Pith. sign in

REVIEW 3 major objections 4 minor 74 references

A single dual action captures the full defect physics of two-dimensional Wigner crystals, from vacancy interactions to a vacancy-dependent phonon speed.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 15:18 UTC pith:BROI3XJH

load-bearing objection A useful duality-based framework for defects in 2D Wigner crystals with new analytic interactions, but the headline phonon-speed prediction rests on an unevaluated drag assumption. the 3 major comments →

arxiv 2607.18477 v1 pith:BROI3XJH submitted 2026-07-20 cond-mat.str-el

Theory of two-dimensional Wigner crystals with defects: Interactions, melting transitions and collective modes

Pawe{\l} Matus , Tobias Holder This is my paper
classification cond-mat.str-el
keywords Wigner crystaldefectsfracton-elasticity dualityvacancy Fermi liquidphonon dispersionmelting transitiondislocationsdisclinations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that vacancies, interstitials, dislocations, and disclinations in a two-dimensional Wigner crystal can all be described by one dual field theory built on elasticity-gauge duality. In that description, every defect type sources elastic and electromagnetic gauge fields, yielding analytic interaction energies between arbitrary defect pairs—a gap in previous work. The central new physical result is the metallic Wigner crystal: when vacancies self-dope the crystal, they form a Fermi liquid and the longitudinal phonon speed becomes a non-monotonic function of vacancy density, peaking at a specific density set by the electron-to-hole mass ratio. The same framework puts the standard incompressible assumption of KTHNY melting on firmer footing and predicts a modest drop in melting temperature upon metallicity. If correct, it gives experimentalists a concrete acoustic probe of the vacancy fluid.

Core claim

The paper's central claim is that the low-energy physics of charged 2D crystals with vacancies/interstitials, dislocations, and disclinations is completely encoded in the dual action (17), where the dislocation current J^disl couples to a tensor gauge field B, the disclination current J^dscl to a vector gauge field C, and the vacancy charge density to the electrostatic potential A. Solving the resulting equations produces the field profiles and analytic defect-pair interaction energies in Table IV, which differ strikingly from neutral crystals—e.g., the bulk stress around a point defect decays as r^-3 without gates and is screened exponentially with gates, while the shear stress decays as r^

What carries the argument

The central mechanical object is the elasticity-gauge duality (fracton-elasticity duality) transformation, which replaces the displacement field u_i by a pair of gauge fields B_iμ and C_μ. Dislocations appear as sources of B (the dislocation current J^disl), disclinations as sources of C (J^dscl), and vacancies/interstitials as scalar charges coupling to the electrostatic potential A. The dual action (17) is the generator of all static and dynamic results: the equations of motion that yield the field profiles in Tables I–III, the interaction energies in Table IV, and the hydrodynamic action (41) for the mixed vacancy-Fermi-liquid/Wigner-crystal system.

Load-bearing premise

The load-bearing premise is that at millikelvin temperatures electron-phonon collisions are so rare that vacancy velocities and lattice velocities do not equilibrate; the paper keeps the factor (1+m/m_h) in n_eff, D, and the couplings, and the predicted sharp maximum in phonon speed depends on it.

What would settle it

Measure the longitudinal phonon speed in a self-doped Wigner crystal (e.g., rhombohedral multilayer graphene) while tuning the vacancy density; if c_ph does not reach a maximum at ρ_v ≈ n₀/(1+m/m_h), or if it varies monotonically, the drag-free assumption (and Eq. 42) is wrong. A simpler check: if the crystal-to-hexatic melting temperature drops by much more than ~10% upon entering the metallic phase, the predicted softening is underestimated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Analytic energy formulas for every defect-pair combination (vacancy–vacancy, dislocation–vacancy, dislocation–dislocation, disclination–vacancy, disclination–dislocation, disclination–disclination) become available, allowing quantitative modeling of defect annealing, pinning, and proliferation in 2D charged crystals.
  • In a self-doped metallic Wigner crystal, the longitudinal phonon speed c_ph is a non-monotonic function of vacancy density with a maximum at ρ_v* = n₀/(1+m/m_h), so measuring c_ph yields the vacancy density and, in combination with independent density measurements, both effective masses.
  • The melting temperature of the crystal-to-hexatic transition drops by about 10% in the metallic phase compared with the pure crystal, though near the quantum melting point the two can look similar; this reconciles apparently conflicting experiments.
  • The framework carries over directly to other translation-symmetry-broken charged systems, including charge density waves and anomalous Hall crystals, where the same defect couplings and collective-mode structure apply.
  • In the hexatic phase, disclination interactions are dominated by the rotational Goldstone mode and are unaffected by the electronic (charged) nature of the crystal, so hexatic-to-liquid melting is governed by neutral-crystal physics.

