Magnetism from multiparticle ring exchange in moir\'e Wigner crystals

Alex Levchenko; Ilya Esterlis

REVIEW 3 major objections 4 minor 52 references

Magnetism from multiparticle ring exchange in moir\'e Wigner crystals

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that a sufficiently strong moiré potential turns a Wigner crystal's dominant ring exchange from three-particle (ferromagnetic) to two-particle (antiferromagnetic), flipping the ground state to a 120-degree Néel…

desk verdict A clean semiclassical action calculation that rationalizes QMC magnetism trends in moiré Wigner crystals; the reported epsilon_c values should be read as order-of-magnitude estimates, not precise predictions. read the letter →

arxiv 2502.00323 v1 pith:GPTLMP7C submitted 2025-02-01 cond-mat.str-el

Ilya Esterlis , Alex Levchenko This is my paper · ORCID

classification cond-mat.str-el

keywords WignercrystalmoirépotentialringexchangemultiparticlesemiclassicalWKB120-degreeNéelordertransitionmetaldichalcogenideheterobilayersferromagnetism

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

This paper asks what controls magnetism in a triangular Wigner crystal sitting in a commensurate moiré potential. The authors argue that the strength of the moiré potential determines which multiparticle exchange process dominates: at weak potential, three-particle exchange wins and the crystal is ferromagnetic, while at stronger potential two-particle exchange wins and the crystal becomes a 120-degree Néel antiferromagnet. Using a semiclassical action calculation, they find the crossover at $\epsilon_c \approx 0.0033$ for a triangular moiré potential and at $\epsilon_c \approx 0.26$ for a honeycomb potential, nearly two orders of magnitude larger. If correct, this simple mechanism explains why triangular moiré potentials stabilize antiferromagnetism while honeycomb potentials stabilize ferromagnetism in generalized Wigner crystals.

What carries the argument

The load-bearing object is the dimensionless instanton action $\tilde S_P(\epsilon)$ defined by Eq. (7), computed along classical paths that permute the electrons in configuration space. The action is minimized numerically by discretizing the path integral with the trapezoid rule using $M=16$ steps, allowing $N_{\rm move} \approx 50$ to $70$ electrons to move while fixing the rest, and evaluating the electrostatic energy with the Ewald method. The paper compares $\tilde S_2$ and $\tilde S_3$; because the couplings depend exponentially on the actions, the smallest action determines the magnetic ground state via the Thouless sign rule that odd permutations are ferromagnetic and even permutations antiferromagnetic.

What would settle it

Compute the Gaussian prefactors $\omega_2$ and $\omega_3$ in Eq. (5) and evaluate the difference $J_2 - J_3$ near $\epsilon_c$; if the ratio $\omega_2/\omega_3$ is large enough to overcome the action difference, the crossing point moves or disappears. A direct quantum Monte Carlo simulation of the same model at $r_s$ around 30 for triangular moiré potentials with $\epsilon$ between 0.004 and 0.1 could also test whether the ground state actually becomes the 120-degree Néel antiferromagnet.

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Extended reading notes

Core claim

The central discovery is that the moiré potential reorders the ring-exchange actions of the Wigner crystal. In the dilute limit the magnetic Hamiltonian is $H_{\rm eff} = \sum_P (-1)^P J_P \hat P$, where odd-particle cyclic permutations give ferromagnetic couplings and even-particle permutations give antiferromagnetic couplings. The exchange coupling is $J_P = \hbar \omega_P e^{-S_P/\hbar}$, so the dominant process is the one with the smallest action $\tilde S_P(\epsilon)$. The paper computes these actions numerically for two-, three-, four-, five-, and six-particle exchanges as functions of $\epsilon = V_M/V_C$. For a triangular moiré potential with one electron per unit cell, the three-particle action $\tilde S_3$ is smallest at $\epsilon=0$, but $\tilde S_2$ crosses below it at $\epsilon_c \approx 0.0033$, signifying a ferromagnet-to-antiferromagnet transition. For a honeycomb potential with half filling, the same crossing occurs at $\epsilon_c \approx 0.26$, because the three-particle path can stay near the potential minima while the two-particle path is strongly squeezed. The same qualitative behavior appears at $\nu=1/3$ filling of a triangular potential, with crossing near $\epsilon \approx 0.0026$.

