Moir\'e magnetism in a bilayer Ising model

Anders W. Sandvik; Ryan Flynn

REVIEW 3 major objections 5 minor 24 references

Moir\'e magnetism in a bilayer Ising model

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Moiré-modulated magnetic textures in a bilayer Ising model are smooth crossovers, not thermodynamic phase transitions.

desk verdict Useful minimal-model paper with a credible crossover picture, but the layer-symmetry test is statistically mis-specified and needs correction. read the letter →

arxiv 2601.18955 v3 pith:HYGROURU submitted 2026-01-26 cond-mat.stat-mech

Ryan Flynn , Anders W. Sandvik This is my paper · ORCID

classification cond-mat.stat-mech MSC 82B2082B2782-05 PACS 75.10.Hk75.40.Mg

keywords moirémagnetismbilayerIsingmodeldomaintexturecrossoveruniversalityclassicalMonteCarlotwisteddifferentialstrain

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 claims that in a minimal classical bilayer Ising model with a moiré-modulated interlayer coupling, the paramagnet-to-ordered transition remains in the ordinary two-dimensional Ising universality class even when the low-temperature ordered state is a domain texture. It further claims that the change from a uniform ferromagnet to a domain-textured state is not a thermodynamic phase transition but a smooth crossover, governed by a simple geometric energy balance between interlayer exchange and intralayer domain-wall costs. This matters because experimental observations of moiré-induced magnetic textures are often read as evidence of new phases; the paper shows that the emergence of textures alone need not imply a distinct thermodynamic phase.

What carries the argument

The moiré-modulated interlayer coupling function Φ(u) = Φ0 + Σ_a cos(b_a·u), truncated to the two lowest harmonics, encodes the effect of twist or strain as a spatially alternating ferromagnetic/antiferromagnetic exchange while keeping the spins on identical square lattices. The analysis relies on finite-size scaling of the Binder cumulant U₂ and the layer-polarization structure factor P, together with the geometric energy balance J′ > z/L_M between bulk interlayer exchange and intralayer domain-wall cost.

What would settle it

Measure the layer-polarization plateau ⟨P²⟩ on a truly quasiperiodic twisted bilayer (spins at rotated lattice positions with incommensurate boundary conditions) or at larger N_M than simulated; a nonzero extrapolated plateau, or a Binder-cumulant crossing that sharpens with system size at the crossover, would indicate a genuine layer-symmetry-breaking phase transition rather than a smooth crossover.

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Extended reading notes

Core claim

The central discovery is that moiré-modulated interlayer exchange—alternating ferromagnetic and antiferromagnetic regions at the moiré scale—does not create a new thermodynamic phase. Using classical Monte Carlo simulations, the authors show that the Binder cumulant of the total magnetization for a domain-textured ordered state collapses onto the standard 2D Ising critical point, with correlation-length exponent 1/ν = 0.97 ± 0.02. They also show that the layer-polarization order parameter in twisted bilayers scales as 1/N_M (where N_M is the number of moiré unit cells) and extrapolates to zero in the thermodynamic limit, meaning no spontaneous layer-symmetry breaking. The ferromagnet-to-doma

Load-bearing premise

For twisted bilayers, the model replaces the physical rotation of the layers with a periodic moiré-modulated coupling map overlaid on identical square lattices; if genuine incommensurability affects ordering or crossover behavior, the twisted-bilayer results may not carry over to real systems.

Editorial extensions

If this is right

Experimental observation of moiré domain textures in magnetic bilayers does not by itself establish a new phase; textures can emerge smoothly without additional symmetry breaking.
The crossover boundary is predicted to scale as J′ ∝ (a−1)/a for differential strain and J′ ∝ tan φ for twist, giving a testable relation for when domain textures appear in real materials.
Even in the physically relevant regime of weak interlayer coupling (J′ ≪ J), domain textures can occur provided the moiré unit cell is sufficiently large (L_M ≫ 1/J′).
The universality class of the thermal ordering transition is robust to long-wavelength moiré modulation and competing ferromagnetic/antiferromagnetic interlayer interactions.