Where Pith is reading between the lines

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

  • Because vacancies inherit the r^-3 (or exponentially screened) interaction found here rather than a bare Coulomb r^-1, the low-density vacancy fluid should remain a Fermi liquid at all accessible temperatures—no secondary Wigner crystallization of vacancies—a prediction that could be tested by heat-capacity or compressibility measurements.
  • The predicted non-monotonic c_ph(ρ_v) is, in effect, a built-in falsifier of the drag-free assumption: measuring the phonon speed as a function of doping in rhombohedral multilayer graphene would locate the maximum (confirming the (1+m/m_h) factor) or miss it (pointing to collision-drag cancellation).
  • The same duality construction should extend to finite magnetic fields, suggesting that the field theory can serve as a base for studying Wigner-crystal phonons under quantum Hall conditions or in moiré systems where self-doping has been observed.
  • Since the vacancy Fermi-liquid action in Eq. (41) is written in hydrodynamic form, the framework invites a direct computation of transport coefficients (resistivity, thermopower) of the metallic Wigner crystal, which would give further experimental contact beyond sound velocity.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops an elasticity-gauge duality (fracton-elasticity duality) description of a two-dimensional Wigner crystal with electrostatic interactions, including vacancies/interstitials, dislocations, and disclinations. From the dual action (Eq. 17), it derives the fields and interaction energies for all defect pairs (Tables I–IV), studies dislocation proliferation and the hexatic phase, and considers a metallic Wigner crystal with a dilute Fermi liquid of vacancies. The central new results are the analytical defect interaction formulas, the claim that the bulk modulus can be treated as effectively infinite for KTHNY melting of the pure crystal, and a predicted longitudinal phonon dispersion (Eq. 44) whose speed depends nonmonotonically on vacancy density, with a maximum at ρ_v* = n0/(1 + m/m_h). The paper proposes that measuring the sound speed can extract vacancy density and hole effective mass.

Significance. If the central derivations hold, the paper provides a substantial and largely analytic framework for defect physics in two-dimensional charged crystals, going beyond prior work that treated only dislocations or used purely numerical estimates. The defect interaction tables and the identification of the metallic Wigner crystal's long-wavelength modes are useful organizing results. The work also makes a concrete, falsifiable experimental proposal via the sound-speed dependence on vacancy density. Among its strengths are that the defect-sector calculations are parameter-free within the stated model (the only inputs being measured or independently computed parameters such as n0, m, ε, C_s, C_b, m_h), and that several earlier ad hoc assumptions (e.g., infinite bulk modulus in KTHNY) are shown to be justified. However, the most striking prediction—the nonmonotonic phonon speed and its maximum—rests on an adiabatic no-drag assumption that is asserted but not quantitatively established.

major comments (3)
  1. [§V (text below Eq. 42); Fig. 3]
  2. [§V, Eqs. (41)–(44); Supplemental Material S1]
  3. [§V (paragraph after Eq. 42)]
minor comments (4)
  1. [Eq. (44)]
  2. [Fig. 3]
  3. [Sec. IV, text near Eq. (38)]
  4. [Sec. V, text below Eq. (42)]

Circularity Check

0 steps flagged

No significant circularity: the defect-action derivation is self-contained, and the metallic-phonon prediction is a model consequence rather than a fitted or self-cited input.

full rationale

The central derivation begins with the explicit elastic + Coulomb action Eq. (4), then performs Hubbard-Stratonovich/gauge dualization (Eqs. (10)-(17)); defect fields and energies (Tables I-IV) are obtained by solving the equations of motion (20), not by assuming the answers. The metallic-sector action Eq. (41) is built in App. S1 by matching charge continuity and Euler equations of the vacancy Fermi liquid to a scalar field theory; coefficients in Eq. (42) are expressed in terms of externally supplied n0, rho_v, m_h, epsilon_F, and the longitudinal dispersions Eq. (44) are found by diagonalizing Eq. (43). No speed-of-sound data are used as input, so the c_ph(rho_v) maximum in Fig. 3 is not a renamed fit. The same-author citations (e.g., [56], and [55] for fracton details) are methodological pointers; the 2D dual action and all results used here are re-derived within this paper, so the argument does not lean on an unverified self-citation. The collision-drag caveat below Eq. (42) is a stated physical assumption: the paper explicitly notes that Refs. [69,70] would replace n_eff by n0 - rho_v in the drag-dominated regime and shifts the predicted maximum; that is an acknowledged external model-robustness risk, not an equivalence-by-construction circularity.

