Interaction-Induced Topological Phase Transition in Magnetic Weyl Semimetals

Aur\'elien Manchon; Konstantinos Sourounis

REVIEW 3 major objections 5 minor 89 references

Interaction-Induced Topological Phase Transition in Magnetic Weyl Semimetals

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper predicts that electron-magnon interaction can drive a topological phase transition in magnetic Weyl semimetals below the Curie temperature, with the fate of the Weyl nodes controlled by their spin chirality.

desk verdict Plausible new mechanism for chirality-dependent Weyl-node destabilization by electron-magnon coupling, but the quantitative predictions need self-consistent checking before they are taken as reliable. read the letter →

arxiv 2412.17044 v2 pith:H2F6LWRG submitted 2024-12-22 cond-mat.mes-hall

Konstantinos Sourounis , Aurélien Manchon This is my paper · ORCID

classification cond-mat.mes-hall

keywords Weylsemimetalelectron-magnoninteractiontopologicalphasetransitionanomalousHalleffectspinchiralitymany-bodyinvariantmagneticfluctuationsHeuslercompounds

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

The paper predicts that magnetic fluctuations, the magnons ubiquitous in a magnetic Weyl semimetal, can dismantle the Weyl nodes themselves and trigger a topological phase transition at a temperature below the Curie temperature. The effect is chirality-dependent: the electron-magnon self-energy softens the effective spin splitting when the nodes have trivial chirality ($m>0$), pulling the Weyl points together until they annihilate, while for inverted chirality ($m<0$) the same interaction strengthens the splitting and keeps the nodes alive at high temperature. Using the many-body topological number $N_3$ and the anomalous Hall conductivity computed from interacting Green's functions, the paper shows that the topological response vanishes in the trivial regime well below $T_c$ but survives in the inverted regime. If this is right, the giant anomalous Hall effect used to identify magnetic Weyl semimetals can be quenched before the magnetic order collapses, and the temperature at which it disappears encodes the chirality of the underlying nodes.

What carries the argument

The engine of the argument is the electron-magnon self-energy $\Sigma^{R/A}(k,\epsilon,T)$ given by Eqs. (5)-(6), computed in a one-shot Born approximation from the spin-flip electron-magnon coupling obtained by a standard magnon-boson transformation. This self-energy renormalizes the spin splitting into $\Delta_{\mathrm{eff}} = 2\Delta + \Sigma_{\downarrow} - \Sigma_{\uparrow}$, and the sign of $\Sigma_{\downarrow} - \Sigma_{\uparrow}$ relative to the spin chirality of the eigenstates (set by $m$) decides whether thermal magnons pull the Weyl points together or push them apart. The topological verdict is carried by the $k_z$-resolved many-body topological number $N_3$ (Eq. 8), a Green's-function generalization of the Chern number that stays meaningful in the interacting system, and by the intrinsic anomalous Hall conductivity from the Green's function formula (Eq. 9), whose temperature dependence is the main transport signature.

What would settle it

Measure the separation of the two Weyl points as a function of temperature by photoemission or optical spectroscopy in a magnetic Weyl semimetal whose chirality (sign of $m$) is known from band-structure calculations: the paper requires that trivial-chirality nodes move together and merge at a temperature strictly below the Curie point, while inverted-chirality nodes stay separated nearly up to the Curie point. Seeing the same temperature dependence for both chiralities, or no merger below the Curie point, would refute the claim.

Watch

Extended reading notes

Core claim

The central claim is that electron-magnon interaction makes the topology of a magnetic Weyl semimetal strongly temperature dependent below the Curie temperature. In the noninteracting picture the two Weyl nodes sit at fixed positions set by the exchange splitting; when electrons scatter off magnons, the self-energy shifts the spin-dependent bands by an amount whose sign depends on the mass parameter $m$, which sets the spin chirality of the nodes. For $m=+t$ (trivial chirality) the shift softens the effective splitting $\Delta_{\mathrm{eff}} = 2\Delta + \Sigma_{\downarrow} - \Sigma_{\uparrow}$, so as temperature rises the Weyl points move toward each other and annihilate before the magnetization itself collapses; both the many-body topological number $N_3$ (a Green's-function generalization of the Chern number) and the anomalous Hall conductivity fall to zero. For $m=-t$ (inverted chirality) the same self-energy increases the effective splitting in the region between the nodes, counteracting the softening; the Weyl points survive to higher temperature and the anomalous Hall conductivity remains finite. The paper states this as a chirality-controlled topological phase transition, with trivial-chirality nodes the more sensitive and inverted-chirality nodes the resilient ones.