Reading between the lines

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

If the periodic-coupling-map approximation is faithful, the smooth-crossover picture should extend to Heisenberg or XY bilayer models, where noncollinear textures replace Ising domains—a direct extension suggested by the same energy-balance argument.
The geometric balance J′ > z/L_M could be used to predict a 'texture threshold' in specific materials once the constant Φ0 is fixed by first-principles calculations.
A quasiperiodic implementation that physically rotates the layers (rather than overlaying a periodic coupling map) could reveal whether incommensurability sharpens the crossover into a true transition; this is a testable consequence of the paper's main approximation.
The crossover mechanism may generalize to other spatially modulated interactions, such as strain-engineered lattices or artificially patterned exchange, beyond magnetic bilayers.

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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

Colleague — quick take on arXiv:2601.18955. The central claim is likely right: in a minimal classical Ising bilayer with moiré-modulated interlayer coupling, the domain-textured state is a smooth crossover from the ferromagnet, not a new thermodynamic phase. The model is clean, the Monte Carlo evidence for 2D Ising universality at the ordering transition is plausible, and the geometric energy balance J′ > z/LM gives a satisfying account of the crossover boundary. That part of the paper deserves a serious referee.

But there is a real problem in the layer-symmetry analysis for twisted bilayers (Fig. 3). The paper says that if each moiré cell independently chooses which layer hosts the domain wall, ⟨P²⟩ should scale as 1/NM. That is not right. With NM² cells, the Fourier amplitudes entering P give a binomial sum of NM² independent choices, so the null scaling is 1/NM², not 1/NM. The data reportedly follow 1/NM, which means either the text is describing the wrong null model or the cell choices are correlated, with an effective number of independent domains O(NM) rather than O(NM²). Either way, the stated test does not support the claim as written. The extrapolation to zero still goes through, so the qualitative no-symmetry-breaking conclusion may survive, but the argument needs to be redone.

Other soft spots, in proportion: the Ising universality claim is generalized to 'all moiré parameters' from a single Binder-cumulant analysis (one strain, one LM, one Φ0). That is a modest overreach; one or two more parameter sets or a scaling argument would fix it. The crossover boundary is fit with an unspecified constant z — not a flaw, but it is a scaling law with a fitted prefactor, not a parameter-free prediction. The twisted bilayer is modeled by a periodic overlay rather than an actual rotation; the paper is honest about this, but it does leave incommensurability effects unexamined. No code or data are provided, which makes checking the numerics harder than it should be.

On balance, this is a worthwhile paper for the moiré-magnetism audience. It gives a minimal classical framework for why domain textures in twisted CrI3 and similar systems can be crossovers. I would send it to peer review, with the layer-symmetry scaling analysis flagged for a required revision. The authors are clearly serious about the physics; the flaws are in the presentation of one statistical test, not in the core idea.

Referee Report

3 major / 5 minor

Summary. The paper studies a classical bilayer Ising model with a moiré-modulated interlayer coupling, generated either by differential strain or by a periodicized twist. Using Monte Carlo simulations, it claims three main results: (i) the paramagnetic-to-ordered transition remains in the conventional two-dimensional Ising universality class even when the low-temperature state is domain-textured; (ii) in the twisted bilayer there is no spontaneous breaking of layer symmetry, with the layer-polarization order parameter extrapolating to zero; and (iii) the low-temperature ferromagnet-to-domain-texture change is a smooth crossover, not a thermodynamic phase transition, whose location follows the geometric balance J′ > z/LM. The authors propose that moiré-induced magnetic textures in real systems can emerge without a new phase transition.

