Topological metal-insulator transition within the ferromagnetic state

Domain: chemistry
Published: 2026-02-27
Editorial outcome: major_revision

Abstract
A major challenge in condensed matter physics is integrating topological phenomena with correlated electron physics to leverage both types of states for next-generation quantum devices. Metal-insulator transitions are central to bridging these two domains while simultaneously serving as on-off switches for electronic states. Here, we demonstrate how the prototypical material of K2Cr8O16 undergoesa ferromagnetic metal-insulator transition accompanied by a change in band topology. Through inelastic x-ray and neutron scattering experiments combined with first-principles theoretical calculations, we show that this transition is not driven by a Peierls mechanism, given the lack of phonon softening. Instead, we establish the transition as a topological metal-insulator transition within the ferromagnetic phase with potential axionic properties, where electron correlations play a key role in stabilizing the insulating state. These results reveal how a metal-insulator transition provides a pathway through which magnetism, topology, and electronic correlations interact.

Introduction
A major milestone in condensed matter physics is bridging the gap between topological phenomena and correlated electron physics to harness both magnetic and topological states for next-generation quantum devices. Central to this effort are metal-insulator transitions (MITs), which serve as both conceptual and functional bridges between these two domains while simultaneously acting as ideal on-off switches for manipulating electronic states, an essential mechanism for device engineering.

While most known MITs occur in antiferromagnetic states with no net magnetization, ferromagnetic MITs, which maintain net magnetization across the transition, offer unique functionality for quantum information and spintronics. However, such transitions are exceptionally rare and poor understood, particularly when involving topological character. Most well-studied topological phase transitions (TPTs) rely on the closing and reopening of a local gap at a Weyl point, characteristic of single-particle Weyl semimetals1–3. In contrast, a more profound form of TPT involves a correlation-driven MIT, where the gap closes without reopening – a phenomenon largely unexplored in topological systems. Such ferromagnetic MITs expand the frontier of topologically driven material manipulation, presenting unprecedented opportunities for innovation in systems where strong electron correlations intersect with topological phenomena4–10.

In a typical Weyl semimetal, Weyl points exhibit chiral symmetry, meaning points of opposite chirality do not interact11. Pairing interactions that break this symmetry, such as charge density waves (CDWs), can induce new phases with topological Goldstone modes known as axions. When qCDW nests Weyl points of opposite chirality near the Fermi level11, it can drive a TPT from a Weyl semimetal to a topological axion insulating phase, marking a MIT.

Traditional models, such as the Mott-Hubbard model, have successfully described MITs in various materials12,13, including high-temperature cuprate superconductors14 or Verwey transitions in magnetites15. However, understanding the temperature-dependent ferromagnetic-metal-to-ferromagnetic-insulator transition (FM-MIT) has proven particularly challenging. K2Cr8O16 is renowned for exhibiting this FM-MIT. It has a Curie temperature TC = 167 K with a MIT (TMIT = 95 K) that retains the FM order16. Since K2Cr8O16 is considered a quasi-1D compound (Fig. 1), the MIT has been explained by a 1D CDW17, also known as a Peierls transition, accompanied by a structural distortion from a metallic tetragonal phase to an insulating monoclinic phase (Fig. 1) with qCDW = (1/2, 1/2, 0)17,18. However, a recent study19 suggesting the presence of Weyl fermions raises new questions regarding the role of topological effects in this transition. Related compounds, such as RbCr4O8, have also been shown to host Weyl points20, demonstrating the robustness of these topological features in this family of compounds.

Here, we show that the ferromagnetic MIT in K2Cr8O16 is associated with a change in band topology. Using a combination of neutron diffraction, inelastic x-ray scattering, and first-principles calculations, we demonstrate that no phonon condensation is observed across the transition. Instead, Weyl fermions of opposite chirality are nested by the CDW wavevector. These results are inconsistent with a Peierls driven mechanism and instead identify the transition as a correlation-driven MIT occurring within the ferromagnetic phase, with possible axionic character.

Results
Magnetic response across the transition
As the positions of the Weyl points are highly dependent on the FM ordering19, we investigated the magnetic response across the phase transitions using powder and single-crystal neutron scattering. The collected ND data and the corresponding Rietveld refinement at T = 10 K are shown in Fig. 2a. Upon cooling below TC = 167 K, a strong enhancement of the {1,2,1} reflections is observed, consistent with additional scattering resulting from long-range FM ordering with qFM = (0,0,0) where qFM is the magnetic propagation vector. In order to identify the magnetic structure, the group-subgroup relationships for the given magnetic propagation were analyzed (Sec. S2 of Supplementary Materials (SM)), and the solution of C2' = m' (#12.62) was obtained (Fig. 1(b)). The FM spin polarization reduces the symmetry from the paramagnetic space group I4/m (#87) to the magnetic subgroup C2' = m' (#12.62), which is monoclinic. The moment direction within the ab-plane could not be determined due to the powder average, an ambiguity that could not be resolved even with single-crystal ND due to the formation of magnetic domains. The magnetic {1,2,1} peaks could in principle be inequivalent below the temperature of the MIT (TMIT), due to the monoclinic distortion. However, temperature-dependent measurements of the peaks (1,2,1) and (2,1,1) do not show a difference (Fig. 2b). To resolve the magnetic order within the ab-plane, angle dependent magnetization measurements in the ab plane as a function of magnetic field at T = 120 K is presented in Fig. 1c. Fitting the data to a sinusoidal function yields an order direction tilted ~22.5° relative to the principal axis. Together, these results show that the magnetic order remains unchanged across the MIT, while the moment orientation at 120 K lies slightly off-axis within the ab-plane.

To further study the spin correlations, inelastic neutron scattering (INS) spectra were measured on powder at T = 5 and 130 K (Fig. 2c, d). Both spectra contained the same essential features, confirming the absence of any significant change in the exchange interactions across the MIT. Figure 1b defines the most relevant exchange parameters for the compound: J1 and J2 (between nearest neighbor separated by ~2.9 Å) are both intra-chain interactions within the double chain of edge-sharing octahedra (Fig. 1(a), shaded red), whereas J3 (between third-nearest neighbors separated by ~3.4 Å) is the inter-chain interaction between corner-sharing octahedra (Fig. 1a, shaded blue). In this paper, we use the following conventions for the classical Heisenberg Hamiltonian, H, defined as H = -1/2 \sum J_{ij} (e_i \cdot e_j), where e_i is the unit vector in the direction of the i-th site magnetization and J_{ij} is the isotropic exchange interaction parameter multiplied with the square of the magnetic moment of the i-th site. With this convention, we note that positive exchange values (J) favor FM alignment21.

Using linear spin wave theory22 and the magnetic order derived from ND, the exchange parameters, J1 = 6.0, J2 = 0.6 and J3 = 10 meV are obtained by fitting the experimental INS spectra to the Heisenberg Hamiltonian (Fig. 2e, f and Sec. S2 of SM), consistent with FM spin alignment. The data also confirm that the MIT has little or no effect on the exchange couplings. To verify the experimentally obtained values, we used density functional theory (DFT) and the magnetic force theorem23,24 to calculate J from first principles. The theoretically calculated exchange parameters are also smaller between the nearest neighbors, J1 and J2, than they are for the third exchange parameter, J3 (Fig. S10(a) SM), in agreement with our experimental data and previous calculations17. These results confirm that the magnetic building blocks of the quasi-1D structure are the groups of four corner sharing chains between next-nearest neighbors (referred to as chimneys17; Fig. 1(a), (b) in shaded blue), instead of a one-dimensional linear chain (Fig. 1(a), (b)). This questions the previous conclusions that K2Cr8O16 experiences Peierls instability at the MIT, a notion predicated on K2Cr8O16 being 1D17 and is further weakened by the absence of experimental evidence. We will return to this point later in this paper.

Finally, we found a clear t^2/U dependence in the calculated exchange interactions, where t is the strength of electron hopping between sites and U is the Hubbard repulsion of electrons at the same site. This quadratic scaling indicates that the average interactions in K2Cr8O16 arise from a superexchange mechanism25,26, rather than double exchange17,18,27–29–for which J_{DE} \propto t–or direct exchange type30. While superexchange is often associated with antiferromagnetism, in this case the dominant couplings are ferromagnetic, consistent with the Goodenough-Kanamori-Anderson rules. The microscopic origin is outlined in Sec. S3C of the SM.

Topological phase transition
Using structural17,31 and magnetic information derived from experiments, we analyzed the electronic properties by performing DFT calculations and employing a group theory analysis. We start our discussion with the paramagnetic tetragonal metallic phase, with space group of I4/m (#87). In terms of atomic arrangement and structural periodicity, all Cr are symmetrically equivalent. The mirror symmetry relates their d_{xz} and d_{yz} orbitals to each other (Fig. 3a), i.e., in the paramagnetic phase all bands have an equal amount of d_{xz} and d_{yz} characters (Fig. 3(c, d)) (as pointed out in Ref. 27). In the present work, the spatial orientation of the d-orbitals are defined with respect to the local z-axis of the Cr-O octahedron (as defined in Fig. 3(a)), which due to the crystal symmetry, always lies parallel to the ab-plane. The symmetry-equivalent d_{xz} and d_{yz} orbitals are higher in energy than the d_{xy} orbital due to compression along the z-axis of the distorted Cr-O octahedra (Fig. 3(a, c, d)). Since Cr in K2Cr8O16 has a nominal charge of +3.75, only (6−3.75=) 2.25 of its t_{2g}-like (i.e., d_{xy}, d_{yz} and d_{xz}) orbitals are occupied. The fractional occupation of these states guarantees that the tetragonal phase is metallic or at least filling-enforced semi-metallic (Fig. 3(c)).

In order to evaluate the topological nature of the bands, we used the DFT wave functions to construct a downfolded tight-binding model with the Cr d_{xy}, d_{yz} and d_{xz} Wannier functions as basis states. By calculating the Berry curvature of the magnetic metallic phase (#12.62), we found pairs of Weyl points located close to k_c = ± π/c nodal plane 0.1 eV above the Fermi level (see Fig. 3(b, c) and see Table S3 SM for their precise locations)), confirming the topological nature of the electronic structure. For each pair, Weyl points of opposite chiralities are related by inversion symmetry, consistent with the magnetic space group #12.62. Furthermore, since all irreducible co-representations of this group are one-dimensional, the Weyl points are accidental crossings near the k_c = ± π/c plane, which explains why the Weyl points are not located at high-symmetry k-points in the BZ (see Table S3 SM for the locations). In addition, we find that the positions of the nodal points are robust. Regardless of the direction of the magnetic moment, we found that the nodal points always form a cross-like feature about the nodal plane, such that nested nodal pairs are always present in the (0.5, 0.5, 0.0) and (−0.5, −0.5, 0.0) directions of the tetragonal Brillouin zone (SM Fig. S8).

These Weyl points exhibit remarkable characteristics. In particular, the nesting vectors connecting Weyl points of opposite chirality, q_{nest} = (0.75, 0.75, 0.0) and (−0.75, −0.75, 0.0) (Fig. 3(b)), closely match the lattice distortion wave vector observed in single-crystal X-ray diffraction experiments17,18. This suggests that the nesting of Weyl points may be coupled to the lattice distortion, thereby breaking chiral symmetry and allowing the emergence of axion-like excitations32–34. A definitive confirmation of this scenario remains an important direction for future investigation.

