Pith. sign in

REVIEW 5 major objections 4 minor 52 references

Finite temperature activates an anisotropic exchange that competes with and can reverse altermagnetic magnon splitting.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

At finite temperature, anisotropic spin exchange renormalized by magnon-magnon interactions competes with isotropic exchange to control—and potentially reverse—chiral magnon splitting and spin current in altermagnets.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Plausible finite-T mechanism for altermagnon splitting, but the paper needs a defined T_c and a stability check on the HF solutions before the quantitative claims can be trusted. the 5 major comments →

arxiv 2511.03922 v3 pith:NRNLDZ5I submitted 2025-11-05 cond-mat.str-el

Competitive Orders in Altermagnetic Chiral Magnons

Congzhe Yan Zhijun Jiang Jinyang Ni , Guoqing Chang This is my paper
classification cond-mat.str-el
keywords altermagnetismmagnonschiral splittinganisotropic spin exchangespin-wave theoryHartree-Fockspin Seebeck effectfinite temperature
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 claims that in d-wave altermagnets — collinear magnets whose opposite-spin sublattices are related by a rotation instead of inversion, producing momentum-dependent spin splitting — the chiral splitting of magnon bands at finite temperature is not set once by the material's alternating isotropic exchange (ISE). Once magnon-magnon interactions are included, a long-range anisotropic spin exchange (ASE) term is activated, and it generates its own d-wave splitting with the opposite sign. The two contributions compete: the ASE-induced splitting grows roughly linearly with temperature while the ISE-induced splitting stays nearly constant, so the net splitting can shrink, vanish, or even invert. Because the thermal spin conductivity is proportional to this splitting, the same competition can reverse the sign of the spin current as temperature rises. The authors support this with a renormalized spin-wave calculation, and show that the ratio of anisotropic to isotropic exchange, together with the spin quantum number S, controls when and whether the cancellation happens.

Core claim

The key claim is that long-range anisotropic spin exchange, which makes no contribution at the level of linear spin-wave theory, becomes an effective source of altermagnetic magnon splitting at finite temperature once the quartic magnon terms are decoupled in the Hartree-Fock approximation. In this treatment the ASE self-energy couples magnons at the X/Y points to the thermally populated Γ-point magnons through a d-wave form factor, weighted by the difference in magnon occupation between the two sublattices. The resulting splitting is opposite in sign to the splitting produced by alternating isotropic exchange and grows with temperature, so for sufficiently large $J_{2z}/\Delta J$ the altermagnetic gap

What carries the argument

The central object is the Hartree-Fock self-energy generated by the quartic anisotropic-exchange interaction: $H_a^{2,\mathrm{HF}} = \frac{1}{N} \sum_{k,k'} 2J_{2z} f^d_{k'-k} \left( \langle n^a_{k'} \rangle n^a_k - \langle n^b_{\bar{k}'} \rangle n^b_{\bar{k}} \right)$. This term is what turns a parameter invisible in linear spin-wave theory into a temperature-dependent competitor with the isotropic exchange. It works by connecting the X/Y magnon modes to the Γ-point magnon population through the d-wave form factor $f^d$, so its magnitude grows with the thermal magnon density and its sign is fixed by $J_{2z}$. The competition between this self-energy and the bare ISE splitting is what produces the vanishing, inversion, and spin-current reversal described

Load-bearing premise

The calculation relies on a Hartree-Fock decoupling that replaces four-magnon anisotropic-exchange interactions by products of two-magnon density averages, and in particular assumes the sublattice magnon populations $\langle n^a \rangle$ and $\langle n^b \rangle$ differ; if the symmetric population is the stable self-consistent solution, or if non-Hartree-Fock fluctuations dominate near $T \sim J$, the predicted ASE splitting and sign reversal do not follow.