Significance. If correct, the paper provides a clean minimal framework for understanding moiré magnetism as a crossover phenomenon and shows that long-wavelength periodic modulation does not change the 2D Ising universality class of the ordering transition. The model is simple and the numerical methods are appropriate; the use of both Metropolis and cluster updates, combined with simulated annealing, is a strength. The geometric energy balance is attractive and gives a falsifiable scaling prediction. However, the support for the generalized universality claim rests on one parameter set, the layer-polarization null model has a scaling error, and the crossover boundary is fit rather than quantitatively predicted. These issues are fixable and do not obviously invalidate the central qualitative conclusions, but they require substantial clarification and additional analysis before the paper can be accepted.

major comments (3)

[Sec. III, Fig. 2 and Sec. IV] The universality claim is supported for a single parameter set. Binder cumulant crossings and the extrapolated exponent 1/ν = 0.97 ± 0.02 are shown only for a strained bilayer with LM = 10, Φ0 = 0.5, and large J′. The text nonetheless asserts that the transition is Ising 'for all values of the moiré parameters' and repeats this in the conclusions. No Binder analysis is shown for the twisted bilayer, for other Φ0, for other LM, or for small J′. This overgeneralization is load-bearing for the first central claim. Either add systematic data across the parameter space or qualify the claim to the representative case.
[Sec. IV, Fig. 3, Eq. (7)] The finite-size null model for layer polarization is mis-specified. In the twisted bilayer there are N_d = N_M^2 moiré cells. If each cell independently chooses the minority layer, the Fourier amplitude entering S_l(Q_M) sums over all cells with the same phase, so S_l ∝ (N_{l,cells}/N_d)^2. The polarization then reduces to P ≈ (N_2 - N_1)/N_d, whose RMS fluctuation is 1/√N_d = 1/N_M. The correct binomial null is therefore ⟨P²⟩ ∝ 1/N_M², not 1/N_M. The observed 1/N_M scaling for N_M > 8 is a factor N_M larger than this independence prediction, meaning either the null comparison is wrong or the domain-cell choices are correlated. The thermodynamic conclusion ⟨P²⟩→0 may still hold, but the stated binomial test and the interpretation of independence must be corrected; otherwise the no-layer-symmetry-breaking evidence is ambiguous.
[Sec. IV, Fig. 4, Eq. (8)] The crossover boundary is fit, not predicted. The scaling argument equates J′L_M² with J L_M and yields J′ ∝ 1/L_M, but the constant z is left unspecified, and no fit parameters, residuals, or error bars are given for the dashed line in Fig. 4. There is also no comparison showing the same boundary for the twisted bilayer or for different Φ0. Since the geometric energy balance is presented as a central result, the authors should either provide quantitative support (e.g., tabulated boundary points, fits, dependence on Φ0) or clearly label Eq. (8) as a scaling estimate rather than a demonstrated law.

minor comments (5)

[Sec. II] The displacement field u, the reciprocal vectors b_a, the cutoff radius r_c, and the exponential screening length are not specified numerically or in a table. Please provide these details or state typical values.
[Eq. (1)] The relation LM = a/(a−1) = 1/tanφ assumes a specific convention for φ and strain. Please state explicitly how φ enters the displacement field and whether the strain is biaxial.
[Sec. IV, Fig. 3] The lower panel would benefit from stating the axes, whether the plot is log-log, and how the plateau values are extracted and averaged. Error bars on the plateau values and the fitted slope would help assess the 1/N_M claim.
[Sec. IV] The crossing point J′_c is used in the description of Fig. 3 before it is defined. Define J′_c explicitly, perhaps by the peak position or an operational criterion.
[Sec. III] The normalization of U2 and the size-pair fitting procedure for 1/ν* are described briefly; please give the exact functional form used in the extrapolation and define the error bars in the inset.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central claims are independent numerical results benchmarked against the known 2D Ising exponent; the only self-citation is a standard finite-size-scaling method that is not load-bearing.

