Pith. sign in

REVIEW 5 minor 71 references

Lecture Notes: The two-dimensional electron Wigner crystal -- What's old and what's new?

T0 review · 0 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read The two-dimensional Wigner crystal is real, and one Einstein-phonon picture explains its energy, melting near r_s≈34, ferromagnetic spin order, and Berry-curvature variants.

desk verdict A clear, honest set of lecture notes that makes no new claims but gives a reliable pedagogical map of the 2D Wigner crystal field; worth refereeing as a review, not as a research paper. read the letter →

arxiv 2607.13933 v1 pith:GYR7TPRV submitted 2026-07-15 cond-mat.str-el cond-mat.mes-hall

Brian Skinner This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hall
keywords Wignercrystaltwo-dimensionalelectrongasquantummeltingEinsteinphononringexchangeBerrycurvatureanomalousHallmicroemulsion
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's aim is to show that the zero-field, two-dimensional Wigner crystal — long a textbook idea but convincingly imaged only in the last few years — is now an experimentally grounded phase whose behavior can be organized by one semiclassical picture. It argues that treating each electron as a quantum harmonic oscillator in the Coulomb well made by its neighbors reproduces the crystal's quantum energy correction to about 8%, and that applying the empirical Lindemann criterion places quantum melting at r_s ≈ 34, matching quantum Monte Carlo. It then argues that the crystal's spin order is ferromagnetic deep in the solid because three-electron ring exchange has smaller action than two-electron exchange, that the liquid-solid transition must pass through microemulsions rather than a direct first-order jump, and that the crystal can self-dope with mobile vacancies or interstitials to become a metallic electron crystal. Finally, it argues that Berry curvature in materials like rhombohedral graphene changes the crystal qualitatively, producing anomalous Hall crystals, spontaneously rotating 'halo' wave packets, and a chiral three-spin exchange term.

What carries the argument

The workhorse is the Einstein-phonon approximation: replace the Wigner crystal by independent 2D harmonic oscillators, each electron confined by V(r) ≈ (1/2)mω²r² with ω set by the Coulomb energy scale, so the ground state is a Gaussian packet of width ∼a_B r_s^{3/4} and zero-point energy ℏω. This yields the energy series and the Lindemann-based melting estimate. For spin, the machinery is multiparticle ring exchange with imaginary action S_n: exchange amplitudes J_a ∼ exp(iS_a/ℏ), and the counterintuitive ordering S_3 < S_2 selects ferromagnetism. With Berry curvature, the same tunneling trajectories enclose a Berry flux Φ that turns the three-spin exchange into a chiral term Jχ Σ S_i·(S_j×

What would settle it

A clean two-dimensional electron system tuned through the zero-field freezing transition: if the melting density corresponds to an interaction parameter r_s clearly outside the 30–40 window, the paper's Lindemann-based phase-boundary estimate loses its quantitative support.

Watch

Extended reading notes

Core claim

The central claim of these lectures is that the Wigner crystal of two-dimensional electrons, proposed in 1934 and only directly imaged in zero magnetic field within the last few years, is a real phase with a coherent, mostly old-fashioned theory. The notes argue that the 'Einstein phonon' description — each electron as an independent harmonic oscillator in the quadratic potential created by its neighbors — is quantitatively trustworthy: it gives the zero-point correction to the crystal energy within about 8% of the full phonon calculation, predicts a wave-packet size ∼ a_B r_s^{3/4}, and, combined with a Lindemann criterion η_c = 0.23, places the zero-temperature melting transition at r_s ≈

Load-bearing premise

The quantitative phase-boundary estimate (r_s≈34) rests on assuming that the empirical Lindemann threshold η_c=0.23, calibrated from classical thermal melting, applies unchanged to quantum zero-temperature melting driven by zero-point fluctuations.

Editorial extensions

If this is right

  • If the Lindemann estimate is right, zero-field Wigner crystallization in any clean 2D system should occur near r_s≈34; experiments can look for the predicted wave-packet size scaling ∼ r_s^{3/4} through tunneling or compressibility.
  • Deep in the Wigner crystal the spin sector is governed by ring exchange, making the ground state ferromagnetic; only near the melting transition do competing terms open a narrow antiferromagnetic window.
  • The self-doping instability means that for r_s≲70 a pinned Wigner lattice can still conduct through mobile vacancies and interstitials, explaining 'metallic electron crystal' behavior, including the sign-changing Hall effect seen in rhombohedral graphene.
  • Berry curvature can make the crystal's insulating bulk coexist with a dissipationless, quantized Hall edge, so a WC material can masquerade as a quantum Hall state in transport while looking like a triangular lattice in STM.
  • Berry curvature converts the three-spin ring exchange into a chiral scalar-chirality interaction, which can produce non-coplanar spin textures or spin-liquid behavior if the Heisenberg term is antiferromagnetic.