Secondly, the Weyl points lie near the nodal planes at the Brillouin-zone boundary, k_c = ± π/c, where every Bloch state is symmetry-enforced to be doubly degenerate. This degeneracy arises from the magnetic screw rotation TC_{1/2[001]}, which combines a 180° rotation about the [001] axis with time-reversal symmetry. The operation squares to a lattice translation along c with a phase factor, (TC_{1/2[001]})^2 = e^{ik_c c}, so at k_c = ± π/c, it imposes (TC_{1/2[001]})^2 = -1, enforcing degeneracy. This protection is a direct consequence of the base-centering in the conventional unit cell, which introduces a fractional translation in the symmetry operation. As a result, the k_c = ± π/c planes host symmetry-enforced nodal degeneracies that constrain the electronic band structure and promote the formation of Weyl points.

Having discussed the metallic phase, we now proceed to analyze the insulating phase, which belongs to the structural space group P2/c (#14). In this phase, the degeneracy between the d_{xz} and d_{yz} orbitals is lifted due to monoclinic distortions. Moreover, all four Cr atoms within each Cr4O6 chimney become inequivalent, further splitting the d-orbital levels into twelve non-degenerate states. As a result, the d electrons can redistribute unevenly among the Cr sites. A nominal filling of 2.25 electrons per Cr corresponds to nine out of 12 available t_{2g}-like states being occupied. This is consistent with the calculated (2.1 μB) and experimentally measured (2.2 μB) Cr magnetic moments. Moreover, the total electron count and the total spin moment per primitive cell are both integers, a condition necessary for the opening of a band gap.

Since the magnetic moment directions are confined to the ab-plane, we deduced the highest symmetry magnetic subgroup to be P2'/c' (#14.79). This subgroup is different from metallic phase magnetic group C2' = m' (#12.62) in one distinct way: the monoclinic subgroup of the insulating phase is non-symmorphic, and its primitive unit cell coincides with the conventional one. Consequently, the screw rotation’s fractional translation does not arise from lattice centering, and the symmetry-enforced degeneracy on the k_c = ± π/c planes originates directly from the non-symmorphic symmetry, not from band folding. This reduction of translation symmetry is accompanied by a change in topological character. In the metallic phase, accidental crossings of the Cr t_{2g}-derived bands, allowed by the magnetic space group (#12.62), produce Weyl points near the k_c = ± π/c planes. In the insulating phase, the Weyl points are removed due to the combination of monoclinic crystal distortion and orbital splitting. We further note that the nodal planes are topologically uncharged: they are symmetry-enforced by TC_{1/2[001]}, and inversion symmetry renders their Chern number trivial. By calculating the parity eigenvalues at all time-reversal invariant momentum points35, we confirm that the insulating monoclinic phase is a trivial insulator (i.e., Z4 index = 0). In summary, the disappearance of Weyl points is a direct consequence of translation symmetry reduction due to the tetragonal-to-monoclinic structural distortion, establishing the MIT as a topological phase transition within the FM state.

Phonon soft mode analysis
We next focus on the driving mechanism behind the FM-MIT. The Peierls instability proposition is based on the experimental observation of superstructure peaks at qCDW = (0.5, 0.5, 0) below TMIT17,18. We confirm the existence of these by plotting the temperature dependence of the integrated elastic peak intensity at q = hh0 for h = 4.50(5) in Fig. 4b. In a Peierls transition36, the phonon energy of a strictly 1D material softens via the electron-phonon coupling due to divergence in the electronic susceptibility. This divergence causes the phonon energy to renormalise and collapse onto the elastic line (also known as a Kohn anomaly or phonon condensation), forming new Bragg peaks.

In order to determine the role of phonons in a FM-MIT, we present IXS spectra collected at three different temperatures (115, 95 and 20 K): above, around and below the TMIT, for a wavevector close to qCDW (q = hh2 where h = 4.45) in Fig. 4a. Apart from the elastic peak, the energy scans contain inelastic Stokes and the smaller anti-Stokes pairs, attributed to a phonon mode. The observed phonon mode around qCDW reveals no temperature dependence (~10 meV for all temperatures close to qCDW, Sec. S1). This demonstrates the absence of phonon softening at qCDW during the MIT, firmly establishing that the phase transition is not a Peierls transition.

The absence of phonon softening is more clearly shown in Fig. 4c, d, where the phonon dispersion of q-vectors across qCDW is shown over a wider range. The transverse acoustic phonon branch dispersed linearly with q from the Γ-point for all temperatures. A similar linear dispersion was observed for the longitudinal acoustic phonon, albeit at a larger gradient with respect to q. Nonetheless, irrespective of phonon polarization, we were unable to detect any phonon condensation at qCDW. This is further corroborated by our first principles phonon calculations. The calculated phonon dispersion for the tetragonal structure from Γ to M is superposed with experimental data points obtained from IXS with good agreement (Fig. 4e). These results are indirectly supported by a high-pressure experiment, which shows an antiferromagnetic ordering stabilized under pressure37, instead of a FM-like ordering as theoretically predicted29 for a Peierls insulating FM compound.

We confirm that there is no drastic phonon softening from a higher-energy optical phonons through our Density-Functional Perturbation Theory (DFPT) calculation. Instead, phonon modes in the insulating monoclinic phase exhibit some minor softening for the low-energy Cr phonons relative to the metallic phase (Fig. S11e), leading to higher vibrational entropy. At low temperatures, the insulating phase is favored over the metallic phase as the former will have a lower Gibbs free energy. The fully calculated phonon spectra containing all phonon branches are included in Fig. S11a, b).

Finally, a hard x-ray photoelectron spectroscopy (HAXPES) measurements38 have showed that in the metallic phase, the valence of Cr is not static at the nominal valence value of 3.75. Instead, a dynamic noninteger Cr4+-Cr3+ valence fluctuations was observed. In fact, similar valence fluctuations have indeed mediated phase transitions, such as structural instabilities in YbPb, where lattice-valence fluctuations coupling allowed a structural instability39. One may speculate that there exists a valence-lattice coupling in K2Cr8O16 as well, from which the low temperature structure is derived.

Methods
Samplesynthesis
The difficulty in determining the underlying physics of K2Cr8O16 is partly related to the small size of the available single crystals, which arises from the required high-pressure synthesis conditions. To address this, new cell designs for a high-pressure Walker-type multi-anvil system were developed at the Max Planck Institute for Solid State Research, enabling growth of larger, higher-quality crystals. The new 18/11 and 25/15 multianvil cell assemblies provide more uniform pressure and temperature distributions, resulting in increased sample volumes (up to several mm in size) while maintaining stable synthesis conditions up to ~12 GPa and 1500 °C. With these improved cells the CrO2 impurity phase is reduced to below 2% in powder, and the obtained crystals are significantly larger (up to 0.5 × 0.5 × 1 mm3) with TMIT ≃ 100 K. Although, we have referenced the value of TMIT = 95 K in this work, based on previously reported values.

For this work, two single crystals (0.5 × 0.5 × 1 mm3 and 0.1 × 0.1 × 0.2 mm3) with a TMIT ≃ 100 K were successfully prepared. These crystals were grown using the high-pressure Walker-type multianvil apparatus. A stoichiometric mixture of K2Cr2O7:Cr2O3 = 1:3 was loaded into multianvil cell assemblies and compressed to pressures of approximately 7 GPa. The mixture was heated to temperatures up to 1273 K, held at the synthesis temperature for 1 h, and then cooled to room temperature before decompression to ambient pressure. Single crystals were recovered from the reaction products and selected for the experiments.

InelasticX-rayscatteringmeasurements
The crystals were mounted onto separate copper sample holders and aligned to measure the transverse and longitudinal phonons using IXS. The IXS data were collected at the high resolution beamline BL35XU at the SPring-8 synchrotron source in Japan using Ei = 21.74 keV40. The inelastic peaks were fitted using a damped harmonic oscillator convoluted with the experimental resolution (~1.7 meV). The peaks were fitted using the software Dave41.

Magnetizationmeasurements
Angle-dependent magnetization measurements were performed on a single crystal of K2Cr8O16 (mass m = 0.00148 g) using a Quantum Design MPMS3 magnetometer in DC mode. Measurements were taken at selected angles between 0° and 135° (37°, 45°, 83°, 96°, and 127°) relative to the crystallographic a- and b-axes at T = 120 K under zero-field-cooled conditions. The initial crystal orientation was determined by Laue diffraction, after which controlled relative rotations were applied. The magnet was demagnetized between each angular measurement to minimize remanent fields. Magnetic fields up to 3.5 T were applied, and the measured magnetic moments were converted to molar magnetization.

Assuming fourfold rotational symmetry, the angular dependence of the magnetization was fitted with the sinusoidal function; M(θ) = A sin(4θ + φ) + M0.

Neutronscatteringmeasurements
Neutron powder diffraction (NPD) was measured at the Spallation Neutron Source (SNS) in Oak Ridge National Lab (ORNL) using the POWGEN instrument42, set at a central wavelength of 0.8 Å and a bandwidth of 1 Å. These measurements were made in high resolution mode. About 500 g of sample was loaded under helium gas for temperature equilibration onto a vanadium container. Single-crystal neutron diffraction (ND) was performed at the Paul Scherrer Institut (PSI) using the ZEBRA instrument with wavelength = 1.178 Å on a single crystal (0.5x0.5x1 mm3). A 4 circle goniometer with a closed cycle refrigerator was used in order to measure selected peaks from 10 K up to room temperature. The inelastic neutron scattering (INS) was measured at Japan Proton Accelerator Research Complex (J-PARC) using the 4SEASON instrument on a powder sample of ~2 g with a resolution of 5% of Ei43–45. Given the uniqueness of the instrument, several Ei (7, 9, 12, 18, 27, 47, 97 and 300 meV) were measured simultaneously at 10, 130 and 200 K and the relevant data is presented in this work. All measured neutron scattering intensities were normalized to the spallation source proton current. The diffraction pattern was analyzed using the Fullprof software package46 while the inelastic spectra were analyzed using SpinW47.