Load-bearing premise

The paper orders the exchange couplings by their computed actions and does not calculate the prefactors $\omega_P$; if those prefactors or finite-density corrections change the ordering near the crossing, the critical potential strengths $\epsilon_c$ would shift.

Editorial extensions

If this is right

In a triangular moiré potential with one electron per unit cell, even a weak potential ($\epsilon > 0.0033$) should select the antiferromagnetic two-particle exchange, making the 120-degree Néel state the ground state in the dilute limit.
In a honeycomb moiré potential, the ferromagnet should remain stable to much larger potential strengths, with the transition only near $\epsilon \approx 0.26$.
The same ferromagnet-to-antiferromagnet crossover occurs at $\nu=1/3$ filling of a triangular potential, near $\epsilon \approx 0.0026$, indicating the mechanism is robust to filling fraction.
Increasing the moiré potential spreads the exchange couplings exponentially apart, strongly reducing magnetic frustration compared with the zero-potential Wigner crystal, so competing spin-liquid phases become less likely.
The semiclassical method is directly applicable to other commensurate moiré Wigner crystals, including stripe and honeycomb charge-ordered states observed at other fillings.

Reading between the lines

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

An experimental test not performed in the paper would be to measure spin susceptibility or magnetotransport in TMD heterobilayers as the moiré potential depth is tuned; one should see a switch from ferromagnetic to antiferromagnetic behavior near the predicted $\epsilon_c$ values.
Because the paper neglects prefactors $\omega_P$, a natural extension is to compute Gaussian fluctuation corrections; if the prefactor ratio $\omega_2/\omega_3$ varies slowly the qualitative picture survives, but if it is steep near the crossing the quantitative $\epsilon_c$ values could shift substantially.
The same action-crossing logic could be applied to other lattice geometries such as square or kagome moiré potentials, where the shortest exchange path may change with potential and produce different magnetic phase boundaries.
The paper notes but does not calculate the interplay with kinetic magnetism from interstitials or vacancies in lightly doped Wigner crystals; a combined treatment of ring exchange and defect dynamics would be a direct next step.

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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

Short version: this is a solid, readable Letter that does something genuinely new—computes how the shape and strength of a commensurate moiré potential changes the multiparticle ring-exchange actions of a triangular Wigner crystal, and shows that triangular potentials kill the three-particle exchange quickly while honeycomb potentials protect it. That asymmetry cleanly rationalizes why QMC sees a 120-degree Néel state in triangular moiré potentials but a robust ferromagnet in honeycomb ones. This is a real insight, and the paper earns its place in the moiré electron crystal discussion.

The calculation itself is a standard instanton/WKB estimate: they minimize the action for 2- through 6-particle cyclic exchanges on truncated paths, using Ewald sums, and report the dimensionless actions as a function of epsilon. The numerics look careful (M=16, ~50-70 mobile particles, 1% accuracy claimed), and the appendix for nu=1/3 adds useful consistency. No physical parameter is fitted to the magnetic phase; the comparison with QMC is external and qualitative.

The soft spot is exactly what the paper admits: prefactors omega_P in Eq. (5) are not computed. For the physical exchange couplings J_P, the condition J2=J3 involves sqrt(rs)(S3-S2)=ln(omega_2/omega_3). At rs=30, a prefactor ratio as small as e shifts the required action difference by ~0.18, which is comparable to the action differences near the triangular crossing at epsilon_c=0.0033. So the specific critical values are not established; they are order-of-magnitude estimates. The qualitative conclusion—weak triangular moiré drives AFM, honeycomb requires a much stronger potential—is robust because it rests on exponential action differences and the distinct geometry of the paths. But the paper should not be cited for "epsilon_c = 0.0033" as a quantitative prediction. The extrapolation from rs~10 QMC to the semiclassical rs->infinity limit is also uncontrolled, though the authors acknowledge this.