Load-bearing premise

The whole prediction depends on the assumption that the effect of magnetic fluctuations on the electrons can be captured in a single pass through a fixed band structure, without letting those fluctuations feed back and change the magnetic order they come from; if electrons and magnons were allowed to adjust to each other self-consistently, the size or even the sign of the induced spin splitting could change, and the finding that inverted-chirality Weyl nodes are the resilient ones could be reversed.

Editorial extensions

If this is right

The anomalous Hall conductivity of a magnetic Weyl semimetal can fall sharply with temperature while the magnetization is still large, because the electron-magnon interaction suppresses the Berry-curvature contribution before the ferromagnetic order itself disappears.
Materials whose Weyl nodes have trivial chirality should show a markedly faster temperature suppression of the anomalous Hall effect than materials with inverted chirality, an asymmetry visible already in the computed phase diagram.
In Weyl semimetals with low Curie temperature, the electron-magnon interaction should dominate over the electron-phonon interaction in controlling the temperature evolution of topology, since the phonon scale is set by the higher Debye temperature.
Interband quantities such as the Nernst effect and the orbital magnetization, which depend on inverse band-energy differences, are expected to show strong temperature dependence near the Weyl nodes and could even change sign, in contrast to the anomalous Hall effect.

Reading between the lines

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

A self-consistent treatment in which the magnons are renormalized by the same electrons, rather than the one-shot bare-magnon calculation used here, could turn the predicted temperature-driven transition into a coupling-strength-driven topological phase transition at zero temperature, with a critical electron-magnon coupling where $N_3$ jumps.
The chirality asymmetry suggests a practical design rule for spintronics and topological transport: compounds with inverted-chirality Weyl nodes at the Fermi level should preserve their anomalous Hall signal to higher temperature, a prediction that could be tested by comparing members of the same Heusler family with different mass terms.
The prediction that Weyl-point separation shrinks with temperature in the trivial regime is directly testable by temperature-dependent photoemission or optical spectroscopy in any known magnetic Weyl semimetal with identified chirality, providing a measurement that could confirm or reject the mechanism before any transport experiment.

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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

You should know two things about this paper. First, it identifies a genuinely new mechanism: electron-magnon scattering shrinks the Weyl-node separation in a magnetic Weyl semimetal, and the effect is chirality-dependent — trivial-chirality nodes annihilate before the Curie temperature while inverted-chirality nodes survive. Second, the calculation that produces this asymmetry is a one-shot self-energy built from non-interacting bands and bare magnons, with the magnetization imposed phenomenologically. The qualitative asymmetry is coherent and probably survives better approximations; the quantitative phase diagram is not yet controlled.

The novelty is real. Prior work on interaction-driven topological transitions used electron-phonon coupling or electron-electron correlations, not electron-magnon scattering. The chirality dependence is a new result, and the paper connects it sensibly to experiments — including the destruction of Weyl points in Co3Sn2S2 well below Tc. It also uses standard topological Green's function methods (N3 invariant, Kubo-Bastin formula) and posts data on Zenodo, which is good practice. The discussion of half-metallic candidates like NiMnSb is honest: those materials suppress electron-magnon scattering at the Fermi level, and the authors flag that caveat.