DFTcalculations
The DFT calculations were performed using the Quantum Espresso48 package. All DFT calculations for band structures, Jij, phonon, Berry curvature and Weyl point analyses were performed in the ferromagnetic spin-polarized state. DFT calculations were carried out with U = 4.0 eV for the insulating monoclinic phase to capture electron-correlation effects, whereas U = 0.0 eV was used for the metallic phase, where metallic screening is expected to strongly reduce the effective on-site interactions. The phonon eigenvectors and eigenvalues were calculated using the finite displacement and supercell approach with Phonopy49. Wannier functions are constructed from the DFT eigenvectors and eigenvalues using Wannier9050. The Cr t2g states were used as initial projections in the local coordinates of the octahedral crystal field. Using these Wannier functions, we use the magnetic force theorem23,24 to map the tight-binding Wannier Hamiltonian onto a classical Heisenberg model using TB2J51. The chiralities of the Weyl points were calculated using WannierTools52 with the same Wannier basis sets. For more technical details, see Sec. S3 of SM.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript is a combined computational and experimental work outlining a topological phase transition that occurs beneath (in temperature) a magnetic phase transition to a magnetic state. The compound, K Cr O is unusual for two 2 8 16 reasons: first the topological transition within an existing FM state and second that the topological nature depends on spin- lattice coupling and could therefore be manipulated by a field. The manuscript argues against a classic Peierls transition into the insulating phase. This is most certainly interesting from a basic science and future device point of view. However, a few things badly need clarification. Cognizant of the fact that there is limited space and some things must necessarily be relegated to SM, it is still necessary to make clear in the main text what is being done. And, in the SM where there is plenty of room, it should be crystal clear. First, it is not possible to tell which of the calculations are done spin-polarized (\"magnetic\"), if any. The supplemental material describes the paramagnetic tetragonal and paramagnetic monoclinic phases. As far as I can tell in the literature, there is no paramagnetic monoclinic phase since the monoclinic phase occurs fully within the FM state and remains after the MIT - a main interest point of this material. Were both of these phases calculated non-magnetically (un-spin-polarized) in DFT? If so, its very hard to draw any conclusions about the true band structure. The authors rigidly shift the bands to mimic the experimental gap, indicating that, again, this is within the FM state and should be calculated that way. Could exchange splitting itself contribute to the gap? Spin simply can't be ignored. And if it has not been ignored, it needs to be made far, far clearer what is happening in the calculations. The SM mentions noncollinear calculations which indicate that maybe magnetism was, in fact, explored. This could, in principle, be used to resolve the in-plane magnetic orientation, but after the single instance in the methodology part, this concept is never mentioned again. Are there moments? Are they non-collinear? On a similar note, what *are* the calculated moments of the Cr atoms in the insulating phase? They should be integer (it is gapped), but this would require non-uniform spins. This would be a very interesting application of DFT but does not seem to have been done. Again with respect to magnetism, the authors claim that the Peierl's mechanism is not operative because of the absence of a phonon soft mode at the CDW wavevector. But once again, the phonon calculations were done within the non-magnetic phase (by definition since methods to do it in the FM phase are currently lacking) and the transition of interest occurs in the FM phase. Given that there is certainly some and likely fairly strong spin-lattice coupling, calculations that lack spin-polarization do not reveal the true underlying physics. Finally, the phase transition from tetragonal to monoclinic is initially described as \"a structural distortion from a metallic tetragonal phase to an insulating monoclinic phase (Fig. 1) with qCDW = (1/2, 1/2, 0)\". Fig. 1 shows a larger monoclinic phase that does indeed look to be based on a q=(1/2, 1/2, 0) supercell. But later, when calculating the Z4 invariant, the authors state \"we confirm that the monoclinic phase is indeed a trivial insulator (i.e., Z4 index is 0 in the absence of CDW\". I dug through the SM and thought I found the answer: the authors found a magnetic space group that is monoclinic but not the same as the monoclinic cell resulting from the CDW, but then this sentence is included \"Note that for simplicity and consistency with the literature, we will refer to both the paramagnetic and FM states by the lattice type of their structural space group, i.e., as the tetragonal phases of K2Cr8O16, even though the magnetic lattice of latter is actually monoclinic\". Are there two monoclinic cells? One ferromagnetic and metallic and the other ferromagnetic and insulating, the latter being the CDW cell? If it is not the CDW (post-MIT) cell that is calculated to be non-trivial, how can we understand that this is truly a topological transition? I hope that these problems can be solved with better wording, but if all the calculations were done non-magnetically, I fear they are not useful for elucidating the interesting physics of this material. In conclusion, much clarification is needed to justify the computational portion of this manuscript before publication is warranted.", "reviewer_2": "This manuscript reports the combined experimental and computational study of the hollandite compound K2Cr8O16 that shows a metal-insulator transition within the ferromagnetically ordered phase. The authors revisit the mechanism of this transition and rule out the hitherto widely accepted Peierls scenario. They argue that the transition is topological in nature and driven by a nesting between the Weyl points. This work makes at least two advances in understanding K2Cr8O16. First, it uses inelastic neutron scattering to identify \"chimneys\" as the magnetic building blocks of the material. Second, it shows the absence of phonon softening near the transition, thus ruling out the conventional Peierls mechanism. K2Cr8O16 is an important model system for a metal-insulator transition in the presence of magnetic order. Moreover, the idea of a metal-insulator transition driven by topological features of the band structure is certainly interesting, even if not thoroughly proved in the present version of the manuscript. Therefore, I am generally supportive of publication in Nature Comm. but only if stronger evidence for the occurrence of the Weyl points and their role in the transition could be demonstrated. 1. Leading magnetic interactions and the absence of phonon softening are convincingly proved experimentally. By contrast, the discussion of the Weyl points and their nesting fully relies on DFT and lacks any direct support from the experiment. K2Cr8O16 is probably a correlated material that may not be fully described by DFT or even by DFT+U. In this context, ARPES measurements would be ideal in order to pinpoint the Weyl points. Alternatively, the dependence of the proposed Weyl points on the 'U' parameter should be discussed in detail. Are the appearance of these Weyl points near the Fermi level and their nesting robust features that do not depend on the computational parameters? Fig. 3c,d shows band structures with the Weyl points, but it is not quite clear whether those are calculated with or without 'U'. This information should be available directly on the graphs or in the figure caption, and the dependence on 'U' should be analyzed. 2. While the absence of phonon softening indicates an electronic origin of the transition, it is not quite clear why this transition is attributed to the nesting of the Weyl points and not to some other features of the electronic structure. A Fermi surface nesting would be a more natural scenario in this case. This possibility should be explored in detail, e.g., by calculating Lindhard susceptibility. On general grounds, it is not quite plausible that some features lying 0.1 eV away from the Fermi level trigger such a major transformation in the material. 3. It is highly desirable to pin down the magnetic moment direction, because it determines the exact positions of the Weyl points and the separation between them. This can be done by angle-dependent magnetization measurements on a single crystal. 4. While the metal-insulator transition in K2Cr8O16 is dubbed topological, there seems to be no direct evidence for its topological nature. The formation of an axion insulator in the low-temperature monoclinic phase is speculated but never proved. In my opinion, the claims of the topological transition and axion-insulator phase are exaggerated and should be removed unless they can be directly supported by DFT calculations or experiments. 5. Previous studies have shown that the tetragonal structure of K2Cr8O16 is stable in PBE, whereas imaginary phonons appear in PBE+U [Phys. Rev. B 90, 045124 (2014)]. Therefore, a phonon-driven transition becomes possible when correlations are sufficiently strong. While it does not seem to be the case experimentally, the agreement between the experimental and calculated phonon frequencies (Fig. 4e) is mediocre. It is not quite clear whether DFT or DFT+U was used in the phonon calculations, and it will be desirable to compare the experimental phonon spectrum with both calculations. Which calculation allows a better agreement with the experiment? 6. The INS fits are not very convincing because there is very little contrast in the experimental data. Please, show the constant-energy cuts along with the spin-wave fits in order to confirm the accuracy of the proposed model. 7. Calculated magnetic interactions are proportional to t^2/U, but the reasons for it are not explained in detail. The t^2/U dependence is typical for antiferromagnetic superexchange, but J1-J3 are all ferromagnetic. Which hopping parameters are considered in this case? I could imagine that FM superexchange follows the t^2/U dependence if 't' is the t2g-eg hopping, but is it really the case? What about the t2g-t2g hoppings? They must lead to antiferromagnetic superexchange interactions. 8. Ref. [19] is not displayed in the manuscript. I presume it is New J. Phys. (2020)", "reviewer_3": "K₂Cr₈O₁₆ ferromagnetic hollandite has been studied for almost two decades for presenting the prototype ferromagnetic metal-insulator transition, a rare phenomenon. In two seminal PRLs in 2009 and 2011 (cited as refs. 16, 17 in the manuscript), and also in another paper the year after (ref. 18), the following description was given: K₂Cr₈O₁₆ was determined to be a quasi-1D system with Peierls distortion in the ferromagnetic insulating state. Peierls here is due to the dimerization of tetramers within the unit cell with no doubling of the lattice, but it does not bring along charge ordering on the Cr sites. Ab initio calculations describe a double exchange mechanism between double chains via corner-shared oxygens, with exchange constants of 5.3 meV within the double chains but 10.6 meV between the double chains. This results in a Fermi surface of strongly quasi-1D character with a nesting vector of (0,0,1) along the c direction. The low-temperature structure was described in the P112₁/a space group. In general, there is a mixed Cr³⁺/Cr⁴⁺ valence. This picture has become well accepted. A decade later, two ab initio studies—one cited as ref. 22 from NJP in 2020, the other from the same group in PRL the year before, but for some reason not cited (the manuscript refers to ref. 19 for a similar study, but it is actually missing from the reference list, probably a typo) — already described Weyl fermions in the system. The present manuscript describes a different picture for the MIT. It presents experimental arguments against a Peierls instability and argues that the Weyl fermions bring along a correlation-induced topological phase transition that results in the MIT. The low-temperature magnetic structure is described in the C2'/m' space group, surprisingly, since it doesn't seem to be the best choice for ferromagnetic ordering. The manuscript also argues against the interaction being double exchange, based on calculations of the dependence of the exchange on the hopping integral, and instead argues for a uniform Cr3.75+ valence state with superexchange-like interactions between them. Yet, the manuscript proposes exchange values of 6 and 10 meV, quite similar to those from 2011. Therefore, the current manuscript proposes a drastically different chemical situation from the original picture and a dramatically different origin for the MIT. A particularly glaring omission in the manuscript is that it strongly argues for the much improved quality of the samples, based on a modified high-pressure synthesis, but does not mention a single word about the modification. The original PRL from 2009 describes the high-pressure synthesis conditions in detail. The manuscript proposes to set up a strong and interesting polemic in the literature. The pros and cons should be more clearly and antagonistically described; the arguments in the current text are quite scattered. However, and more importantly, there are both experimental and ab initio arguments in the current study. The ab initio results don't seem to be qualitatively different from the 2019–20 results. The experimental results of the inelastic X-ray scattering show a lack of phonon condensation. This is employed as an argument against the Peierls instability. However, the Peierls distortion proposed in 2011 is quite unusual. Would it necessarily be accompanied by drastic changes in the phonon spectrum? The inelastic neutron scattering shows a lack of drastic change in the spin-wave excitations across the MIT. It is unclear—although the text makes it look as such—how this relates to the lack of a Peierls instability."