No code or data are provided, which makes it harder to check the convergence claims, but the method is described well enough to reproduce.

Who this is for: anyone working on generalized Wigner crystals in TMD heterobilayers, and people who care about multiple-spin exchange in 2D. It deserves a serious referee: the mechanism is physically interesting, the calculation is transparent, and the limitations are honestly stated. My recommendation: send it to review, with the expectation that the referee asks the authors to either compute prefactors at least at the saddle-point level or soften the quantitative claims about epsilon_c. If they do that, it's a good Letter.

Referee Report

3 major / 4 minor

Summary. The manuscript studies ring-exchange-mediated magnetism in triangular Wigner crystals subject to commensurate triangular and honeycomb moiré potentials. Using a semiclassical WKB/instanton approach, the authors numerically compute the dimensionless actions for n = 2–6 particle exchange paths as a function of the potential strength parameter ε. They find that the two-particle action becomes smaller than the three-particle action above a critical ε_c, implying, in the large-rs limit, a transition from a ferromagnet to a 120° Néel antiferromagnet. The critical ε is reported as ≈ 0.0033 for the triangular moiré potential (ν = 1 and ν = 1/3) and ≈ 0.26 for the honeycomb potential. The authors argue that this asymmetry rationalizes recent QMC results for moiré Wigner crystals.

Significance. The paper gives a clean and physically motivated framework for understanding magnetism in generalized Wigner crystals, with the important strength that no physical parameter is fitted to the magnetic phase result: the actions are computed directly from the Coulomb Hamiltonian and the external potential, and the comparison with QMC is an external consistency check. The predicted qualitative asymmetry between triangular and honeycomb potentials is a falsifiable statement that can be tested in future QMC or experiments. If confirmed, the semiclassical action crossing would provide a simple design principle for engineering magnetic ground states in moiré systems. However, the quantitative predictions (ε_c, J3/J2) depend on an uncomputed prefactor and on the numerical resolution of small action differences, so the current manuscript establishes the qualitative mechanism but not the precise phase boundary.

major comments (3)

[Eq. (5) and the Results paragraphs for Figs. 2 and 4] The central quantitative outputs, ε_c ≈ 0.0033 and 0.26, are obtained by comparing only the actions S2 and S3 while neglecting the prefactors ω_P in Eq. (5). Since JP = ℏω_P exp(−SP/ℏ), the physical condition J2 = J3 is sqrt(rs)(S3−S2) = ln(ω2/ω3). At rs = 30, a prefactor ratio of order e shifts the required action difference by ≈ 0.18, which is comparable to the action-difference changes that set the triangular crossing at ε ≈ 0.0033. The reported ε_c values and the J3/J2 estimate are therefore not established for finite rs. The authors should either compute the prefactor ratio within the same semiclassical scheme or explicitly reformulate the claims as statements about action ordering at rs → ∞, with the finite-rs transition point left undetermined.
[Numerical paragraph after Eq. (7)] The authors state that the actions are accurate to within 1% for M = 16 and Nmove ≈ 50–70. Near the S2/S3 crossing at ε ≈ 0.0033, the difference S3−S2 vanishes, so resolving the crossing requires an accuracy for the difference that is far better than 1% of the individual actions (each of order unity). The reported ε_c is thus sensitive to the numerical errors in the two actions. The authors should provide convergence tests for the difference S3−S2 (e.g., as a function of M and Nmove) and an estimate of the uncertainty in ε_c.
[Numerical paragraph after Eq. (7) and Figs. 2 and 4] The exchange path is computed with only Nmove ≈ 50–70 electrons allowed to move while all others are fixed at their equilibrium positions. The paper does not report how the action difference or the crossing point depends on Nmove or on the overall system size used for the Ewald summation. Since the crossing condition compares two nearly equal actions, finite-size effects in the fixed-electron region could distort the relative ordering. The authors should demonstrate that the crossing points are stable with respect to the number of moving electrons and the system size.

