Interacting Dirac magnons in the van der Waals ferromagnet CrBr$_3$

Saikat Banerjee; Stephan Humeniuk

REVIEW 3 major objections 4 minor 60 references

Interacting Dirac magnons in the van der Waals ferromagnet CrBr$_3$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Second-order magnon self-energy removes the predicted van Hove singularities in CrBr3 and yields distinct $T^3$ (optical) and $T^2$ (acoustic) renormalization.

desk verdict A likely correct qualitative retraction of the 2018 Dirac-magnon van Hove prediction, but the paper's numerical core is undermined by an unsupported kinematic-factor identity and missing convergence data. read the letter →

arxiv 2506.07650 v3 pith:M7ZERK47 submitted 2025-06-09 cond-mat.mtrl-sci

Saikat Banerjee , Stephan Humeniuk This is my paper

classification cond-mat.mtrl-sci MSC 82D40 PACS 75.30.Ds75.50.Dd78.70.Nx

keywords DiracmagnonsCrBr3magnon-magnoninteractionsvanHovesingularitysecond-orderself-energythermalmagnonapproximationhoneycomblatticespin-waverenormalization

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 tries to establish that the van Hove-like singularities previously predicted in the interacting magnon spectrum of the honeycomb ferromagnet CrBr3 are artifacts of the thermal magnon approximation, and that a full second-order perturbation treatment removes them. It also claims that the renormalized optical and acoustic magnon branches obey distinct temperature power laws, $T^3$ and $T^2$ respectively, at second order, while the Hartree term alone is roughly $T^2$ for both. The authors argue that this agrees with recent inelastic neutron scattering, which saw no singular features, and that the same full treatment applied to a triangular-lattice ferromagnet also gives no singularities but a $T^2\log T$ trend. If the paper is right, earlier predictions need revision and the temperature exponent of each magnon branch becomes a sharper test of interaction theory.

What carries the argument

The load-bearing object is the second-order (sunset) self-energy $\Sigma_S(\omega,\mathbf{k})$ of Eq. (8), computed from the five momentum-resolved scattering channels of the quartic Holstein-Primakoff Hamiltonian. The crucial step is evaluating the four-dimensional momentum integrals over the full Brillouin zone with Simpson's rule and the full Bose-Einstein occupation factor $F_{\mathbf{k};\mathbf{q},\mathbf{p}}$, without the thermal magnon approximation, and then feeding the Hartree shift back into the bare energies self-consistently. The momentum-dependent vertex phases $\phi_{\mathbf{k}}=\arg\gamma_{\mathbf{k}}$ enter through the matrix elements, and the calculation's treatment of the Dirac points, where $\gamma_{\mathbf{k}}=0$, is part of what the argument depends on.

What would settle it

Recompute the kinematic factors and decay rates of Eqs. (11)-(24) with a documented grid-refinement study and explicit delta-function regularization: if van Hove-like peaks reappear or the fitted exponents $\beta_u=3$ and $\beta_d=2$ shift with grid size, the central claim fails. Alternatively, a neutron experiment with enough resolution along the optical band near the $M$ point could check directly whether the renormalization scales as $T^3$ rather than $T^2$.

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Extended reading notes

Core claim

The central claim is that retaining the full momentum-dependent interaction vertices and evaluating the four-dimensional self-energy integrals, rather than assuming one magnon sits thermally near the band bottom, removes the previously predicted van Hove-like singularities near the $M$ points and the Brillouin-zone corners. In this calculation the second-order (sunset) self-energy produces a $T^2$ temperature dependence for the acoustic (down) band and a $T^3$ dependence for the optical (up) band. The absence of singular features is stated to be consistent with the INS data of Ref. [26], and a similar full calculation for the triangular-lattice ferromagnet MnBi$_2$Te$_4$ gives no dip or van Hove features, with a $T^2\log T$ correction. The paper also finds that the second-order correction can become negative near the $\Gamma$ point, which limits the validity of perturbation theory to low temperatures.

Load-bearing premise

The whole negative result and the extracted exponents rest on the claim that the numerical four-dimensional integrals are converged and accurate, including the handling of vertex phases where $\gamma_{\mathbf{k}}=0$ and the regularization of the delta functions; the paper reports no grid sizes, convergence checks, or broadening scheme.