What would settle it

Measure the temperature-dependent energy difference between the two magnon branches at the X and Y points in a d-wave altermagnet engineered to have same-sign $J_{2z}$ and $\Delta J$ with a large ratio (order one). If the splitting does not shrink, vanish, and invert as the temperature approaches the exchange scale, or if the thermal spin conductivity does not change sign accordingly, the ASE-competition picture is falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The altermagnetic magnon splitting becomes temperature-tunable: when J_{2z} and ΔJ have the same sign, the gap shrinks with T, closes at a characteristic temperature, and reopens with opposite sign.
  • The thermal spin conductivity inherits this behavior, so the spin Seebeck signal can change sign as temperature rises — a striking signature of the ASE-ISE competition.
  • Materials with small spin S are more susceptible to the cancellation, since stronger spin fluctuations amplify the ASE self-energy; large-S altermagnets keep their splitting stable to higher temperatures.
  • The ratio J_{2z}/ΔJ, not just the isotropic exchange, becomes the decisive material parameter for predicting the finite-temperature spin response.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This mechanism suggests that a temperature-swept measurement of the magnon gap could serve as a quantitative probe of anisotropic exchange, since the temperature at which the gap vanishes directly encodes J_{2z}/ΔJ.
  • If the generalization to f-, g-, and i-wave altermagnets holds, temperature could act as a universal switch for chiral magnon transport across a whole family of materials, not just the d-wave case.
  • Near the cancellation temperature, fluctuations beyond Hartree-Fock may become important, so the precise numerical predictions for the collapse point may shift; the qualitative sign-reversal, however, should survive as long as the ASE self-energy is d-wave and thermally activated.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper develops a renormalized spin-wave theory (RSWT) for a monolayer d-wave square altermagnet with nearest-neighbor and second-nearest-neighbor isotropic exchange (ISE), second-nearest-neighbor anisotropic exchange (ASE), and single-ion anisotropy. The linear spin-wave part is summarized in Eq. (7). The authors include quartic magnon terms and treat them at Hartree-Fock level; Eq. (11) gives the d-wave ASE self-energy. Solving self-consistently, they find that ASE induces an altermagnetic magnon splitting at finite temperature with sign opposite to the ISE-induced splitting, so the two contributions compete. Depending on the ratio J2z/ΔJ, spin number S, and temperature, the net splitting can vanish or invert, and the thermal spin conductivity in Eq. (12) can change sign. DFT estimates for NiF2, CoF2, and MnF2 are presented in Table I.

Significance. If established, the result is significant: it identifies a finite-temperature, interaction-driven contribution to altermagnetic magnon splitting beyond linear spin-wave theory and predicts a testable temperature-induced sign reversal of the magnon spin current. The paper has clear strengths: the LSWT eigenvalues in Eq. (7) are analytically concise, Eq. (11) explicitly displays the ASE-induced d-wave self-energy, and the material-specific DFT-based predictions make the claims falsifiable. The significance is however conditional on the Hartree-Fock treatment being representative of the true finite-temperature state, which is not fully demonstrated in the present manuscript.

