pith. sign in

arxiv: 2511.15461 · v2 · pith:7XGB5T2Bnew · submitted 2025-11-19 · ❄️ cond-mat.str-el

On flat bands in the J₁-J₂-J₃ XXZ sawtooth chain

Vadim Ohanyan , Johannes Richter , Michael Sekania , Lucas Giambattista , Alexei Andreanov , Marcus Kollar This is my paper

Pith reviewed 2026-05-22 12:52 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords flat magnon bandssawtooth chainXXZ modelDzyaloshinskii-Moriya interactionslocalized magnonsmagnetoelectric couplingalgebraic constraintsone-magnon spectrum
0
0 comments X

The pith

Algebraic constraints on bond-dependent exchanges produce a flat band in the one-magnon spectrum of the generalized sawtooth XXZ chain.

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

The paper examines a sawtooth spin chain where each triangle carries distinct symmetric Heisenberg exchanges, antisymmetric Dzyaloshinskii-Moriya terms, and axial anisotropies. It derives algebraic relations among these constants that force one branch of the single-magnon energy to become completely independent of momentum. The resulting flat band supports localized magnon states whose wave functions vanish outside specific bonds or plaquettes. The same conditions are mapped onto parameters that can be tuned by an external electric field through magnetoelectric coupling.

Core claim

Distinct values of the three symmetric, three antisymmetric, and three anisotropic exchanges on the bonds of each triangle permit algebraic constraints whose solutions make one eigenvalue of the one-magnon Hamiltonian strictly constant across the Brillouin zone. The corresponding eigenstates are compact localized magnons. These flat-band conditions translate directly onto the Katsura-Nagaosa-Balatsky set of parameters, opening a route to electric-field control.

What carries the argument

Algebraic constraints on the nine bond-dependent exchange constants that nullify the wave-vector dependence of one eigenvalue in the one-magnon Hamiltonian.

If this is right

  • Localized magnon states appear with explicit spatial support confined to individual triangles or bonds.
  • The flat-band conditions map onto the Katsura-Nagaosa-Balatsky parameters, permitting electric-field tuning via magnetoelectric coupling.
  • Electric-field-driven flat bands become realizable in materials exhibiting the required magnetoelectric response.

Where Pith is reading between the lines

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

  • Similar algebraic constraints may be sought in other frustrated lattices that share the same triangle motif.
  • The flat-band states could serve as a starting point for constructing exact many-body eigenstates at higher magnon fillings.
  • Dynamical probes such as neutron scattering would directly reveal the momentum-independent magnon energy if the constraints are met.

Load-bearing premise

The spectrum is fully captured by the one-magnon Hamiltonian without higher-order magnon interactions or extra terms that would lift the flatness.

What would settle it

Exact diagonalization or numerical solution of the full spin Hamiltonian at finite magnon density showing whether the constant-energy band remains dispersionless once magnon-magnon scattering is restored.

Figures

Figures reproduced from arXiv: 2511.15461 by Alexei Andreanov, Johannes Richter, Lucas Giambattista, Marcus Kollar, Michael Sekania, Vadim Ohanyan.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Sawtooth chain with different cou [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Zero-temperature exact diagonaliza [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) The one-magnon spectrum for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Zero-temperature exact diagonaliza [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Zero-temperature magnetization [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Zero-temperature magnetization [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Zero-temperature magnetization plots [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We consider a generalization of the XXZ model on the sawtooth spin chain with Dzyaloshinskii-Moriya interactions in which all exchange constants (symmetric, antisymmetric, and axial anisotropy) are different for the three different bonds of each triangle. We derive and resolve algebraic constraints on the exchange constants ensuring the appearance of a flat band in the one-magnon spectrum. The properties of the corresponding flat magnon bands and localized magnon states are analyzed. We further construct the mapping of the flat-band conditions for the Dzyaloshinskii-Moriya constants onto the Katsura-Nagaosa-Balatsky parameters. Based on the mapping, the possibility of the electric-field-driven flat bands with the aid of the magnetoelectric coupling is examined.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript considers a generalization of the XXZ model on the sawtooth spin chain in which the three distinct bonds per triangular unit cell have independent symmetric exchanges J_i, Dzyaloshinskii-Moriya vectors D_i, and axial anisotropies Δ_i. Algebraic constraints on these parameters are derived and solved so that one eigenvalue of the one-magnon Bloch Hamiltonian becomes strictly k-independent. The resulting flat-band wave functions are shown to be localized, their properties are analyzed, and the flat-band conditions on the DM constants are mapped onto the Katsura-Nagaosa-Balatsky magnetoelectric parameters to explore electric-field control of the flat band.

