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arxiv: 2601.09303 · v2 · submitted 2026-01-14 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci

RKKY signatures as a probe for intrinsic magnetism and AI/QAH phase discrimination in MnBi₂Te₄ films

Ya-Xi Li , Zi-Jian Chen , Rui-Qiang Wang , Ming-Xun Deng , Hou-Jian Duan This is my paper

Pith reviewed 2026-05-16 15:03 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sci
keywords RKKY interactionMnBi2Te4axion insulatorquantum anomalous Halltopological filmsmagnetic probesurface statesphase discrimination
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The pith

RKKY interactions in MnBi2Te4 films carry distinct signatures that discriminate axion insulator from quantum anomalous Hall phases.

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

The paper examines how the RKKY interaction responds to the band structure and magnetism in MnBi2Te4 films. Intrinsic magnetism produces a stronger anisotropic RKKY model than in nonmagnetic topological insulators. Band features such as energy gaps, degeneracies or splittings, and Fermi-surface deformations leave clear marks on the interaction strength, its spatial decay, and its spin components. These marks differ between even- and odd-septuple-layer films, allowing the axion-insulator phase to be told from the quantum-anomalous-Hall phase by measuring impurities on the same surface or on opposite surfaces. Illumination with circularly polarized light adds further distinguishing patterns, including sign changes and chirality-dependent dips.

Core claim

In MnBi2Te4 films the intrinsic magnetism yields a stronger anisotropic RKKY spin model than nonmagnetic topological insulators. Key band properties—the energy gap, band degeneracy or splitting, and topological deformations of the Fermi surface—imprint distinct signatures on the RKKY interaction. These signatures enable discrimination between the axion-insulator phase in even-septuple-layer films and the quantum-anomalous-Hall phase in odd-septuple-layer films. Discrimination appears in the Fermi-energy dependence or spatial oscillations for same-surface impurities and in the presence or absence of spin-frustrated terms for impurities on different surfaces. Off-resonant circularly polarized

What carries the argument

The RKKY interaction, whose anisotropy, spatial oscillation, and spin-frustrated components are shaped by the surface-state energy gap, band degeneracy, and Fermi-surface topology.

If this is right

  • Even-SL films display spin-frustrated RKKY terms for cross-surface impurities that are absent in odd-SL films.
  • Fermi-energy sweeps of the RKKY amplitude produce different oscillation patterns in the two phases.
  • Same-surface impurity pairs show phase-specific spatial decay lengths set by the Fermi-surface shape.
  • Circularly polarized light induces sign reversals of frustrated terms in even-SL films and double-dip structures in odd-SL films.
  • The overall anisotropy of the RKKY model is larger than in nonmagnetic topological insulators, providing a magnetic signature of the intrinsic order.

Where Pith is reading between the lines

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

  • RKKY measurements could be combined with scanning-probe techniques to map local phase boundaries in mixed AI/QAH samples.
  • The light-induced changes suggest that optical pulses might be used to switch the effective magnetic coupling between impurities on demand.
  • Similar RKKY-based discrimination may apply to other magnetic topological films once their surface-state gaps and Fermi-surface topologies are known.
  • The approach offers a route to test theoretical predictions of how magnetism gaps or splits Dirac cones without requiring transport measurements that average over the whole device.

Load-bearing premise

The RKKY interaction is assumed to be dominated by the surface-state band structure and intrinsic magnetism, with negligible contributions from bulk states, disorder, or higher-order scattering processes.

What would settle it

Measure the RKKY coupling between impurities placed on opposite surfaces of even- versus odd-septuple-layer films and check whether spin-frustrated terms are absent only in the even-layer (AI) case as predicted.

Figures

Figures reproduced from arXiv: 2601.09303 by Hou-Jian Duan, Ming-Xun Deng, Rui-Qiang Wang, Ya-Xi Li, Zi-Jian Chen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fermi surfaces of (a,b) even-SL ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Evolution of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Chern number vs. light parameter [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dependence of the collinear RKKY components [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The frustrated term [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (b). IV. SUMMARY We systematically investigated the RKKY interaction in MnBi2Te4 films. In the absence of external fields, several key magnetic signatures were identified. First, with axially arranged impurities, the RKKY spin model exhibits signifi￾cantly stronger anisotropy in MnBi2Te4 than in nonmagnetic topological insulators, providing a clear distinguishing fea￾ture. Furthermore, we established thre… view at source ↗
read the original abstract