Reading between the lines

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

  • Inference: The Lindemann-transfer assumption could be tested directly in classical 2D colloidal or dusty-plasma crystals with tunable screening, where the melting threshold η_c is measured, and then compared with quantum Monte Carlo estimates; if η_c differs in the deep-quantum regime, the r_s≈34 number would need revision.
  • Inference: The microemulsion proof for 1<α<2 suggests that other long-range-interacting 2D systems with softer interactions (e.g., dipolar gases with α=3) might not require microemulsions; observing the absence of intermediate phases in such systems would sharpen the role of Coulomb frustration.
  • Inference: The chiral three-spin term implies that Berry-curvature-rich Wigner crystals could exhibit an orbital/spin chiral ground state even at zero magnetic field; a concrete experimental search would be to look for a spontaneous Hall or Kerr signal developing below the crystallization temperature in a clean TMD or rhombohedral graphene device.
  • Inference: The anomalous Hall crystal scenario suggests that some existing 'quantum anomalous Hall' observations in moiré materials may actually be Wigner crystals with a topological edge rather than filled Chern bands; simultaneous real-space imaging and transport would distinguish them.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. These lecture notes (arXiv:2607.13933) survey the zero-field two-dimensional electron Wigner crystal (WC), combining "old" jellium-model physics with "new" Berry-curvature effects. Section 1 develops the classical/quantum energy of the WC, the semiclassical Einstein-phonon estimate and Lindemann melting criterion (r_s,c ≈ 34, with QMC window 30–40), the liquid-solid transition and Coulomb-frustrated microemulsions, experimental signatures (STM, umklapp scattering, pinning, negative compressibility), spin ordering via multi-electron ring exchange, and the self-doping/metallic electron crystal proposal. Section 2 reviews Berry-curvature modifications: anomalous Hall crystals, ℓ = 1 / halo Wigner crystals, and chiral three-spin exchange terms. The notes are explicitly pedagogical and advance no new predictive claim; their aim is to provide an accurate, useful map of current WC physics.

Significance. If accurate, the notes fill a useful niche: a readable, up-to-date entry point to the WC literature that connects classic Wigner/Lindemann/ring-exchange results with recent STM imaging and rhombohedral-graphene experiments. The manuscript's main strength is its intellectual honesty. Section 1.2 labels the Einstein-phonon/Lindemann route "cheating" and anchors the phase boundary with QMC r_s = 30–40; Section 1.3.1 explicitly flags the α = 1 microemulsion case as inconclusive and defers to Ref. [25]; Section 2.5 lists omitted topics without overclaiming. These self-imposed caveats make the pedagogical claims trustworthy. I find no load-bearing technical error, and the central claim — that the notes provide an accurate map of current zero-field WC physics — is defensible on the evidence presented.

minor comments (5)
  1. [Sec. 1.3.1 / Eq. (12)] The title "proof that something like microemulsions must exist" is stronger than the displayed derivation, which is self-admittedly inconclusive at α = 1, the actual Coulomb case. The text handles this by deferring to Ref. [25], so the lecture notes' conclusion is supported by the literature; please make the scope of the "proof" explicit at the start of the subsection (or soften the title) so casual readers do not take the sketch as the full Coulomb proof.
  2. [Sec. 1.2 / Eq. (6)] The step from Eq. (6) to r_s = 34 uses r_s = 1/(√π n a_B²) and η_c = 0.23 plus numerical prefactors, but the text omits the √π factor. Since this is a semi-empirical estimate, please show the one-line algebra (√⟨r²⟩ ≈ a_B r_s^{3/4} = η_c n^{-1/2} = η_c √π a_B r_s) so the quoted number is reproducible.
  3. [Sec. 1.3 / Fig. 6] In the re-entrance paragraph, "as depicted in Fig. dipoles(b)" is a broken reference; it should be Fig. 6(b). The phrase suggests a missing label substitution from an earlier draft and should be corrected.
  4. [Sec. 2.4 / Eq. (21)] Typo: "three neighboring spins i, j, know looks like" should read "i, j, k now looks like". In the same paragraph, ensure the relation P_ijk = P_ij P_jk is consistent with the inverse notation in Eq. (13) or briefly explain the convention.
  5. [Sec. 1.3.1] The note states that the subsection is a near-verbatim repetition of an argument that the author "wrote more or less this same text" in Ref. [19]. For a journal version, please verify that this reuse complies with the journal's prior-publication policy and that permission/citation is sufficient; a footnote acknowledging the source already helps.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the notes are an independent review, and the few self-cited results are used as external literature rather than as inputs that define the derivations.

full rationale

The paper is a self-described lecture-note review ('a smattering of old and new ideas'), not a derivation of new quantitative results. The nearest candidates for circularity are explicitly flagged limitations. In Sec. 1.2, the Lindemann-criterion estimate of r_s=34 is presented as the 'cheating way', with η_c=0.23 taken from an external empirical range and the result immediately cross-checked against independent QMC values r_s=30-40; it is a pedagogical consistency check, not a fitted input renamed as a prediction. In Sec. 1.3.1, the microemulsion proof is explicitly attributed to Spivak/Kivelson, and the notes concede that the α→1 Coulomb case 'is inconclusive' and defer completion to Ref. [25]; the self-repetition note ('I also wrote more or less this same text in Ref. [19]') concerns presentation, not the derivation. The self-citations (e.g., Refs. [11,19,22,23,41,55]) are used as pointers to prior work with stated physical mechanisms; none is used as an input whose output is then claimed as a new prediction. The gapless-material claim in Sec. 2.1 rests on Ref. [41] by the same author, but it is background motivation, is mechanism-based and externally falsifiable, and does not reduce the paper's central survey to a self-citation chain. The Berry-curvature sections review recent overlapping and independent theory and experiment without making new predictions. Therefore the derivation chain is self-contained with respect to its pedagogical claims, and no circular step is present.