The soft spots are the same ones the stress-test note raises. The self-energy is computed with a step-function electron occupation, the frequency dependence of the self-energy is dropped, and the magnetization (T) is put in by hand with the same law in interacting and non-interacting calculations. There is no back-action of electron-magnon scattering on the magnon spectrum or the local moments. Near the transition, where the self-energy is large enough to close the Weyl gap, the starting point of the perturbative expansion is questionable. A self-consistent Born calculation could shift the transition temperatures and might alter the apparent robustness of the inverted phase. I don't think it will reverse the chirality ordering, because that asymmetry comes from the sign of the mass term m, but that is an inference, not something this paper proves. The self-energy matrix elements are delegated to the Supplemental Material, so a referee can't check them from the main text alone. The parameter set is also narrow: the phase diagram varies m, but everything else is fixed at one hand-picked point.

Who is this for? Researchers working on magnetic Weyl semimetals, anomalous Hall transport at finite temperature, or electron-magnon coupling in topological materials. It is a credible model calculation with a testable qualitative prediction. It deserves a serious referee, not a desk reject, but the referee should push for a self-consistent calculation or at least a clear statement of the regime where one-shot is reliable, plus access to the Supplemental Material. I would not cite the numerical transition temperatures yet, but I would cite the mechanism.

Referee Report

3 major / 5 minor

Summary. The manuscript studies a four-band tight-binding model of a magnetic Weyl semimetal coupled to magnons. The authors compute the one-shot electron self-energy from electron-magnon scattering, the interacting spectral function, the many-body topological invariant N3(kz), and the anomalous Hall conductivity via a Kubo-Bastin-type formula. Their central claim is that electron-magnon interactions annihilate Weyl nodes of trivial chirality (m>0) at temperatures below the Curie temperature, whereas inverted-chirality nodes (m<0) remain robust, producing a chirality-dependent topological phase transition and a corresponding suppression of the anomalous Hall effect.

Significance. If the predictions hold, the paper identifies a concrete mechanism by which magnetic fluctuations control topology: the electron-magnon interaction acts as a chirality-selective renormalization of the effective exchange splitting, allowing the anomalous Hall conductivity to vanish before magnetic order disappears. This is a falsifiable prediction with direct relevance to recent experiments on Co3Sn2S2, Co2Mn(Ga,Al) and NiMnSb, and it complements earlier work on electron-phonon-driven topological transitions. The paper also develops a convenient Green's-function framework for evaluating N3 and sigma_xy that accounts for broadening, and it makes the numerical data available on Zenodo. The main caveat is that the calculation is one-shot; the strength of the claim is therefore not yet fully matched by the level of approximation.

major comments (3)

[Electron-Magnon Interaction, Eqs. (5)-(6)] The self-energy is evaluated in a one-shot manner using non-interacting bands and bare magnons, with the electron occupation approximated by a step function and the frequency dependence of the self-energy dropped (explicitly stated after Eq. (9) as partial_epsilon/eta Sigma_k(epsilon,T) approximately 0). At the predicted transition for m=+t (T approximately 0.75Tc, Fig. 3(b)), the self-energy is strong enough to close the Weyl gap, so the validity of the non-interacting starting point is unclear. Because the central chirality asymmetry is precisely the sign of the induced spin splitting Delta_eta_eff, a self-consistent Born calculation (or at least an estimate of the second-order correction) is needed to show that the sign and magnitude are robust. Without this, the predicted ordering of transition temperatures for the two chiralities is not yet controlled.
[Electron-Magnon Interaction, phenomenological <Sz>] The paper accounts for magnon-magnon interactions only through the phenomenological law <Sz> = S(1 - T/Tc)^(1/3) with Tc = 8J, applied identically in the interacting and non-interacting calculations. This neglects the back-action of electron-magnon scattering on the localized moment and on the magnon spectrum, even though the paper's own narrative treats the electron-magnon interaction as cooperatively softening the magnetization. The phase diagram in Fig. 4 therefore mixes a self-consistent quenching of <Sz> with a one-shot electron self-energy; the transition temperature of the interacting system is not determined self-consistently. Please estimate the magnitude of this feedback effect or justify its neglect.
[Interacting Topology and Transport, Eq. (9)] The anomalous Hall conductivity is computed under the approximation partial_epsilon/eta Sigma_k(epsilon,T) approximately 0, while the spectral function in Eq. (7) retains the frequency dependence implicitly. Since the approximation is applied to both chiralities, it may not bias the comparison, but the paper should explicitly verify this, for instance by computing sigma_xy with the full frequency derivative at a representative temperature and showing that the qualitative chirality ordering and the transition temperature are unchanged.