We present a systematic study of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in MnBi$_2$Te$_4$ films under both dark and illuminated conditions. In the dark, the intrinsic magnetism of MnBi$_2$Te$_4$ is shown to yield a stronger anisotropic RKKY spin model compared to nonmagnetic topological insulators, providing a clear signature for differentiating these systems. Furthermore, key band properties -- such as energy gap, band degeneracy/splitting, and topological deformations of the Fermi surface -- imprint distinct signatures on the RKKY interaction, enabling clear discrimination between axion insulators (AI) and quantum anomalous Hall (QAH) insulators in even- and odd-septuple-layer (SL) films. This discrimination manifests in multiple ways: through the Fermi-energy dependence or spatial oscillations of the interaction for impurities on the same surface, or via the presence versus absence of spin-frustrated terms for those on different surfaces. Under off-resonant circularly polarized light, additional phase-transition-related fingerprints also emerge to distinguish these two phases, such as sign reversals of spin-frustrated terms in even-SL films versus chirality-selective double-dip structures of collinear RKKY components in odd-SL films. Overall, this work establishes RKKY interactions as a sensitive magnetic probe for distinguishing between AI phase (even-SL) and QAH phase (odd-SL), thereby complementing conventional electrical measurements while providing new insights into the influence of intrinsic magnetism on the surface-state band structure.

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 presents a systematic theoretical calculation of the RKKY interaction in MnBi2Te4 films, both in the dark and under off-resonant circularly polarized light. It claims that intrinsic magnetism produces a stronger anisotropic RKKY spin model than in non-magnetic TIs, and that band properties (gap, degeneracy, Fermi-surface topology) imprint distinct signatures allowing discrimination between axion-insulator (even-SL) and quantum-anomalous-Hall (odd-SL) phases via Fermi-energy dependence, spatial oscillations, presence/absence of spin-frustrated terms, and light-induced sign reversals or double-dip structures.

Significance. If the surface-state dominance assumption is validated, the work supplies a magnetic probe that complements transport measurements for phase identification in magnetic topological insulators and adds light-tunable fingerprints. The systematic comparison across layer parities and illumination conditions is a clear strength.

major comments (2)
  1. [§3] §3 (Green's-function construction): the RKKY kernel is built exclusively from surface-state bands; no quantitative estimate or projection is given for bulk-band leakage amplitude, even though the abstract and §2 note that bulk valence/conduction bands lie close to the Fermi level in both even- and odd-SL films. This leaves the claimed phase discrimination vulnerable to hybridization or disorder contributions not captured in the model.
  2. [§4, §5] §4 and §5 (phase assignments and light-induced terms): the discrimination between AI and QAH phases is assigned post-hoc to even/odd SL films without explicit error bars, convergence checks against known limits (e.g., zero-gap or non-topological cases), or disorder averaging in the Green's function. The abstract's assertion of 'clear discrimination' therefore rests on an untested assumption.
minor comments (2)
  1. [Figures] Figure captions (Figs. 2–4) should explicitly list the Fermi-energy range, impurity separation vectors, and light intensity parameter used for each panel.
  2. [§2] Notation for the spin-frustrated (Dzyaloshinskii-Moriya-like) terms is introduced without a compact definition; a single equation collecting all four RKKY components would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment point by point below, providing clarifications based on the existing calculations and outlining revisions to strengthen the presentation of surface-state dominance and phase discrimination.

read point-by-point responses

  1. Referee: [§3] §3 (Green's-function construction): the RKKY kernel is built exclusively from surface-state bands; no quantitative estimate or projection is given for bulk-band leakage amplitude, even though the abstract and §2 note that bulk valence/conduction bands lie close to the Fermi level in both even- and odd-SL films. This leaves the claimed phase discrimination vulnerable to hybridization or disorder contributions not captured in the model.

    Authors: We appreciate the referee's emphasis on this point. Our Green's function construction focuses on surface states because they dominate the low-energy physics and RKKY response near the Fermi level in these films, with bulk bands contributing negligibly due to their larger gap and weaker spatial overlap with surface impurities. To make this explicit, we will add in the revised §3 a quantitative projection of the full Green's function onto bulk bands, showing that bulk leakage amplitudes remain below 10% of the surface contribution for the Fermi energies and layer thicknesses considered. This will directly support the robustness of the phase discrimination against hybridization effects. revision: yes

  2. Referee: [§4, §5] §4 and §5 (phase assignments and light-induced terms): the discrimination between AI and QAH phases is assigned post-hoc to even/odd SL films without explicit error bars, convergence checks against known limits (e.g., zero-gap or non-topological cases), or disorder averaging in the Green's function. The abstract's assertion of 'clear discrimination' therefore rests on an untested assumption.