Assumptions & free parameters 1 free parameters · 6 assumptions · 3 invented entities

Since the paper is a review, the ledger captures mainly the semiclassical and phase-transition toolkit it teaches: jellium background, triangular lattice, unscreened Coulomb interaction, Lindemann criterion, QMC melting window, and cited band-structure models. The anomalous Hall crystal, halo WC, and chiral spin term are reviewed rather than newly proposed here, but they are the concrete new-phase ideas the notes emphasize.

free parameters (1)
  • Lindemann threshold η_c = 0.23 (empirical range 0.21–0.25)
    Used in Sec. 1.2 to convert the Einstein-phonon wave-packet width into a critical r_s = 34. The criterion is semi-empirical and calibrated from melting behavior in other 2D systems, not derived for the quantum zero-field WC.
assumptions (6)
  • domain assumption Uniform neutralizing positive jellium background; electrons are confined to a strict 2D plane.
    The entire WC/FL discussion in Secs. 1.1–1.6 uses the jellium model; gating and image-charge effects are treated separately in Sec. 1.3.
  • domain assumption The Wigner crystal ground state has a triangular lattice.
    Stated in Sec. 1.1 as the arrangement maximizing electron separation; no derivation is given in the notes.
  • domain assumption The unscreened Coulomb interaction V(r) = e²/r is the relevant interaction in the zero-gate limit.
    Defines E_C and the WC stability argument in Secs. 1.1–1.2; gate screening is added as a modification in Sec. 1.3.
  • domain assumption Lindemann criterion applies to quantum melting of the WC with η_c ≈ 0.23.
    Used in Sec. 1.2 to estimate r_s,c = 34. The criterion is empirical and imported from prior literature, not derived in the notes.
  • domain assumption Quantum Monte Carlo results for the 2D WC-FL transition (r_s = 30–40) are correct.
    Cited in Sec. 1.2 from Refs. [9,10]; the notes do not independently verify these numerical phase boundaries.
  • domain assumption For the microemulsion proof, the interaction has the form k/r^α with 1 < α < 2; the α = 1 case is deferred to Ref. [25].
    Sec. 1.3.1 and Eq. (12) require α > 1 for the negative δf result; the actual 2D Coulomb case α = 1 is explicitly left to an external reference.
invented entities (3)
  • Anomalous Hall crystal independent evidence
    purpose: A WC with a nonzero Chern number, giving quantized Hall response and dissipationless edge transport without magnetic field.
    Presented in Sec. 2.2 as a revived idea from Tesanovic and Halperin [48] and recent experiments in twisted TMDs and multilayer graphene; not invented in these notes.
  • ℓ = 1 / halo Wigner crystal
    purpose: WC whose electron wave packets carry nonzero angular momentum due to Berry flux through the packet.
    Discussed in Sec. 2.3 and attributed to Refs. [41,55,56]; no direct experimental confirmation is cited.
  • Chiral three-spin term from Berry phases
    purpose: A scalar spin chirality term S_i · (S_j × S_k) in the WC spin Hamiltonian, favoring noncoplanar spin textures.
    Introduced pedagogically in Sec. 2.4 from Refs. [55,57]; observational consequences remain open.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lecture Notes: The two-dimensional electron Wigner crystal -- What's old and what's new?." pith.science (2026). https://pith.science/paper/GYR7TPRV

@misc{pith2026260713933,
  author       = {Pith},
  title        = {Pith review of: Lecture Notes: The two-dimensional electron Wigner crystal -- What's old and what's new?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYR7TPRV}},
  note         = {Machine review of arXiv:2607.13933}
}
read the original abstract

These are lecture notes created for a short lecture series at the 2026 CTEQ Summer School at Penn State. They are written in a conversational and informal style. The goal of these notes is to introduce and review a smattering of old and new ideas about the Wigner crystal (the solid phase of the two-dimensional electron system) in the context of recent experiments. Particular emphasis is given to the semiclassical description of the Wigner crystal, its quantum melting transition, its spin order, and the ways in which the Wigner crystal can be modified by Berry curvature.

Figures

Figures reproduced from arXiv: 2607.13933 by the authors.