Editorial extensions

If this is right

If the central claim is correct, the van Hove singularities predicted for CrBr3 in Ref. [21] are artifacts of assuming one thermally excited magnon, and future spin-wave analyses should retain full momentum-resolved vertices.
The distinct exponents imply that high-resolution neutron scattering on the optical branch could discriminate the $T^3$ prediction from the earlier universal $T^2$ expectation.
For monolayer MnBi$_2$Te$_4$, the full second-order calculation predicts $T^2\log T$ renormalization with no van Hove dips, a signature that inelastic neutron scattering could test directly.
Because the second-order correction is negative near the $\Gamma$ point and grows with temperature, the perturbative regime is bounded; at higher temperatures methods beyond second-order perturbation are required.

Reading between the lines

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

The numerical robustness of the negative result could be checked by repeating the 4D integrals with explicit grid refinement and controlled delta-function broadening; the paper itself reports no such convergence tests.
If the absence of singularities is genuinely caused by full momentum dependence, the same suppression should appear for other honeycomb-lattice magnets, making the effect a generic property of non-Bravais bosonic Dirac systems rather than a CrBr3-specific detail.
The $T^3$ optical exponent is plausibly a generic consequence of the gap in the optical branch, which suppresses its thermal occupancy by one extra power of $T$; comparing with gapped optical branches in other materials would test this.

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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

Let me give you the short version: this paper probably does the right thing qualitatively. Dropping the thermal magnon approximation removes the van Hove singularities that one of the authors predicted in PRX 2018, and that did not survive in the 2022 INS data on CrBr3. That is a real and useful correction, and the contrast with the triangular lattice suggests a Bravais vs non-Bravais difference in temperature scaling I have not seen elsewhere. If the numerics hold up, I would want this in the literature.

The problems are not minor. The paper asserts that the kinematic factors for processes 2, 3, and 4 are identical and shows a single curve in Fig. 4(b). The delta functions in Eqs. (14), (17), and (20) are not equivalent: they put different band energies in the conservation condition, and the up vs down energies differ by a momentum-dependent |gamma| term. I cannot see any symmetry that would make those three four-dimensional integrals equal, and the paper offers no proof. Those kinematic factors feed directly into the decay rates and the real self-energy for the processes that dominate the upper-band renormalization. So the T^3 exponent and even the claimed smooth profiles are not established. This is an internal contradiction with the paper's own equations, and it is load-bearing.

Second, Appendix B, the derivation of the central self-energy expression in Eq. (8), is corrupted beyond reading—it is full of garbled, machine-generated text. Right now the key equation has no verifiable derivation in the manuscript.

Third, there is no code, no data, no grid sizes, no convergence tests, and no broadening scheme for the delta functions in what is a 4D Simpson quadrature. For a negative result whose entire force is "we looked everywhere and found no singularity", that is not acceptable. The reader's conditional verdict is fair.

Two minor points. The abstract's T^3 claim sits awkwardly with Section III, which correctly says the Hartree T^2 term dominates the observable shift, making T^3 subleading. And the model parameters come from the same experimental paper used for comparison—not fatal for the qualitative claim, but worth flagging.

Who is this for? Anyone who cares about interacting Dirac magnons and the theory-experiment discrepancy in CrBr3. With a careful rewrite—proven or recomputed kinematic factors, a readable appendix, and full numerical documentation—this could become a solid paper. Right now, the quantitative results are not trustworthy.

Recommendation: send it to peer review, because the question is important and the qualitative direction is probably right. A referee should demand that the kinematic factors be recomputed separately and that the numerical evidence be made available.

Referee Report

3 major / 4 minor

Summary. The paper revisits the problem of magnon-magnon interactions in the honeycomb ferromagnet CrBr3 and, as a comparison, the triangular-lattice ferromagnet MnBi2Te4. The authors perform second-order perturbation theory for the magnon self-energy, retaining full momentum-dependent interaction vertices and evaluating the resulting four-dimensional integrals numerically, without the 'thermal magnon approximation' used in earlier work [PRX 8, 011010 (2018)]. They report that the previously predicted van Hove-like singularities in the renormalized spectrum and decay rates are absent, consistent with inelastic neutron scattering data, and that the optical magnon branch renormalizes as T^3 while the acoustic branch renormalizes as T^2. They also claim a distinction between non-Bravais (honeycomb) and Bravais (triangular) lattices in the temperature dependence. The central claims are based on numerical evaluations of five scattering processes, whose self-energy integrals are given in Eqs. (8)-(24).