}, "author_response": "Report of the first Referee\nThis manuscript is a combined computational and experimental work outlining a topological\nphase transition that occurs beneath (in temperature) a magnetic phase transition to a magnetic\nstate. The compound, K2Cr8O16 is unusual for two reasons: first the topological transition within\nan existing FM state and second that the topological nature depends on spin-lattice coupling and\ncould therefore be manipulated by a field. The manuscript argues against a classic Peierls\ntransition into the insulating phase. This is most certainly interesting from a basic science and\nfuture device point of view.\nHowever, a few things badly need clarification. Cognizant of the fact that there is limited space\nand some things must necessarily be relegated to SM, it is still necessary to make clear in the\nmain text what is being done. And, in the SM where there is plenty of room, it should be crystal\nclear.\nAuthor Reply:\nWe thank the referee for the careful reading, constructive suggestions, and interest in our work.\nIn the revised manuscript and Supplementary Material (SM), we have streamlined the language\nto improve clarity while maintaining technical accuracy. We now explicitly state the calculation\nconfigurations: All calculations were performed using spin-polarized settings.\nRegarding the crystal structures, while the parent crystal structures are tetragonal (metallic) and\nmonoclinic (insulating), the inclusion of ferromagnetic (FM) polarization breaks the tetragonal\nsymmetry and results in a monoclinic symmetry. As a result, the system exhibits two distinct\nmonoclinic configurations: one corresponding to a magnetic metallic state, and the other to a\nmagnetic insulating state. While both share monoclinic symmetry, they are structurally and\nelectronically distinct. We understand that our original phrasing may have caused confusion by\nnot clearly distinguishing between crystallographic symmetry and symmetry breaking due to\nmagnetic ordering. We have clarified the language throughout the manuscirpt and now explicitly\nindicate which symmetry/phase is being referred to in each case.\n1. First, it is not possible to tell which of the calculations are done spin-polarized (\"magnetic\"), if\nany. The supplemental material describes the paramagnetic tetragonal and paramagnetic\nmonoclinic phases. As far as I can tell in the literature, there is no paramagnetic monoclinic phase\nsince the monoclinic phase occurs fully within the FM state and remains after the MIT - a main\ninterest point of this material. Were both of these phases calculated non-magnetically (un-spin-\npolarized) in DFT? If so, its very hard to draw any conclusions about the true band structure. The\nauthors rigidly shift the bands to mimic the experimental gap, indicating that, again, this is within\nthe FM state and should be calculated that way. Could exchange splitting itself contribute to the\ngap? Spin simply can't be ignored. And if it has not been ignored, it needs to be made far, far\nclearer what is happening in the calculations.\nAuthor Reply 1:\nhighlighting the difference between a spin-polarized calculation and a paramagnetic calculation.\nIn the SM, we used the term paramagnetic in the context of our group theory analysis. It denotes\nthe structural parent cell prior to any symmetry reduction imposed by spin polarization. We used\nit only as a crystallographic parent to extract symmetry relations. The “paramagnetic monoclinic”\ncell belongs to space group #14, while the paramagnetic tetragonal cell belongs to #87. These\nare subsequently reduced to the magnetic subgroup of #14.79 and #12.62 once the magnetic\nmoments are taken into account. All first-principles calculations (band structures, J , phonon,\nij\nWeyl points analysis) are fully spin-polarized, for both the insulating and metallic phases.\nRevisions\nWe understand that our original phrasing may have caused confusion by not clearly\ndistinguishing between crystallographic and symmetry lowering due to magnetic ordering. In the\nrevised manuscript and Supplementary Material, we have clarified the language and now\nexplicitly state which symmetry is being referred to in each case.\nWe have also added the following clarifications in the SM Sec. S3A:\n“All first-principles calculations (bandstructures, J , phonon and Weyl points analyses) were\nij\nperformed in the ferromagnetic spin-polarized state, for both the insulating and metallic phases.”\nWe also added elaborations in the “Methods” Section of the Main Text:\n\"All DFT(+U)+SOC calculations for band structures, J , phonon and Weyl points analyses were\nij\nperformed in the ferromagnetic spin-polarized state. \"\n2. The SM mentions noncollinear calculations which indicate that maybe magnetism was, in fact,\nexplored. This could, in principle, be used to resolve the in-plane magnetic orientation, but after\nthe single instance in the methodology part, this concept is never mentioned again. Are there\nmoments? Are they non-collinear?\nAuthor Reply 2:\nWe thank the referee for the opportunity for clarification. For the Weyl points calculations of the\nmetallic phase, we performed noncollinear DFT+SOC calculations. The spin directions are\nunconstrained, but after self-consistency was achieved in the DFT calculation, a collinear\nconfiguration was found to be energetically most favorable. This configuration is also what was\nfound from experiment (neutron diffraction). The calculated ferromagnetic moment of Cr in the\nmetallic phase is 1.9 µ The ordered moment was experimentally measured to be 1.6 µ at 110\nB. B\nK. By extrapolating using the mean-field–type expression m(T) = m (1 − T/T )β with β = 0.22 and\nT = 170 K, we estimate the zero-temperature moment to be m ≈ 2.0 μB, consistent with the\nC 0\ncalculated value.\naffect the nodal positions. Even though the exact positions of the Weyl points depend on the\ndirections of the magnetic moment (as also reported by New Journal of Physics 22, 073062\n(2020)), we found that the nodal points always form a cross-like feature about the nodal plane,\nsuch that nested nodal pairs are always present in the (0.5,0.5,0.0) and (-0.5,-0.5,0.0) directions\nof the tetragonal structure Brillouin zone (SM Fig. S8).\nOn a similar note, what *are* the calculated moments of the Cr atoms in the insulating phase?\nThey should be integer (it is gapped), but this would require non-uniform spins. This would be a\nvery interesting application of DFT but does not seem to have been done.\nWe thank the referee for raising this insightful question.\nFor the insulating monoclinic phase, we carried out fully noncollinear DFT+U+SOC calculations\nwith unconstrained spin directions. After reaching self-consistency, the system relaxes back to a\ncollinear ferromagnetic alignment, confirming that the collinear configuration is energetically\npreferred, consistent with neutron diffraction results.\nIn the metallic FM phase, each Cr ion carries a nominal charge of +3.75, corresponding to 2.25\nelectrons in its t₂ -like orbitals (dₓ , dₓ , d ). This fractional occupation ensures that the bands\ng y z yz\ncannot be fully filled, resulting in a metallic or semi-metallic state.\nIn the insulating FM phase, two key structural changes allow for gap formation. First, the\nmonoclinic strcuture distortion lifts the degeneracy between the dₓ and d orbitals. Second, the\nz yz\nfour Cr atoms in each Cr₄O₆ chimney become crystallographically inequivalent, further splitting\nthe d-orbital levels into twelve non-degenerate states. As a result, the nine electrons from four\nCr atoms (4 × 2.25) can redistribute unevenly among the twelve orbitals. This redistribution\nenables nominal integer orbital fillings, which allows a band gap to open.\nThe calculated magnetic moment per Cr atom in the insulating phase is 2.1 μ , in close agreement\nB\nwith the experimental value of 2.2 μ measured at 10 K. This is consistent with the nominal\nB\nmixed-valence expectation of 2.25 μ /Cr: [2Cr3+ (d3) + 6Cr4+ (d2)]/8. The small reduction in the\nB\ncalculated value, compared with the ionic value, reflects Cr–O covalency and inter-site\nhybridization, which transfer part of the spin density onto the oxygen ligands and slightly\ndelocalize it across Cr sites. Importantly, the total electron count per primitive cell remains\ninteger, as required for an insulating state, and the total magnetic spin moment per unit cell is\nalso an integer.\nRevision:\nWe have updated the manuscript to include the following sentence regarding the moment size:\n“This is consistent with the calculated (2.1 μ ) and experimentally measured (2.2 μ ) Cr magnetic\nB B\n4\nmoments. Moreover, the total electron count and the total spin moment per primitive cell are\nboth integers, a condition necessary for the opening of a band gap.”\nWe have also included the following to the main text:\n“In addition, we find that the positions of the nodal points are robust. Regardless of the direction\nof the magnetic moment, we found that the nodal points always form a cross-like feature about\nthe nodal plane, such that nested nodal pairs are always present in the (0.5,0.5,0.0) and (-0.5,-\n0.5,0.0) directions of the tetragonal Brillouin zone (Figs. 3(b) and SM Fig. S8).”\nFigure 1. Cross-like nodal points and their dependence on the directions of the in-plane magnetic moment. (a-h)\nEvolution of the nodal points (gapless points around the k = ±π/c plane) as a function of the magnetization direction.\nc\nHere, the angle defines the direction of the magnetic moment relative to the a -lattice vector of the monoclinic\nM\nsupercell, which increases counterclockwise within the a b -plane (as defined in Fig. 1(a) of Main Text). Although\nM M\nvarying the direction of the magnetic moment shifts the positions of the nodal lines and position of the nodes, the\noverall cross-like feature persists, ensuring that nested nodal pairs persist along the directions of [0.5, 0.0, 0.0] and\n[0.0, 0.5, 0.0] in the monoclinic BZ (corresponding to [0.5, −0.5, 0.0] and [0.5, 0.5, 0.0] in the tetragonal BZ). Circle\nradii at the nodal points are proportional to log(gap), while color intensity scales as [log(gap)]2 using the colormap\nfrom (a), highlighting regions with smaller band gaps.\n3. Again with respect to magnetism, the authors claim that the Peierl's mechanism is not\noperative because of the absence of a phonon soft mode at the CDW wavevector. But once again,\nthe phonon calculations were done within the non-magnetic phase (by definition since methods\nto do it in the FM phase are currently lacking) and the transition of interest occurs in the\nFM phase. Given that there is certainly some and likely fairly strong spin-lattice coupling,\ncalculations that lack spin-polarization do not reveal the true underlying physics.\nAuthor Reply 3:\nWe thank the referee for raising this important point and for the opportunity to clarify the role\nof spin polarization in our phonon calculations.\ncarried out in a spin-polarized state. All phonon dispersions reported in this work were obtained\nusing the finite-displacement supercell method with spin-polarized DFT, based on Quantum\nESPRESSO force calculations in the experimentally observed ferromagnetic configuration. This\napproach captures the spin-lattice coupling within the framework of DFT.\nSpecifically, for the metallic tetragonal phase, phonons were computed with the following\nparameters (see Supplementary S3D and Table S4):\n• Supercell: 1 × 1 × 4\n• Displacement amplitude: 0.01 Å\n• k-point grid for forces: shifted 2 × 2 × 2\n• Hubbard U: 0 eV\n• Spin configuration: collinear ferromagnetic\n• Plane-wave cutoff: 40 Ry\n• Method: finite-displacement phonons using Phonopy + Quantum ESPRESSO\nThis calculation reveals no imaginary phonon modes, including at the CDW wavevector q = (0.5,\n0.5, 0). The results are directly compared with inelastic x-ray scattering measurements in Fig. 4e\nof the main text, showing good agreement without any anomalous softening near the transition.