minor comments (4)

[Introduction] The introduction states that the WKB results are 'in agreement' with recent QMC studies, but the body of the paper appropriately limits this to a 'rationalization.' To avoid overstatement, the introduction should use language such as 'consistent with' or 'rationalize' rather than 'in agreement.'
[Reference [29]] Reference [29] lists the authors in an unusual order; it should be S. Chakravarty, S. Kivelson, and K. Voelker.
[Fig. 2(a) and Fig. 6(a)] The axis label 'ε (×10^2)' in Fig. 2(a) and Fig. 6(a) may confuse readers; since the actual range is 0 to about 0.0053, the tick values are in units of 10^-2, and the label should be 'ε (×10^-2)' or the axes should be relabeled to avoid ambiguity.
[Appendix A] The paper does not provide a table of the numerical values of the actions or the crossing points; including such a table in the appendix would aid reproducibility and quantitative comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exchange actions are computed directly from the Hamiltonian, with no fitted parameter targeting the magnetic phases.

full rationale

The paper's central quantitative claims are the crossing points of the dimensionless actions S2 and S3 computed by direct numerical minimization of Eq. (7), which follows from the Hamiltonian (1)-(2) with the Coulomb interaction and the external moire potential as explicit inputs. No physical parameter is fitted to the ferromagnetic-to-antiferromagnetic phase result: the actions are independent outputs of the semiclassical calculation, and the comparison with QMC results is made after the fact as external consistency, not used as a constraint. The semiclassical ring-exchange framework of Eqs. (4)-(6) is imported from the prior literature, but it is neither tailored to reproduce the target phase diagram nor does it smuggle in the crossing; the same framework gives S3<S2 at epsilon=0, consistent with the known ferromagnetic Wigner crystal. The acknowledged neglect of the prefactors omega_P in Eq. (5) means the physical ordering of J2 and J3 at finite rs is not fully secured, but that is an approximation and quantitative-correctness concern, not a circularity: the prefactors are simply not computed, and they are certainly not adjusted to force the reported crossing. The self-citations present in the manuscript (e.g., Refs. [36] and [43]) concern related but separate calculations and are not load-bearing for the present derivation. No step in the derivation reduces by construction to its own inputs, so no circular step is identified.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No physical parameter is fitted to the target result; the only tuned inputs are numerical convergence settings. The calculation imports the semiclassical exchange framework and the known 120-degree Néel ground state of the triangular Heisenberg model, and it does not introduce new entities. The main unpaid input is the omitted prefactor.

free parameters (1)

Numerical path discretization (M, Nmove) = M=16 (M=32 for ν=1/3 at large epsilon), Nmove approximately 50 to 70
Chosen by hand for convergence of the action integral in Eq. (7); the authors state accuracy within 1 percent, but these settings are not derived and enter all reported actions.

assumptions (4)

domain assumption Exchange couplings are given by the semiclassical WKB formula JP = hbar omega_P exp(-S_P/hbar), with Gaussian prefactors subleading at large rs.
Invoked after Eq. (5); the entire action comparison and magnetic phase prediction relies on this approximation, which is valid only as rs approaches infinity.
standard math The effective magnetic Hamiltonian sums over cyclic n-particle permutations with the Thouless sign rule, Eq. (4).
Standard multiple-spin exchange theory from refs [24-27,29,30], accepted without re-derivation.
ad hoc to paper Minimal action paths can be approximated by discretizing Eq. (7) with M steps and moving only Nmove electrons while all others remain fixed.
Numerical approximation described in the methods paragraph; it is not exact and could affect the location of the S2/S3 crossing.
domain assumption The magnetic ground state is determined by the dominant exchange process, i.e., the smallest action, in the large-rs limit.
Used to infer ferromagnet versus 120-degree Néel antiferromagnet from the S3/S2 crossing; frustration and subdominant exchanges may matter at finite rs.