Significance. If correct, the paper would resolve a longstanding discrepancy between the theoretical prediction of van Hove singularities in interacting Dirac magnons and the smooth spectra measured by inelastic neutron scattering. The branch-dependent T^3 (optical) versus T^2 (acoustic) scaling is a concrete, falsifiable prediction, and the comparison between honeycomb and triangular lattices is a useful contribution. The methodological step of abandoning the thermal magnon approximation and keeping the full momentum dependence in a fully numerical calculation is in principle valuable. However, the paper's numerical evidence is not transparently documented, and at least one internal inconsistency in the presentation of the kinematic factors directly affects the reported results.

major comments (3)

[Sec. II B, Eqs. (14), (17), (20) and Fig. 4] The paper states that the kinematic factors for the second, third, and fourth processes are identical and shows a single curve for all three in Fig. 4(b). This is contradicted by the definitions: Eq. (14) conserves epsilon^u_k + epsilon^d_q = epsilon^d_p + epsilon^d_{k+q-p}, Eq. (17) conserves epsilon^u_k + epsilon^d_q = epsilon^u_p + epsilon^d_{k+q-p}, and Eq. (20) conserves epsilon^u_k + epsilon^u_q = epsilon^d_p + epsilon^u_{k+q-p}. Since epsilon^u_k - epsilon^d_k = 2JS|gamma_k| is nonzero except at isolated points, the three energy-conservation surfaces in the four-dimensional integration domain are generically different. The numerical results for processes 2-4, including the decay rates W^(2), W^(3), W^(4) and the real self-energies in Fig. 4(d,e,f,i,j,k), are presented as derived from these integrals. Unless the authors prove a nontrivial identity, for example by a change of variables that maps the band labels, the reported results for these processes are not supported by the stated equations, and the upper-band T^3 exponent (obtained by summing processes 2-5) is not secure.
[Sec. II B, Eq. (8) and Figs. 4-5] The four-dimensional integrals in Eq. (8) are the sole basis for the central negative result (absence of van Hove singularities) and the extracted temperature exponents, but the manuscript gives no grid size, no convergence test, and no specification of the delta-function broadening scheme for the on-shell condition (the parameter delta in Eq. (8) is not defined). The claim that the renormalized spectrum and decay rates are free of singular features is a statement about the non-analytic behavior of these integrals; without evidence that the numerical discretization resolves the Brillouin zone sufficiently finely, the smooth profiles in Fig. 4 could be numerical artifacts. The authors should report the numerical parameters and provide convergence checks, for example by varying the Simpson grid size and the broadening delta.
[Sec. II, Eq. (5) and Fig. 1] The vertex amplitudes in Eq. (4) depend on the phase phi_k = arg gamma_k through the angles zeta, theta, and kappa defined in Eq. (5). At the Dirac points K and K', gamma_k = 0 and the phase is ill-defined. The paper presents results along the Gamma-M-K path, including the K point, but does not explain how this phase singularity is regularized in the numerical integration. This is a potential source of error in the matrix elements and could affect the decay rates and self-energies near the Dirac points.

minor comments (4)

[Fig. 4 caption] The caption gives the single-ion anisotropy as A = 0.028 meV, whereas the text and Eq. (1) use A = -0.028 meV; the sign is important for the magnon gap.
[Appendix B] The manuscript contains a long corrupted passage of unreadable characters immediately after Eq. (B2), interrupting the Matsubara summation derivation; this must be repaired before publication.
[Sec. II B(b)] The phrase 'the following Eq. (4)' should be rephrased, for example as 'the following from Eq. (4)'.
[Sec. IV A] The word 'Feynmann' should be 'Feynman' in the heading of Section IV A.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild benchmark circularity: J and A are adopted from the same INS experiment used for the Fig. 7 comparison, but the central T^2/T^3 and no-singularity claims remain computed outputs rather than fitted inputs.

fitted input called prediction [Sec. II (Model) and Sec. III (Contrast with INS experiment)]
"We adopt the parameters from Ref. [26] and choose J = 1.494 meV and A = −0.028 meV. ... the overall trend and the magnitudes match surprisingly well with the experimental results [26]."