\nIn light of this, we respectfully clarify that the statement \"by definition, methods to do [phonon\ncalculations] in the FM phase are currently lacking\" is not correct. In fact, two established first-\nprinciples frameworks allow phonon calculations in magnetically ordered phases:\n1. The finite-displacement method (used here) is always compatible with spin-polarized\nstates, since its forces are evaluated from fully self-consistent spin-polarized calculations.\n2. Density-functional perturbation theory (DFPT) with spin polarization has been available\nsince the early 2000s (e.g., Baroni et al., Review of Modern Physics 73, 515 (2001)), and\nis implemented in several DFT packages including Quantum ESPRESSO. Spin-polarized\nDFPT is routinely used in modern studies of magnetic materials, including phonon\ndispersions in bcc Fe and Fe-Co alloys (e.g., Nature Communications, 6, 9961 (2015);\nPhysical Review B 109, 184306 (2024)).\nIn summary, the spin-lattice coupling relevant to the FM state of K₂Cr₈O₁₆ is already captured by\nour finite-displacement calculations. The absence of a soft mode at the CDW wavevector in this\nfully spin-polarized framework rules out a conventional Peierls mechanism as the driver of the\nmetal-insulator transition. This strengthens our conclusion that the transition arises from an\nelectronic (topological) instability rather than a structural one.\nRevision:\nWe have revised the Methods section of the main text to explicitly address this point. In addition,\nas part of our response to Question 1, we have added further elaboration in the “Methods”\nsection of the main text:\n6\n\"All DFT calculations for band structures, J , phonon, Berry curvature and Weyl points analyses\nij\nwere performed in the ferromagnetic spin-polarized state.\"\n4. Finally, the phase transition from tetragonal to monoclinic is initially described as \"a structural\ndistortion from a metallic tetragonal phase to an insulating monoclinic phase (Fig. 1) with qCDW\n= (1/2, 1/2, 0)\". Fig. 1 shows a larger monoclinic phase that does indeed look to be based on a\nq=(1/2, 1/2, 0) supercell. But later, when calculating the Z4 invariant, the authors state \"we\nconfirm that the monoclinic phase is indeed a trivial insulator (i.e., Z₄ index is 0 in the absence of\nCDW\".\nI dug through the SM and thought I found the answer: the authors found a magnetic space group\nthat is monoclinic but not the same as the monoclinic cell resulting from the CDW, but then this\nsentence is included \"Note that for simplicity and consistency with the literature, we will refer to\nboth the paramagnetic and FM states by the lattice type of their structural space group, i.e., as\nthe tetragonal phases of K2Cr8O16, even though the magnetic lattice of latter is actually\nmonoclinic\". Are there two monoclinic cells? One ferromagnetic and metallic and the other\nferromagnetic and insulating, the latter being the CDW cell? If it is not the CDW (post-MIT) cell\nthat is calculated to be non-trivial, how can we understand that this is truly a topological\ntransition?\nAuthor Reply 4:\nWe thank the referee for the careful reading of our manuscript. The referee is correct that there\nare two monoclinic magnetic cells: one corresponding to the ferromagnetic–metallic state (space\ngroup #12.62) and the other to the ferromagnetic–insulating state (space group #14). The\nunderlying crystallographic parent structures, before including magnetic ordering, are metallic\ntetragonal (space group #87) and insulating monoclinic (space group #14). Once ferromagnetic\norder is imposed, the tetragonal parent cell is reduced to monoclinic symmetry. For our\ncalculations, we employed the same supercell lattice—specifically, the unit cell of the insulating\nphase, which is a (2×2×1) supercell for the metallic crystal structure. While both phases adopt a\nmonoclinic lattice, they are not equivalent: the monoclinic lattices belong to different space\ngroups, since atomic displacements in the insulating phase break additional symmetries and alter\nthe atomic positions relative to the metallic phase.\nIn the statement, “we confirm that the monoclinic phase is indeed a trivial insulator (i.e., Z₄ index\nis 0 in the absence of CDW),” we are referring to the insulating monoclinic cell (post-MIT). Our\ncalculations show that its band topology is trivial. Meanwhile, our Berry curvature and Weyl point\nanalyses of the metallic phase (see Fig. 3b) confirm that its band topology in the magnetic\nmonoclinic phase is non-trivial. The transition between these two phases coincides with the MIT,\nconfirming that it is indeed a topological transition.\nRevision:\nWe understand that this was not clear and we have revised the manuscript to address this point.\nIn the revised manuscript and Supplementary Material, we have clarified the language and now\nexplicitly state which symmetry is being referred to in each case. Specifically, we now reference\n7\nthese phases as “metallic phase” or “insulating monoclinic phase” etc. to be more explicit. We\nhave also added the following sentence to the main text. “The FM spin polarization reduces the\nsymmetry from the paramagnetic space group I4/m (#87) to the magnetic subgroup C2'/m'\n(#12.62), which is monoclinic. “\n5. I hope that these problems can be solved with better wording, but if all the calculations were\ndone non-magnetically, I fear they are not useful for elucidating the interesting physics of this\nmaterial.\nIn conclusion, much clarification is needed to justify the computational portion of this manuscript\nbefore publication is warranted.\nAuthor Reply 5:\nWe are grateful to the referee for giving us this opportunity to clarify the details of the\ncalculations. The comments have greatly helped improve the quality and clarity of our manuscript.\nWe explicitly mention the correct symmetry/phase for clarity, and all calculations in this work\nwere done spin polarised, which is now clearly stated in both the Main text and the SM.\nThis manuscript reports the combined experimental and computational study of the hollandite\ncompound K2Cr8O16 that shows a metal-insulator transition within the ferromagnetically\nordered phase. The authors revisit the mechanism of this transition and rule out the hitherto\nwidely accepted Peierls scenario. They argue that the transition is topological in nature and\ndriven by a nesting between the Weyl points.\nThis work makes at least two advances in understanding K2Cr8O16. First, it uses inelastic neutron\nscattering to identify \"chimneys\" as the magnetic building blocks of the material. Second, it shows\nthe absence of phonon softening near the transition, thus ruling out the conventional Peierls\nmechanism. K2Cr8O16 is an important model system for a metal-insulator transition in the\npresence of magnetic order. Moreover, the idea of a metal-insulator transition driven by\ntopological features of the band structure is certainly interesting, even if not thoroughly proved\nin the present version of the manuscript. Therefore, I am generally supportive of publication in\nNature Comm. but only if stronger evidence for the occurrence of the Weyl points and their role\nin the transition could be demonstrated.\nAuthor Reply:\nWe appreciate the referee’s thoughtful assessment and their general support for the publication\nof this work in Nature Communications. To address the referee’s primary concern —the presence,\nlocation, and role of the Weyl points — we have performed angle-dependent magnetization\nmeasurements to precisely determine the direction of the magnetic order. Based on these results,\nwe recalculated the band structure, incorporating the refined spin orientation. The updated band\nstructure reveals Weyl points connected by a nesting vector of approximately (0.75, 0.75, 0),\nwhich, while not exactly commensurate, closely aligns with the experimentally observed lattice\ndistortion vector (0.5, 0.5, 0). In addition, we find that the positions of the nodal points are robust.\nRegardless of the direction of the magnetic moment, we found that the nodal points always form\na cross-like feature about the nodal plane, such that nested nodal pairs are always present in the\n(0.5,0.5,0.0) and (-0.5,-0.5,0.0) directions of the tetragonal Brillouin zone. Importantly, aside\nfrom these nodal points, there are no indications of conventional Fermi surface nesting or other\nelectronic instabilities along this direction. This makes the correlation between the Weyl point\nnesting and the structural distortion particularly compelling.\nAdditionally, we now include preliminary ARPES data in this response letter to demonstrate\nconsistency between the calculated band structure and experimental observations. These data\nsuggest a Weyl point nesting vector of approximately (0.48, y, 0.02), which is consistent with the\nlattice distortion vector, although the y-component cannot be precisely determined at this stage.\nWe hope this can be further clarified in a follow-up study, which would require a careful photon\nenergy scan.\nWe wish to however emphasize that while our results show that the metal-insulator transition is\na topological phase transition within a magnetically ordered state — connecting a Weyl\nsemimetal and a topologically trivial insulating state — this study does not directly demonstrate\nthat the Weyl points drive the system into an axion-insulating ground state. While the compound\n9\nis certainly a good candidate, we have intentionally left this as an open question for future\nresearch. In the revised manuscript, we have clarified this point further. For example, it is\nmentioned that K₂Cr₈O₁₆ is exhibiting “potential axionic properties”.\nWe thank the referee for the opportunity to clarify these issues and for their valuable input,\nwhich has helped us strengthen the manuscript.\n1. Leading magnetic interactions and the absence of phonon softening are convincingly proved\nexperimentally. By contrast, the discussion of the Weyl points and their nesting fully relies on\nDFT and lacks any direct support from the experiment. K2Cr8O16 is probably a correlated\nmaterial that may not be fully described by DFT or even by DFT+U. In this context, ARPES\nmeasurements would be ideal in order to pinpoint the Weyl points. Alternatively, the\ndependence of the proposed Weyl points on the 'U' parameter should be discussed in detail. Are\nthe appearance of these Weyl points near the Fermi level and their nesting robust features that\ndo not depend on the computational parameters? Fig. 3c,d shows band structures with the Weyl\npoints, but it is not quite clear whether those are calculated with or without 'U'. This information\nshould be available directly on the graphs or in the figure caption, and the dependence on 'U'\nshould be analyzed.\nAuthor Reply 1:\nWe thank the referee for this important and insightful point. As the referee correctly notes, the\ndegree of electronic correlation in K₂Cr₈O₁₆ remains an open question. While there is\nexperimental evidence suggesting moderate correlation effects—such as those indicated by\nHAXPES measurements [Physical Review X 5, 041004 (2015)]—the extent to which standard DFT\nor DFT+U captures these correlations remains an area of active discussion.\nNonetheless, we find that standard DFT (GGA) performs remarkably well in describing several\nkey properties of K₂Cr₈O₁₆, supporting its use in this context:\n• Magnetic interactions: The exchange interactions calculated via DFT are consistent with\nour neutron scattering results, showing that J and J (nearest-neighbor couplings) are\n1 2\nsmaller than J (the third-nearest-neighbor coupling). This agreement indicates that DFT\n3\ncaptures the essential magnetic physics observed experimentally.\n• Curie temperature: While DFT predicts a Curie temperature of ~300 K (Fig. 2 below)—\nhigher than the experimental value of ~170 K—this level of overestimation is not\nunexpected. It is well known that DFT, particularly in the mean-field treatment of\nmagnetism, often yields elevated T values. Importantly, the qualitative agreement\nC\n(ferromagnetic ordering and scale of magnetic interactions) supports the reliability of the\nmagnetic picture obtained.\n• Moment size: The calculated ferromagnetically ordered magnetic moment of the Cr of\ninsulating monoclinic phase is 2.1 µ (Fig. 2 below) which is close to the experimental\nB\nvalue of 2.2 µ\nB.\n10\n• Band structure: The DFT band structure agrees well with our preliminary ARPES data\npresented below (Fig. 4), further validating the reliability of DFT in capturing the low-\nenergy electronic states relevant to the Weyl physics.\nWhile we agree that a full treatment of dynamic correlations (e.g., via DMFT or GW) would be\nnecessary to describe certain high-energy features or the Mott physics, the physical phenomena\ndiscussed in this work—namely, the topology of the metallic state and its relation to the observed\nstructural distortion—are predominantly governed by the symmetry and geometry of the band\nstructure, which are already well captured by DFT. We believe, therefore, that DFT provides a\nreliable framework for identifying and analyzing the Weyl points and their nesting in K₂Cr₈O₁₆.\nFigure 2. Calculated magnetization as a function of temperature. See Fig. 1 in Ref. Nakao, A. et al. ”Observation of\nStructural Change in the Novel Ferromagnetic Metal–Insulator Transition of K2Cr8O16.” J. Phys. Soc. Jpn. 81, 054710\n(2012) for measured magnetization as a function of temperature.