Weyl nodes in CeRu$_4$Sn$_6$ studied by dynamical mean-field theory

Frank T. Ebel; Jor\=unas Dobilas; Karsten Held; Martin Brass; Silke Paschen

REVIEW 3 major objections 6 minor 44 references

Weyl nodes in CeRu$_4$Sn$_6$ studied by dynamical mean-field theory

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A DFT+DMFT calculation finds 64 Weyl nodes in CeRu4Sn6, the closest only 0.5 meV below the Fermi level, establishing the compound as a correlated Weyl semimetal.

desk verdict A credible DFT+DMFT prediction of 64 Weyl nodes in CeRu4Sn6, but the meV-scale proximity of the closest node demands an error analysis the paper does not provide. read the letter →

arxiv 2507.12944 v2 pith:PSC7FFPK submitted 2025-07-17 cond-mat.str-el

Jorūnas Dobilas , Martin Brass , Frank T. Ebel , Silke Paschen , Karsten Held This is my paper · ORCID

classification cond-mat.str-el

keywords CeRu4Sn6WeylsemimetalKondoinsulatorheavyfermionDFT+DMFTBerrycurvaturenonlinearHalleffecttopologicalbandcrossings

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the heavy-fermion compound CeRu4Sn6 is a topological Weyl semimetal, and answers yes on the basis of a correlated electronic-structure calculation. The authors combine density functional theory with dynamical mean-field theory (DFT+DMFT), the standard tool when electron correlations are strong, and they track the Berry curvature through the renormalized quasiparticle bands. They find five inequivalent Weyl nodes (64 symmetry-related copies in total), both type I and type II, located inside the Kondo-hybridization gap, with the closest node only 0.5 meV below the Fermi level. If this is right, the compound becomes a concrete material platform where the Kondo effect itself generates topological band crossings, which is what experiments on the nonlinear Hall effect and specific heat had suggested.

What carries the argument

The load-bearing construction is the renormalized quasiparticle Hamiltonian $H_{\rm qp}(\mathbf{k}) = \sqrt{Z}\,(H(\mathbf{k}) + \mathrm{Re}\,\Sigma(0) - \mu I - V_{\mathrm{DC}})\sqrt{Z}$, built from the DFT tight-binding Hamiltonian $H(\mathbf{k})$ and the DMFT self-energy $\Sigma(\omega)$ for the Ce $4f_{5/2}$ orbitals. The quasiparticle weight $Z = [1 - \partial \mathrm{Re}\,\Sigma(\omega)/\partial\omega|_{\omega=0}]^{-1}$ and the zero-frequency real part of the self-energy carry all the correlation renormalization; the search algorithm then follows the Berry curvature field lines of this Hamiltonian to points where the integrated Chern number is $\pm 1$, identifying each Weyl node and its topological charge. The direct but not indirect Kondo gap of about 5 meV is what the nodes bridge.

What would settle it

Recompute $H_{\rm qp}(\mathbf{k})$ with $\mathrm{Re}\,\Sigma(0)$ for $m_j = \pm 1/2$ shifted by the uncertainty range of the maximum-entropy continuation and re-run the Berry-curvature search: if the node at fractional coordinates [0.244, -0.127, 0.060], 0.5 meV below the Fermi level, disappears, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that CeRu4Sn6 is a strongly correlated Weyl semimetal in which the Weyl nodes are a product of correlation physics, not of the bare band structure. Treating the Ce 4f electrons with DMFT at 23 K produces a direct Kondo gap of about 5 meV across the Brillouin zone, but the gap is bridged by 64 Weyl nodes that arise from the hybridization of the correlated f states with conduction bands. Five of these nodes are inequivalent; two sit at high-symmetry positions and have eight replicas each, three have sixteen each, and the Chern numbers balance at +32 and -32. The closest node lies only 0.5 meV (about 6 K) below the Fermi level, and the geometric-mean Weyl velocities are one to two orders of magnitude smaller than in weakly correlated Weyl semimetals. The paper argues these features explain the low-temperature spontaneous nonlinear Hall effect and the $T^{3}$ specific-heat contribution observed in experiments, and they position CeRu4Sn6 as a model for studying interaction-driven band topology.