Significance. The explicit algebraic solution for parameter constraints that enforce a flat one-magnon band supplies a concrete, falsifiable route to engineering flat magnon bands in a frustrated one-dimensional geometry. The mapping to KNB parameters and the discussion of magnetoelectric tuning add a link to possible experimental realizations. Because the result is confined to the single-magnon sector and the derivation is algebraic rather than numerical, the work offers a clean, parameter-free prediction once the constraints are verified.

major comments (2)
  1. §3 (one-magnon Bloch matrix): the explicit 2×2 or 3×3 matrix elements in momentum space and the subsequent polynomial system obtained by nulling all k-dependent coefficients must be written out in full; without these equations the claim that the algebraic constraints have been “resolved” cannot be independently checked.
  2. §4 (localized magnon states): the normalization and orthogonality of the flat-band eigenvectors should be demonstrated explicitly for the solved parameter set, because localization is asserted but the overlap integrals with neighboring unit cells are not shown.
minor comments (2)
  1. A figure labeling the three distinct bonds (J1, J2, J3 and corresponding D, Δ) of the sawtooth unit cell should be added at the beginning of the model section to fix notation.
  2. The mapping from the DM constants to the KNB parameters (Eq. (X)) would be clearer if presented as a short table rather than inline text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments that will improve the clarity and verifiability of the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: §3 (one-magnon Bloch matrix): the explicit 2×2 or 3×3 matrix elements in momentum space and the subsequent polynomial system obtained by nulling all k-dependent coefficients must be written out in full; without these equations the claim that the algebraic constraints have been “resolved” cannot be independently checked.

    Authors: We agree that the explicit matrix elements and the resulting algebraic system are essential for independent verification. In the revised manuscript we will display the full one-magnon Bloch Hamiltonian (a 2×2 matrix for the two-site unit cell of the sawtooth chain) in momentum space, write out the three independent polynomial equations obtained by requiring all k-dependent coefficients of the characteristic polynomial to vanish, and show the step-by-step solution of this system that yields the flat-band constraints on the exchange parameters. revision: yes

  2. Referee: §4 (localized magnon states): the normalization and orthogonality of the flat-band eigenvectors should be demonstrated explicitly for the solved parameter set, because localization is asserted but the overlap integrals with neighboring unit cells are not shown.

    Authors: We appreciate this suggestion. In the revised version we will explicitly construct the flat-band eigenvectors for the solved parameter sets, compute their normalization constants, and evaluate the overlap integrals between a given unit cell and its nearest neighbors, demonstrating that these overlaps vanish identically. This will provide a direct, analytic confirmation of the strict localization of the flat-band states. revision: yes

Circularity Check

0 steps flagged

Algebraic derivation of flat-band conditions is self-contained with no circular reduction

full rationale

The central derivation constructs the one-magnon Bloch matrix for the generalized XXZ sawtooth chain, sets the k-dependent coefficients of the dispersion to zero, and solves the resulting algebraic system for the bond-dependent J, D, and Delta parameters. This procedure directly enforces flatness by construction of the eigenvalue problem and does not rely on fitted inputs, self-citations as load-bearing premises, or imported uniqueness theorems. The subsequent mapping to Katsura-Nagaosa-Balatsky parameters is presented as an explicit re-expression of the already-derived conditions rather than a foundational assumption. No step reduces the claimed result to its own inputs by definition or statistical forcing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard quantum spin Hamiltonian for the XXZ model plus Dzyaloshinskii-Moriya terms and the assumption that the one-magnon sector can be diagonalized independently.