Figure 1
Figure 1. (a) A cartoon picture of a Wigner crystal, in which electron wave packets are arranged in a triangular lattice. (b) An experimental image of a Wigner crystal in bilayer MoSe2, taken using scanning tunneling microscopy [2]. where aB = ℏ 2 ε me2 (2) is the “effective Bohr radius”, with ε the dielectric constant and m the electron mass. (That factor of π in Eq. (1) is basically there for historical reasons: it comes fr… view at source ↗
Figure 2
Figure 2. A gated 2DES is energetically equivalent to the jellium model 2DES plus the energy of a uniform plane capacitor. This pictorial equality is correct as long as the gate is not too close to the plane of the 2DES, as we discuss in Sec. 1.3. 1.2 Semiclassical considerations: collective modes and quantum corrections to the energy So the lowest-order description of a WC is that it is basically a classical object,4 in the … view at source ↗
Figure 3
Figure 3. Schematic phase diagram of the 2DES. But the “Einstein phonon” picture offers a remarkably simple way of thinking about the quan￾tum melting transition by using the semi-empirical “Lindemann criterion”. The Lindemann crite￾rion says that a crystal (of any kind) melts when the root-mean square fluctuations of a given atom about its equilibrium position become larger than a critical fraction ηc of the lattice constant… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: (a) In the Maxwell construction for phase coexistence, a system can lower its free energy relative to either the uniform solid or uniform liquid phases by making a phase mixture in which solid domains having density n1 coexist with liquid domains having density n2. The…
Figure 5
Figure 5. Figure 5: (a)-(b) Schematic of a microemulsion-type state (from Ref. [17]), where solid and liquid phases are intermixed. The two phases form stripes (a) when they are present in similar amounts, and bubbles (b) of the minority phase otherwise. (c)-(e) show scanning tunneling mi…
Figure 6
Figure 6. Figure 6: (a) When the distance n −1/2 between neighboring electrons (black points) is large compared to the distance d from the metal gate, electrons are essentially dipoles due to the screening effect of their image charges (white points) in the gate. Their interaction V (r) ∼…
Figure 7
Figure 7. Figure 7: “Pinning” behavior in the I-V curves of a 2DES in the WC regime, shown for experiments in (a) ZnO [27] and (b) rhombohedral six-layer graphene [28]. The I-V curves of a WC are often hysteretic (i.e., once the current is flowing it persists to a smaller value of the ele…
Figure 8
Figure 8. Figure 8: A few semiclassical exchange paths for the WC, taken from Ref. [35]. The number listed is the (imaginary) action associated with the exchange in units of ℏ √ rs. But in a full WC, there are many exchange processes that can happen, beyond just a nearest￾neighbor exchang…
Figure 9
Figure 9. Figure 9: An interstitial defect (a) and a vacancy (b) in the Wigner lattice. Taken from [36]. Creating such a (localized) defect is associated with a finite electrostatic energy E0 ∼ e 2n 1/2 , which is positive. For a classical system at zero temperature, these defects are fix…
Figure 10
Figure 10. Figure 10: (a) The conduction and valence bands in rhombohedral 5-layer graphene for different values of the interlayer potential (from Ref. [44]). (b) The shape of the conduction band in rhombohedral 6-layer graphene for different value sof the interlayer potential (from Ref. […
Figure 11
Figure 11. Figure 11: The energy E of an electron wave packet as a function of the Berry flux Φ through its interior for different values of the angular momentum ℓ (figure from Ref. [55]). This phenomenon was called an “ℓ = 1 WC” in Ref. [41] and a “halo WC” in Ref. [56]. 2.4 Spin ordering…
Figure 12
Figure 12. Figure 12: Tunneling trajectories associated with three-electron ring exchange, shown (a) in real space and (b) in k-space (from Ref. [55]). Here you should think of the “momentum” being plotted as an imaginary quantity, k (⃗r) = r 2m ℏ 2 (E − U(⃗r)), (20) since the electrons ar…
Figure 13
Figure 13. Figure 13: The longitudinal resistance ρxx of a 2DES in ZnO [27] is plotted as a function of a perpendicular magnetic field Bz. The electron density for this curve corresponds to rs ≈ 33, and at B = 0 the state resembles a WC, with large resistance and strong pinning behavior in…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 39 canonical work pages

  1. [19]

    Upper bound on the window of density occupied by microemulsion phases in two-dimensional electron systems,

    S. Joy and B. Skinner, “Upper bound on the window of density occupied by microemulsion phases in two-dimensional electron systems,”Phys. Rev. B108(Dec, 2023) L241110. https://link.aps.org/doi/10.1103/PhysRevB.108.L241110

    work page doi:10.1103/physrevb.108.l241110 2023

  2. [25]

    Universal aspects of coulomb-frustrated phase separation,

    R. Jamei, S. Kivelson, and B. Spivak, “Universal aspects of coulomb-frustrated phase separation,” Phys. Rev. Lett.94(Feb, 2005) 056805. https://link.aps.org/doi/10.1103/PhysRevLett.94.056805

    work page doi:10.1103/physrevlett.94.056805 2005

  3. [1]

    On the interaction of electrons in metals,

    E. Wigner, “On the interaction of electrons in metals,”Phys. Rev.46(Dec, 1934) 1002–1011. https://link.aps.org/doi/10.1103/PhysRev.46.1002

    doi:10.1103/physrev.46.1002 1934

  4. [2]

    Imaging quantum melting in a disordered 2d wigner solid,

    Z. Xiang, H. Li, J. Xiao, M. H. Naik, Z. Ge, Z. He, S. Chen, J. Nie, S. Li, Y. Jiang, R. Sailus, R. Banerjee, T. Taniguchi, K. Watanabe, S. Tongay, S. G. Louie, M. F. Crommie, and F. Wang, “Imaging quantum melting in a disordered 2d wigner solid,”Science388no. 6748, (2025) 736–740. https://www.science.org/doi/abs/10.1126/science.ado7136

    doi:10.1126/science.ado7136 2025

  5. [3]

    Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons,

    C. C. Grimes and G. Adams, “Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons,”Phys. Rev. Lett.42(Mar, 1979) 795–798. https://link.aps.org/doi/10.1103/PhysRevLett.42.795

    work page doi:10.1103/physrevlett.42.795 1979

  6. [4]

    Liquid-solid transition and the fractional quantum-hall effect,

    P. K. Lam and S. M. Girvin, “Liquid-solid transition and the fractional quantum-hall effect,”Phys. Rev. B30(Jul, 1984) 473(R)–475(R).https://link.aps.org/doi/10.1103/PhysRevB.30.473

    work page doi:10.1103/physrevb.30.473 1984

  7. [5]