\nRobustness of Weyl points and role of U:\nThe Weyl points we report are consistent with prior DFT studies [e.g., New Journal of Physics 22,\n073062 (2020)] and remain robust under small variations of magnetic configuration and\nexchange-correlation functionals. To directly address the referee’s concern, we have performed\nadditional PBE+U calculations with U ranging from 1 eV to 4 eV (Fig. 3). We found that other than\nthe slight increase in the bandwidths, the qualitative features of electronic band structure\nremains largely the same:\n• The overall electronic structure remains qualitatively similar across this range.\n• Certain segments of the nodal plane begin to gap, while the cross-shaped nodal features\nand their corresponding nesting vectors near (±0.5, ±0.5, 0) are preserved for U ≤ 3 eV.\n• Some partial gapping appears for U = 2 – 3 eV, but the essential topology and nesting\nbehavior remain intact.\n• Only for U > 4 eV does strong localization lead to a complete gapping of the nodal\nfeatures.\nand physically reasonable range of U.\nFigure 3. Wannier bandstructure of K Cr O for (a) U = 0eV, (b) U = 1eV, (c) U = 2 eV and (d) U = 3 eV. (e) The\n2 8 16\npositions of the nodal points for Hubbard U for 0 < U < 3 eV. The cross-like feature remains robust even under the\ninfluence of Hubbard U for 0 < U < 3 eV.\nJustification of U = 0:\nWith this in mind, we employed DFT-PBE (U = 0) for the metallic phase in this work. This choice\nis both standard practice and physically well justified in the context of strongly correlated\nmaterials exhibiting a metal-insulator transition, such as VO2 and V2O3 [Physics Review Research\n3, 033286 (2021); Science 362, Issue 6418 pp. 1037-1040 (2021); The Journal of Chemical Physics\n160, 134101 (2024)]. In metallic systems, the d-electrons are itinerant and the Coulomb\ninteraction is strongly screened by conduction electrons. Consequently, the effective on-site\ninteraction is significantly reduced, and the electronic properties of the metallic phase are\nalready well captured by conventional DFT. As noted in a recent article [Science 362, Issue 6418\npp. 1037-1040 (2021)], the metallic rutile phase of VO is described as having “little electronic\n2\ncorrelation,” in contrast to the correlated insulating monoclinic phase.\nApplying a static Hubbard U in a metallic phase can introduce unphysical artifacts. Several studies\nhave shown that it can over-localize the electrons, artificially open a gap, or induce incorrect\nmagnetic ordering [The Journal of Chemical Physics 160, 134101 (2024)]. Similarly, hybrid\nfunctionals, which introduce a screened Fock exchange equivalent to a static U, have been shown\nto spuriously drive the metallic phase into an insulating state or cause it to relax to the low-\ntemperature phase [Physical Review B 86, 081101(R) (2012); Journal of Computational Chemistry\n41, 1 (2019); Physical Review B 103, 075134 (2021)]. These results caution against using DFT+U\nindiscriminately in metallic regimes. Theoretically, the Hubbard U in DFT+U is a static\napproximation to the frequency-dependent self-energy. A more accurate description of the\n12\ndynamically screened Coulomb interaction in the metallic state requires a many-body approach\nsuch as GW or DMFT [Physical Review Research 3, 033286 (2021)]. However, these methods are\nbeyond the scope of the present study, which focuses on the low-energy band topology\naccessible within static mean-field theory.\nGiven that the focus of this work is on the topology of the metallic state, which is largely governed\nby symmetry and geometry of the band structure, DFT without U offers a physically justified and\ncomputationally stable framework.\nARPES measurements:\nFollowing the referee’s suggestion to provide experimental support for the Weyl points, we have\nconducted initial ARPES measurements on K₂Cr₈O₁₆. Identifying the exact location of the Weyl\npoints would have been most straightforward through band structure measurements on the\n(0,0,1) surface as a function of k . However, due to the quasi-one-dimensional crystal structure\nz\nof K₂Cr₈O₁₆, the material cleaves preferentially along the {0,1,0} planes, which limits direct access\nto the ideal geometry. Moreover, the quasi-1D nature of the compound leads to an exponential\nsuppression of photoemission intensity near the Fermi level, further constraining the\nmeasurements.\nDespite these limitations, we are able to partially reconstruct the electronic structure by probing\nthe (k , k ) plane (see Fig. 5). In this geometry, we observe Fermi surface features that is\nx z\nconsistent with DFT predictions. A clear nesting feature is manifested along (0,0,1), which is seen\nin our and earlier DFT work [PRL 107, 266402 (2011)] (although, as known, the structure\ndistortion is along (0.5, 0.5, 0)). Using circularly polarized light (left and right), we selectively\nprobe band crossings with different chiralities. These measurements reveal intensity\nmodulations, which are switched on and off depending on the polarization, at the expected Weyl\npoint locations, in agreement with the predicted chiral band structure.\nTaken together with symmetry considerations, the observed patterns are consistent with the\npresence of nested Weyl points with approximate nesting vectors of ≈ (0.48, y, 0.02) and (0.44,\ny, 0), in agreement with the DFT results. While the limited data prevent precise determination of\nthe y-component of the nesting vector, the overall experimental observations support the\ntheoretical picture of Weyl points and their nesting along (0.5, 0.5, 0). Further ARPES\nmeasurements, in particular a more detailed photon energy scan, will be necessary to confirm\nthe y component of the nesting vector more definitively.\nx z\npolarised light, respectively. The dashed box indicates the Brillouin zone. The dark spot on the right side is an artifact\ncaused by the analyzer deflector.\nRevision:\nWe have updated the figures to directly include the value of U used. Moreover, we added in the\nMethods section of the Main Text:\n“DFT calculations were carried out with U = 4.0 eV for the insulating monoclinic phase to capture\nelectron-correlation effects, whereas U = 0 eV was used for the metallic phase, where metallic\nscreening is expected to strongly reduce the effective on-site interactions.”\n2. While the absence of phonon softening indicates an electronic origin of the transition, it is not\nquite clear why this transition is attributed to the nesting of the Weyl points and not to some\nother features of the electronic structure. A Fermi surface nesting would be a more natural\nscenario in this case. This possibility should be explored in detail, e.g., by calculating Lindhard\nsusceptibility. On general grounds, it is not quite plausible that some features lying 0.1 eV away\nfrom the Fermi level trigger such a major transformation in the material.\nAuthor Reply 2:\nWe thank the referee for this insightful comment. Indeed, a conventional Fermi surface nesting\nat the vector (0.5, 0.5, 0) is absent in our and previously calculated Fermi surfaces (e.g. Fig. 6 in\nPhysical Review B 80, 024416 (2009)). While a 3D-like pocket near the center of the Brillouin zone\nappears in the calculations (but only very weakly in the experimental data; Figure 5, reply 1),\nsuch a 3D structure is generally difficult to nest effectively.\nEarly works [Physical Review Letters 107, 266402 (2011)] presents calculated Fermi surface\n(comparable to our calculations) with a nesting feature along (0, 0, 1), which was the foundation\nbehind the suggestion of a Peierls transition. However, crucially, the observed crystal distortion\nis along (0.5, 0.5, 0) [Physical Review Letters 107, 266402 (2011)], which is perpendicular to (0, 0,\n1). Thus, the nesting along (0, 0, 1) does not drive the lattice modulation. Instead, our key finding\nis that Weyl points are located near (0.5, 0.5, 0), which coincides with the experimentally\nobserved lattice distortion vector. While this suggests a possible connection between Weyl point\n14\nnesting and the transition, we acknowledge that direct evidence for the Weyl points driving the\ndistortion is currently lacking. We have softened the language accordingly.\nLindhard susceptibility:\nWe agree that an explicit calculation of the Lindhard susceptibility χ (q,ω) could be informative.\nthis quantity (see Review of Modern Physics 89, 015003 (2017), Giustino). Below we explain how\nthe phonon spectrum already embeds the relevant electronic response, and why we do not\nexpect a qualitative change in the Lindhard susceptibility when U is varied moderately.\nIn short, the phonon spectrum incorporates the full dielectric screening, extending beyond the\nLindhard function and even beyond the random phase approximation (RPA). Thus, the phonon\ndispersion indirectly mirrors the behavior of the noninteracting Lindhard susceptibility, χ : an\n0\nenhancement of χ (q,0) (and its dressed χ) reduces ε(q,0), increases the magnitude of Cel(q),\n0\nwhich is the electronic screening contribution to the dynamical matrix, and results in a phonon\nsoftening at the corresponding wavevector.\nTo begin, we introduce the (noninteracting) Lindhard susceptibility that controls the basic\nelectron-hole response.\nHere N is the number of k points, f are Fermi occupations, ε are band energies, η is a positive\nk nk nk\ninfinitesimal, and the matrix element couples Bloch states at k and k+q.\nIncluding Hartree screening yields the RPA response:\nHere v(q) is the bare Coulomb kernel. DFT further improves this by adding the exchange-\ncorrelation kernel f , giving:\nxc\nwhich is precisely the level that DFPT phonons probe in the static limit.\nIn DFPT, phonon frequencies are the eigenvalues of the dynamical matrix, which contains an\nelectronic term built from the screened density response χ. This term makes the connection\nbetween dielectric screening and lattice dynamics explicit.\nκ\nionic mass, Cion is the bare ionic contribution, and Cel is the electronic screening contribution.\nThe electronic part can be written as a bilinear form of the screened response χ with respect to\nthe linear variation of the self-consistent potential induced by ionic displacements:\nwhere χ(ω=0) is the static screened response that includes v and f , and the integrals run over\nxc\nreal space. Here, g is the electron-phonon perturbation that follows from the linear variation of\nκα\nthe self-consistent Kohn-Sham potential, which itself depends on the variation of the self-\nconsistent potential via,\nwhere V is the self-consistent potential and δn is the induced charge density. We see that the\nSCF\nsame Hartree term v and the exchange-correlation kernel f that dress χ and define χ also\nc xc 0\nappear in the electron-phonon vertex.\nThus, the same Hartree and exchange-correlation kernels that dress χ appear directly in both χ\n0\nand g , establishing that Cel samples the fully screened χ rather than the bare χ . In the finite-\nκα 0\ndisplacement supercell method, the same screening physics is captured implicitly, since forces\nare computed self-consistently with spin-polarized DFT. Forces are taken from self-consistent\nDFT calculations for small displacements, so δn and δV are obtained by solving the Kohn-Sham\nSCF\nresponse with the same Hartree and exchange-correlation kernels. The resulting force-constant\nmatrix C recovers the same Cel as DFPT in the static limit.\nThese relations show that if a strong nesting or divergent electronic response existed at a given\nq, ε(q,0) would be reduced and the electronic contribution Cel(q) would become strongly negative,\nwhich would appear as a softening or an imaginary phonon at that q. Our spin-polarized phonon\ndispersions show no softening at q = (0.5, 0.5, 0). This disfavors a conventional Fermi-surface-\ndriven Peierls mechanism.\nFinally, Eq. (1) highlights that χ depends on band energies and matrix elements. The band\n0\nstructures in Fig. S3a-d show that increasing U mainly narrows the bandwidth while keeping the\nnear-Fermi topology similar. The phase space for electron-hole excitations and the matrix-\n16\nelement structure therefore remain comparable, so χ (q,0) should not change qualitatively for\n0\nmoderate U. No new sharp nesting instabilities are expected to arise from bandwidth\nrenormalization alone.\n3. It is highly desirable to pin down the magnetic moment direction, because it determines the\nexact positions of the Weyl points and the separation between them. This can be done by angle-\ndependent magnetization measurements on a single crystal.\nAuthor Reply 3:\nWe thank the referee for the valuable suggestions. Following this, we have performed angle-\ndependent magnetization measurements on single crystal K₂Cr₈O₁₆. Specifically, we measured\nthe magnetization as a function of applied magnetic field at various angles within the ab plane.\nThese measurements indicate that the magnetic moments are oriented approximately 22.5°\naway from the a/b crystallographic axis. We have included these new results in the main text (Fig.\n1(c) in main text) and of the revised manuscript.