axioms (1)
  • domain assumption The system is described by a nearest-neighbor XXZ Hamiltonian with additional Dzyaloshinskii-Moriya and single-ion anisotropy terms on each bond of the sawtooth lattice.
    Standard starting point for the model; invoked throughout the derivation of the one-magnon spectrum.

pith-pipeline@v0.9.0 · 5686 in / 1346 out tokens · 38151 ms · 2026-05-22T12:52:42.037080+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Electric-field driven flat bands in the distorted sawtooth chain via the Katsura-Nagaosa-Balatsky mechanism

    cond-mat.str-el 2025-11 unverdicted novelty 5.0

    Electric fields induce flat one-magnon bands in the distorted sawtooth chain when DM couplings follow the KNB magnetoelectric form, with explicit field strengths derived for bond-aligned fields.

Reference graph

Works this paper leans on

111 extracted references · 111 canonical work pages · cited by 1 Pith paper

  1. [1]

    As the flat-band constraints are symmetric with respect to permutations ofρ 2 andρ 3, we present here only one of the cases

    andρ 2 ̸=ρ 1 (ρ3 ̸=ρ 1). As the flat-band constraints are symmetric with respect to permutations ofρ 2 andρ 3, we present here only one of the cases. Tak- ing into account the conditionσ= 1, the solution for the positive quantityρ3 is given by: ρ3 = ±ρ1ρ2p J2 1 ∆2 1 −ρ 2 1 +ρ 2 2 ∓J 1∆1 .(3.23) Then the localized states take the following form: |1j⟩=ρ 1e−...

    work page

  2. [2]

    + 2ξ∆1J2J3(1−d 2d3) =J 2 2 J2 3 (1−d 2d3)2 , ξ= sign (J 1J2J3). (3.41) Thus, formally one can conclude that the values ofd2 ≡x andd 3 ≡ycorresponding to the flat band are lying on a quartic planar curve [76, 77]: x2y2 −A 2x2 −B 2y2 −2Cxy−D= 0,(3.42) where A= J1 J3 , B= J1 J2 , C= 1−ξ∆ 1 J2 1 J2J3 , D= J2 1eR J2 2 J2 3 −1, eR=J 2 2 +J 2 3 + 2ξ∆1J2J3.(3.43)...

    work page

  3. [3]

    These flat bands correspond to the formal solution of Eq

    However, in this case one has to fix the value of one of the DM interaction parametersD2 orD 3 in addition to the values ofJa and∆ 1. These flat bands correspond to the formal solution of Eq. (3.42) with respect to eitherx ory: x± = Cy± p C2y2 + (y2 −A 2) (B2y2 +D) y2 −A 2 , y̸=±A,(3.50) or y± = Cx± p C2x2 + (x2 −B 2) (A2x2 +D) x2 −B 2 , x̸=±B.(3.51) Thus...

    work page

  4. [4]

    J2∆2 +J 3∆3 − J2 2 +J 2 3 J # ,(4.66) B0 = 1 2

    DM interactions can be im- plemented via electric field driven flat band [34]. Note the absence of the plateau atM= 0. 0 1 2 3 4 B 0.0 0.2 0.4 0.6 0.8 1.0 M N = 24 N = 28 N = 32 N = 36 FIG. 6. (Color online) Zero-temperature magnetization curves for finite system with uniform exchange couplings, J1 =J 2 =J 3 = 1and∆ 1 = 1, corresponding to the sec- ond li...

    work page

  5. [5]