Axiom & Free-Parameter Ledger

11 free parameters · 7 axioms · 0 invented entities

No new particles, forces, or conserved quantities are postulated. The dual gauge fields B_iμ and C_μ come from the established fracton-elasticity duality, and the scalar field φ in the metallic phase is a hydrodynamic variable constructed from the Fermi-liquid model. The quantitative predictions depend on several experimentally inherited or hand-picked parameters, listed above.

free parameters (11)
  • n0 (electron density) = 0.3–0.5 ×10^12 cm^-2
    Taken from rhombohedral graphene experiment [8]; sets all length and energy scales in the model.
  • m (electron effective mass) = ≈3 m_e
    Input from [8].
  • epsilon_r (dielectric constant) = ≈4
    Input from [8]; used in V(k) and l_sc.
  • d (gate distance) = 25 nm
    Taken from [18]; controls screening length k* and the parameter alpha.
  • C_s (shear modulus) = 0.57 e^2 n0^2 a/(4πε)
    From lattice calculation [62]; used in defect energies, KTHNY T_c, and phonon speed.
  • C_b (bulk modulus) = 0.39 e^2 n0^2 a/(4πε)
    From [62]; same use as C_s.
  • m_h (hole effective mass) = 0.05 m
    From [18]; heavily influences n_eff, B, F, D and Eq. (44).
  • rho_v (vacancy density) = up to 0.15 n0 (variable in Fig. 3)
    Not measured independently in this paper; the speed-of-sound curve is plotted against it and proposed as a measurement target.
  • a_core (defect core size) = ~a
    Used in Eq. (27) to estimate single-vacancy energy; order-one unknown.
  • m1, m2, g, |psi0|^2 (hexatic action constants) = unspecified
    Phenomenological parameters in Eq. (36); the hexatic transverse sound speed depends on g|psi0|^2/m1 but no values are given.
  • gamma (quasiparticle decay rate) = arbitrary (10^11 Hz in Fig. 2)
    Chosen for illustration; authors state spectral weight does not depend on gamma.
axioms (7)
  • domain assumption The Wigner crystal is described by continuum quadratic elasticity with hexagonal elastic tensor and displacement field u_i (Eq. 4).
    Fundamental starting model; assumes long-wavelength elastic description, neglecting microscopic lattice-scale effects except through core parameters.
  • domain assumption Electrostatic interaction between electrons in the slab is screened by two gates at z=±d, V(k)=tanh(kd)/(2εk) (Eq. 3).
    Used throughout to compute defect fields and energies; gate screening determines exponential vs power-law behavior.
  • domain assumption The fracton-elasticity duality action (17) with couplings B_iμ J^disl_iμ and C_μ J^dscl_μ is the complete low-energy effective action for defects.
    The duality maps defects to gauge sources; completeness of this minimal coupling is assumed, following Refs. [45–56], and is not proven here.
  • domain assumption KTHNY melting criterion (34) applies with the renormalized shear modulus C_s(1−α).
    Used to estimate T_c; known to fail near the quantum regime, which the paper acknowledges.
  • domain assumption Dilute vacancies form an ideal parabolic Fermi liquid with no residual interactions (Eqs. S12–S17).
    Needed for the hydrodynamic action (41) and phonon speed Eq. (44); assumes r^-3 interaction is weak even at ρ_v up to 0.15 n0.
  • ad hoc to paper The adiabatic factor (1+m/m_h) in Eq. (42) is not canceled by collision drag at the relevant temperatures.
    The paper argues e-ph scattering is suppressed at ~100 mK (Refs. [69–71]), but the opposite limit is well established in metals; this choice directly controls n_eff and the phonon-speed curve.
  • domain assumption Elastic moduli scale as C_s,C_b ~ e^2 n0^2 a/(4πε) and take values from Ref. [62] (Eq. 9).
    Quantitative melting and sound-speed numbers rely on these lattice-calculated constants.