major comments (5)
  1. [Many body effects, Eq. (11) and Fig. 2(c,d)] The paper does not demonstrate that the self-consistent Hartree-Fock equations have a unique, converged, physically stable solution with the plotted ASE-induced splitting. For the ASE-only case shown in Fig. 2(c,d), the solution branches are not analyzed and no free-energy comparison is provided. Because the central claim is that ASE produces a d-wave splitting that competes with ISE, the selected self-consistent solution must be shown to be the stable physical fixed point. Please provide the HF free-energy functional and compare the relevant branch with any competing branch, and report convergence criteria.
  2. [Many body effects, Eqs. (8)-(11)] The Hartree-Fock decoupling is applied at T∼J, where the magnon density is not parametrically small and the quasiparticle picture can break down. The manuscript offers no benchmark—for example, comparison with small-system exact diagonalization or quantum Monte Carlo, or an estimate of neglected higher-order/non-HF contractions. This is load-bearing because the sign reversal in Fig. 4(b) occurs in this high-temperature regime. Please quantify the validity of the HF approximation (e.g., report the self-consistent ⟨n⟩/S and compare with a numerically exact small-cluster calculation) or otherwise bound the neglected terms.
  3. [Figs. 3-4 and Table I] The temperature axis is given as t=T/T_c, but T_c is never defined. Without a definition (e.g., classical ordering temperature, self-consistent HF transition temperature, or DFT-derived Néel temperature) and numerical values, the temperature dependence in Figs. 3-4 and the material-specific columns in Table I (ϕ_{0.1}, ϕ_{0.3}, ϕ_{0.5}) cannot be reproduced or quantitatively assessed. Please define T_c explicitly and list its values for the materials considered.
  4. [Eqs. (8)-(11) and Supplemental Material] The derivation of Eq. (11) requires the complete set of Hartree-Fock contraction channels for all quartic terms in Eq. (8). Only one example is shown in Eq. (10), and the manuscript repeatedly defers the remaining contractions and the formula for Ĉ_k to the Supplemental Material, which is not available in the posted manuscript. The completeness and sign of Eq. (11) therefore cannot be checked. Please include the full contraction list and the explicit expression for Ĉ_k in the main text or an accessible appendix.
  5. [Table I and DFT parameters] The DFT-derived exchange parameters ΔJ and J2z in Table I are presented without error bars or computational details. The sign and magnitude of J2z/ΔJ are decisive for the predicted competition, gap closure, and spin-current reversal. A sensitivity statement (e.g., basis set, U value, or magnetic configuration dependence) is needed before the material-specific predictions can be considered robust.
minor comments (4)
  1. [General model, Eqs. (2)-(3)] The notation is easy to confuse: J2 is defined as (J2+ + J2-)/2 while J2z denotes the anisotropic exchange; later J2/J1 is also used. Please collect all exchange parameters in one table or clearly distinguish them throughout.
  2. [Spin wave theory, Eqs. (6)-(7)] Eq. (6) is written as H_{1k}/S, but Eq. (7) simply gives E. Please clarify whether the eigenvalues in Eq. (7) are per spin, and state the value of S used for the numerical figures.
  3. [Fig. 2 caption] The caption says '2NN spin exchanges exhibit d-wave behavior (J2+ ≠ J2-)', but the text defines ΔJ=-(J2+ - J2-)/2. Please unify these conventions.
  4. [Discussion] The sentence 'ASE, though suppressed near the zero temperature' is vague. It would be clearer to state that ASE does not enter at the LSWT level and first appears through the quartic terms in the HF treatment.

Circularity Check

0 steps flagged

No circularity: finite-temperature ASE splitting is obtained from the stated Hamiltonian via a standard Hartree-Fock decoupling; no fitted output or load-bearing self-citation is used.

full rationale

The claimed derivation is self-contained: the Hamiltonian is specified in Eqs. (1)-(3), mapped to bosons in Eq. (4), and solved first at LSWT level in Eqs. (6)-(7), then with quartic terms in Eqs. (8)-(9). The central ASE-induced splitting is Eq. (11), obtained by explicit Hartree-Fock contraction (Eq. (10)) of the ASE quartic term; it is not imposed or fitted, and the subsequent gap/order-parameter curves and the spin-conductivity sign reversal are computed from the renormalized spectrum using Eq. (12). DFT values in Table I are external inputs, not outputs used to tune parameters. Self-citations (e.g., Ref. [40]) appear only as background/transport references and are not load-bearing. Concerns about the undefined T_c, the deferred SM contraction details, or the uncontrolled HF regime at T~J are validity/completeness issues, not evidence that any prediction reduces by construction to an input. Hence no circular step is present.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central claim is a model-calculation result, so the ledger is dominated by chosen couplings and standard spin-wave assumptions. The key free parameters are the d-wave ISE strength ΔJ and the d-wave ASE strength J2z, with S, J2/J1, K/J1 and τ0 as additional model choices. No new particles, mediators, or conserved quantities are introduced.