    Most solutions presented in the Tables I and II do not satisfy the condition (5.87): it can be shown that KNB interpretation is only possible for large negative values ofα

    However, the KNB interpretation is only possible for a <1, leading to θ= arctan √ 1−a 2 |a| , E=± 1 |a| . Most solutions presented in the Tables I and II do not satisfy the condition (5.87): it can be shown that KNB interpretation is only possible for large negative values ofα. The case from the Table II withα=−6which is illustrated in Fig. 7 (J1 =J 2 =J ...

    work page

  6. [6]

    21AG-1C047 and 23AA-1C032)

    and CS RA MESCS (Grants No. 21AG-1C047 and 23AA-1C032). V.O. and M.K. acknowledge partial fund- ing by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –TRR360 – 492547816. This work was supported by the Institute for Basic Science, Project Code (Project No. IBS-R024-D1). [1]Introduction to Frustrated Magnetism. Materials, Ex- periments, ...

    work page 2011

  7. [7]

    Kubo, Phys

    K. Kubo, Phys. Rev. B48, 10552 (1993)

    work page 1993

  8. [8]

    Nakamura, K

    T. Nakamura, K. Kubo, Phys. Rev. B53, 6393 (1996)

    work page 1996

  9. [9]

    D. Sen, B. S. Shastry, R. E. Walsted, and R. J. Cava, Phys. Rev. B53, 6401 (1996)

    work page 1996

  10. [10]

    Rousochatzakis, J

    Zhihao Hao, Yuan Wan, I. Rousochatzakis, J. Wilde- boer, A. Seidel, F. Mila, and O. Tchernyshyov, Phys. Rev. B84, 094452 (2011)

    work page 2011

  11. [11]

    S. A. Blundell, and M. D. Núñez-Regueiro, Eur. Phys. J. B31, 453 (2003)

    work page 2003

  12. [12]

    Kaburagi, J

    T, Tonegawa, and M. Kaburagi, J. Magn. Magn. Mater. 272-276, 898 (2004)

    work page 2004

  13. [13]

    Kaburagi, T

    M. Kaburagi, T. Tonegawa, and M. Kang, J. Appl. Phys.97, 10B306 (2005)

    work page 2005

  14. [14]

    Jian-Jun Jiang, Yong-Jun Liu, Fei Tang, Cui-Hong Yang, Yu-Bo Sheng, Physica B463, 30 (2015)

    work page 2015

  15. [15]

    Paul, and A

    S. Paul, and A. K. Ghosh, Eur. Phys. J. B92, 40 (2019)

    work page 2019

  16. [16]

    V. Ya. Krivnov, D. V. Dmitriev, S. Nishimoto, S.- L. Drechsler, and J. Richter, Phys. Rev. B90, 014441 (2014)

    work page 2014

  17. [17]

    D. V. Dmitriev and V. Ya. Krivnov, Phys. Rev. B92, 184422 (2015)

    work page 2015

  18. [18]

    D. V. Dmitriev and V. Ya. Krivnov, J. Phys.: Condens. Matter28, 506002 (2016)

    work page 2016

  19. [19]

    V. Ya. Krivnov, D. V. Dmitriev, J. Richter, and J. Schnack, Phys. Rev. B99, 094410 (2019)

    work page 2019

  20. [20]

    D. V. Dmitriev, V. Ya. Krivnov, J. Schnack, and J. Richter, Phys. Rev. B101, 054427 (2020)

    work page 2020

  21. [21]

    Rausch, M

    R. Rausch, M. Peschke, C. Plorin, J. Schnack, and C. Karrasch, SciPost Phys.14, 052 (2023)

    work page 2023

  22. [22]

    Rausch and C

    R. Rausch and C. Karrasch, Phys. Rev. B111, 045154 (2025)

    work page 2025

  23. [23]

    Ohanyan, Condens

    V. Ohanyan, Condens. Matter Phys.12, 343 (2009)

    work page 2009

  24. [24]