pith-pipeline@v1.3.0-alltime-deepseek · 17743 in / 23127 out tokens · 204194 ms · 2026-08-01T15:18:14.934531+00:00 · methodology

0 comments
read the original abstract

The physics of Wigner solids is characterized by an interplay of elasticity and long-range electrostatics, endowing crystal defects with properties distinct from those in charge-neutral crystals. Recent experiments observing lattice melting and Wigner-crystal-adjacent phases in two dimensions necessitate an examination of how defects affect the long-wavelength properties of such solids. Here, we use duality techniques to construct a comprehensive framework for studying the contribution of vacancies and interstitials, dislocations, and disclinations to the effective action of two-dimensional charged crystals. This allows for a systematic investigation of the interaction energies for the different combinations of defect pairs. We further study melting transitions due to defect proliferation, assessing and justifying some of the assumptions present in the literature. In the metallic Wigner crystal phase, characterized by a finite ground state density of vacancies, we find phonons with a dispersion relation that varies in an unusual way with the vacancy density. Consequently, we discuss how thermodynamic properties of such a vacancy Fermi liquid can be probed in measurements of the melting temperature and speed of sound. The field theory developed here can serve as a starting point in the study of anomalous Hall crystals and charge density waves with defects.

Figures

Figures reproduced from arXiv: 2607.18477 by Pawe{\l} Matus, Tobias Holder.

Figure 1
Figure 1. Figure 1: Crystal defects and deformations. (a) Vacancy [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectral function of the longitudinal mode [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The speed of sound of the two modes in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

74 extracted references · 2 canonical work pages

  1. [1]

    Wigner, On the interaction of electrons in metals, Phys

    E. Wigner, On the interaction of electrons in metals, Phys. Rev.46, 1002 (1934)

    1934

  2. [2]

    Wigner, Effects of the electron interaction on the en- ergy levels of electrons in metals, Trans

    E. Wigner, Effects of the electron interaction on the en- ergy levels of electrons in metals, Trans. Faraday Soc.34, 678 (1938)

    1938

  3. [3]

    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, 795 (1979)

    1979

  4. [4]

    J. Yoon, C. C. Li, D. Shahar, D. C. Tsui, and M. Shayegan, Wigner crystallization and metal-insulator transition of two-dimensional holes in gaas atB= 0, Phys. Rev. Lett.82, 1744 (1999)

    1999

  5. [5]

    Smole´ nski, P

    T. Smole´ nski, P. E. Dolgirev, C. Kuhlenkamp, A. Popert, Y. Shimazaki, P. Back, X. Lu, M. Kroner, K. Watanabe, T. Taniguchi, I. Esterlis, E. Demler, and A. Imamo˘ glu, Signatures of wigner crystal of electrons in a monolayer semiconductor, Nature595, 53–57 (2021)

    2021

  6. [6]

    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 dichalco- genide heterostructure, Nature595, 48–52 (2021)

    2021

  7. [7]

    A. M. Seiler, M. Statz, C. Eckel, I. Weimer, J. P¨ ohls, K. Watanabe, T. Taniguchi, F. Zhang, and R. T. Weitz, Signatures of sliding wigner crystals in bilayer graphene at zero and finite magnetic fields, Nature Communica- tions16, 8921 (2025)

    2025

  8. [8]

    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 crys- tal in rhombohedral graphene (2026), arXiv:2604.00113 [cond-mat.mes-hall]

    arXiv 2026

  9. [9]

    Xiang, H

    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, Science388, 736 (2025), https://www.science.org/doi/pdf/10.1126/science.ado7136

    work page doi:10.1126/science.ado7136 2025

  10. [10]

    Z. Wang, R. Song, Y. Jiang, Q. Sun, M. Zhao, L. Yin, J. Shen, and C. Gao, Intrinsic heavy wigner crystal forged by transferred 4felectrons, Phys. Rev. Lett.135, 266502 (2025)

    2025

  11. [11]

    E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu- 10 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

  12. [12]

    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, Imag- ing two-dimensional generalized wigner crystals, Nature 597, 650 (2021)

    2021

  13. [13]

    Padhi, R

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

    2021

  14. [14]