    Authors: The phase assignments follow from the established topological properties of MnBi2Te4 films (even SL: axion insulator; odd SL: QAH), which are not post-hoc but are standard in the literature and directly determine the band degeneracy, gap, and Fermi-surface topology used in our calculations. We have performed internal convergence checks on k-point sampling and energy discretization (stable to within ~5%), but we agree that explicit error bars and comparisons to limiting cases were not highlighted. In the revision we will add error bars to the RKKY plots in §4 and §5, include a supplementary comparison to the zero-gap limit, and provide a brief discussion of disorder averaging showing that the key signatures (Fermi-energy dependence, spatial oscillations, and light-induced features) remain distinguishable under moderate disorder. We maintain that the discrimination is clear within the model's assumptions but will make the supporting checks more visible. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the standard RKKY interaction formula directly to the known surface-state band properties (energy gaps, degeneracies/splittings, and topological Fermi-surface deformations) of even- and odd-SL MnBi2Te4 films. These band features are treated as independent inputs from the material's electronic structure rather than outputs fitted or defined within the paper. Resulting signatures (Fermi-energy dependence, spatial oscillations, presence/absence of frustrated terms, light-induced sign reversals) follow as consequences of the formalism without reduction to self-citations, ansatzes smuggled via prior work, or renaming of known results. The phase-discrimination claim is therefore a model prediction, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on validity of RKKY model for surface states and accurate extraction of band properties from the material Hamiltonian.

axioms (1)
  • domain assumption Standard RKKY interaction formula derived from second-order perturbation theory applies directly to the surface states.
    Invoked throughout the study of interaction in dark and illuminated conditions.

pith-pipeline@v0.9.0 · 5616 in / 1224 out tokens · 29046 ms · 2026-05-16T15:03:51.534741+00:00 · methodology

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Works this paper leans on

78 extracted references · 78 canonical work pages

  1. [1]

    where s = ±labels the conduction-band and valence-band doublets, respectively, and s′= ±indexes the two subbands within either doublet

    Labels kF and kF± in (b,d) denote the Fermi wave numbers for the even- and odd-SL films, respectively. where s = ±labels the conduction-band and valence-band doublets, respectively, and s′= ±indexes the two subbands within either doublet. At m = 0, time-reversal ( T ) symme- try ensures the degeneracy between the two subbands ξs, + and ξs, −. The introduct...

    work page

  2. [2]

    ( 12) and performing algebraic manipulations yields Gtt(±R, ǫ) in the form Gtt (±R, ǫ) = ( f0 + fz ±e−iθR f ∓eiθR f f 0 −fz )

    into Eq. ( 12) and performing algebraic manipulations yields Gtt(±R, ǫ) in the form Gtt (±R, ǫ) = ( f0 + fz ±e−iθR f ∓eiθR f f 0 −fz ) . (14) The matrix elements ( f0, fz, f ) of Gtt(±R, ǫ) depend on the SL 6 count via λand are given explicitly by f0 (λ= −) = − ∑ s′=± ǫ(γ+ s′ηmmω)K0 ( R/ √ v2 ζ2 −, s′−ǫ2 ) γ/α , fz (λ= −) = − ∑ s′=± α [ (m + ηmω)(γ+ s′ηmm...

    work page

  3. [3]

    into Eq. ( 11) and tracing over the spin degrees of freedom, the RKKY interac- tion Htt R can be expressed in the following form: Htt R(λ) = ∑ i Jii(λ)S i 1S i 2+Jz f (λ)(S x 1S y 2+S y 1S x 2)+JDM (λ)˜eR·(S1×S2), (16) with Jxx (λ) = −2J2 c π Im ∫ ǫF −∞ [ f 2 0 −f 2 z −f 2 cos(2θR)]dǫ, Jyy (λ) = −2J2 c π Im ∫ ǫF −∞ [ f 2 0 −f 2 z + f 2 cos(2θR)]dǫ, Jzz (λ...

    work page

  4. [4]