full rationale

The paper's main results—Ising universality of the ordering transition, absence of a thermodynamic transition between ferromagnet and domain-textured states, and zero layer polarization in the thermodynamic limit for twisted bilayers—are obtained by direct Monte Carlo measurements and compared with external benchmarks. The 1/nu = 0.97 ± 0.02 value is an independent fit to Binder-cumulant slopes, not an input; citing Shao et al. [24] for the slope-extrapolation method is a self-citation (Sandvik is a coauthor), but the method is a standard, externally checkable finite-size-scaling relation and is not doing circular work. Similarly, the crossover boundary J' > z/L_M in Eq. (8) comes from a dimensional/geometric energy balance; the constant z is fitted, but the functional form J' ∝ (a-1)/a is not forced by that fit and is independently consistent with the color map of Fig. 4. The layer-polarization argument uses a binomial null model and measures ⟨P^2⟩ directly; although the stated 1/N_M null scaling is questionable (with N_M^2 moiré cells a binomial count would give 1/N_M^2), both scalings extrapolate to zero, so the no-layer-symmetry-breaking conclusion is not equivalent by construction to the test. No load-bearing derivation reduces to its own input.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The model contributes one fitted crossover constant z plus the free bias Φ0, while the rest is standard Ising statistical mechanics. No new physical entities (particles, forces, dimensions) are introduced; the moiré-modulated Hamiltonian is a model construction, not an invented entity in the sense used here.

free parameters (3)

Φ0 = 0.5 in Fig. 2; otherwise free
Constant ferromagnetic bias in the interlayer coupling function Eq. (3). Treated as a free parameter; the paper says it only changes Tc and domain size, and it factors into the crossover constant z.
z = not stated; absorbed into the linear fit of Fig. 4
System-dependent constant in Eq. (8), J′ > z/LM, absorbing ⟨Φ(u)⟩, Φ0, and domain-wall prefactors. The boundary in Fig. 4 is fit with J′ ∝ (a−1)/a, so z is effectively determined by the simulation data rather than computed a priori.
rc and exponential screening length = not specified
Interlayer exchange in the strained bilayer is defined for spin pairs within a cutoff radius rc with an exponential screening exp(−|rij|/rc). Values are never reported, making the coupling map not fully reproducible.

assumptions (6)

standard math Binder cumulant finite-size scaling and dU2/dT ∝ L^{1/ν}
The correlation-length exponent extraction relies on accepted finite-size scaling relations from Shao et al. [24].
domain assumption Interlayer coupling Φ(u) = Φ0 + Σ cos(b_a·u) with only lowest harmonics
Sec. II states higher harmonics are neglected because 'they do not qualitatively affect the resulting magnetic textures.' This is a modeling assumption about the essential physics.
domain assumption Twisted bilayer implemented by overlaying a periodic coupling map on unrotated lattices
Sec. II replaces the physically quasiperiodic twisted bilayer with a periodic coupling map to preserve boundary conditions. This assumes the periodic approximation does not change ordering or crossover behavior.
domain assumption Biaxial strain preserves square-lattice rotational symmetry and creates a perfectly periodic moiré pattern
Sec. II constructs the strained bilayer with different lattice constants and a cutoff-based interlayer coupling; this is a specific deformation model, not a general strain treatment.
domain assumption Energy balance J′ > z/LM with bulk energy ∝ J′ L_M² and domain-wall energy ∝ J L_M
Sec. IV assumes sharp domain walls and ignores entropic or fluctuation contributions. The resulting linear scaling is then fit to the numerical crossover boundary.
standard math Layer polarization ⟨P²⟩ ∝ 1/N_M if domains independently choose layers
Sec. IV models each moiré domain as independently forming in either layer, giving a binomial distribution of the polarization. This is the null model used to conclude no layer symmetry breaking.

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Cite this review

Pith. "Pith review of Moir\'e magnetism in a bilayer Ising model." pith.science (2026). https://pith.science/paper/HYGROURU