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crys- tals in rhombohedral multilayer graphene. i. interaction- driven chern bands and fractional quantum hall states at zero magnetic field, Physical Review Letters133, 10.1103/physrevlett.133.206503 (2024)

    doi:10.1103/physrevlett.133.206503 2024

  15. [15]

    Soejima, J

    T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zale- tel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohedral multilayer graphene. ii. gen- eral mechanism and a minimal model, Phys. Rev. B110, 205124 (2024)

    2024

  16. [16]

    Tan and T

    T. Tan and T. Devakul, Parent berry curvature and the ideal anomalous hall crystal, Physical Review X14, 10.1103/physrevx.14.041040 (2024)

    work page doi:10.1103/physrevx.14.041040 2024

  17. [17]

    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. B109, 235130 (2024)

    2024

  18. [18]

    J. Dong, T. Soejima, D. E. Parker, and A. Vishwanath, Crystals caught doping: Metallic wigner crystals in rhombohedral graphene (2026), arXiv:2604.00114 [cond- mat.str-el]

    arXiv 2026

  19. [19]

    J. Feng, Z. Han, M. P. Zaletel, and Z. Dong, Self- doped crystal from preempted band-inversion transitions (2026), arXiv:2604.09820 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  20. [20]

    B. I. Halperin and D. R. Nelson, Theory of two- dimensional melting, Phys. Rev. Lett.41, 121 (1978)

    1978

  21. [21]

    K. J. Strandburg, Two-dimensional melting, Rev. Mod. Phys.60, 161 (1988)

    1988

  22. [22]

    R. H. Morf, Temperature dependence of the shear mod- ulus and melting of the two-dimensional electron solid, Phys. Rev. Lett.43, 931 (1979)

    1979

  23. [23]

    Muto and H

    S. Muto and H. Aoki, Crystallization of a classical two- dimensional electron system: Positional and orienta- tional orders, Phys. Rev. B59, 14911 (1999)

    1999

  24. [24]

    W. J. He, T. Cui, Y. M. Ma, Z. M. Liu, and G. T. Zou, Phase transition in a classical two-dimensional electron system, Phys. Rev. B68, 195104 (2003)

    2003

  25. [25]

    B. K. Clark, M. Casula, and D. M. Ceperley, Hexatic and mesoscopic phases in a 2d quantum coulomb system, Phys. Rev. Lett.103, 055701 (2009)

    2009

  26. [26]

    G. M. Bruun and D. R. Nelson, Quantum hexatic order in two-dimensional dipolar and charged fluids, Phys. Rev. B89, 094112 (2014)

    2014

  27. [27]

    E. H. Hwang and S. Das Sarma, Plasmon dispersion in dilute two-dimensional electron systems: Quantum- classical and wigner crystal–electron liquid crossover, Phys. Rev. B64, 165409 (2001)

    2001

  28. [28]

    Deville, A

    G. Deville, A. Valdes, E. Y. Andrei, and F. I. B. Williams, Propagation of shear in a two-dimensional electron solid, Phys. Rev. Lett.53, 588 (1984)

    1984

  29. [29]

    S. Kim, J. Bang, C.-y. Lim, S. Y. Lee, J. Hyun, G. Lee, Y. Lee, J. D. Denlinger, S. Huh, C. Kim, S. Y. Song, J. Seo, D. Thapa, S.-G. Kim, Y. H. Lee, Y. Kim, and S. W. Kim, Quantum electron liquid and its possible phase transition, Nature Materials21, 1269 (2022)

    2022

  30. [30]

    Barabanov, L

    A. Barabanov, L. Maksimov, and A. Mikheyenkov, On the wigner crystal state with point defects, Physics Let- ters A201, 81 (1995)

    1995

  31. [31]

    Barraza, L

    N. Barraza, L. Colletti, and M. Tosi, Vacancies in quan- tal wigner crystals near melting, Solid State Communi- cations112, 261 (1999)

    1999

  32. [32]

    Pankov and V

    S. Pankov and V. Dobrosavljevi´ c, Self-doping instability of the wigner-mott insulator, Phys. Rev. B77, 085104 (2008)

    2008

  33. [33]

    D. S. Fisher, B. I. Halperin, and R. Morf, Defects in the two-dimensional electron solid and implications for melting, Phys. Rev. B20, 4692 (1979)

    1979

  34. [34]