free parameters (5)
  • ΔJ/J1 (d-wave isotropic exchange) = 0.2 in model calculations; DFT values in Table I
    Sets the LSWT ISE chiral splitting; central to the competition mechanism.
  • J2z/J1 (d-wave anisotropic exchange) = 0.6 in model calculations; DFT values in Table I
    Source of the ASE-induced finite-temperature splitting; its ratio to ΔJ controls sign and reversal.
  • S (spin quantum number) = 3/2 in the generic model; 1, 3/2, 5/2 for NiF2, CoF2, MnF2
    Controls the strength of quantum spin fluctuations and the temperature dependence of the gap (Fig. 4a).
  • J2/J1 and K/J1 = -1 and -0.01
    Model choices to maintain collinear easy-axis order; K sets the anisotropy gap and stabilizes finite-T order.
  • τ0 (magnon lifetime scale) = ℏ/J1
    Inserted into the Kubo spin-conductivity formula; affects the magnitude of σ but not the sign-reversal mechanism.
axioms (5)
  • domain assumption Holstein-Primakoff expansion truncated at quartic order with small magnon density ⟨n⟩≪S
    Used in Eq. (4) and the low-temperature statements; questionable at T∼J where the HF calculation is applied.
  • domain assumption Hartree-Fock factorization of quartic boson terms
    Eqs. (10)–(11); uncontrolled at T∼J and the paper does not benchmark against other methods.
  • domain assumption ⟨Sz⟩ is conserved and DMI is forbidden, reducing ASE to diagonal J2z S^z_i S^z_j
    Stated in 'General spin model'; excludes off-diagonal or spin-orbit-coupled exchange that could alter the splitting sign and temperature dependence.
  • domain assumption Collinear easy-axis magnetic order persists at finite temperature in 2D up to T/Tc = 0.5
    Relies on the easy-axis anisotropy K<0; no derivation or definition of Tc is provided.
  • standard math Standard Bogoliubov transformation and Kubo formula for spin conductivity
    Used to obtain magnon eigenvalues (Eq. 7) and the spin Seebeck conductivity (Eq. 12).

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Competitive Orders in Altermagnetic Chiral Magnons." pith.science (2026). https://pith.science/paper/NRNLDZ5I

@misc{pith2026251103922,
  author       = {Pith},
  title        = {Pith review of: Competitive Orders in Altermagnetic Chiral Magnons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRNLDZ5I}},
  note         = {Machine review of arXiv:2511.03922}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

The magnons in altermagnets exhibit chiral splitting even in the absence of spin-orbit coupling and external magnetic fields. Typically, this chiral splitting behavior can be well described by alternating isotropic spin exchanges (ISE) near the zero temperature. However, its finite-temperature dynamics, particularly when incorporating spin-orbit coupling effects, remains elusive. In this study, we reveal that, when including magnon-magnon interactions, long-range anisotropic spin exchange (ASE) can also induce chiral splitting of magnons at a finite temperature. Crucially, the chiral splitting induced by ASE competes with that arising from ISE, leading to a pronounced temperature-dependent modulation of the altermagnetic chiral splitting. Moreover, this competition is intimately connected to spin fluctuations, and can reverse the spin current driven by the band splitting as temperature increases. Our work uncovers the intrinsic competition governing collective spin excitations in altermagnets, providing new insights into their finite-temperature dynamical behavior.

Figures

Figures reproduced from arXiv: 2511.03922 by Congzhe Yan Zhijun Jiang Jinyang Ni, Guoqing Chang.