    Bellucci and V

    S. Bellucci and V. Ohanyan, Eur. Phys. J. B75, 531 (2010)

    work page 2010

  25. [25]

    Bellucci, and V

    S. Bellucci, and V. Ohanyan, Eur. Phys. J. B86, 446 (2013). 17

    work page 2013

  26. [26]

    Schmidt, Phys

    J.Schulenburg, A.Honecker, J.Schnack, J.Richter, and H.-J. Schmidt, Phys. Rev. Lett.88, 167207 (2002)

    work page 2002

  27. [27]

    Richter, J

    J. Richter, J. Schulenburg, A. Honecker, J. Schanck, and H.-J. Schmidt, J. Phys.: Condens. Matter16, S779 (2004)

    work page 2004

  28. [28]

    Richter, Low Tepm

    J. Richter, Low Tepm. Phys.31, 695 (2005)

    work page 2005

  29. [29]

    Zhitomirsky, H

    M.E. Zhitomirsky, H. Tsunetsugu, Phys. Rev. B70, 100403(R) (2004)

    work page 2004

  30. [30]

    Derzhko, J

    O. Derzhko, J. Richter, Phys. Rev. B70, 104415 (2004)

    work page 2004

  31. [31]

    Derzhko, J

    O. Derzhko, J. Richter, Eur. Phys. J. B52, 23 (2006)

    work page 2006

  32. [32]

    Derzhko, J

    O. Derzhko, J. Richter, A. Honecker, and H.-J. Schmidt, Low Temp. Phys.33, 745 (2007)

    work page 2007

  33. [33]

    Richter, O

    J. Richter, O. Derzhko, A. Honecker, Int. J. Mod. Phys. B22, 4418 (2008)

    work page 2008

  34. [34]

    Derzhko, J

    O. Derzhko, J. Richter, M. Maksymenko, Int. J. Mod. Phys. B29, 1530007 (2015)

    work page 2015

  35. [35]

    Metavitsiadis, C

    A. Metavitsiadis, C. Psaroudaki, and W. Brenig, Phys. Rev. B101, 235143 (2020)

    work page 2020

  36. [36]

    Derzhko, J

    O. Derzhko, J. Schnack, D. V. Dmitriev, V. Ya. Krivnov, J. Richter, Eur. Phys. J. B93, 161 (2020)

    work page 2020

  37. [37]

    Richter, J

    J. Richter, J. Schulenburg, D. V. Dmitriev, V. Ya. Krivnov, J. Schnack, Condens, Matter Phys.23, 43713 (2020)

    work page 2020

  38. [38]

    Acevedo, P

    S. Acevedo, P. Pujol, and C.A. Lamas, Phys. Rev. B 102, 195139 (2020)

    work page 2020

  39. [39]

    Richter, V

    J. Richter, V. Ohanyan, J. Schulenburg, and J. Schnack, Phys. Rev. B105, 054420 (2022)

    work page 2022

  40. [40]

    Johannesmann, J

    F. Johannesmann, J. Eckseler, H. Schlüter, and J. Schnack, Phys. Rev. B108, 064304 (2023)

    work page 2023

  41. [41]

    Reichert, H

    N. Reichert, H. Schlüter, T. Heitmann, J. Richter, R. Rausch, and J. Schnack, Zeitschrift für Nat. (2024)

    work page 2024

  42. [42]

    Eckseler, and J

    J. Eckseler, and J. Schnack, Phys. Rev. Research7, 013178 (2025)

    work page 2025

  43. [43]

    M. E. Zhitomirsky and H. Tsunetsugu, Prog. Theor. Phys. Suppl.160, 361 (2005)

    work page 2005

  44. [44]

    Schnack, J

    J. Schnack, J. Schulenburg, A. Honecker, and J. Richter Phys. Rev. Lett.125, 117207 (2020)

    work page 2020

  45. [45]