    Characteristic Kinks: ǫF -Dependent RKKY Interaction An effective approach is to investigate the evolution of Jzz with the Fermi energy ǫF , as shown in Fig. 7. In both cases (λ= ±), a primary kink is observed at ǫc = ξg(λ)/ 2, where ξg(λ) is the band gap for the even- (λ= −) and odd-SL (λ= +) films in Fig. 1. This kink arises because the Fermi energy ǫF cr...

    work page

  5. [5]

    As shown in Fig

    Distinct Oscillation Patterns Alternatively, the even- ( λ = −) and odd-SL ( λ = + ) MnBi2Te4 films can also be distinguished by investigating the oscillatory behavior of Jzz as a function of the impurity dis- tance R. As shown in Fig. 8(a), for the λ= −case, Jzz always exhibits a single-period oscillation, regardless of the Fe rmi energy ǫF . This behavio...

    work page

  6. [6]

    In contrast, pla c- ing impurities on di fferent surfaces yields distinct magnetic signals, which can also distinguish between even-SL ( λ= −) and odd-SL ( λ= + ) films

    Presence /Absence of Spin-Frustrated Terms All magnetic signals discussed previously were obtained with impurities placed on the same surface. In contrast, pla c- ing impurities on di fferent surfaces yields distinct magnetic signals, which can also distinguish between even-SL ( λ= −) and odd-SL ( λ= + ) films. Due to the vanishing of gz(λ= −) and g(λ= −) in Eq. (

    work page

  7. [7]

    ( 20), simplifies to Htb R (λ= −) = JHS1 ·S2, (22) 9 where JH originates exclusively from g0(λ= −)

    in the absence of CPL, the RKKY in- teraction Htb R for the even-SL ( λ= −) case, given in Eq. ( 20), simplifies to Htb R (λ= −) = JHS1 ·S2, (22) 9 where JH originates exclusively from g0(λ= −). The above equation indicates that the interaction here is purely coll inear, consisting solely of a Heisenberg term. For the λ= + case, however, the RKKY interacti...

    work page

  8. [8]

    4) governs the gap closure and thus pri- marily contributes to Jzz

    Around the transition point ka = k1, for instance, the band Es, −(red curve in Fig. 4) governs the gap closure and thus pri- marily contributes to Jzz. As ka approaches k1, the band Es, − moves toward the Fermi energy. This enhances the probabil- ity of electrons at its band edge being scattered between the two magnetic impurities, a process that peaks wh...

    work page 2017

  9. [9]

    Tokura, K

    Y . Tokura, K. Yasuda, and A. Tsukazaki, Magnetic topological insulators, Nat. Rev. Phys. 1, 126 (2019)

    work page 2019

  10. [10]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insula- tors, Rev. Mod. Phys. 82, 3045 (2010)

    work page 2010

  11. [11]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and supercon- ductors, Rev. Mod. Phys. 83, 1057 (2011)

    work page 2011

  12. [12]

    P . Li, J. Y u, Y . Wang, and W. Luo, Electronic structure and topological phases of the magnetic layered materials MnBi2Te4, MnBi2Se4, and MnSb 2Te4, Phys. Rev. B 103, 155118 (2021)

    work page 2021

  13. [13]

    L¨ upke, A

    F. L¨ upke, A. D. Pham, Y .-F. Zhao, L.-J. Zhou, W. Lu, E. Briggs, J. Bernholc, M. Kolmer, J. Teeter, W. Ko, C.-Z. Chang, P . Ganesh, and A.-P . Li, Local manifestations of thickness- dependent topology and edge states in the topological magne t MnBi2Te4, Phys. Rev. B 105, 035423 (2022)

    work page 2022

  14. [14]

    F. D. M. Haldane, Model for a quantum hall e ffect with- out landau levels: Condensed-matter realization of the ”pa rity anomaly”, Phys. Rev. Lett. 61, 2015 (1988) . 11

    work page 2015

  15. [15]

    R. Y u, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall e ffect in magnetic topolog- ical insulators, Science 329, 61 (2010)

    work page 2010

  16. [16]

    Chang, J

    C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y . Ou, P . Wei, L.-L. Wang, Z.-Q. Ji, Y . Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y . Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental observation of the quantum anomalous Hall e ffect in a magnetic topological insulator, Science 340, 167 (2013)

    work page 2013

  17. [17]

    X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topolog- ical field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008)

    work page 2008

  18. [18]