\nIn response to the referee’s inquiry about the impact of moment direction on Weyl points, we\nconducted a systematic theoretical study of how varying the spin orientation affects the positions\nof the Weyl nodes. Consistent with prior reports (New Journal of Physics 22, 073062 (2020)), the\nexact Weyl point locations shift with the magnetic moment direction. However, importantly, the\nnodal points consistently form a characteristic cross-like structure on the nodal plane, with\nnested pairs robustly present along the {±0.5, ±0.5, 0.0} directions in the tetragonal Brillouin zone\n(see Figs. 3b and SM Fig. S8).\nRevision:\nIn the Main Text, we added the following line:\n“In addition, we find that the positions of the nodal points are robust. Regardless of the direction\nof the magnetic moment, we found that the nodal points always form a cross-like feature about\nthe nodal plane, such that nested nodal pairs are always present in the (0.5,0.5,0.0) and (-0.5,-\n0.5,0.0) directions of the tetragonal Brillouin zone (Fig. 3(b) and SM Fig. S8)”\n4. While the metal-insulator transition in K2Cr8O16 is dubbed topological, there seems to be no\ndirect evidence for its topological nature. The formation of an axion insulator in the low-\ntemperature monoclinic phase is speculated but never proved. In my opinion, the claims of the\ntopological transition and axion-insulator phase are exaggerated and should be removed unless\nthey can be directly supported by DFT calculations or experiments.\nAuthor Reply 4:\nWe agree with the referee that firm claims of a topological transition to an axion-insulator phase\nwould be exaggerated. Our main claim, however, is that the MIT is a topological phase transition\nas we have shown that the metallic phase is a Weyl semi metal (Fig. 3(b)), as is also known in and\nhighly reproducible from literature [New Journal of Physics. 22, 073062 (2020)] and that the\n17\ninsulating phase is at least topologically trivial. A transition between these two phases with\ndistinct topological character naturally qualifies as a topological phase transition.\nIn addition, our work provides novel evidence that the Weyl points of opposite chirality are\nnested by wavevectors q = [0.75, 0.75, 0.0], which correspond closely to distortion lattice\nwavevectors observed experimentally via single-crystal X-ray diffraction [Physical Review Letters\n107, 266402 (2011); Journal of the Physical Society of Japan 81, 054710 (2012)]. This nesting\noffers a plausible microscopic mechanism for the MIT linked to electronic topology. However, a\nnesting of Weyl points do not guarantee an axion insulating ground state. Importantly,\nthroughout the manuscript and abstract, we have carefully phrased the possibility of an axion\ninsulating state only as a potential scenario or “possibly axionic in nature,” without making\nconclusive claims. We have now further clarified in the revised manuscript that the axion\ninsulator is one of several candidate ground states, and that future theoretical and experimental\nstudies are necessary to confirm or refute this.\nWe hope this clarifies that our primary focus remains on demonstrating the topological character\nof the MIT, without overstating the evidence for an axion insulating phase.\n5. Previous studies have shown that the tetragonal structure of K2Cr8O16 is stable in PBE,\nwhereas imaginary phonons appear in PBE+U [Phys. Rev. B 90, 045124 (2014)]. Therefore, a\nphonon-driven transition becomes possible when correlations are sufficiently strong. While it\ndoes not seem to be the case experimentally, the agreement between the experimental and\ncalculated phonon frequencies (Fig. 4e) is mediocre. It is not quite clear whether DFT or DFT+U\nwas used in the phonon calculations, and it will be desirable to compare the experimental phonon\nspectrum with both calculations. Which calculation allows a better agreement with the\nexperiment?\nAuthor Reply 5:\nWe thank the referee for highlighting the excellent phonon study carried out using DFT+U,\nas\nreported in Ref. [Physical Review B 90, 045124 (2014)]. We clarify that the phonon calculations\nshown in Fig. 4e were performed using DFT-PBE with U = 0 eV. As also outlined in reply 1, this\nchoice was made for both physical and practical reasons:\n1. Physical justification: In the high-temperature metallic phase of K₂Cr₈O₁₆, the electrons\nare itinerant and screening is strong, effectively reducing the impact of local Coulomb\ninteractions. Applying a static U in this regime without proper justification may often\nintroduces unphysical artifacts, such as spurious gap openings or magnetic order, as\ndiscussed in the context of similar correlated metals [Physical Review Research 3, 033286\n(2021); Science 362, Issue 6418 pp. 1037-1040 (2021); The Journal of Chemical Physics\n160, 134101 (2024)].\n2. Consistency with experiment: The experimental phonon spectrum does not show any\nsoft modes or imaginary frequencies that would indicate a phonon-driven instability. In\ncontrast, Ref. [Physical Review B 90, 045124 (2014)] shows that including a Hubbard U\n18\ncan introduce such imaginary modes in the tetragonal phase — a result that appears to\nconflict with our experimental findings.\nBased on these considerations, we chose to rely on DFT (U = 0), which gives a phonon spectrum\nthat, while not in perfect agreement with experiment, qualitatively reproduces the key features\nand does not introduce unphysical instabilities. We conclude that U = 0 is more appropriate for\ndescribing the phonon behavior in the metallic phase of K₂Cr₈O₁₆.\n6. The INS fits are not very convincing because there is very little contrast in the experimental\ndata. Please, show the constant-energy cuts along with the spin-wave fits in order to confirm the\naccuracy of the proposed model.\nAuthor Reply 6:\nWe thank the referee for this valuable comment. We agree that the contrast in the INS data is\nlimited, which is primarily due to powder averaging effects. Additionally, some spurious signals\nin the data further complicate the comparison between experiment and model. However,\ndespite these challenges, we believe there is sufficient contrast in the measured spectra to\ndemonstrate the validity of the fitted spin-wave model.\nTo address the referee’s suggestion, we have added constant-energy and Q cuts along with the\ncorresponding spin-wave fits in the revised SM (Fig. S6). These additional figures more clearly\nillustrate the agreement between the experimental data and our model, and support the\nreliability of the extracted exchange parameters (which are comparable to the calculated values).\n7. Calculated magnetic interactions are proportional to t^2/U, but the reasons for it are not\nexplained in detail. The t^2/U dependence is typical for antiferromagnetic superexchange, but\nJ1-J3 are all ferromagnetic. Which hopping parameters are considered in this case? I could\nimagine that FM superexchange follows the t^2/U dependence if 't' is the t2g-eg hopping, but is\nit really the case? What about the t2g-t2g hoppings? They must lead to antiferromagnetic\nsuperexchange interactions.\nAuthor Reply 7:\nWe thank the referee for raising this important point regarding the origin of ferromagnetic\nexchange and its relation to the t2/U dependence. While it is true that the t2/U scaling is typically\nassociated with antiferromagnetic superexchange (particularly for t -t or e -e interactions),\n2g 2g g g\nferromagnetic exchange can also exhibit a similar energy dependence under certain conditions,\nas described by the Goodenough-Kanamori-Anderson (GKA) rules.\nK₂Cr₈O₁₆ is a mixed-valent compound containing both Cr³⁺ (d³) and Cr⁴⁺ (d²) ions in octahedral\ncoordination. The relevant electronic states arise predominantly from the t g manifold, as the e\n2 g\nlevels are significantly higher in energy and are unoccupied. The Cr-O-Cr bond angles are close to\n97°, leading to nearly orthogonal orbital overlap between neighboring Cr ions and strongly\nsuppressing direct t -t hopping.\n2g 2g\ndue to Hund’s coupling on the intermediate oxygen ligand—specifically, the oxygen p orbitals.\nThis mechanism, sometimes referred to as Hund’s-rule-driven ferromagnetic superexchange,\nfavors parallel spin alignment when electrons hop from a half-filled orbital (Cr³⁺) to an empty\norbital (Cr⁴⁺) via an oxygen p orbital, especially when the orbital overlap is weak or orthogonal.\nAlthough this exchange differs from conventional AFM superexchange, it remains a second-order\nvirtual hopping process. As such, the resulting exchange interaction retains a quadratic\ndependence on the hopping amplitude, scaling as t2/Δ, where Δ is an effective excitation energy\nthat includes the oxygen site energy and Hund’s exchange. Therefore, while the J₁–J₃ interactions\nare ferromagnetic in sign, they still follow a t2/U-like dependence, as they are governed by the\nsame underlying second-order perturbative framework.\nRevision:\nWe have now added expanded the explanations mainly to the supplementary materials (Sec.\nS3C), in which the main text references.\n8. Ref. [19] is not displayed in the manuscript. I presume it is New J. Phys. (2020)\nAuthor Reply 8:\nWe thank the referee for pointing out this mistake. Reference 19 is indeed New Journal of Physics\n(2020). We have corrected this mistake.\nK₂Cr₈O₁₆ ferromagnetic hollandite has been studied for almost two decades for presenting the\nprototype ferromagnetic metal-insulator transition, a rare phenomenon. In two seminal PRLs in\n2009 and 2011 (cited as refs. 16, 17 in the manuscript), and also in another paper the year after\n(ref. 18), the following description was given:\nK₂Cr₈O₁₆ was determined to be a quasi-1D system with Peierls distortion in the ferromagnetic\ninsulating state. Peierls here is due to the dimerization of tetramers within the unit cell with no\ndoubling of the lattice, but it does not bring along charge ordering on the Cr sites. Ab initio\ncalculations describe a double exchange mechanism between double chains via corner-shared\noxygens, with exchange constants of 5.3 meV within the double chains but 10.6 meV between\nthe double chains. This results in a Fermi surface of strongly quasi-1D character with a nesting\nvector of (0,0,1) along the c direction. The low-temperature structure was described in the\nP112₁/a space group. In general, there is a mixed Cr³⁺/Cr⁴⁺ valence. This picture has become well\naccepted.\nA decade later, two ab initio studies—one cited as ref. 22 from NJP in 2020, the other from the\nsame group in PRL the year before, but for some reason not cited (the manuscript refers to ref.\n19 for a similar study, but it is actually missing from the reference list, probably a typo) — already\ndescribed Weyl fermions in the system.\nThe present manuscript describes a different picture for the MIT. It presents experimental\narguments against a Peierls instability and argues that the Weyl fermions bring along a\ncorrelation-induced topological phase transition that results in the MIT. The low-temperature\nmagnetic structure is described in the C2'/m' space group, surprisingly, since it doesn't seem to\nbe the best choice for ferromagnetic ordering.\nThe manuscript also argues against the interaction being double exchange, based on calculations\nof the dependence of the exchange on the hopping integral, and instead argues for a uniform\nCr3.75+ valence state with superexchange-like interactions between them.\nYet, the manuscript proposes exchange values of 6 and 10 meV, quite similar to those from 2011.\nTherefore, the current manuscript proposes a drastically different chemical situation from the\noriginal picture and a dramatically different origin for the MIT.\nAuthor Reply:\nWe thank the referee for the accurate historical overview, as well as pointing out the reference\ntypo. Reference 19 is indeed New Journal of Physics (2020). We have corrected the syntax error.\nIn addition, we have included citation of the 2019 work from the same group in the Main Text:\n“Related compounds such as RbCr O , have also shown to host Weyl points [Physical Review\n4 8\nLetters 122, 057205 (2019)], demonstrating the robustness of these topological features in this\nfamily of compounds. “\nstudies. We have addressed the questions raised, point by point, below.\n1. A particularly glaring omission in the manuscript is that it strongly argues for the much\nimproved quality of the samples, based on a modified high-pressure synthesis, but does not\nmention a single word about the modification. The original PRL from 2009 describes the high-\npressure synthesis conditions in detail.\nAuthor Reply 1:\nWe thank the referee for pointing this out. We have updated the manuscript to include the\nexplanations about the modifications. New designs of 18/11 and 25/15 multi-anvil high-pressure\ncells were developed at Max Planck Institute, which provide stable experimental conditions:\nunder 8GPa at 1500°C in 25/15 cell and 12 GPa at 1500°C in the 18/11 cell, allowing for larger\nsample volumes. We have included the following sentences to the manuscript:\n“In particular, new cell designs for a high-pressure Walker-type multi-anvil system were\ndeveloped at the Max Planck Institute for Solid State Research, enabling growth of larger, higher-\nquality crystals. The new 18/11 and 25/15 multi-anvil cell assemblies provide more uniform\npressure and temperature distributions, resulting in increased sample volumes while maintaining\nstable synthesis conditions up to ~12 GPa and 1500 °C. With these improved cells the CrO\n2\nimpurity phase is reduced to below 2 % in powder, and the obtained crystals are significantly\nlarger (up to 0.5×0.5×1 mm3). The synthesis conditions otherwise follow those reported\npreviously [Physical Review Letters 103, 146403 (2009)].”