minor comments (5)

[Figure 1 caption] The caption states the band structure is shown at T=0 K, but the arrows indicate the energy shift induced by the self-energy, which at T=0 includes zero-point magnon effects; please clarify whether the arrows are schematic or computed at T=0.
[General notation] The symbol N3 is used both as a label and a mathematical quantity; please define N_3 consistently with a subscript in all equations and text.
[Eqs. (5)-(6)] The notation for the matrix elements Phi is inconsistent: the first equation has Phi_{eta down, eta' down} and the second has Phi_{eta up, eta' up}, while the text defines Phi_{eta sigma, eta' sigma'}; please align the notation with the Supplemental Material.
[Discussion] In the sentence about NiMnSb, the transport is attributed to crossings away from the Fermi level where both spin chiralities exist; please make the mapping between these crossings and the model's m=+t and m=-t regimes explicit.
[Introduction and Discussion] The phrases 'trivial chirality' and 'inverted chirality' are used to label the m>0 and m<0 cases; please define them operationally in terms of the spin texture shown in Fig. 1 to avoid confusion with chirality used for Weyl nodes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chirality-dependent topological transition is a computed consequence of the stated model and one-shot self-energy, with no fitted parameter tied to the target result.

full rationale

The derivation chain is explicit and self-contained: the model Hamiltonian is fixed by Eqs. (1)-(4), the electron-magnon self-energy by Eqs. (5)-(6), and the topology and Hall response by Eqs. (8)-(9). The central claim, that trivial-chirality Weyl nodes are annihilated below Tc while inverted-chirality nodes survive, is obtained by evaluating these expressions for m=+t and m=-t; no parameter is fitted to reproduce that asymmetry. The phenomenological magnetization law <S_z>=S(1-T/Tc)^(1/3) is an input taken from prior work [61] and applied identically to the interacting and noninteracting cases, so it is not a hidden way of building in the predicted ordering. The appearance of a coauthor's earlier work in Ref. [71] for the Kubo-Bastin-type Hall formula is not load-bearing: that formula is a standard, externally checkable Green's function expression and is not used to exclude alternative mechanisms. The acknowledged approximations (step-function electron occupations, bare magnons, one-shot self-energy, and dropping dSigma/depsilon) are accuracy and robustness limitations that could affect the magnitude or even sign of the effect, but they do not reduce the conclusion to its inputs. Therefore no circular step is established.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a model Hamiltonian with hand-chosen parameters, a phenomenological magnetization law, and a one-shot perturbative self-energy. No new particles or forces are introduced. The main unverified inputs are the form of the electron-magnon coupling, the truncation of magnon interactions to a mean-field magnetization, and the neglect of self-consistency and vertex corrections.

free parameters (10)

t = 1 (energy unit)
Sets the energy scale; all other energies are quoted in units of t.
Delta = 2t
Exchange coupling between itinerant and localized spins; chosen to put the system in the Weyl semimetal phase with two Weyl points at the Fermi level.
lambda = t
Spin-orbit coupling strength; chosen to match the model of Refs. [24,58,59].
S = 1
Localized spin magnitude; chosen.
J = t/100
Heisenberg exchange; sets Tc=8J and controls the magnon bandwidth and thermal occupation.
K = J/10
Uniaxial anisotropy; chosen to stabilize the ferromagnetic order in the model.
m = +t and -t
Mass parameter; the two values define the trivial (m>0) and inverted (m<0) chirality regimes that the central claim compares.
alpha = 1/3
Exponent in the phenomenological magnetization law <S_z>=S(1-T/Tc)^alpha; chosen ad hoc to mimic magnon-driven quenching.
Gamma = 0.2t
Homogeneous broadening used in the Hall conductivity calculation, Eq. (9).
0+ broadening = 0.1t
Numerical broadening in the retarded self-energy.