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Pith. "Pith review of Magnetism from multiparticle ring exchange in moir\'e Wigner crystals." pith.science (2026). https://pith.science/paper/GPTLMP7C

@misc{pith2026250200323,
  author       = {Pith},
  title        = {Pith review of: Magnetism from multiparticle ring exchange in moir\'e Wigner crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPTLMP7C}},
  note         = {Machine review of arXiv:2502.00323}
}

abstract

We investigate the multiparticle ring exchange couplings of the two-dimensional triangular Wigner crystal in external commensurate triangular and honeycomb potentials, using a semiclassical approach valid in the regime where Coulomb interactions dominate over electronic kinetic energy. In this limit, increasing the strength of the potential drives a transition from a ferromagnet to a $120^\circ$ N\'eel antiferromagnet for both external potential types. In the triangular case, we find that the transition occurs already for a weak potential, whereas in the honeycomb case, it occurs when the potential is nearly two orders of magnitude larger. Our results are relevant to the magnetism of generalized Wigner crystal phases observed at certain rational fillings of the moir\'e superlattice in transition-metal dichalcogenide heterobilayers.

Figures

Figures reproduced from arXiv: 2502.00323 by the authors.

**Figure 4.** FIG. 4. Dimensionless actions [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** FIG. 5. Exchange paths for 2-particle exchange (left) and 3- [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

Figures from the paper (1 more)

**Figure 6.** Figure 6: FIG. 6. Dimensionless actions [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

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Works this paper leans on

52 extracted references · 39 canonical work pages

[1]

E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu- migeta, M. Blei, J. D. Carlstr¨ om, K. Watanabe, T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and F. Wang, Mott and generalized Wigner crystal states in WSe2/WS2 moir´ e superlattices, Nature579, 359 (2020)

2020
[2]

Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated in- sulating states at fractional fillings of moir´ e superlattices, Nature 587, 214 (2020)

2020
[3]

Huang, T

X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, S.- F. Shi, and Y.-T. Cui, Correlated insulating states at fractional fillings of the WS2/WSe2 moir´ e lattice, Nature Physics 17, 715 (2021)

work page 2021
[4]

C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu, K. Watanabe, T. Taniguchi, J. C. Hone, L. Fu, J. Shan, and K. F. Mak, Stripe phases in WSe 2/WS2 moir´ e su- perlattices, Nature Materials 20, 940 (2021)

work page 2021
[5]

kinetic magnetism

lattices have been observed, respectively. Given the relative simplicity of the approach, an extension of the present calculations to this broader class of moir´ e WCs certainly seems warranted. Various generalizations that are relevant to real ex- perimental devices are also easily incorporated in the semiclassical approach. These include screening of th...

work page
[6]

H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe,et al., Imaging two-dimensional generalized Wigner crystals, Nature 597, 650 (2021)

work page 2021
[7]

T. Li, J. Zhu, Y. Tang, K. Watanabe, T. Taniguchi, V. Elser, J. Shan, and K. F. Mak, Charge-order-enhanced capacitance in semiconductor moir´ e superlattices, Nature Nanotechnology 16, 1068 (2021)

work page 2021
[8]

H. Li, Z. Xiang, E. Regan, W. Zhao, R. Sailus, R. Baner- jee, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, Mapping charge excita- tions in generalized Wigner crystals, Nature Nanotech- nology 19, 618 (2024)

work page 2024

Show all 52 references

[9]

Shabani, D

S. Shabani, D. Halbertal, W. Wu, M. Chen, S. Liu, J. Hone, W. Yao, D. N. Basov, X. Zhu, and A. N. Pasu- pathy, Deep moir´ e potentials in twisted transition metal dichalcogenide bilayers, Nature Physics 17, 720 (2021)

2021
[10]

Hubbard, Generalized Wigner lattices in one dimen- sion and some applications to tetracyanoquinodimethane (TCNQ) salts, Phys