\n2. The manuscript proposes to set up a strong and interesting polemic in the literature. The\npros and cons should be more clearly and antagonistically described; the arguments in the current\ntext are quite scattered. However, and more importantly, there are both experimental and ab\ninitio arguments in the current study. The ab initio results don't seem to be qualitatively different\nfrom the 2019-20 results.\nAuthor Reply 2:\nWe thank the referee for this thoughtful comment and for pointing out the need to more clearly\narticulate the contrast with earlier work. We agree that our manuscript sets up an important\ndiscussion in the literature, and we have revised the text to highlight this polemic more explicitly\nand in a more structured way.\nWhile our ab initio calculations share some qualitative similarities with the 2019-20 results\n[Physical Review Letters 122, 057205 (2019); New Journal of Physics 22, 073062 (2020)], our work\ndiffers in several key respects:\n1. Experimental anchor. In our study, the experiments take the lead role, while first-\nprinciples calculations play a supporting role. With the aid of neutron scattering results,\nwe carried out a group-theory analysis of the magnetically ordered states. This\nexperimental anchor allows us to move beyond purely computational predictions and to\n22\nground our symmetry analysis in experimentally verifiable measurements. For example,\nwithout experimental guidance, we would not have known that the magnetic space group\nof the metallic phase is #12.62, which laid the basis for our theoretical group-theory\nanalysis.\n2. Peierls transition. A central pillar of our work is the experimental disproof of the Peierls\nmechanism, which has been widely assumed in the literature. Earlier theoretical work did\nnot test this assumption [Physical Review Letters 109, 076401 (2012); New Journal of\nPhysics 22, 073062 (2020)], which constrained their assumptions and, consequently, their\nconclusions.\n3. Exchange interactions. It was widely accepted that double-exchange interaction is the\nFM exchange interaction in the insulating phase [Physical Review Letters 109, 076401\n(2012)], based on the assumed Peierls transition. With Peierls transition disproved by our\nIXS measurements, the double-exchange scenario can also be questioned. We compared\nthe exchange parameters J of nearest-, next-nearest-, and next-next-nearest neighbors\nwith experiment, finding good agreement. By computing J vs. t/U, we find a quadratic\ndependence, allowing us to conclude that superexchange interactions govern the\nexchanges in both the metallic and insulating phases.\n4. Relation to topology. The New Journal of Physics 22, 073062 (2020) study established\nWeyl fermions in the metallic phase but did not consider the effect of the MIT. By contrast,\nwe note that in the monoclinic phase, the combination of structural distortion and orbital\nsplitting removes the Weyl points present in the ferromagnetic metallic phase. Our\ncombined experimental–theoretical analysis thus demonstrates that the MIT links a Weyl\nsemimetallic state to a trivial insulating state, establishing it as a correlation-driven\ntopological MIT.\nIn the revised manuscript, we have reorganized parts of the discussion to make these contrasts\nclearer.\n3. The experimental results of the inelastic X-ray scattering show a lack of phonon condensation.\nThis is employed as an argument against the Peierls instability. However, the Peierls distortion\nproposed in 2011 is quite unusual. Would it necessarily be accompanied by drastic changes in the\nphonon spectrum?\nAuthor Reply 3:\nWe thank the referee for this thoughtful question. As the referee notes, the Peierls instability—\noriginally described by Peierls in the 1930s—refers to a transition in which 1D metallic chains\nbecome unstable toward lattice distortions due to electron-phonon coupling. This distortion\nleads to a gap opening at the Fermi surface and a transition to an insulating state. Crucially, such\na transition is inherently accompanied by changes in the phonon spectrum (as the electron\nphonon coupling is the foundation behind the transition), typically manifested as phonon\nsoftening or condensation at a characteristic wavevector.\nthose in higher dimensions (2D and 3D), where electronic susceptibility does not diverge and only\npartial gap openings occur. In these cases, the resulting ground state is not necessarily insulating.\nThe term \"Peierls transition\" is typically reserved for the 1D case, where the distortion is tightly\ncoupled to the divergence in electronic susceptibility and phonon condensation plays a central\nrole, resulting in the insulating state.\nEven in more modern theories of CDW formation—such as those involving momentum-\ndependent electron-phonon coupling—phonon anomalies are still expected, as the underlying\nmechanism remains electron-phonon coupling. Therefore, we argue that the lack of any clear\nphonon softening or anomaly in our inelastic X-ray scattering (IXS) data provides strong evidence\nagainst a Peierls-like transition in K₂Cr₈O₁₆.\nWe acknowledge, however, that a scenario involving the softening of an optical phonon might\nnot result in full phonon condensation. Still, one would expect some anomaly in the phonon\nspectrum. However, previous theoretical work [Physical Review B 90, 045124 (2014)] suggests\nthat phonon condensation in K₂Cr₈O₁₆ is predicted for sufficiently high values of U, where an\nacoustic phonon softens at the q = (0.5, 0.5, 0) point. However, as also noted by Referee 2 in\nCDW\ntheir Comment 5, our experimental data do not show any such anomaly.\nWe would also like to comment on the original proposal of a Peierls transition in K₂Cr₈O₁₆\n[Physical Review Letters 107, 266402 (2011)], which we believe contains a conceptual\ninconsistency. The nesting vector observed in that study is along the c-axis (qₙₑₛₜ = (0, 0, 1)), a\nresult we also confirm in our calculations. However, the same study observed lattice distortion\npropagates in the ab-plane, with q = (0.5, 0.5, 0). This means that the nesting and distortion\nCDW\nvectors are perpendicular to one another. Thus, the observed nesting does not drive the lattice\nmodulation. To argue for a Peierls mechanism, one would need to identify nesting along q ,\nCDW\nbut the Fermi surface in that direction is strongly 3D-like (see e.g. in Physical Review B 80, 024416\n(2009)). As discussed earlier, CDWs in three dimensions do not typically drive a full insulating\nstate and are, by definition, not Peierls transitions.\nWe did not include these critiques and mistakes of earlier work in the main text, as our focus was\nto emphasize our own findings rather than highlight the shortcomings of previous studies.\n4. The inelastic neutron scattering shows a lack of drastic change in the spin-wave excitations\nacross the MIT. It is unclear—although the text makes it look as such—how this relates to the\nlack of a Peierls instability.\nAuthor Reply 4:\nWe thank the referee for raising this important point. We agree that the inelastic neutron\nscattering (INS) results, on their own, do not directly demonstrate the absence of a Peierls\ninstability. Our intention was to emphasize that the nature of the underlying structural units—\nbeing chimney-like rather than quasi-one-dimensional—provides an initial indication against the\npresence of a Peierls mechanism. Since a Peierls transition typically requires a quasi-1D electronic\n24\nstructure, the dimensionality inferred from the INS data serves as a crucial contextual element\nfor interpreting the IXS results.\nWe have revised the manuscript to clarify this point and to ensure that the connection between\nthe INS findings and the broader discussion of the Peierls instability is not overstated.\nTo summarize, our argument against a Peierls instability in K₂Cr₈O₁₆ is supported by three key\nobservations:\n1. The crystal structure consists of chimney-like units rather than a 1D or quasi-1D lattice.\n2. The previously proposed nesting vector (qₙₑₛₜ = (0, 0, 1)) is perpendicular to the observed\nCDW vector (q = (0.5, 0.5, 0)).\nCDW\n3. There is no clear Fermi surface nesting along (0.5, 0.5 ,0)\n4. There is no evidence of phonon softening or condensation associated with the transition.\nThese observations, taken together, strongly support the conclusion that a Peierls mechanism is\nnot operative in K₂Cr₈O₁₆.\n25"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am very pleased to report that all of my concerns were indeed addressed by the authors of this manuscript. My primary concern was that wording including \"paramagnetic\" was included frequently and I had thought it was being used to describe calculations that were non-magnetic. As it turns out, there are various crystallographic and magnetic phases and two of them are monoclinic and experimentally there are paramagnetic phases, but all calculations were magnetic. Likely, having performed all of this work, the authors understood what they were referring to and where throughout the manuscript, but from an outsider's point of view, it was quite confusing. The authors have taken the time to extensively explain what has been done in their response to my report and then, crucially, have included relevant wording in the manuscript and SM. I can now endorse publication of this work as it is indeed interesting and will surely have an impact on the community.", "reviewer_2": "Most of my previous concerns have been addressed in the revised version. However, I have some additional queries that are inspired by the clarified computational methodology: 1. Both abstract and introduction put a lot of weight onto the correlated electron physics, but I am tempted to ask what the manifestations of the correlated physics are. In fact, the authors argue that the metallic phase of K2Cr8O16 is best described using U=0, so they essentially conclude on its uncorrelated nature. One especially odd statement is the claim of the \"correlation-induced topological phase transition\" in the abstract, while the rest of the manuscript never mentions correlations at all. In my opinion, the claim of an MIT driven by topology instead of correlations would be a much better framework for this study, at least in its current form (but see #2 below). I would also advise the authors to completely re-write the abstract and highlight the concrete experimental findings, such as the absence of phonon softening, instead of the vague and arbitrary concepts like \"quantum devices\", which do not occur in this study at all. 2. On a related note, the authors argue that the INS spectra do not change across the MIT, so the exchange couplings must be similar in the metallic and insulating phases. This is indeed the case in Fig. S9b,c if the U=0 results are compared, but it would then indicate that the insulating phase must be uncorrelated too. On a closer look, the spin-wave fits are done with J1=6 meV and J3=10 meV, the values that seemingly correspond to U=2 eV in Fig. S9b. In contrast, the exchange couplings of up to 40 meV, as shown in Fig. S9a, should be inconsistent with the experiment. All of this is quite confusing and in fact casts doubt on the idea that the metallic phase is uncorrelated. Would not it be better to apply the same U-value to the metallic and insulating phases, considering that their exchange couplings are similar? 3. A comparison of the band structures calculated with different 'U' is shown in Fig. 3 of the response letter, but it does not appear anywhere in the manuscript. In my opinion, it should be included. I am also confused by the panel 'e' of that figure. What does it show exactly? What is the color scheme? 4. I would still recommend updating the captions of Figs. 3 and S7 and writing explicitly the U-values used in each calculation. This will facilitate the reading. 5. If feels a bit strange that the magnetic couplings are in the range of 10 meV, but the INS data are mostly shown up to 4 meV only. The data obtained with the incident energy of 97 meV are also presented in Fig. S4, but without any comments or analysis, although these data clearly show more features than the low-energy data (note, for example, the flat band around 40 meV). Are the high-energy INS data equally well described by the spin model? 6. Fig. S5 is hard to read because of the many black lines that look more like \"spaghetti's\". One could use different shades of gray for each line, and the values of J2 should be given either in the graph or in the figure caption."}, "author_response": ""}, {"round": 3, "reviewer_comments": {"reviewer_2": "My previous concerns have been addressed, and I gratefully recommend publication"}, "author_response": ""}]

Decision letter