assumptions (7)

domain assumption The magnetic subsystem is described by the nearest-neighbor Heisenberg Hamiltonian with uniaxial anisotropy and treated in linear spin-wave theory via the Holstein-Primakoff transformation.
Standard approximation for magnons; truncates multi-magnon interactions and assumes small spin deviations. Invoked in Section 'Electron-Magnon Interaction' and SM [64].
domain assumption The electron-magnon coupling is a local spin-flip interaction of the form Delta sqrt(1/2) sum_i (S_i^+ sigma^- + S_i^- sigma^+) tensor tau0, with the same coupling constant Delta as the mean-field exchange splitting.
Microscopic form is plausible for a local exchange model, but its momentum independence and orbital structure are assumed without derivation from a specific material.
domain assumption The electron self-energy is evaluated in a one-shot perturbative expansion from the non-interacting Green's functions and bare magnon propagators, and the frequency derivative of the self-energy is neglected.
This ignores self-consistency, vertex corrections, and magnon damping; stated implicitly in Eqs. (5)-(7) and the sentence 'partial_epsilon/eta Sigma_k approx 0' before Eq. (9).
domain assumption The electron occupation is approximated by a step function at the temperatures considered.
The paper states 'the distribution of electrons can be approximated by a step function' in the Section 'Electron-Magnon Interaction'.
ad hoc to paper The temperature dependence of the magnetization follows the phenomenological law <S_z> = S(1-T/Tc)^(1/3) with Tc=8J, which is used to account for magnon-magnon interaction.
The specific exponent and the proportionality Tc=8J are chosen rather than derived in this work; the same law is applied to both interacting and non-interacting cases.
standard math The many-body topological number N3 and the Kubo-Bastin expression for the anomalous Hall conductivity correctly characterize the topology and transport of the interacting system.
Established formalisms (Refs. [41-43,71]); used without proof in Section 'Interacting Topology and Transport'.
domain assumption The Weyl points of the interacting system can be located from the spectral function and their annihilation signaled by N3 and sigma_xy vanishing.
Because the interacting Green's functions are broadened, the definition of a Weyl point is not as sharp as in the non-interacting limit; the paper does not provide a quantitative criterion for annihilation beyond visual inspection of Figs. 2 and 3.

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Cite this review

Pith. "Pith review of Interaction-Induced Topological Phase Transition in Magnetic Weyl Semimetals." pith.science (2026). https://pith.science/paper/H2F6LWRG

@misc{pith2026241217044,
  author       = {Pith},
  title        = {Pith review of: Interaction-Induced Topological Phase Transition in Magnetic Weyl Semimetals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2F6LWRG}},
  note         = {Machine review of arXiv:2412.17044}
}

read the original abstract

Despite the tremendous interest raised by the recent realization of magnetic Weyl semimetals and the observation of giant anomalous Hall signals, most of the theories used to interpret experimental data overlook the influence of magnetic fluctuations, which are ubiquitous in such materials and can massively impact topological and transport properties. In this work, we predict that in such magnetic topological systems, the interaction between electrons and magnons substantially destabilizes the Weyl nodes, leading to a topological phase transition below the Curie temperature. Remarkably, the sensitivity of the Weyl nodes to electron-magnon interaction depends on their spin chirality. We find that Weyl nodes with a trivial chirality are more sensitive to electron-magnon interactions than Weyl nodes presenting an inverted chirality, demonstrating the resilience of the latter compared to the former. Our results open perspectives for the interpretation of the transport signatures of Weyl semimetals, especially close to the Curie temperature.

Figures

Figures reproduced from arXiv: 2412.17044 by the authors.

**Figure 2.** FIG. 2. The spectral function of the electrons (color) given by [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. The phase diagram of the anomalous Hall conduc [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: FIG. 3. The non-interacting and interacting many-body topo [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

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In addition, switching the sign ofmchanges the spin chirality of Weyl nodes, as illustrated by the color scale in Fig. 1. In the following, we refer tom >0 (m <0) as the trivial (inverted) regime. When the chemical potential is set to zero,µ= 0, this model is equivalent to a stack of 2D Chern Insulators along thek z direction [63]. In this case, the anoma...

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