The Hamiltonian parameters J and A are taken directly from the same INS study that later serves as the quantitative benchmark in Sec. III. Because the magnon bandwidth and the interaction vertex scale are set by these input parameters, the agreement in Fig. 7 is partly inherited from the data set being used as validation rather than being an independent prediction. This is a mild benchmarking circularity. It does not make the central qualitative claims circular: the absence of van Hove singularities and the extracted exponents beta_u = 3 and beta_d = 2 are properties of the numerical self-energy integrals, not parameters fitted to those experimental features.

full rationale

The core derivation is self-contained: starting from the Heisenberg model of Eq. (1), the paper evaluates the sunset self-energy of Eq. (8) with full momentum-dependent vertices, and the temperature exponents are fits to the computed curves, not inputs. The self-citations to Ref. [21] are used for derivation scaffolding (the interaction Hamiltonian and the general self-energy expressions) and are reproduced in the appendices; they are not invoked as a uniqueness theorem or to forbid alternatives. The main circularity concern is the benchmark one: J and A come from Ref. [26], and the same Ref. [26] is used as the experimental comparison, so the quantitative agreement is not fully independent. Separately, the assertion in Sec. II B that K^(2)_k = K^(3)_k = K^(4)_k is not proven and appears inconsistent with the different delta-function arguments in Eqs. (14), (17), and (20); if false, this is a correctness risk affecting the summed beta_u result, but it is not a fitted-input circularity. Overall, the central claims retain independent computational content, so the score reflects only the minor benchmark circularity.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The calculation is a standard Heisenberg plus single-ion-anisotropy model at second order in 1/S perturbation theory; it introduces no new entities and no ad hoc parameters beyond the material parameters taken from prior experiment. The ledger records the exchange/anisotropy inputs, the perturbative truncation axioms, and the numerical-analysis choices that the central claims rest on.

free parameters (5)

J (nearest-neighbor exchange, CrBr3) = 1.494 meV
Adopted from Ref. [26], the same inelastic neutron scattering experiment used as the benchmark in Sec. III, so quantitative agreement in Fig. 7 partially inherits a fit to the target data.
A (single-ion anisotropy, CrBr3) = -0.028 meV
Adopted from ferromagnetic resonance measurements (Ref. [38]); sets the gap -2AS = 0.084 meV that the negative second-order corrections are compared against.
J1 (nearest-neighbor exchange, MnBi2Te4) = 0.2496 meV
Adopted from Ref. [35]; J2 = -0.024 meV and J3 = -0.010 meV are dropped, though the model is nominally for monolayer MnBi2Te4.
Eg (SIA gap, MnBi2Te4) = 0.144 meV (text), A = -0.028 meV in figure captions
Sec. IV states Eg = -2AS = 0.144 meV with S = 5/2, implying A = -0.0288 meV, while Figs. 10-11 print A = -0.028 meV; a minor internal inconsistency.
Temperature power-law exponents = alpha = 2.08 (Hartree); beta_u = 3, beta_d = 2 (sunset)
Obtained by polynomial fits to the authors' own numerics at 14 temperatures from 5-31 K; the fit window, the sign changes of Re Sigma (called numerical artifacts), and the absence of reported fit uncertainties affect the extracted exponents.

assumptions (7)