Figure 1
Figure 1. Figure 1: FIG. 1. The illustration of the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Magnon bands of the monolayer [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Altermagnetic splitting of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The dependence of the ISE/ASE ratio ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 1 linked inside Pith

  1. [1]

    Jungwirth, X

    T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Antiferromagnetic spintronics, Nat. Nanotechnol.11, 231 (2016)

    2016

  2. [2]

    Baltz, A

    V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics, Rev. Mod. Phys.90, 015005 (2018)

    2018

  3. [3]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging re- search landscape of altermagnetism, Phys. Rev. X12, 040501 (2022)

    2022

  4. [4]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X12, 031042 (2022)

    2022

  5. [5]

    Hayami, Y

    S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- dependent spin splitting by collinear antiferromagnetic ordering, J. Phys. Soc. Jpn.88, 123702 (2019)

    2019

  6. [6]

    ˇSmejkal, R

    L. ˇSmejkal, R. Gonz´ alez-Hern´ andez, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets, Sci. Adv.6, eaaz8809 (2020)

    2020

  7. [7]

    I. I. Mazin, K. Koepernik, M. D. Johannes, R. Gonz´ alez- Hern´ andez, and L.ˇSmejkal, Prediction of unconventional magnetism in doped FeSb 2, Proc. Nati. Acad. Sci.118, e2108924118 (2021)

    2021

  8. [8]

    L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y. Mokrousov, and Y. Yao, Altermagnetism: Exploring new frontiers in mag- netism and spintronics, Adv. Funct. Mater.34, 2409327 (2024)

    2024

  9. [9]

    Z. Zhou, X. Cheng, M. Hu, R. Chu, H. Bai, L. Han, J. Liu, F. Pan, and C. Song, Manipulation of the al- termagnetic order in CrSb via crystal symmetry, Nature 638, 645 (2025)

    2025

  10. [10]

    Zhang, H

    Y. Zhang, H. Bai, J. Dai, L. Han, C. Chen, S. Liang, Y. Cao, Y. Zhang, Q. Wang, W. Zhu,et al., Electri- cal manipulation of spin splitting torque in altermagnetic RuO2, Nat. Commun.16, 5646 (2025)

    2025

  11. [11]

    X. Duan, J. Zhang, Z. Zhu, Y. Liu, Z. Zhang, I. ˇZuti´ c, and T. Zhou, Antiferroelectric altermagnets: Antiferro- electricity alters magnets, Phys. Rev. Lett.134, 106801 (2025)

    2025

  12. [12]

    C. Song, H. Bai, Z. Zhou, L. Han, H. Reichlova, J. H. Dil, J. Liu, X. Chen, and F. Pan, Altermagnets as a new class of functional materials, Nat. Rev. Mater. , 1 (2025)

    2025

  13. [13]

    P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Spin-group symmetry in magnetic materials with negligible spin- orbit coupling, Phys. Rev. X12, 021016 (2022)

    2022

  14. [14]

    R. He, D. Wang, N. Luo, J. Zeng, K.-Q. Chen, and L.- M. Tang, Nonrelativistic spin-momentum coupling in an- tiferromagnetic twisted bilayers, Phys. Rev. Lett.130, 046401 (2023)

    2023

  15. [15]

    X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C. Li, and Q. Liu, Enumeration and representa- 6 tion theory of spin space groups, Phys. Rev. X14, 031038 (2024)

    2024

  16. [16]

    Jiang, Z

    Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of spin-space groups: Toward a complete description of symmetries of magnetic orders, Phys. Rev. X14, 031039 (2024)

    2024

  17. [17]

    Z. Xiao, J. Zhao, Y. Li, R. Shindou, and Z.-D. Song, Spin space groups: Full classification and applications, Phys. Rev. X14, 031037 (2024)

    2024

  18. [18]

    Y. Liu, J. Yu, and C.-C. Liu, Twisted magnetic van der Waals bilayers: An ideal platform for altermagnetism, Phys. Rev. Lett.133, 206702 (2024)

    2024

  19. [19]

    P. Das, V. Leeb, J. Knolle, and M. Knap, Realizing al- termagnetism in Fermi-Hubbard models with ultracold atoms, Phys. Rev. Lett.132, 263402 (2024)

    2024

  20. [20]