Load-bearing premise

The argument depends on the analytic continuation of the DMFT self-energy being reliable near zero frequency: the paper states that the $m_j = \pm 1/2$ component has numerical glitches near $\omega = 0$, and the closest Weyl node lies only 0.5 meV below the Fermi level, so a small error in $\mathrm{Re}\,\Sigma(0)$ could shift that node or remove it.

Editorial extensions

If this is right

CeRu4Sn6 becomes a concrete, ab initio example of a correlated Weyl semimetal, giving a materials-specific origin for the observed spontaneous nonlinear Hall effect.
The smallest energy scale, 0.5 meV between the closest Weyl node and the Fermi level, sets a low-temperature scale of about 6 K below which topological transport should be most visible.
The calculated Weyl velocities, roughly 7 to 51 km/s, are orders of magnitude below those of noninteracting Weyl semimetals, which would show up as enhanced low-temperature specific heat and amplified nonlinear responses.
The DMFT picture refines the 'failed Kondo insulator' description: the material has a direct hybridization gap but no indirect gap, because both trivial bands and Weyl nodes cross the Fermi level.
Because the Weyl nodes are not pinned to the Fermi energy by filling constraints, their energy offsets (from -3.8 to +22.1 meV relative to the Fermi level) predict a temperature- or doping-dependent evolution of the transport signatures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the two closest opposite-charge nodes (node III) sit only $|\Delta k_{\min}| = 0.0021$ Å$^{-1}$ apart, small strain or disorder could pair-annihilate them; a strain-dependent calculation would give a testable prediction for how the low-temperature Hall signal degrades with sample quality.
Sweeping the Coulomb interaction $U$ around the chosen 5.5 eV would show how robust the 0.5 meV node is; if its energy crosses zero under a modest change, CeRu4Sn6 would sit near a doping- or pressure-driven topological transition.
The same approach could be applied to other 'failed' Kondo insulators, with $\mathrm{Re}\,\Sigma(0)$ near zero frequency as the controlling quantity; wherever the analytic continuation is trustworthy, similar correlation-generated Weyl nodes should appear.
A calculation beyond DMFT that includes non-local correlations could test whether the predicted Fermi-liquid Weyl nodes survive the Kondo-destruction quantum criticality hinted at by neutron-scattering $\omega/T$ scaling below 10 K.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Plainly: this is a well-executed DFT+DMFT study that produces a concrete, falsifiable prediction of 64 Weyl nodes in CeRu4Sn6, and it is the first full DMFT inventory for this material. The main thing you should know is that the closest node sits only 0.5 meV below the Fermi level, and the paper itself flags numeric glitches in the analytic continuation of the mj=±1/2 self-energy — the very object used to build the quasiparticle Hamiltonian. There are no error bars on ReΣ(0) and no sensitivity test of node positions. That is a legitimate soft spot, not a nitpick.

What is genuinely new: the complete set of five inequivalent nodes with coordinates, types, velocities, and Ce-4f spectral weights; the demonstration that the direct Kondo gap is bridged by these nodes; and the stark velocity reduction compared to weakly correlated Weyl semimetals. The method is appropriate, the open-source codes are standard, and the calculation is not circular — nodes are computed, not fitted. This is a clear step beyond the earlier DFT+Gutzwiller prediction.

The weakness is real but not fatal. The admitted glitches near ω=0 in the mj=±1/2 channel matter because that channel carries 77% of the spectral weight at the closest node. An uncompensated error of just a few meV in ReΣ(0) could shift or annihilate that node. The authors need to quantify this sensitivity. The temperature mismatch is a second, softer issue: the Weyl search is done at 23 K in a Fermi-liquid description, while the experiment sees quantum criticality below 10 K. The authors note this and do not overclaim, but it leaves open whether the zero-temperature topology is the one computed.