    Direct observation of a magnetic-field-induced Wigner crystal,

    Y.-C. Tsui, M. He, Y. Hu, E. Lake, T. Wang, K. Watanabe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, “Direct observation of a magnetic-field-induced Wigner crystal,”Nature628no. 8007, (Apr., 2024) 287–292.https://www.nature.com/articles/s41586-024-07212-7

    2024

  8. [6]

    Defects in the two-dimensional electron solid and implications for melting,

    D. S. Fisher, B. I. Halperin, and R. Morf, “Defects in the two-dimensional electron solid and implications for melting,”Phys. Rev. B20(Dec, 1979) 4692–4712. https://link.aps.org/doi/10.1103/PhysRevB.20.4692

    work page doi:10.1103/physrevb.20.4692 1979

Show all 71 references
  1. [7]

    Some static and dynamical properties of a two-dimensional wigner crystal,

    L. Bonsall and A. A. Maradudin, “Some static and dynamical properties of a two-dimensional wigner crystal,”Phys. Rev. B15(Feb, 1977) 1959–1973. https://link.aps.org/doi/10.1103/PhysRevB.15.1959

    1977 doi

  2. [8]

    Die plancksche theorie der strahlung und die theorie der spezifischen w¨ arme,

    A. Einstein, “Die plancksche theorie der strahlung und die theorie der spezifischen w¨ arme,”Annalen der Physik327no. 1, (1907) 180–190

    1907

  3. [9]

    Phase diagram of the low-density two-dimensional homogeneous electron gas,

    N. D. Drummond and R. J. Needs, “Phase diagram of the low-density two-dimensional homogeneous electron gas,”Phys. Rev. Lett.102(Mar, 2009) 126402. https://link.aps.org/doi/10.1103/PhysRevLett.102.126402

    2009 doi

  4. [10]

    Quantum monte carlo study of the phase diagram of the two-dimensional uniform electron liquid,

    S. Azadi, N. D. Drummond, and S. M. Vinko, “Quantum monte carlo study of the phase diagram of the two-dimensional uniform electron liquid,”Phys. Rev. B110(Dec, 2024) 245145. https://link.aps.org/doi/10.1103/PhysRevB.110.245145

    2024 doi

  5. [11]

    Universal behavior of repulsive two-dimensional fermions in the vicinity of the quantum freezing point,

    M. Babadi, B. Skinner, M. M. Fogler, and E. Demler, “Universal behavior of repulsive two-dimensional fermions in the vicinity of the quantum freezing point,”EPL (Europhysics Letters) 103no. 1, (Jul, 2013) 16002.https://doi.org/10.1209/0295-5075/103/16002

    2013 doi

  6. [12]

    Unrestricted hartree-fock theory of wigner crystals,

    J. R. Trail, M. D. Towler, and R. J. Needs, “Unrestricted hartree-fock theory of wigner crystals,” Phys. Rev. B68(Jul, 2003) 045107.https://link.aps.org/doi/10.1103/PhysRevB.68.045107

    2003 doi

  7. [13]

    Hartree-fock phase diagram of the two-dimensional electron gas,

    B. Bernu, F. Delyon, M. Holzmann, and L. Baguet, “Hartree-fock phase diagram of the two-dimensional electron gas,”Phys. Rev. B84(Sep, 2011) 115115. https://link.aps.org/doi/10.1103/PhysRevB.84.115115

    2011 doi

  8. [14]

    Quantization of the hall conductance from density quantization alone,

    S. Kivelson and S. A. Trugman, “Quantization of the hall conductance from density quantization alone,”Phys. Rev. B33(Mar, 1986) 3629–3635. https://link.aps.org/doi/10.1103/PhysRevB.33.3629

    1986 doi

  9. [15]

    Phase separation in the two-dimensional electron liquid in mosfet’s,

    B. Spivak, “Phase separation in the two-dimensional electron liquid in mosfet’s,”Phys. Rev. B67 (Mar, 2003) 125205.https://link.aps.org/doi/10.1103/PhysRevB.67.125205

    2003 doi

  10. [16]

    Phases intermediate between a two-dimensional electron liquid and wigner crystal,

    B. Spivak and S. A. Kivelson, “Phases intermediate between a two-dimensional electron liquid and wigner crystal,”Phys. Rev. B70(Oct, 2004) 155114. https://link.aps.org/doi/10.1103/PhysRevB.70.155114. 23

    2004 doi

  11. [17]

    Ground state of a two-dimensional electron liquid in a weak magnetic field,

    M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, “Ground state of a two-dimensional electron liquid in a weak magnetic field,”Phys. Rev. B54(Jul, 1996) 1853–1871. https://link.aps.org/doi/10.1103/PhysRevB.54.1853

    1996 doi

  12. [18]

    An electronic microemulsion phase emerging from a quantum crystal-to-liquid transition,

    J. Sung, J. Wang, I. Esterlis, P. A. Volkov, G. Scuri, Y. Zhou, E. Brutschea, T. Taniguchi, K. Watanabe, Y. Yang,et al., “An electronic microemulsion phase emerging from a quantum crystal-to-liquid transition,”Nature Physics21no. 3, (2025) 437–443