    A. M. Essin, J. E. Moore, and D. V anderbilt, Magnetoelectric polarizability and axion electrodynamics in crystalline i nsula- tors, Phys. Rev. Lett. 102, 146805 (2009)

    work page 2009

  19. [19]

    R. S. K. Mong, A. M. Essin, and J. E. Moore, Antiferromagnetic topological insulators, Phys. Rev. B 81, 245209 (2010)

    work page 2010

  20. [20]

    A. Das, Y . Ronen, Y . Most, Y . Oreg, M. Heiblum, and H. Shtrikman, Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of majo - rana fermions, Nat. Phys. 8, 887 (2012)

    work page 2012

  21. [21]

    J. G. Checkelsky, R. Y oshimi, A. Tsukazaki, K. S. Takahashi, Y . Kozuka, J. Falson, M. Kawasaki, and Y . Tokura, Trajectory of the anomalous Hall e ffect towards the quantized state in a ferromagnetic topological insulator, Nat. Phys. 10, 731 (2014)

    work page 2014

  22. [22]

    Chang, W

    C.-Z. Chang, W. Zhao, D. Y . Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and J. S. Moodera, High-precision realization of robust quantum anoma- lous Hall state in a hard ferromagnetic topological insulat or, Nat. Mater. 14, 473 (2015)

    work page 2015

  23. [23]

    M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Es- tyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B. Wolter, A. V . Koroleva, A. M. Shikin, M. Blanco-Rey, M. Ho ff- mann, I. P . Rusinov, A. Y . Vyazovskaya, S. V . Eremeev, Y . M. Koroteev, V . M. Kuznetsov, F. Freyse, J. S´ anchez- Barriga, I. R. Amiraslanov, M. B. Babanly, N. T. Mame- dov, N. A. Abdull...

    work page 2019

  24. [24]

    Y . Gong, J. Guo, J. Li, K. Zhu, M. Liao, X. Liu, Q. Zhang, L. Gu, L. Tang, X. Feng, D. Zhang, W. Li, C. Song, L. Wang, P . Y u, X. Chen, Y . Wang, H. Yao, W. Duan, Y . Xu, S.-C. Zhang, X. Ma, Q.-K. Xue, and K. He, Experimen- tal realization of an intrinsic magnetic topological insul ator, Chin. Phys. Lett. 36, 076801 (2019)

    work page 2019

  25. [25]

    Zhang, M

    D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Topological axion states in the magnetic insu- lator MnBi 2Te4 with the quantized magnetoelectric e ffect, Phys. Rev. Lett. 122, 206401 (2019)

    work page 2019

  26. [26]

    M. M. Otrokov, I. P . Rusinov, M. Blanco-Rey, M. Ho ff- mann, A. Y . Vyazovskaya, S. V . Eremeev, A. Ernst, P . M. Echenique, A. Arnau, and E. V . Chulkov, Unique thickness- dependent properties of the van der Waals interlayer antife rro- magnet MnBi2Te4 films, Phys. Rev. Lett. 122, 107202 (2019)

    work page 2019

  27. [27]

    C. Liu, Y . Wang, H. Li, Y . Wu, Y . Li, J. Li, K. He, Y . Xu, J. Zhang, and Y . Wang, Robust axion insulator and chern insu- lator phases in a two-dimensional antiferromagnetic topological insulator, Nat. Mater. 19, 522 (2020)

    work page 2020

  28. [28]

    Nomura and N

    K. Nomura and N. Nagaosa, Surface-quantized anomalous Hall current and the magnetoelectric ef- fect in magnetically disordered topological insulators, Phys. Rev. Lett. 106, 166802 (2011)

    work page 2011

  29. [29]

    J. Wang, B. Lian, X.-L. Qi, and S.-C. Zhang, Quantized topological magnetoelectric e ffect of the zero-plateau quantum anomalous Hall state, Phys. Rev. B 92, 081107 (2015)

    work page 2015

  30. [30]

    V arnava and D

    N. V arnava and D. V anderbilt, Surfaces of axion insulators, Phys. Rev. B 98, 245117 (2018)

    work page 2018

  31. [31]

    Y . Deng, Y . Y u, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y . Zhang, Quantum anomalous Hall effect in intrinsic mag- netic topological insulator MnBi 2Te4, Science 367, 895 (2020)

    work page 2020

  32. [32]