Minor: the tight-binding basis and the static treatment of the f7/2 manifold are standard but not tested for their effect on the node energies.

Who this is for: researchers working on correlated topology, heavy-fermion semimetals, or DMFT-based topological prediction. It deserves peer review. If I were refereeing, I would request an error analysis on the analytic continuation, a sensitivity check of node energies to the glitch, and a brief discussion of extrapolation to T→0. The central claim is plausible and the paper is an honest, reproducible advance.

Referee Report

3 major / 6 minor

Summary. The manuscript reports a DFT+DMFT study of the heavy-fermion compound CeRu4Sn6. The authors construct a 90-orbital tight-binding model from DFT, solve the correlated Ce 4f5/2 problem with DMFT at several temperatures, extract quasiparticle parameters Z and ReΣ(0) from the DMFT self-energy, and use the resulting renormalized quasiparticle Hamiltonian to search for Weyl nodes via Berry-curvature monopoles. They report five inequivalent Weyl nodes (64 in total after accounting for D2d symmetry and time reversal), with node energies between -3.8 meV and +22.1 meV relative to the Fermi level, including one node only 0.5 meV below the Fermi level. The nodes are interpreted as bridging the direct Kondo-insulating gap, and the reduced Weyl velocities are argued to be consistent with the experimentally observed nonlinear Hall effect.

Significance. If the central result holds, this would be a valuable step beyond previous DFT+Gutzwiller treatments: it would provide a DFT+DMFT-based identification of Weyl nodes in CeRu4Sn6 and connect the Kondo hybridization directly to the low-energy topology. The calculation has notable strengths: the Weyl-node positions are an emergent output rather than a fit to experiment, the chemical potential is set self-consistently, the symmetry analysis of the 64 nodes is explicit, and the use of publicly available DMFT and analytic-continuation codes supports reproducibility. The predicted very small node energy (-0.5 meV) and low Weyl velocities are falsifiable statements that can be confronted with transport and photoemission experiments. However, the quantitative precision of the main claim is not yet established, because the quasiparticle Hamiltonian that determines the node positions is built from a self-energy whose analytic continuation is explicitly reported to have glitches near ω=0 in the very mj=±1/2 channel used for the Weyl search.

major comments (3)

[Section III A/B, Eq. (2), Table I/II] The manuscript states in Section III A that the mj=±1/2 component of the self-energy exhibits "minor glitches near ω = 0, which likely arise from numerical artifacts in the analytic continuation." This is the same component whose ReΣ(0)=1.013 eV and Z are inserted into Eq. (2) to build Hqp(k), and the closest Weyl node (node III in Table II) lies only 0.5 meV below the Fermi level. Because ReΣ(0) enters Eq. (2) linearly, an unresolved uncertainty of even 1 meV in the analytic continuation shifts the f-level and hence the node energies by a comparable amount, which is enough to move node III across the Fermi level or qualitatively change the central claim. The paper provides no error bar on ReΣ(0) or Z and no sensitivity test. Please add such a test, for example by varying ReΣ(0) and Z within the plausible MaxEnt uncertainty and reporting the resulting changes in Table II, or by cross-checking ReΣ(ω) with an independent analytic-continuation method such as Padé.
[Section III B, Eq. (2), Fig. 9] The Weyl-node search is performed entirely within the linearized quasiparticle Hamiltonian Hqp(k), whose parameters Z and ReΣ(0) are evaluated at ω=0 only. The self-energy has strong frequency dependence with poles on the scale of an eV (Fig. 5), while the Weyl nodes span a window of roughly ±22 meV. Figure 9 validates the quasiparticle bands against the DMFT spectral function for only two of the five inequivalent nodes. The authors should quantify the error of the quasiparticle approximation at the momenta of all five nodes, or include the next-order frequency dependence of ReΣ in Eq. (2), to ensure that the nodes and their type-I/type-II classification are not artifacts of the linearization.
[Table II, Section III B] Node III has an energy of only -0.5 meV and a minimum distance of |Δkmin|=0.0021 Å^-1 between symmetry-related nodes of opposite Chern number. These are extremely small scales compared with the direct gap of about 5 meV and the input uncertainties of the calculation. The manuscript should demonstrate that node III survives under small variations of the Hubbard U, the double-counting correction, and the chemical potential, and should report how the node energy changes in such a robustness check. Without this, the "0.5 meV" statement is not yet supported at the stated precision.