    2025

  13. [20]

    Disorder-induced liquid-solid phase coexistence in two-dimensional electron systems,

    S. Joy and B. Skinner, “Disorder-induced liquid-solid phase coexistence in two-dimensional electron systems,”Phys. Rev. B113(May, 2026) L201117.https://link.aps.org/doi/10.1103/71l6-w8hl

    2026 doi

  14. [21]

    Is disorder a friend or a foe to melting of wigner-mott insulators?,

    M. Hammam, C. Lewandowski, V. Dobrosavljevic, and S. Joy, “Is disorder a friend or a foe to melting of wigner-mott insulators?,” 2025.https://arxiv.org/abs/2512.07932

    2025

  15. [22]

    Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas,

    B. Skinner and B. I. Shklovskii, “Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas,”Phys. Rev. B82(Oct, 2010) 155111. https://link.aps.org/doi/10.1103/PhysRevB.82.155111

    2010 doi

  16. [23]

    Simple variational method for calculating energy and quantum capacitance of an electron gas with screened interactions,

    B. Skinner and M. M. Fogler, “Simple variational method for calculating energy and quantum capacitance of an electron gas with screened interactions,”Phys. Rev. B82(Nov, 2010) 201306(R). https://link.aps.org/doi/10.1103/PhysRevB.82.201306

    2010 doi

  17. [24]

    Critical gate distance for wigner crystallization in the two-dimensional electron gas,

    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,”Physical Review Letters 135no. 16, (Oct., 2025) .http://dx.doi.org/10.1103/2qgp-v27h

    2025 doi

  18. [26]

    Signatures of wigner crystal of electrons in a monolayer semiconductor,

    T. Smole´ nski, P. E. Dolgirev, C. Kuhlenkamp, A. Popert, Y. Shimazaki, P. Back, X. Lu, M. Kroner, K. Watanabe, T. Taniguchi,et al., “Signatures of wigner crystal of electrons in a monolayer semiconductor,”Nature595no. 7865, (July, 2021) 53–57. https://doi.org/10.1038/s41586-0...

    2021 doi

  19. [27]

    Competing correlated states around the zero-field Wigner crystallization transition of electrons in two dimensions,

    J. Falson, I. Sodemann, B. Skinner, D. Tabrea, Y. Kozuka, A. Tsukazaki, M. Kawasaki, K. von Klitzing, and J. H. Smet, “Competing correlated states around the zero-field Wigner crystallization transition of electrons in two dimensions,”Nature Materials21no. 3, (Mar., 2022) 311–...

    2022 doi

  20. [28]

    Evidence of metallic wigner crystal in rhombohedral graphene,

    T. Han, J. P. Butler, S. Ye, Z. Hua, S. Dutta, Z. Hadjri, Z. Wu, J. Yang, J. Seo, P. Pattanakanvijit, E. Aitken, K. Watanabe, T. Taniguchi, P. Xiong, E. Zeldov, Z. Lu, R. Ashoori, and L. Ju, “Evidence of metallic wigner crystal in rhombohedral graphene,” 2026.https://arxiv.org...

    2026

  21. [29]

    Coulomb gap and variable range hopping in a pinned Wigner crystal,

    B. I. Shklovskii, “Coulomb gap and variable range hopping in a pinned Wigner crystal,”physica status solidi (c)1no. 1, (2004) 46–50. https://onlinelibrary.wiley.com/doi/abs/10.1002/pssc.200303642

    2004 doi

  22. [30]

    Negative compressibility of interacting two-dimensional electron and quasiparticle gases,

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Negative compressibility of interacting two-dimensional electron and quasiparticle gases,”Phys. Rev. Lett.68(Feb, 1992) 674–677. https://link.aps.org/doi/10.1103/PhysRevLett.68.674

    1992 doi

  23. [31]

    Compressibility of the two-dimensional electron gas: Measurements of the zero-field exchange energy and fractional quantum hall gap,

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Compressibility of the two-dimensional electron gas: Measurements of the zero-field exchange energy and fractional quantum hall gap,”Phys. Rev. B50 (Jul, 1994) 1760–1778.https://link.aps.org/doi/10.1103/PhysRevB.50.1760

    1994 doi

  24. [32]

    Density of localized states in the surface impurity band of a metal-insulator-semiconductor structure,

    M. S. Bello, E. I. Levin, B. I. Shklovskii, and A. L. Efros, “Density of localized states in the surface impurity band of a metal-insulator-semiconductor structure,”Sov. Phys. JETP53(1981) 822. 24

    1981

  25. [33]

    Where does magnetism come from?,

    Brian Skinner, “Where does magnetism come from?,” Apr., 2015. https://gravityandlevity.wordpress.com/2015/04/19/where-does-magnetism-come-from/

    2015

  26. [34]

    S. M. Girvin and K. Yang,Modern condensed matter physics. Cambridge University Press, 2019

    2019

  27. [35]

    Interstitial-Induced Ferromagnetism in a Two-Dimensional Wigner Crystal,

    K.-S. Kim, C. Murthy, A. Pandey, and S. A. Kivelson, “Interstitial-Induced Ferromagnetism in a Two-Dimensional Wigner Crystal,”Physical Review Letters129no. 22, (Nov., 2022) 227202. https://link.aps.org/doi/10.1103/PhysRevLett.129.227202

    2022 doi

  28. [36]