domain assumption Holstein-Primakoff expansion truncated at quartic order; S = 3/2 (CrBr3) and S = 5/2 (MnBi2Te4) treated as large.
Sec. II, Eq. (2); the entire perturbative scheme and the neglect of higher-order (six-boson) vertices require small magnon density at T up to 31 K.
domain assumption Only nearest-neighbor exchange J is kept for CrBr3; J2 = 0.077 meV and J3 = -0.068 meV are ignored, as are DM interactions; single-ion anisotropy is dropped from the interaction vertex (J >> A).
Sec. II: 'the other longer-range hopping terms... are also ignored'; the paper argues J >> J2, J3, A gives 'good quantitative agreement' but does not quantify the error. DM controversy with Ref. [47] is acknowledged and set aside.
domain assumption On-shell evaluation of the self-energy: omega replaced by epsilon_k in Eq. (8); Hartree correction inserted into bare or self-consistent propagators.
Sec. II B: 'we utilize the on-shell condition i.e., replace omega by epsilon_k'; this is the standard perturbative choice but is not tested against a full self-consistent Dyson solution.
standard math Wick contractions assume a ferromagnetic ground state and drop anomalous <u^† d> averages.
Appendix A, Eq. (A3): 'Since the ferromagnetic ground state preserves the total number of magnons, we ignore <u^†_k d_k>.' Standard at this order, but it fixes the Hartree form in Eq. (6).
ad hoc to paper The five quartic vertices in Eq. (4) exhaust the two-boson scattering channels; the asserted identity K^(2) = K^(3) = K^(4) holds.
Sec. II B; the completeness and the identity claim are asserted without proof, and the appendix derivation meant to support them is corrupted. Substituting Eqs. (14)-(20) into the delta functions gives different arguments (e.g., at k = Gamma), so this axiom needs checking.
domain assumption Mermin-Wagner: a tiny single-ion anisotropy suffices to stabilize 2D ferromagnetic order across the studied T range (5-31 K for CrBr3, 3-9 K for MnBi2Te4).
Sec. I: 'incorporate a small on-site single-ion anisotropy to reconcile the observed long-range magnetic order at finite temperature with the Mermin-Wagner theorem'. The gap (-2AS = 0.084 meV for CrBr3) also bounds the negative second-order corrections near Gamma.
standard math Standard Matsubara summation identities for bosonic propagators are used to evaluate the sunset diagram.
Appendix B; the final temperature factor F in Eq. (B7) matches the standard sunset form, but the intermediate manipulations are garbled in the preprint text.

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Pith. "Pith review of Interacting Dirac magnons in the van der Waals ferromagnet CrBr$_3$." pith.science (2026). https://pith.science/paper/M7ZERK47

@misc{pith2026250607650,
  author       = {Pith},
  title        = {Pith review of: Interacting Dirac magnons in the van der Waals ferromagnet CrBr$_3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7ZERK47}},
  note         = {Machine review of arXiv:2506.07650}
}

abstract

We study the effects of magnon-magnon interactions in the two-dimensional van der Waals ferromagnet CrBr$_3$ focusing on its honeycomb lattice structure. Motivated by earlier theoretical predictions of temperature-induced spectral shifts and van Hove singularities in the magnon dispersion~[S. S. Pershoguba \textit{et al}., Dirac Magnons in Honeycomb Ferromagnets, \href{https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.011010}{Phys. Rev. X {\textbf{8}}, 011010 (2018)}], we go beyond the commonly used thermal magnon approximation by applying second-order perturbation theory in a fully numerical framework. Our analysis uncovers significant deviations from previous analysis: in particular, the predicted singularities are absent, consistent with recent inelastic neutron scattering measurements~[S. E. Nikitin \textit{et al}., Thermal Evolution of Dirac Magnons in the Honeycomb Ferromagnet CrBr$_3$, \href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.127201}{Phys. Rev. Lett. {\textbf{129}}, 127201 (2022)}]. Moreover, we find that the temperature dependence of the renormalized magnon spectrum exhibits a distinct $T^3$ behavior for the optical magnon branch, while retaining $T^2$ behavior for the acoustic or down magnon band. This feature sheds new light on the collective dynamics of Dirac magnons and their interactions. We further compare the honeycomb case with a triangular Bravais lattice, relevant for ferromagnetic monolayer MnBi$_2$Te$_4$, and show that both systems lack singular features while displaying quite distinct thermal trends.

Figures

Figures reproduced from arXiv: 2506.07650 by the authors.

**Figure 3.** FIG. 3. (a) Temperature-dependent Hartree spectrum, calcu [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** FIG. 5. Comparison between bare and self-consistent second [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

Figures from the paper (4 more)

**Figure 6.** Figure 6: FIG. 6. The best fit to the temperature power law for the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) The temperature-dependent scaled energy parameter as defined in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 9.** Figure 9: FIG. 9. (a) The evolution of the Hartree correction (see [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 11.** Figure 11: FIG. 11. A comparison between the bare (solid line) and [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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