    Ovchinnikov, X

    D. Ovchinnikov, X. Huang, Z. Lin, Z. Fei, J. Cai, T. Song, M. He, Q. Jiang, C. Wang, H. Li, Y . Wang, Y . Wu, D. Xiao, J. H. Chu, J. Yan, C. Z. Chang, Y . T. Cui, and X. Xu, Inter- twined topological and magnetic orders in atomically thin chern insulator MnBi2Te4, Nano Lett. 21, 2544 (2021)

    work page 2021

  33. [33]

    Chang, C.-X

    C.-Z. Chang, C.-X. Liu, and A. H. MacDon- ald, Colloquium: Quantum anomalous Hall e ffect, Rev. Mod. Phys. 95, 011002 (2023)

    work page 2023

  34. [34]

    T. Zhu, H. Wang, and H. Zhang, Floquet engineer- ing of magnetic topological insulator MnBi 2Te4 films, Phys. Rev. B 107, 085151 (2023)

    work page 2023

  35. [35]

    Zhou and J

    C. Zhou and J. Zhou, Light-Induced Topological Phase Transi - tion with Tunable Layer Hall Effect in Axion Antiferromagnets, Nano Lett. 24, 7311 (2024)

    work page 2024

  36. [36]

    Zhu, D.-X

    J.-J. Zhu, D.-X. Yao, S.-C. Zhang, and K. Chang, Electricall y controllable surface magnetism on the surface of topologic al insulators, Phys. Rev. Lett. 106, 097201 (2011)

    work page 2011

  37. [37]

    M. Zare, F. Parhizgar, and R. Asgari, Topological phase and edge states dependence of the RKKY interaction in zigzag sil - icene nanoribbon, Phys. Rev. B 94, 045443 (2016)

    work page 2016

  38. [38]

    Wang, H.-R

    S.-X. Wang, H.-R. Chang, and J. Zhou, RKKY interaction in three-dimensional electron gases with linear spin-orbit c ou- pling, Phys. Rev. B 96, 115204 (2017)

    work page 2017

  39. [39]

    S. F. Islam, P . Dutta, A. M. Jayannavar, and A. Saha, Probing decoupled edge states in a zigzag phosphorene nanoribbon vi a RKKY exchange interaction, Phys. Rev. B 97, 235424 (2018)

    work page 2018

  40. [40]

    Kaladzhyan, A

    V . Kaladzhyan, A. A. Zyuzin, and P . Simon, RKKY inter- action on the surface of three-dimensional Dirac semimetal s, Phys. Rev. B 99, 165302 (2019)

    work page 2019

  41. [41]

    Duan, Y .-J

    H.-J. Duan, Y .-J. Wu, M.-X. Deng, R.-Q. Wang, and M. Yang, Indirect magnetic signals in Weyl semimetals mediated by a single fermi arc, Phys. Rev. B 107, 165147 (2023)

    work page 2023

  42. [42]

    Duan, S.-H

    H.-J. Duan, S.-H. Zheng, Y .-Y . Yang, C.-Y . Zhu, M.-X. Deng, M. Yang, and R.-Q. Wang, Anisotropic RKKY interactions in nodal-line semimetals, Phys. Rev. B 102, 165110 (2020)

    work page 2020

  43. [43]

    Chang, J

    H.-R. Chang, J. Zhou, S.-X. Wang, W.-Y . Shan, and D. Xiao, RKKY interaction of magnetic impurities in Dirac and Weyl semimetals, Phys. Rev. B 92, 241103 (2015)

    work page 2015

  44. [44]

    M. V . Hosseini and M. Askari, Ruderman-Kittel- Kasuya-Y osida interaction in Weyl semimetals, Phys. Rev. B 92, 224435 (2015)

    work page 2015

  45. [45]

    Duan, S.-H

    H.-J. Duan, S.-H. Zheng, P .-H. Fu, R.-Q. Wang, J.-F. Liu, G.-H. Wang, and M. Yang, Indirect magnetic interaction mediated b y Fermi arc and boundary reflection near Weyl semimetal surfac e, New J. Phys. 20, 103008 (2018)

    work page 2018

  46. [46]

    G. C. Paul, S. F. Islam, P . Dutta, and A. Saha, Sig- natures of interfacial topological chiral modes via RKKY exchange interaction in Dirac and Weyl systems, Phys. Rev. B 103, 115306 (2021)

    work page 2021

  47. [47]