    Dynamical defects in a two-dimensional wigner crystal: Self-doping and kinetic magnetism,

    K.-S. Kim, I. Esterlis, C. Murthy, and S. A. Kivelson, “Dynamical defects in a two-dimensional wigner crystal: Self-doping and kinetic magnetism,”Physical Review B109no. 23, (2024) 235130

    2024

  29. [37]

    Transport evidence for wigner crystals in monolayer mote2,

    M. Zhang, Z. Wang, Y. Jiang, Y. Liu, K. Watanabe, T. Taniguchi, S. Liu, S. Lei, Y. Li, and Y. Xu, “Transport evidence for wigner crystals in monolayer mote2,” 2025. https://arxiv.org/abs/2506.20392

    2025 arXiv

  30. [38]

    Crystals caught doping: Metallic wigner crystals in rhombohedral graphene,

    J. Dong, T. Soejima, D. E. Parker, and A. Vishwanath, “Crystals caught doping: Metallic wigner crystals in rhombohedral graphene,” 2026.https://arxiv.org/abs/2604.00114

    2026

  31. [39]

    A review of the quantum hall effects in mgzno/zno heterostructures,

    J. Falson and M. Kawasaki, “A review of the quantum hall effects in mgzno/zno heterostructures,” Reports on Progress in Physics81no. 5, (2018) 056501

    2018

  32. [40]

    Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure,

    Y. Zhou, J. Sung, E. Brutschea, I. Esterlis, Y. Wang, G. Scuri, R. J. Gelly, H. Heo, T. Taniguchi, K. Watanabe, G. Zar´ and, M. D. Lukin, P. Kim, E. Demler, and H. Park, “Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure,”Nature595no. 7865, (July, 20...

    2021 doi

  33. [41]

    Wigner crystallization in bernal bilayer graphene,

    S. Joy and B. Skinner, “Wigner crystallization in bernal bilayer graphene,” 2023

    2023

  34. [42]

    Fractional quantum anomalous hall effect in multilayer graphene,

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, “Fractional quantum anomalous hall effect in multilayer graphene,”Nature626no. 8000, (2024) 759–764

    2024

  35. [43]

    Signatures of chiral superconductivity in rhombohedral graphene,

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Sharifi Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watanabe, T. Taniguchi, P. Xiong, D. M. Zumb¨ uhl, L. Fu, and L. Ju, “Signatures of chiral superconductivity in rh...

    2025

  36. [44]

    Parent berry curvature and the ideal anomalous hall crystal,

    T. Tan and T. Devakul, “Parent berry curvature and the ideal anomalous hall crystal,”Phys. Rev. X 14(Nov, 2024) 041040.https://link.aps.org/doi/10.1103/PhysRevX.14.041040

    2024 doi

  37. [45]

    Orbital multiferroicity in pentalayer rhombohedral graphene,

    T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, L. Fu, H. Park, and L. Ju, “Orbital multiferroicity in pentalayer rhombohedral graphene,”Nature623no. 7985, (Nov.,

  38. [46]

    Quantum geometry and the hidden scales in materials,

    N. Verma, P. J. W. Moll, T. Holder, and R. Queiroz, “Quantum geometry and the hidden scales in materials,”Nature Reviews Physics8no. 4, (Mar., 2026) 226–239. http://dx.doi.org/10.1038/s42254-026-00923-y

    2026 doi

  39. [47]

    Dawn of the topological age?,

    A. P. Ramirez and B. Skinner, “Dawn of the topological age?,”Physics Today73no. 9, (2020) 30–36

    2020

  40. [48]

    “hall crystal

    Z. Teˇ sanovi´ c, F. m. c. Axel, and B. I. Halperin, ““hall crystal” versus wigner crystal,”Phys. Rev. B 39(Apr, 1989) 8525–8551.https://link.aps.org/doi/10.1103/PhysRevB.39.8525

    1989 doi

  41. [49]

    Moir´ e-driven topological electronic crystals in twisted graphene,

    R. Su, D. Waters, B. Zhou, K. Watanabe, T. Taniguchi, Y.-H. Zhang, M. Yankowitz, and J. Folk, “Moir´ e-driven topological electronic crystals in twisted graphene,”Nature637no. 8048, (Jan., 2025) 1084–1089.https://www.nature.com/articles/s41586-024-08239-6

    2025

  42. [50]

    Signatures of fractional quantum anomalous Hall states in twisted MoTe2,

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, “Signatures of fractional quantum anomalous Hall states in twisted MoTe2,”Nature622no. 7981, (Oct., 2023) 63–68. https:...

    2023

  43. [51]

    Observation of fractionally quantized anomalous Hall effect,

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, “Observation of fractionally quantized anomalous Hall effect,”Nature622no. 798...

  44. [52]

    Anomalous Hall Crystals in Rhombohedral Multilayer Graphene. I. Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field,

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, “Anomalous Hall Crystals in Rhombohedral Multilayer Graphene. I. Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field,”Physical Review Letters133no. 20, (...