    Richter, O

    J. Richter, O. Derzhko, and J. Schulenburg, Phys. Rev. Lett.93, 107206 (2004)

    work page 2004

  46. [46]

    M. E. Zhitomirsky and A. Honecker, J. Stat. Mech. (2004) P07012

    work page 2004

  47. [47]

    Schmidt, J

    H.-J. Schmidt, J. Richter, and R. Moessner, J. Phys. A.: Math. Gen. 39, 10673 (2006)

    work page 2006

  48. [48]

    Heinze, H

    L. Heinze, H. O. Jeschke, I. I. Mazin, A. Metavit- siadis, M. Reehuis, R. Feyerherm, J.-U. Hoffmann, M. Bartkowiak, O. Prokhnenko, A. U. B. Wolter, X. Ding, V. S. Zapf, C. Corvalán Moya, F. Weickert, M. Jaime, K. C. Rule, D. Menzel, R. Valentí, W. Brenig, S. Süllow, Phys. Rev. Lett.126, 207201 (2021)

    work page 2021

  49. [49]

    Heinze, T

    L. Heinze, T. Kotte, R. Rausch, A. Demuer, S. Luther, R. Feyerherm, E. L. Q. N. Ammerlaan, U. Zeitler, D. I. Gorbunov, M. Uhlarz, K. C. Rule, A. U. B. Wolter, H. Kühne, J. Wosnitza, C. Karrasch, S. Süllow, Phys. Rev. Lett.134, 216701 (2025)

    work page 2025

  50. [50]

    J. L. Allen, L. Heinze, R. A. Mole, S. Süllow, O. Jan- son, S. Nishimoto, R. A. Lewis, and K. C. Rule,The Magnetic Ground State of Atacamite Cu 2Cl(OH)3: The Crucial Role of Frustrated Zigzag Chains Revealed by Inelastic Neutron Scattering, arXiv:2508.02201 (2025)

    work page arXiv 2025

  51. [51]

    Kikuchi, Y

    H. Kikuchi, Y. Fujii, D. Takahashi, M. Azuma, Y. Shi- makawa, T. Taniguchi, A. Matsuo, and K. Kindo, J. Phys.: Conf. Ser.320, 012045 (2011)

    work page 2011

  52. [52]

    V. O. Garlea, L. D. Sanjeewa, M. A. McGuire, P. Ku- mar, D. Sulejmanovic, J. He, S.-J Hwu, Phys. Rev. B 89, 014426 (2014)

    work page 2014

  53. [53]

    V. P. Gnezdilov, Yu. G. Pashkevich, V. S. Kurnosov, O. V. Zhuravlev, D. Wulferding, P. Lemmens, D. Menzel, E. S. Kozlyakova, A. Yu. Akhrorov, E. S. Kuznetsova, P. S. Berdonosov, V. A. Dolgikh, O. S. Volkova, and A. N. Vasiliev, Phys. Rev. B99, 064413 (2019)

    work page 2019

  54. [54]

    Nhalil, R

    H. Nhalil, R. Baral, B. O. Khamala, A. Cosio, S. R. Singamaneni, M. Fitta, D. Antonio, K. Gofryk, R. R. Zope, T. Baruah, B. Saparov, and H. S. Nair, Phys. Rev. B99, 184434 (2019)

    work page 2019

  55. [55]

    Feihao Pan, Xunwu Hu, Jiale Huang, Bingxian Shi, Jinchen Wang, Juanjuan Liu, Hongxia Zhang, Daye Xu, Hongliang Wang, Lijie Hao, Peng Cheng, Dao-Xin Yao, Phys. Rev. B107, 224423 (2023)

    work page 2023

  56. [56]

    H. C. Mandujano, A. Metta, N. Barišić, Qiang Zhang, W. Tabiś, N. K. C. Muniraju, and H. S. Nair, Phys. Rev. Materials7, 024422 (2023)

    work page 2023

  57. [57]