J. Hubbard, Generalized Wigner lattices in one dimen- sion and some applications to tetracyanoquinodimethane (TCNQ) salts, Phys. Rev. B 17, 494 (1978)

1978
[11]

H. Pan, F. Wu, and S. Das Sarma, Quantum phase di- agram of a moir´ e-Hubbard model, Phys. Rev. B 102, 201104 (2020)

2020
[12]

Padhi, R

B. Padhi, R. Chitra, and P. W. Phillips, Generalized Wigner crystallization in moir´ e materials, Phys. Rev. B 103, 125146 (2021)

2021
[13]

Matty and E.-A

M. Matty and E.-A. Kim, Melting of generalized Wigner crystals in transition metal dichalcogenide heterobilayer Moir´ e systems, Nature Communications13, 7098 (2022)

2022
[14]

Kaushal, N

N. Kaushal, N. Morales-Dur´ an, A. H. MacDonald, and E. Dagotto, Magnetic ground states of honeycomb lattice Wigner crystals, Communications Physics 5, 289 (2022)

2022
[15]

Morales-Dur´ an, P

N. Morales-Dur´ an, P. Potasz, and A. H. MacDon- ald, Magnetism and quantum melting in moir´ e-material Wigner crystals, Phys. Rev. B 107, 235131 (2023)

2023
[16]

Y. Zhou, D. N. Sheng, and E.-A. Kim, Quantum melt- ing of generalized Wigner crystals in transition metal dichalcogenide moir´ e systems, Phys. Rev. Lett. 133, 156501 (2024)

2024
[17]

Y. Yang, M. A. Morales, and S. Zhang, Metal-insulator transition in a semiconductor heterobilayer model, Phys. Rev. Lett. 132, 076503 (2024)

2024
[18]

Y. Yang, M. A. Morales, and S. Zhang, Ferromagnetic semimetal and charge-density wave phases of interact- ing electrons in a honeycomb moir´ e potential, Physi- cal Review Letters 133, 10.1103/physrevlett.133.266501 (2024)

2024 doi
[19]

Kumar, C

A. Kumar, C. Lewandowski, and H. J. Changlani, Origin and stability of generalized wigner crystallinity in tri- angular moir´ e systems (2024), arXiv:2409.13814 [cond- mat.str-el]

2024
[20]

Tanatar and D

B. Tanatar and D. M. Ceperley, Ground state of the two- dimensional electron gas, Phys. Rev. B 39, 5005 (1989)

1989
[21]

Attaccalite, S

C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Correlation energy and spin polarization in the 2d electron gas, Phys. Rev. Lett. 88, 256601 (2002)

2002
[22]

N. D. Drummond and R. J. Needs, Phase diagram of the low-density two-dimensional homogeneous electron gas, Phys. Rev. Lett. 102, 126402 (2009)

2009
[23]

Azadi, N

S. Azadi, N. D. Drummond, and S. M. Vinko, Quan- tum Monte Carlo study of the phase diagram of the two-dimensional uniform electron liquid (2024), arXiv:2405.00425 [cond-mat.str-el]

2024
[24]

Smith, Y

C. Smith, Y. Chen, R. Levy, Y. Yang, M. A. Morales, and S. Zhang, Ground state phases of the two-dimension electron gas with a unified variational approach (2024), arXiv:2405.19397 [cond-mat.str-el]

2024 arXiv
[25]

D. J. Thouless, Exchange in solid 3He and the Heisenberg Hamiltonian, Proceedings of the Physical Society 86, 893 (1965)

1965
[26]

Roger, J

M. Roger, J. H. Hetherington, and J. M. Delrieu, Mag- netism in solid 3He, Rev. Mod. Phys. 55, 1 (1983)

1983
[27]

Roger, Multiple exchange in 3He and in the Wigner solid, Phys

M. Roger, Multiple exchange in 3He and in the Wigner solid, Phys. Rev. B 30, 6432 (1984)

1984
[28]

Katano and D

M. Katano and D. S. Hirashima, Multiple-spin exchange 6 in a two-dimensional Wigner crystal, Phys. Rev. B 62, 2573 (2000)