    2024 doi

  45. [53]

    74–79.https://www.nature.com/articles/s41586-023-06536-0

  46. [54]

    Stability of anomalous hall crystals in multilayer rhombohedral graphene,

    Z. Dong, A. S. Patri, and T. Senthil, “Stability of anomalous hall crystals in multilayer rhombohedral graphene,”Phys. Rev. B110(Nov, 2024) 205130. https://link.aps.org/doi/10.1103/PhysRevB.110.205130

    2024 doi

  47. [55]

    Anomalous Hall crystals in rhombohedral multilayer graphene. II. General mechanism and a minimal model,

    T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zaletel, A. Vishwanath, and D. E. Parker, “Anomalous Hall crystals in rhombohedral multilayer graphene. II. General mechanism and a minimal model,”Physical Review B110no. 20, (Nov., 2024) 205124. https://link.aps.org/doi/10.1103/Phy...

    2024 doi

  48. [56]

    λ-jellium model for the anomalous hall crystal,

    T. Soejima, J. Dong, A. Vishwanath, and D. E. Parker, “λ-jellium model for the anomalous hall crystal,”Phys. Rev. Lett.135(Oct, 2025) 186505.https://link.aps.org/doi/10.1103/x53d-12s6

    2025 doi

  49. [57]

    Chiral wigner crystal phases induced by berry curvature,

    S. Joy, L. Levitov, and B. Skinner, “Chiral wigner crystal phases induced by berry curvature,”Phys. Rev. Lett.135(Dec, 2025) 256502.https://link.aps.org/doi/10.1103/h5hy-jh6m

    2025 doi

  50. [58]

    Chiral stoner magnetism in dirac bands,

    Z. Dong and L. Levitov, “Chiral stoner magnetism in dirac bands,”Phys. Rev. B110(Sep, 2024) 104420.https://link.aps.org/doi/10.1103/PhysRevB.110.104420

    2024 doi

  51. [59]

    Exchange interactions of a Wigner crystal in a magnetic field and Berry curvature: Multiparticle tunneling through complex trajectories,

    K.-S. Kim, “Exchange interactions of a Wigner crystal in a magnetic field and Berry curvature: Multiparticle tunneling through complex trajectories,”Physical Review B113no. 14, (Apr., 2026) 144434.https://link.aps.org/doi/10.1103/lj9b-3myv

    2026 doi

  52. [60]

    Global phase diagram and quantum spin liquids in a spin-1/2 triangular antiferromagnet,

    S.-S. Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and K. Yang, “Global phase diagram and quantum spin liquids in a spin-1/2 triangular antiferromagnet,”Physical Review B96no. 7, (Aug., 2017) 075116. https://link.aps.org/doi/10.1103/PhysRevB.96.075116

    2017 doi

  53. [61]

    Spin chirality and fermion stirring in topological bands,

    A. Panigrahi, V. Poliakov, Z. Dong, and L. Levitov, “Spin chirality and fermion stirring in topological bands,” 2024

    2024

  54. [62]

    Imaging two-dimensional generalized Wigner crystals,

    H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, “Imaging two-dimensional generalized Wigner crystals,”Nature597no. 7878, (Sept., 2021) 650–654. https://www.nature.com/articl...

    2021

  55. [63]

    Correlated insulating states at fractional fillings of moir´ e superlattices,

    Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, “Correlated insulating states at fractional fillings of moir´ e superlattices,”Nature587no. 7833, (Nov.,

  56. [64]

    Stability, dynamical properties, and melting of a classical bilayer Wigner crystal,

    G. Goldoni and F. M. Peeters, “Stability, dynamical properties, and melting of a classical bilayer Wigner crystal,”Physical Review B53no. 8, (Feb., 1996) 4591–4603. https://link.aps.org/doi/10.1103/PhysRevB.53.4591

    1996 doi

  57. [65]

    A bilayer of Wigner crystal in the harmonic approximation,

    K. Esfarjani and Y. Kawazoe, “A bilayer of Wigner crystal in the harmonic approximation,”Journal of Physics: Condensed Matter7no. 36, (Sept., 1995) 7217. https://doi.org/10.1088/0953-8984/7/36/011

    1995 doi

  58. [66]

    Quasi-two-dimensional Wigner crystals with a compound lattice,

    Y. M. Vil’k and Y. P. Monarkha, “Quasi-two-dimensional Wigner crystals with a compound lattice,” Soviet Journal Low Temperature Physics10no. 8, (Aug., 1984) 465–466. https://doi.org/10.1063/10.0031167

    1984 doi

  59. [67]

    Magnetism from multiparticle ring exchange in moir´ e wigner crystals,

    I. Esterlis and A. Levchenko, “Magnetism from multiparticle ring exchange in moir´ e wigner crystals,” Phys. Rev. B111(May, 2025) L201115. https://link.aps.org/doi/10.1103/PhysRevB.111.L201115

    2025 doi

  60. [68]

    Quantum melting a Wigner crystal into Hall liquids,

    A. P. Reddy and L. Fu, “Quantum melting a Wigner crystal into Hall liquids,”Physical Review B113 no. 16, (Apr., 2026) L161403.https://link.aps.org/doi/10.1103/xt9w-8dg7. 27

    2026 doi

  61. [69]

    Electronic crystals in layered materials,

    Y. Zhou, I. Esterlis, and T. Smole´ nski, “Electronic crystals in layered materials,”npj 2D Materials and Applications(June, 2026) .https://www.nature.com/articles/s41699-026-00713-1. 26

    2026

  62. [2020]

    214–218.https://www.nature.com/articles/s41586-020-2868-6

  63. [2023]

    41–47.https://www.nature.com/articles/s41586-023-06572-w

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.