    Oshima, H

    Y. Oshima, H. Nojiri, J. Schnack, P. Kögerler, and M. Luban, Phys. Rev. B85, 024413 (2012)

    work page 2012

  58. [58]

    Baniodeh, N

    A. Baniodeh, N. Magnani, Y. Lan, G. Buth, C. E. An- son, J. Richter, M. Affronte, J. Schnack, and A. K. Pow- ell, npj Quantum Mater.3, 10 (2018)

    work page 2018

  59. [59]

    Katsura, N

    H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett.95, 057205 (2005)

    work page 2005

  60. [60]

    C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B74, 224444 (2006)

    work page 2006

  61. [61]

    Solovyev, R

    I. Solovyev, R. Ono, and S. Nikolaev, Phys. Rev. Lett. 127, 187601 (2021)

    work page 2021

  62. [62]

    I. V. Solovyev, Condens. Matter10, 21 (2025)

    work page 2025

  63. [63]

    Khomskii, Physics2, 20 (2009)

    D. Khomskii, Physics2, 20 (2009)

    work page 2009

  64. [64]

    Cheong and M

    S.-W. Cheong and M. Mostovoy, Nature Materials6, 13 (2007)

    work page 2007

  65. [65]

    Erenstein, N

    W. Erenstein, N. D. Mathur, and J. F. Scott, Nature 442, 760 (2006)

    work page 2006

  66. [66]

    Tokura and S

    Y. Tokura and S. Seki, Adv. Mater.22, 1554 (2010)

    work page 2010

  67. [67]

    Tokura, S

    Y. Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77, 076501 (2014)

    work page 2014

  68. [68]

    Brockmann, A

    M. Brockmann, A. Klümper, and V. Ohanyan, Phys. Rev. B87, 054407 (2013)

    work page 2013

  69. [69]

    Thakur, P

    P. Thakur, P. Durganandini, Phys. Rev. B97, 064413 (2018)

    work page 2018

  70. [70]

    Yi, W.-L

    T.-Ch. Yi, W.-L. You, N. Wu, and A. M. Oleś, Phys. Rev. B100, 024423 (2019)

    work page 2019

  71. [71]

    Ohanyan, Condens

    V. Ohanyan, Condens. Matter Phys.23, 43704 (2020)

    work page 2020

  72. [72]

    Menchyshyn, V

    O. Menchyshyn, V. Ohanyan, T. Verkholyak, T. Krokhmalskii, and O. Derzhko, Phys. Rev. B 92, 14427 (2015)

    work page 2015

  73. [73]

    Sznajd, Phys

    J. Sznajd, Phys. Rev. B97, 214410 (2018)

    work page 2018

  74. [74]

    Sznajd, J

    J. Sznajd, J. Magn. Magn. Mater.479, 254 (2019)

    work page 2019

  75. [75]

    Baran, V

    O. Baran, V. Ohanyan, and T. Verkholyak, Phys. Rev. B98, 064415 (2018)

    work page 2018

  76. [76]

    You, G.-H

    W.-L. You, G.-H. Liu, P. Horsch, and A. M. Oleś, Phys. Rev. B90, 094413 (2014)

    work page 2014

  77. [77]

    Strečka, L

    J. Strečka, L. Gálisová, and T. Verkholyak, Phys. Rev. E101, 012103 (2020)

    work page 2020

  78. [78]

    D. C. Cabra, A. O. Dobry, C. J. Gazza, and G. L. Rossini, Phys. Rev. B100, 161111(R) (2019)

    work page 2019

  79. [79]

    Baran, Ukr

    O. Baran, Ukr. J. Phys.66, 890 (2021)

    work page 2021

  80. [80]

    D. V. Dmitriev and V. Ya. Krivnov, J. Phys.: Condens. Matter30, 385803 (2016)

    work page 2016

Showing first 80 references.