2000
[29]

Ashizawa and D

H. Ashizawa and D. S. Hirashima, Wkb calculation of multiple spin exchange in monolayer solid 3He, Phys. Rev. B 62, 9413 (2000)

2000
[30]

C. N. Sudip Chakravarty, Steven Kivelson and K. Voelker, Wigner glass, spin liquids and the metal- insulator transition, Philosophical Magazine B 79, 859 (1999), https://doi.org/10.1080/13642819908214845

1999 doi
[31]

Voelker and S

K. Voelker and S. Chakravarty, Multiparticle ring ex- change in the Wigner glass and its possible relevance to strongly interacting two-dimensional electron systems in the presence of disorder, Phys. Rev. B 64, 235125 (2001)

2001
[32]

Bernu, L

B. Bernu, L. Cˆ andido, and D. M. Ceperley, Exchange frequencies in the 2d Wigner crystal, Phys. Rev. Lett. 86, 870 (2001)

2001
[33]

Misguich, B

G. Misguich, B. Bernu, C. Lhuillier, and C. Waldtmann, Spin liquid in the multiple-spin exchange model on the triangular lattice: 3He on graphite, Phys. Rev. Lett. 81, 1098 (1998); G. Misguich, C. Lhuillier, B. Bernu, and C. Waldtmann, Spin-liquid phase of the multiple-spin ex...

1998
[34]

Depending on the nature of the moir´ e potential, the WC state may be identified via a metal-insulator (or Mott) transition, as in the case of ν = 1 on the triangular lat- tice, or via the breaking of a discrete symmetry of the external potential, like in the case of ν = 1 /3 ...
[35]

Coleman, The uses of instantons, in The Whys of Subnuclear Physics, edited by A

S. Coleman, The uses of instantons, in The Whys of Subnuclear Physics, edited by A. Zichichi (Springer US, Boston, MA, 1979) pp. 805–941

1979
[36]

One can imagine that rs is changed, for ex- ample, by tuning the electron charge e2 or ℏ (as opposed to tuning the density n, as is usually assumed)

To avoid confusion: we assume the moir´ e potential re- mains commensurate with the triangular lattice WC as rs is varied. One can imagine that rs is changed, for ex- ample, by tuning the electron charge e2 or ℏ (as opposed to tuning the density n, as is usually assumed)
[37]

K.-S. Kim, I. Esterlis, C. Murthy, and S. A. Kivelson, Dynamical defects in a two-dimensional Wigner crystal: Self-doping and kinetic magnetism, Phys. Rev. B 109, 235130 (2024)

2024
[38]

In the deep moir´ e limit corresponding to exceedingly large values of ϵ, a Hubbard model approach the problem is likely to be more appropriate [51]
[39]

Bernu, C

B. Bernu, C. Lhuillier, and L. Pierre, Signature of N´ eel order in exact spectra of quantum antiferromagnets on fi- nite lattices, Phys. Rev. Lett. 69, 2590 (1992); B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Exact spec- tra, spin susceptibilities, and order parame...

1992
[40]

Capriotti, A

L. Capriotti, A. E. Trumper, and S. Sorella, Long-range N´ eel order in the triangular Heisenberg model, Phys. Rev. Lett. 82, 3899 (1999)

1999
[41]

S. R. White and A. L. Chernyshev, Ne´ el order in square and triangular lattice Heisenberg models, Phys. Rev. Lett. 99, 127004 (2007)

2007
[42]

Bonsall and A

L. Bonsall and A. A. Maradudin, Some static and dy- namical properties of a two-dimensional Wigner crystal, Phys. Rev. B 15, 1959 (1977)

1977
[43]

Yang (private communication)

Y. Yang (private communication)
[44]

Valenti, V

A. Valenti, V. Calvera, Y. Yang, M. A. Morales, S. A. Kivelson, I. Esterlis, and S. Zhang, Critical gate distance for Wigner crystallization in the two-dimensional electron gas (2025), arXiv:2501.16430 [cond-mat.str-el]