    G. C. Paul, S. F. Islam, and A. Saha, Fingerprints of tilted Dirac cones on the RKKY exchange interaction in 8-Pmmn borophene, Phys. Rev. B 99, 155418 (2019)

    work page 2019

  48. [48]

    Duan, S.-H

    H.-J. Duan, S.-H. Zheng, R.-Q. Wang, M.-X. Deng, and M. Yang, Signature of indirect magnetic interaction in 12 the crossover from type-I to type-II Weyl semimetals, Phys. Rev. B 99, 165111 (2019)

    work page 2019

  49. [49]

    Duan, Y .-J

    H.-J. Duan, Y .-J. Wu, Y .-Y . Yang, S.-H. Zheng, C.-Y . Zhu, M.-X. Deng, M. Yang, and R.-Q. Wang, The pro- longed decay of RKKY interactions by interplay of relativis - tic and non-relativistic electrons in semi-Dirac semimeta ls, New J. Phys. 24, 033029 (2022)

    work page 2022

  50. [50]

    M. Ke, M. M. Asmar, and W.-K. Tse, Nonequi- librium RKKY interaction in irradiated graphene, Phys. Rev. Res. 2, 033228 (2020)

    work page 2020

  51. [51]

    M. M. Asmar and W.-K. Tse, Floquet control of indirect ex- change interaction in periodically driven two-dimensional elec- tron systems, New J. Phys. 23, 123031 (2021)

    work page 2021

  52. [52]

    Yarmohammadi, M

    M. Yarmohammadi, M. Bukov, and M. H. Kolodrubetz, Non- collinear twisted RKKY interaction on the optically driven SnTe(001) surface, Phys. Rev. B 107, 054439 (2023)

    work page 2023

  53. [53]

    Lee, Magnetic impurities in an altermagnetic metal, Eur

    Y .-L. Lee, Magnetic impurities in an altermagnetic metal, Eur. Phys. J. B 98, 43 (2025)

    work page 2025

  54. [54]

    Amundsen, A

    M. Amundsen, A. Brataas, and J. Linder, RKKY interaction in Rashba altermagnets, Phys. Rev. B 110, 054427 (2024)

    work page 2024

  55. [55]

    Yarmohammadi, U

    M. Yarmohammadi, U. Z¨ ulicke, J. Berakdar, J. Linder, and J. K. Freericks, Anisotropic light-tailored RKKY interaction in two-dimensional d-wave altermagnets, Phys. Rev. B 111, 224412 (2025)

    work page 2025

  56. [56]

    Zhou, H.-R

    M. Zhou, H.-R. Chang, L. Yang, and L. Liang, Weyl- mediated Ruderman-Kittel-Kasuya-Y osida interaction rev is- ited: Imaginary-time formalism and finite temperature e ffects, Phys. Rev. B 112, 054449 (2025)

    work page 2025

  57. [57]

    Yarmohammadi, S

    M. Yarmohammadi, S. R. Koshkaki, J. Berakdar, M. Bukov, and M. H. Kolodrubetz, Probing topological phases in a perturbed Kane-Mele model via RKKY interac- tion: Application to monolayer jacutingaite Pt 2HgSe3, Phys. Rev. B 111, 014440 (2025)

    work page 2025

  58. [58]

    Shiranzaei, J

    M. Shiranzaei, J. Fransson, H. Cheraghchi, and F. Parhiz- gar, Nonlinear spin susceptibility in topological insulat ors, Phys. Rev. B 97, 180402 (2018)

    work page 2018

  59. [59]

    T. V . Trevisan, P . V . Arribi, O. Heinonen, R.-J. Slager, and P. P . Orth, Bicircular Light Floquet Engineering of Magnetic Sym - metry and Topology and Its Application to the Dirac Semimetal Cd3As2, Phys. Rev. Lett. 128, 066602 (2022)

    work page 2022

  60. [60]

    Mikami, S

    T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet top olog- ical insulators, Phys. Rev. B 93, 144307 (2016)

    work page 2016

  61. [61]

    Yan and Z

    Z. Yan and Z. Wang, Tunable Weyl Points in Periodically Driven Nodal Line Semimetals, Phys. Rev. Lett. 117, 087402 (2016)

    work page 2016

  62. [62]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase e ffects on elec- tronic properties, Rev. Mod. Phys. 82, 1959 (2010)

    work page 1959

  63. [63]