minor comments (6)

[Section III B] The notation "mj = (−)1/2" is ambiguous: it should be clarified whether the Weyl search uses both mj=+1/2 and mj=-1/2 (which are degenerate in a paramagnetic calculation) or only one projection, and if only one, why this is sufficient.
[Section III B / Table II] The text calls nodes I and IV "high symmetry points," but their fractional coordinates are not obviously high-symmetry points of the Brillouin zone; this phrase should be rephrased, e.g., as points whose symmetry orbit is reduced because they are invariant under a combined symmetry operation.
[Abstract / Section II A] The paper uses the phrase "ab initio results" while the Hubbard U=5.5 eV, Hund's coupling J=0, and the double-counting correction VDC are taken from previous literature/Anisimov formula; the abstract could be more precise by saying "DFT+DMFT with a literature value of U" to avoid overclaiming a fully parameter-free calculation.
[Fig. 8] The color scale in Fig. 8 is normalized as An(k)=A(k,ω)/max(A(k,ω)), but the color bar label should state that these are normalized spectral weights; currently the units and normalization are clear only from the caption.
[Eq. (2)] In Eq. (2), the role of the term −µI−VDC should be stated explicitly in the main text: it should be clear whether H(k) already includes the chemical potential or whether µI is subtracted to define the band structure relative to the DMFT chemical potential.
[Data and code availability] No data or code availability statement is provided; for reproducibility, it would be helpful to make available the tight-binding parameterization, the DMFT input files, and the Weyl-search settings.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the Weyl nodes are an emergent DFT+DMFT output; the paper's self-citations are methodological precedents, not load-bearing inputs.

full rationale

The central claim—five inequivalent Weyl nodes, 64 in total, bridging the Kondo gap—is obtained by applying the Weyl-node search algorithm to the quasiparticle Hamiltonian Hqp(k) of Eq. (2), whose inputs are the DFT tight-binding Hamiltonian, the DMFT self-energy at T = 23 K, and the double-counting correction. No equation in the paper defines the Weyl-node positions, energies, or Chern numbers in terms of the claimed result, and none of the node properties are fitted to experiment. The Hubbard U = 5.5 eV is taken from prior literature and the chemical potential is fixed self-consistently, so the central output is not a fitted parameter renamed as a prediction. The self-citations ([38] for the TightBindingToolBox search algorithm and [39] for its prior application to Ce3Bi4Pd3) are methodological precedents rather than load-bearing justifications; the algorithm is described in terms of the Berry curvature equations (4)–(6), and the prior application is independent code-reproduced usage. The paper's own admission that the mj = ±1/2 self-energy has 'minor glitches near ω = 0, which likely arise from numerical artifacts in the analytic continuation' (Sec. III A) is a numerical robustness concern for the 0.5 meV claim, but it is not circularity: ReΣ(0) is an input to Hqp(k), and the Weyl-node search is an independent computation from that input. Similarly, the fact that nodes I and IV sit at high-symmetry points and generate only 8 replicas is a symmetry-counting result, not a repackaging of an earlier known result; the paper explicitly distinguishes its DFT+DMFT result from the earlier DFT+Gutzwiller prediction [18]. Overall, the derivation chain is self-contained against stated inputs, and the minor self-citation overlap does not make the central claim circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the DMFT self-energy, which is governed by the Hubbard U parameter and the analytic continuation procedure. The quasiparticle Hamiltonian used for the Weyl search is a low-energy approximation, and the temperature of the calculation (23 K) is higher than the experimental regime (<1 K). No new physical entities are introduced; the Weyl nodes are features of the computed band structure.