    V. Leeb, A. Mook, L. ˇSmejkal, and J. Knolle, Sponta- neous formation of altermagnetism from orbital ordering, Phys. Rev. Lett.132, 236701 (2024)

    2024

  21. [21]

    ˇSmejkal, A

    L. ˇSmejkal, A. Marmodoro, K.-H. Ahn, R. Gonz´ alez- Hern´ andez, I. Turek, S. Mankovsky, H. Ebert, S. W. D’Souza, O. ˇSipr, J. Sinova,et al., Chiral magnons in al- termagnetic RuO2, Phys. Rev. Lett.131, 256703 (2023)

    2023

  22. [22]

    Q. Cui, B. Zeng, P. Cui, T. Yu, and H. Yang, Efficient spin Seebeck and spin Nernst effects of magnons in alter- magnets, Phys. Rev. B108, L180401 (2023)

    2023

  23. [23]

    Hoyer, R

    R. Hoyer, R. Jaeschke-Ubiergo, K.-H. Ahn, L. ˇSmejkal, and A. Mook, Spontaneous crystal thermal Hall effect in insulating altermagnets, Phys. Rev. B111, L020412 (2025)

    2025

  24. [24]

    Z. Liu, M. Ozeki, S. Asai, S. Itoh, and T. Masuda, Chiral split magnon in altermagnetic MnTe, Phys. Rev. Lett. 133, 156702 (2024)

    2024

  25. [25]

    Alaei, P

    M. Alaei, P. Sobieszczyk, A. Ptok, N. Rezaei, A. R. Oganov, and A. Qaiumzadeh, Origin of A-type antifer- romagnetism and chiral split magnons in altermagnetic α-MnTe, Phys. Rev. B111, 104416 (2025)

    2025

  26. [26]

    Q. Sun, J. Guo, D. Wang, D. L. Abernathy, W. Tian, and C. Li, Observation of chiral magnon band splitting in altermagnetic hematite, Phys. Rev. Lett.135, 186703 (2025)

    2025

  27. [27]

    Hoyer, P

    R. Hoyer, P. P. Stavropoulos, A. Razpopov, R. Valent ´ ı, L. ˇSmejkal, and A. Mook, Altermagnetic splitting of magnons in hematiteα-Fe 2O3, Phys. Rev. B112, 064425 (2025)

    2025

  28. [28]

    V. C. Morano, Z. Maesen, S. E. Nikitin, J. Lass, D. G. Mazzone, and O. Zaharko, Absence of altermagnetic magnon band splitting in MnF 2, Phys. Rev. Lett.134, 226702 (2025)

    2025

  29. [29]

    J. L. Lado and J. Fern´ andez-Rossier, On the origin of magnetic anisotropy in two dimensional CrI 3, 2D Mate- rials4, 035002 (2017)

    2017

  30. [30]

    ˇSabani, C

    D. ˇSabani, C. Bacaksiz, and M. V. Miloˇ sevi´ c, Beyond orbitally resolved magnetic exchange in CrI 3 and NiI 2, Phys. Rev. Lett.135, 036704 (2025)

    2025

  31. [31]

    A. Mook, K. Plekhanov, J. Klinovaja, and D. Loss, Interaction-stabilized topological magnon insulator in ferromagnets, Phys. Rev. X11, 021061 (2021)

    2021

  32. [32]

    Garcia-Gaitan, A

    F. Garcia-Gaitan, A. Kefayati, J. Q. Xiao, and B. K. Nikoli´ c, Magnon spectrum of altermagnets beyond linear spin wave theory: Magnon-magnon interactions via time- dependent matrix product states versus atomistic spin dynamics, Phys. Rev. B111, L020407 (2025)

    2025

  33. [33]

    R. Eto, M. Gohlke, J. Sinova, M. Mochizuki, A. L. Chernyshev, and A. Mook, Spontaneous magnon decays from nonrelativistic time-reversal symmetry breaking in altermagnets, Phys. Rev. B112, 094442 (2025)