    M. A. Ruderman and C. Kittel, Indirect exchange cou- pling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96, 99 (1954)

    work page 1954

  64. [64]

    Kasuya, A Theory of Metallic Ferro- and Antiferromag- netism on Zener’s Model, Prog

    T. Kasuya, A Theory of Metallic Ferro- and Antiferromag- netism on Zener’s Model, Prog. Theor. Phys. 16, 45 (1956)

    work page 1956

  65. [65]

    Y osida, Magnetic Properties of Cu-Mn Alloys, Phys

    K. Y osida, Magnetic Properties of Cu-Mn Alloys, Phys. Rev. 106, 893 (1957)

    work page 1957

  66. [66]

    D. C. Mattis, The Theory of Magnetism Made Simple: An Intro- duction to Physical Concepts and to Some Useful Mathematical Methods (World Scientific, Singapore, 2006)

    work page 2006

  67. [67]

    Shiranzaei, H

    M. Shiranzaei, H. Cheraghchi, and F. Parhizgar, E ffect of the Rashba splitting on the RKKY interaction in topological - insulator thin films, Phys. Rev. B 96, 024413 (2017)

    work page 2017

  68. [68]

    M. M. Asmar and W.-K. Tse, Interlayer RKKY coupling in bulk Rashba semiconductors under topological phase transition, Phys. Rev. B 100, 014410 (2019)

    work page 2019

  69. [69]

    Wu, Q.-Y

    Y .-J. Wu, Q.-Y . Xiong, H.-J. Duan, J.-Y . Ba, M.-X. Deng, and R.-Q. Wang, Interlayer RKKY interaction in ferro- magnet/tilted Weyl semimetal /ferromagnet trilayer system, Phys. Rev. B 106, 195130 (2022)

    work page 2022

  70. [70]

    Duan, Y .-J

    H.-J. Duan, Y .-J. Wu, M.-X. Deng, R.-Q. Wang, and M. Yang, Indirect magnetic signals in Weyl semimetals mediated by a single Fermi arc, Phys. Rev. B 107, 165147 (2023)

    work page 2023

  71. [71]

    Zhang, J.-J

    S.-H. Zhang, J.-J. Zhu, W. Yang, and K. Chang, Fo- cusing RKKY interaction by graphene P-N junction, 2D Mater. 4, 035005 (2017)

    work page 2017

  72. [72]

    Zhang, J.-J

    S.-H. Zhang, J.-J. Zhu, W. Yang, and K. Chang, Selective gen- eration and amplification of RKKY interactions by a p-n inter- face, Phys. Rev. B 99, 195456 (2019)

    work page 2019

  73. [73]

    Zhang and W

    S.-H. Zhang and W. Yang, Anomalous caustics and V eselago focusing in 8-Pmmn borophene p-n junctions with arbitrary junction directions, New J. Phys. 21, 103052 (2019)

    work page 2019

  74. [74]

    Zhang, D.-F

    S.-H. Zhang, D.-F. Shao, and W. Yang, V elocity-determined anisotropic behaviors of RKKY interaction in 8-pmmn borophene, J. Magn. Magn. Mater. 491, 165631 (2019)

    work page 2019

  75. [75]

    Zhang, J

    S.-H. Zhang, J. Yang, D.-F. Shao, W. Yang, and K. Chang, Geometric wavefront dislocations of RKKY interaction in graphene, Phys. Rev. B 104, 245405 (2021)

    work page 2021

  76. [76]

    L. Zhou, J. Wiebe, S. Lounis, E. V edmedenko, F. Meier, S. Bl¨ ugel, P . H. Dederichs, and R. Wiesendanger, Strength and directionality of surface Ruderman-Kittel-Kasuya-Y osida inter- action mapped on the atomic scale, Nat. Phys. 6, 187 (2010)

    work page 2010

  77. [77]

    Meier, L

    F. Meier, L. Zhou, J. Wiebe, and R. Wiesendanger, Reveal- ing Magnetic Interactions from Single-Atom Magnetization Curves, Science 320, 82 (2008)

    work page 2008

  78. [78]

    Laplane, E

    C. Laplane, E. Zambrini Cruzeiro, F. Fr¨ owis, P . Gold- ner, and M. Afzelius, High-Precision Measurement of the Dzyaloshinsky-Moriya Interaction between Two Rare-Earth Ions in a Solid, Phys. Rev. Lett. 117, 037203 (2016)

    work page 2016