free parameters (5)

Hubbard U on Ce 4f5/2 = 5.5 eV
Taken from literature [29]; the Weyl node positions and the direct gap size depend on this value.
Hund's coupling J = 0
Set to zero, justified by Ce 4f1 configuration; affects multiplet structure and self-energy.
Double-counting correction VDC = 1.27 eV
Computed via Anisimov formula with <n>=0.73; enters Hqp(k) in Eq. (2).
Chemical potential mu = -0.059 eV at 23 K
Adjusted to fix the Ce 4f occupation; sets the Fermi level against which the 0.5 meV distance is measured.
DMFT temperature for Weyl search = 23 K
Weyl node search uses self-energy at T=23 K; experimental nonlinear Hall effect is observed below 1 K, so the relevant energy scales may differ.

assumptions (5)

domain assumption DMFT is a valid approximation for CeRu4Sn6, capturing local correlations while neglecting non-local ones.
The paper uses DFT+DMFT as the central method; non-local correlations are not included, yet the experimental quantum criticality may require them (Sec. IV).
domain assumption The quasiparticle Hamiltonian Hqp(k) in Eq. (2) captures the low-energy topology of the interacting system.
The Weyl node search uses Hqp(k) built from Z and ReSigma(0); the frequency dependence of the self-energy is neglected, which is not rigorously justified.
domain assumption Maximum entropy analytic continuation reliably provides ReSigma(omega) near omega=0.
The self-energy is continued from Matsubara to real frequencies; the paper notes glitches for mj=±1/2 near omega=0 (Sec. III A).
domain assumption The 90-orbital tight-binding model derived from FPLO captures the relevant bands near the Fermi level.
The basis includes Ce 4f, Ru 4d, Sn 5p orbitals; other orbitals are truncated, which may affect hybridization and Weyl node positions.
standard math The Weyl node search algorithm of Ref. [38] correctly identifies all Weyl nodes in the quasiparticle band structure.
The particle-following method is heuristic; completeness is not formally proven, though it was used successfully for Ce3Bi4Pd3 [39].

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Cite this review

Pith. "Pith review of Weyl nodes in CeRu$_4$Sn$_6$ studied by dynamical mean-field theory." pith.science (2026). https://pith.science/paper/PSC7FFPK

@misc{pith2026250712944,
  author       = {Pith},
  title        = {Pith review of: Weyl nodes in CeRu$_4$Sn$_6$ studied by dynamical mean-field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSC7FFPK}},
  note         = {Machine review of arXiv:2507.12944}
}

abstract

The heavy fermion compound CeRu$_4$Sn$_6$ was recently shown to exhibit a spontaneous nonlinear Hall effect, indicating its topological nature. This is consistent with the lack of inversion symmetry that allows for the existence of Weyl nodes. Here, we employ density functional theory combined with dynamical mean-field theory, which is state-of-the-art for strongly correlated materials, and study the topology of CeRu$_4$Sn$_6$. We find five inequivalent Weyl nodes of either type I or II, each having either eight or sixteen symmetry-related replicas. These Weyl nodes bridge the Kondo insulating gap, which is a direct but not an indirect gap. The Weyl points closest to the Fermi level are situated only 0.5 meV below it, and have a very flat dispersion. Our ab initio results establish CeRu$_4$Sn$_6$ as a model system for investigating the interplay between strong electronic correlations and nontrivial topology. These findings provide a theoretical foundation for future studies of quantum transport and interaction-driven topological phases in heavy-fermion systems.

Figures

Figures reproduced from arXiv: 2507.12944 by the authors.

**Figure 2.** FIG. 2. Self-consistency cycle of dynamical mean-field the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. DMFT spectral functions [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Real (a,b) and imaginary (c,d) parts of the DMFT [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7. Contribution of the different Ce 4 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 9.** Figure 9: FIG. 9. Quasiparticle bands (lines) and DMFT [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

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