    2025

  34. [34]

    Cichutek, P

    N. Cichutek, P. Kopietz, and A. R¨ uckriegel, Quantum fluctuations in two-dimensional altermagnets, Phys. Rev. B112, 174404 (2025)

    2025

  35. [35]

    Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J

    I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids4, 241 (1958)

    1958

  36. [36]

    Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys

    T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev.120, 91 (1960)

    1960

  37. [37]

    Since DMI interaction is forbidden due to the inversion symmetry, the 2NN ASE tensor is symmetric. The con- servation of⟨S z⟩is directly leads to easy-axis anisotropy, which requires that the off-diagonal elements vanish, J xz 2 =J yz 2 =J xy 2 = 0, and in-plane diagonal elements be degenerate,J xx 2 =J yy 2 . Consequently, the 2NN ASE term can be further...

  38. [38]

    Holstein and H

    T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940)

    1940

  39. [39]

    Zhitomirsky and A

    M. Zhitomirsky and A. Chernyshev, Colloquium: Spon- taneous magnon decays, Rev. Mod. Phys.85, 219 (2013)

    2013

  40. [40]

    J. Ni, Y. Jin, Q. Du, and G. Chang, Magnon nonlinear Hall effect in two-dimensional antiferromagnetic insula- tors, Phys. Rev. B112, 054424 (2025)

    2025

  41. [41]

    for more details about

    See Supplemental Material at ... for more details about

  42. [42]

    N. N. Bogoljubov, V. V. Tolmachov, and D. ˇSirkov, A new method in the theory of superconductivity, Fortschr. Phys.6, 605 (1958)

    1958

  43. [43]

    Valatin, Comments on the theory of superconductivity, Il Nuovo Cimento (1955-1965)7, 843 (1958)

    J. Valatin, Comments on the theory of superconductivity, Il Nuovo Cimento (1955-1965)7, 843 (1958)

    1955

  44. [44]

    Sourounis and A

    K. Sourounis and A. Manchon, Impact of magnon in- teractions on transport in honeycomb antiferromagnets, Phys. Rev. B110, 054429 (2024)

    2024

  45. [45]

    Z. Li, T. Cao, and S. G. Louie, Two-dimensional ferro- magnetism in few-layer van der Waals crystals: Renor- malized spin-wave theory and calculations, J. Magn. Magn. Mater.463, 28 (2018)

    2018

  46. [46]

    V. V. Mkhitaryan and L. Ke, Self-consistently renormal- ized spin-wave theory of layered ferromagnets on the hon- eycomb lattice, Phys. Rev. B104, 064435 (2021)

    2021

  47. [47]

    Wei, J.-J

    B. Wei, J.-J. Zhu, Y. Song, and K. Chang, Renor- malization of gapped magnon excitation in monolayer MnBi2Te4 by magnon-magnon interaction, Phys. Rev. B 104, 174436 (2021)

    2021

  48. [48]

    K. Wu, J. Dong, M. Zhu, F. Zheng, and J. Zhang, Magnon splitting and magnon spin transport in alter- magnets, Chin. Phys. Lett. (2025)

    2025

  49. [49]

    Onose, T

    Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Observation of the magnon Hall effect, Science329, 297 (2010)

    2010

  50. [50]

    Cheng, S

    R. Cheng, S. Okamoto, and D. Xiao, Spin Nernst effect of magnons in collinear antiferromagnets, Phys. Rev. Lett. 117, 217202 (2016)

    2016

  51. [51]

    V. A. Zyuzin and A. A. Kovalev, Magnon spin Nernst effect in antiferromagnets, Phys. Rev. Lett.117, 217203 (2016)

    2016

  52. [52]

    Sicheler, R

    N. Sicheler, R. Raimondi, G. Sangiovanni, and L. D. Re, Optically tunable spin transport in bilayer altermagnetic mott insulators (2025), arXiv:2508.06938

    Pith/arXiv arXiv 2025

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.