@misc{pith2026260118955,
  author       = {Pith},
  title        = {Pith review of: Moir\'e magnetism in a bilayer Ising model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYGROURU}},
  note         = {Machine review of arXiv:2601.18955}
}

read the original abstract

Moir\'e patterns in magnetic bilayers generate spatially modulated interlayer exchange interactions that can give rise to nonuniform magnetic textures. We study a minimal classical bilayer Ising model with a moir\'e-modulated interlayer coupling, generated either by relative twist or differential strain between the layers. Using large-scale classical Monte Carlo simulations, we show that the ordering transition remains in the conventional two-dimensional Ising universality class, even when the low-temperature state is domain-textured. At low temperatures, we find a smooth crossover between a uniform ferromagnet and domain-textured state, in which the spins locally follow the sign of the interlayer exchange. We demonstrate that there is no breaking of layer symmetry for twisted bilayers. The location of the crossover is determined by a simple geometric energy balance between bulk interlayer exchange and intralayer domain-wall costs. Our results provide a minimal framework for understanding how moir\'e-modulated magnetic textures can emerge from geometric energetics without requiring a thermodynamic phase transition.

Figures

Figures reproduced from arXiv: 2601.18955 by the authors.

**Figure 2.** FIG. 2. Binder cumulant crossings for the temperature [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. No layer symmetry breaking in the twisted bilayer. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Crossover diagram in the interlayer coupling [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

24 extracted references · 1 linked inside Pith

[1]

Novoselov, A

K. Novoselov, A. Mishchenko, A. Carvalho, and A. Cas- tro Neto,2d materials and van der waals heterostruc- tures, Science353, aac9439 (2016)

2016
[2]

Gibertini, M

M. Gibertini, M. Koperski, A. Morpurgo,et al.,Magnetic 2d materials and heterostructures, Nat. Nanotechnol.14, 408–419 (2019)

2019
[3]

Bistritzer and A

R. Bistritzer and A. MacDonald,Moir´ e bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. U.S.A.108, 12233–12237 (2011)

2011
[4]

Y. Cao, V. Fatemi, A. Demir,et al.,Correlated insulator behaviour at half-filling in magic-angle graphene super- lattices, Nature556, 80–84 (2018)

2018
[5]

Hejazi, Z

K. Hejazi, Z. Luo, and L. Balents,Noncollinear phases in moir´ e magnets, Proc. Natl. Acad. Sci. U.S.A.117, 10721–10726 (2020)

2020
[6]

Sivadas, S

N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and D. Xiao,Stacking-dependent magnetism in bilayer CrI 3, Nano letters18, 7658–7664 (2018)

2018
[7]

K.-M. Kim, D. H. Kiem, G. Bednik, M. J. Han, and M. J. Park,Theory of moir´ e magnets and topological magnons: Applications to twisted bilayer CrI 3, arXiv e- prints , arXiv:2206.05264 (2022)

arXiv 2022
[8]

Cheng, M

G. Cheng, M. Rahman, A. Allcca,et al.,Electrically tunable moir´ e magnetism in twisted double bilayers of chromium triiodide, Nat Electron6, 434–442 (2023)

2023

Show all 24 references

[9]

Y. Xu, A. Ray, Y. Shao,et al.,Coexisting ferromag- netic–antiferromagnetic state in twisted bilayer CrI 3, Nat. Nanotechnol.17, 143–147 (2022)

2022
[10]

M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Coupling of crystal structure and magnetism in the lay- ered, ferromagnetic insulator CrI 3, Chemistry of Materi- als27, 10.1021/cm504242t (2014)

2014 doi
[11]

Samarth,Magnetism in flatland, Nature546, 216–217 (2017)

N. Samarth,Magnetism in flatland, Nature546, 216–217 (2017)

2017
[12]

C. Gong, L. Li, Z. Li,et al.,Discovery of intrinsic fer- romagnetism in two-dimensional van der waals crystals, Nature546, 265–269 (2017)

2017
[13]

K. S. Burch, D. Mandrus, and J. G. Park,Magnetism in two-dimensional van der waals materials, Nature563, 47–52 (2018)

2018
[14]

Park,Opportunities and challenges of two- dimensional magnetic van der waals materials: magnetic graphene?, J

J.-G. Park,Opportunities and challenges of two- dimensional magnetic van der waals materials: magnetic graphene?, J. Phys. Condens. Matter28, 301001 (2016)

2016
[15]

