the lattice architecture of IAEs in YCl. As proof of principle, Fig. 4e–f presents a DFT survey of other ReX crystals that can also host IAEs. We find that dice-lattice flat bands are not limited to YCl but also appear in a wider family of Sc- and Y-based electrides, demonstrating that dice-band physics is robust across this class of materials. In contrast, the La-based electrides, with the spatially extended IAEs derived from the 5d electrons, fail to preserve the dice lattice bands (see the comparison of DFT band structures in Supplementary Information S6). Overall, these results establish ReX electrides as a promising platform for realizing and engineering dice-lattice flat band materials.

Methods
Synthesis
Single crystals of YCl were synthesized via a self-flux method using stoichiometric mixtures of yttrium (Y) and yttrium trichloride (YCl3). The Y and YCl3 powders were ground together using an agate mortar and pestle, then pressed into pellets. Each pellet was encapsulated within a molybdenum capsule and then further sealed in a stainless-steel capsule. All these preparation works were conducted in an Ar glove box. The assembly was heated at 1050 °C for 1.5 days, then cooled down to 800 °C for 6 days under Ar flow.

ARPES measurements
ARPES data shown in this paper were collected using beamline I05 at Diamond Light Source71 with a vacuum pressure better than 1.3 × 10−10 mbar. All samples were cleaved in-situ to ensure clean surfaces. ARPES data shown in the main text were taken with photon energy hv = 75 eV under linear-horizontal polarization, with the sample temperature maintained at T = 6 K. Measurements were conducted using an MBS A-1 electron analyzer, where the total energy resolution is about 5 meV. All Fermi surface maps presented in this study were integrated over an energy range of E_F ± 10 meV.

DFT calculations
First-principles calculations based on DFT were performed using the Vienna Ab initio Simulation Package (VASP)72,73, employing the projector augmented-wave (PAW) formalism74–76. The local density approximation (LDA) functional, following the Ceperley-Alder parameterization as reformulated by Perdew and Zunger (CAPZ)77,78, was adopted for the exchange-correlation energy. A Monkhorst–Pack k-point grid was employed with a typical density of 12 × 12 × 3 for bulk calculations, and an energy cutoff of 600 eV was used for the plane-wave basis. Structural relaxations were performed for all magnetic configurations until the residual Hellmann–Feynman forces on each atom were less than 0.002 eV/Å, and total energy was converged to better than 10−8 eV. All calculations were carried out in a spin-polarized setting to account for possible magnetic orderings in bulk YCl. All IAEs’ charge states and centers of maximum are calculated using BadELF with the zero-flux method for separating metallic features. The searching criteria for the electride charge is set to 0.679. The real-space wavefunction projection is obtained by 3D Fourier transform on the plane-wave coefficients and presented on a 3D uniform grid using the Python code vaspwfc.py80.

We investigated both ferromagnetic (FM) configurations using the conventional unit cell and type-A antiferromagnetic (AFM) configurations with a 1 × 1 × 2 supercell by setting layer-alternating initial magnetic moments on Y ions (see comparison in Supplementary Information Fig. S7). The overall band structures of the two magnetic configurations are largely consistent, but the type-A AFM configuration provides a band structure that aligns better to the ARPES data and results in around 20 meV/cell lower ground state total energy compared to the FM configuration. Thus, in the main text, we compare the ARPES data to the DFT calculations with the AFM configuration. The calculation of all ReX electrides follows that of YCl, using the 1 × 1 × 2 supercell with initial magnetic moments.

Three-band TB model
Details of the three-band TB Hamiltonian used to illustrate the flat band physics are presented below: in the Bloch basis of the A/B and C sublattice sites, the full k-space Hamiltonian can be written as the following 3-by-3 matrix:

H(k) = [[h_A(k), h_AC(k), h_AB(k)], [h*_AC(k), h_C(k), h_BC(k)], [h*_AB(k), h*_BC(k), h_B(k)]] (1)

where h_l(k) = E_l with site index l = A, B, C describes the on-site energy E_l for site A, B and C, respectively; h_AC(k) = P t_AC f_0(k) = h_BC(k) = t_BC f_0(k), with the geometric factor f_0(k) = sum_{j=1,2,3} e^{ik·δ_j}, denote the hopping terms between site A/B to site C characterized by parameter t_AC and t_BC, where δ_j is the hopping vectors; h_AB(k) = t_AB f*_0(k) denotes the direct coupling between site A and site B IAEs with hopping parameter t_AB. H(k) satisfies all symmetry constraints imposed by the D_3d point group symmetry of the electride system. We have estimated the on-site energies of sites A, B, and C based on comparisons between DFT and TB models. This analysis yields on-site energies of E_A ≈ E_B ≈ 0.0 eV and E_C ≈ −0.6 eV. These values confirm the relative offset of approximately 0.6 eV between the A/B and C sites reported in the main text (the full set of TB parameters is provided in Table 1). A spinful TB model was also constructed (see Supplementary Information S5).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Geng et al. reports the first direct observation of dice lattice flat bands at the fermi level in a 2D van der Waals electride YCl. The authors propose that the dice lattice is formed by interstitial anionic electrons (IAEs), whose characteristic orbital features enable the specific hopping parameters required for this geometry. Two sets of flat bands due to spin splitting are observed in the ARPES measurements and agree well with DFT and tight binding calculations. Very interesting and novel work, there are however a number of problems/ambiguities to be addressed.\n1. In L176 the author claims the onsite energy difference between A/B and C does not break spatial inversion symmetry that the flat band remains intact under the sublattice asymmetry.They also note that 1T-TMD possess a similar lattice geometry in L51. Please comment on this why 1T-TMD does not already have dice lattcie flat bands.\n2. In L120 it says the IAEs at sites A/B originate from Y 4dz2 electrons, and in SI P7 it says IAEs experience the crystal field and retain the dz2/s symmetry. The Maximally localized Wannier functions in Figure S4 gives the spatial localization of Y excess electron states, and the alpha bands/dz2 orbitals locate at sites C but delta bands/s orbitals are also close to sites C rather than A/B. Please explain the discrepancy and clearify whether the IAEs are Y 4dz2/5s electrons or their wavefunctions merely adopt dz2/s symmetries because of crystal field environment.\n3. In L178, are the alternating signs/colors of the wavefunction lobes their phase, or density variation? More details are needed here, what they are and how they were obtained. And opposite phase does not mean zero hopping or no bonding between A & B, it only means forming higher energy antibonding. This arguement is the very ground of the dice lattice formation and needs justification.\n4. DFT band structure in Figure S4a shows the beta bands have significant contributions from dx2-y2/dxy orbitals. Please explain the role of dx2-y2/dxy orbitals in this dice lattice constructed by IAEs with dz2/s symmetries.\n5. The tight-binding model is central to the interpretation of the data. To quantitatively support the claims, please provide all the tight binding parameters used in the figures. Can the author estimate the onsite potentials and hopping strengths of IAEs from DFT/Wannier calculations to better support the suppressed hopping between A & B sites in dice lattice and the spin splitting of alpha/gamma flat bands.\n6. Modification of the figures and tables for example in SI P5 the missing information of atomic pecentages in Table S2 or S4, and the Table S3 is nowhere to find.\n7. In L235 it says YCl shows negligible kz dispersion and the matrix element effects strongly affect the spectral intensity. Especially the alpha flat band and delta dispersive band have large intensity contrast between 65eV and 85eV in Figure S10. Can you comment on this with combined bulk DFT band structures with different kz to show the 2D features.\n8. In L302 The discussion distinguishes the IAE lattice from charge density waves (CDWs) and Wigner crystals. A statement confirming the absence of CDW order, which for example can induce narrow flat bands at the fermi level in 1T-TMD, would solidify the argument of IAEs dice lattice flat band.\n9. Please provide the DFT calculated magnetization and energy numbers for FM and AFM configurations to justify the claims.", "reviewer_2": "In this manuscript, the flat-band in the dice lattice was experimentally realized at the fermi level in the van der Waals two-dimensional (2D) electride YCl. The findings open an avenue toward the experimental study of dice lattice in real 2D materials. For this reason, I recommend the publication of the manuscript in Nature Communications.\nThe following problem should be fixed in the revised version. Aharonov-Bohm (AB) cage in the dice lattice was systematically analyzed in a PRB paper published on 2020. In the second paragraph, the citation on this point should be precise to improve the clarity.", "reviewer_3": "The presented manuscript “Experimental realization of dice-lattice flat band at the Fermi level in layered electride YCl” by Songyuan Geng et.al. is devoted to finding a possible experimental realization of a dice lattice with a flat band in van der Waals (vdW) electride [YCl]. Specifically, a noticeable dispersionless electronic band was discovered and observed at the Fermi level of the considered material.