2025
[45]

K.-S. Kim, C. Murthy, A. Pandey, and S. A. Kivelson, Interstitial-induced ferromagnetism in a two-dimensional Wigner crystal, Phys. Rev. Lett. 129, 227202 (2022)

2022
[46]

Kim and V

K.-S. Kim and V. Elser, Itinerant ferromagnetism from one-dimensional mobility (2025), arXiv:2412.03638 [cond-mat.str-el]

2025 arXiv
[47]

Morera, M

I. Morera, M. Kan´ asz-Nagy, T. Smolenski, L. Ciorciaro, A. Imamo˘ glu, and E. Demler, High-temperature kinetic magnetism in triangular lattices, Phys. Rev. Res. 5, L022048 (2023)

2023
[48]

Morera and E

I. Morera and E. Demler, Itinerant magnetism and mag- netic polarons in the triangular lattice hubbard model (2024), arXiv:2402.14074 [cond-mat.str-el]

2024 arXiv
[49]

Y. Tang, L. LI, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan, and K. F. Mak, Simulation of Hubbard model physics in WSe2/WS2 moir´ e superlattices, Nature579, 353 (2020)

2020
[50]

Ciorciaro, T

L. Ciorciaro, T. Smole´ nski, I. Morera, N. Kiper, S. Hies- tand, M. Kroner, Y. Zhang, K. Watanabe, T. Taniguchi, E. Demler, and A. ˙Imamo˘ glu, Kinetic magnetism in tri- angular moir´ e materials, Nature623, 509 (2023)

2023
[51]

Z. Tao, W. Zhao, B. Shen, T. Li, P. Kn¨ uppel, K. Watan- abe, T. Taniguchi, J. Shan, and K. F. Mak, Observation of spin polarons in a frustrated moir´ e Hubbard system, Nature Physics 20, 783 (2024)

2024
[52]

F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub- bard model physics in transition metal dichalcogenide moir´ e bands, Phys. Rev. Lett.121, 026402 (2018)

2018

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu- migeta, M. Blei, J. D. Carlstr¨ om, K. Watanabe, T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and F. Wang, Mott and generalized Wigner crystal states in WSe2/WS2 moir´ e superlattices, Nature579, 359 (2020)

2020

[2] [2]

Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated in- sulating states at fractional fillings of moir´ e superlattices, Nature 587, 214 (2020)

2020

[3] [3]

Huang, T

X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, S.- F. Shi, and Y.-T. Cui, Correlated insulating states at fractional fillings of the WS2/WSe2 moir´ e lattice, Nature Physics 17, 715 (2021)

work page 2021

[4] [4]

C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu, K. Watanabe, T. Taniguchi, J. C. Hone, L. Fu, J. Shan, and K. F. Mak, Stripe phases in WSe 2/WS2 moir´ e su- perlattices, Nature Materials 20, 940 (2021)

work page 2021

[5] [5]

kinetic magnetism

lattices have been observed, respectively. Given the relative simplicity of the approach, an extension of the present calculations to this broader class of moir´ e WCs certainly seems warranted. Various generalizations that are relevant to real ex- perimental devices are also easily incorporated in the semiclassical approach. These include screening of th...

work page

[6] [6]

H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe,et al., Imaging two-dimensional generalized Wigner crystals, Nature 597, 650 (2021)

work page 2021

[7] [7]

T. Li, J. Zhu, Y. Tang, K. Watanabe, T. Taniguchi, V. Elser, J. Shan, and K. F. Mak, Charge-order-enhanced capacitance in semiconductor moir´ e superlattices, Nature Nanotechnology 16, 1068 (2021)

work page 2021

[8] [8]

H. Li, Z. Xiang, E. Regan, W. Zhao, R. Sailus, R. Baner- jee, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, Mapping charge excita- tions in generalized Wigner crystals, Nature Nanotech- nology 19, 618 (2024)

work page 2024