Huang, G

B. Huang, G. Clark, D. Klein,et al.,Electrical control of 2d magnetism in bilayer CrI 3, Nature Nanotech13, 544–548 (2018)

2018
[16]

Cheng, M

G. Cheng, M. Rahman, Z. He,et al.,Emergence of electric-field-tunable interfacial ferromagnetism in 2d an- tiferromagnet heterostructures, Nat Commun13, 7348 (2022)

2022
[17]

Balents,General continuum model for twisted bi- layer graphene and arbitrary smooth deformations, Sci- Post Phys.7, 048 (2019)

L. Balents,General continuum model for twisted bi- layer graphene and arbitrary smooth deformations, Sci- Post Phys.7, 048 (2019)

2019
[18]

Jiang, C

P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z.-Y. Lu, and W. Ji,Stacking tunable interlayer magnetism in bilayer CrI 3, Phys. Rev. B99, 144401 (2019)

2019
[19]

Songet al.,Direct visualization of magnetic domains and moir´ e magnetism in twisted 2d magnets, Science374, 1140–1144 (2021)

T. Songet al.,Direct visualization of magnetic domains and moir´ e magnetism in twisted 2d magnets, Science374, 1140–1144 (2021)

2021
[20]

Soriano, C

D. Soriano, C. Cardoso, and J. Fern´ andez-Rossier,Inter- play between interlayer exchange and stacking in CrI 3 bi- layers, Solid State Communications299, 113662 (2019)

2019
[21]

Escudero, A

F. Escudero, A. Sinner, Z. Zhan, P. A. Pantale´ on, and F. Guinea,Designing moir´ e patterns by strain, Phys. Rev. Res.6, 023203 (2024)

2024
[22]

F. Yao, D. Rossi, I. A. Gabrovski,et al.,Moir´ e mag- netism in CrBr 3 multilayers emerging from differential strain, Nature Communications15, 10377 (2024)

2024
[23]

J. Jung, A. Raoux, Z. Qiao, and A. H. MacDonald,Ab initio theory of moir´ e superlattice bands in layered two dimensional materials, Phys. Rev. B89, 205414 (2014)

2014
[24]

H. Shao, W. Guo, and A. W. Sandvik,Quantum critical- ity with two length scales, Science352, 213 (2016)

2016

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Novoselov, A

K. Novoselov, A. Mishchenko, A. Carvalho, and A. Cas- tro Neto,2d materials and van der waals heterostruc- tures, Science353, aac9439 (2016)

2016

[2] [2]

Gibertini, M

M. Gibertini, M. Koperski, A. Morpurgo,et al.,Magnetic 2d materials and heterostructures, Nat. Nanotechnol.14, 408–419 (2019)

2019

[3] [3]

Bistritzer and A

R. Bistritzer and A. MacDonald,Moir´ e bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. U.S.A.108, 12233–12237 (2011)

2011

[4] [4]

Y. Cao, V. Fatemi, A. Demir,et al.,Correlated insulator behaviour at half-filling in magic-angle graphene super- lattices, Nature556, 80–84 (2018)

2018

[5] [5]

Hejazi, Z

K. Hejazi, Z. Luo, and L. Balents,Noncollinear phases in moir´ e magnets, Proc. Natl. Acad. Sci. U.S.A.117, 10721–10726 (2020)

2020

[6] [6]

Sivadas, S

N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and D. Xiao,Stacking-dependent magnetism in bilayer CrI 3, Nano letters18, 7658–7664 (2018)

2018

[7] [7]

K.-M. Kim, D. H. Kiem, G. Bednik, M. J. Han, and M. J. Park,Theory of moir´ e magnets and topological magnons: Applications to twisted bilayer CrI 3, arXiv e- prints , arXiv:2206.05264 (2022)

arXiv 2022

[8] [8]

Cheng, M

G. Cheng, M. Rahman, A. Allcca,et al.,Electrically tunable moir´ e magnetism in twisted double bilayers of chromium triiodide, Nat Electron6, 434–442 (2023)

2023