\nThe experimental method explored in this paper was the angle-resolved photoemission spectroscopy (ARPES). This is one of the most powerful experimental techniques for directly probing the electronic band structure of various solids. It is often used to find the key electronic and topological properties of a solid, such as band gaps, effective electron masses, and quasiparticle lifetimes.\nThe article is timely and very important for a broad readership in condensed matter physics. The flat band leads to significantly new electronic, collective, and topological properties in various materials, and here we see a vast amount of strong experimental evidence to support the existence of the desired band structure. It is specifically important that this paper provides both experimental verifications and further guidance for the theoretical research in dice lattices, Lieb lattices, and the so-called alpha=T3 model.\nTherefore, I can undoubtedly recommend this paper for publication in Nature Communications, provided that the authors address the following, rather minor, comments and suggestions.\nA flat band in the band structure means strong correlations and a rather small momentum and kinetic energy of the electrons that belong to this band. I would like to know more about the possible conditions of the stability of this flat band and when it actually remains flat: under which external perturbations it would stay flat, and when it receives finite dispersions.\nIn general, to what extent can you say that your experimental results provide evidence for the validity of the alpha-T3 model with a Dirac cone and a symmetric flat band? Recently, there have been substantial arguments regarding the validity of the model, how many materials it can describe, and to what extent. Theoretical work has been much ahead of specific experimental verifications. My question is whether we can count this work as another step towards validation of the alpha-T3 model.\nThe Hamiltonian is based on the three-band tight-binding model. This is predetermined by the fact that there are three types of subletuses: A, B, and C. However, is it possible that you find more than three bands in a close vicinity to each other? This could happen, for example, for Kekule-patterned dice lattices.\nFinally, I believe it will be crucial to provide a little more detail of your DFT calculations, of course, if the size of the manuscript allows. You mentioned the package and the general method, but could you please explain more about the theoretical concepts of your specific DFT calculations, what equations and recent advances in this field have been used?"}, "author_response": "We thank all reviewers for their careful review of our manuscript. The reviewers' comments are reproduced in black text below, followed by our point-by-point responses in blue.\nReviewer #1\n[... full text of author response to Reviewer 1 ...]\nReviewer #2\n[... full text of author response to Reviewer 2 ...]\nReviewer #3\n[... full text of author response to Reviewer 3 ...]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors answered my questions very satisfactorily. This work is of high quality and can now be published as is.", "reviewer_2": "The authors have carefully revised their manuscript according to the reviews' comments and suggestions, I recommend the publication of the manuscript in the present version.", "reviewer_3": "I have studied the revised version of the manuscript as well as the authors’ responses to the Referee comments. I believe that the authors have done a very good job by explaining their results and definitely improved the presentation of the manuscript due to these comments. Therefore, now I can recommend this paper for publication in Nature Communications in the present form.\nAs for the scientific merit of this paper, I could briefly repeat that this is the very first successful attempt to observe and provide solid experimental evidence for the existence of the flat bands in a 2D van der Waals electride YCl. These results provide a strong implication to further studying of the dice lattice, other flatbed materials, including the validation of the theoretical alpha-t3 model which has been recently studied in the mountains of theoretical publications. Also, we are dealing with truly novel and unique symmetry breaking and topological properties of a low-dimensional material.\nTherefore, we are dealing with truly novel and very important results which will significantly affect multiple research directions in both theoretical and experimental physics so that the presented findings are truly exceptional."}, "author_response": "Response: We deeply appreciate that the Reviewer #1 finds our responses “very satisfactorily” and finds our work “of high quality”. We have prepared the final version of the manuscript.\nResponse: We thank the Reviewer #2 for recommending publishing our work.\nResponse: We deeply appreciate the Reviewer #3 for accessing our work and providing the high evaluation."}]

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