    Cockayne and V

    E. Cockayne and V. Elser, Energetics of point defects in the two-dimensional wigner crystal, Phys. Rev. B43, 623 (1991)

    1991

  35. [35]

    Esfarjani, S

    K. Esfarjani, S. T. Chui, and X. Qiu, Dislocation waves in a two-dimensional coulomb lattice, Phys. Rev. B46, 4638 (1992)

    1992

  36. [36]

    Zheng and H

    L. Zheng and H. A. Fertig, Quantum correlated intersti- tials and the hall resistivity of the magnetically induced wigner crystal, Phys. Rev. Lett.73, 878 (1994)

    1994

  37. [37]

    Cha and H

    M.-C. Cha and H. A. Fertig, Topological defects, orien- tational order, and depinning of the electron solid in a random potential, Phys. Rev. B50, 14368 (1994)

    1994

  38. [38]

    E. Frey, D. R. Nelson, and D. S. Fisher, Interstitials, vacancies, and supersolid order in vortex crystals, Phys. Rev. B49, 9723 (1994)

    1994

  39. [39]

    Cˆ andido, P

    L. Cˆ andido, P. Phillips, and D. M. Ceperley, Single and paired point defects in a 2d wigner crystal, Phys. Rev. Lett.86, 492 (2001)

    2001

  40. [40]

    Polini, G

    M. Polini, G. Sica, and M. Tosi, Phonon scattering against point defects in the quantal 2d wigner crystal, Solid State Communications117, 561 (2001)

    2001

  41. [41]

    W. T. M. Irvine, M. J. Bowick, and P. M. Chaikin, Frac- tionalization of interstitials in curved colloidal crystals, Nature Materials11, 948 (2012)

    2012

  42. [42]

    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

  43. [43]

    P. T. Madathil, C. Wang, S. K. Singh, A. Gupta, K. A. V. Rosales, Y. J. Chung, K. W. West, K. W. Baldwin, L. N. Pfeiffer, L. W. Engel, and M. Shayegan, Signatures of correlated defects in an ultraclean wigner crystal in the extreme quantum limit, Phys. Rev. Lett.132, 096502 (2024)

    2024

  44. [44]

    Zhuang and I

    Z. Zhuang and I. Esterlis, Defect liquids in a weakly imbalanced bilayer wigner crystal, Phys. Rev. B111, 205122 (2025)

    2025

  45. [45]

    Kleinert, Duality transformation for defect melting, Physics Letters A91, 295–298 (1982)

    H. Kleinert, Duality transformation for defect melting, Physics Letters A91, 295–298 (1982)

    1982

  46. [46]

    Kleinert, Dual model for dislocation and disclination melting, Physics Letters A96, 302–306 (1983)

    H. Kleinert, Dual model for dislocation and disclination melting, Physics Letters A96, 302–306 (1983)

    1983

  47. [47]

    Zaanen, Z

    J. Zaanen, Z. Nussinov, and S. Mukhin, Duality in 2+1d quantum elasticity: superconductivity and quantum ne- matic order, Annals of Physics310, 181–260 (2004)

    2004

  48. [48]

    A. J. Beekman, J. Nissinen, K. Wu, K. Liu, R.-J. Slager, Z. Nussinov, V. Cvetkovic, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in two dimensions, Physics Reports683, 1–110 (2017). 11

    2017

  49. [50]

    Pretko, The Fracton Gauge Principle, Phys

    M. Pretko, The Fracton Gauge Principle, Phys. Rev. B 98, 115134 (2018)

    2018

  50. [51]

    Pretko, Z

    M. Pretko, Z. Zhai, and L. Radzihovsky, Crystal-to- fracton tensor gauge theory dualities, Phys. Rev. B100, 134113 (2019)

    2019

  51. [52]

    Caddeo, C

    A. Caddeo, C. Hoyos, and D. Musso, Emergent dipole gauge fields and fractons, Physical Review D106, 10.1103/physrevd.106.l111903 (2022)

    doi:10.1103/physrevd.106.l111903 2022

  52. [53]

    Tsaloukidis, J

    L. Tsaloukidis, J. J. Fern´ andez-Melgarejo, J. Molina- Vilaplana, and P. Sur´ owka, Fracton-elasticity duality on curved manifolds, Physical Review B109, 10.1103/Phys- RevB.109.085427 (2024)

    doi:10.1103/phys- 2024

  53. [54]

    Tsaloukidis and P

    L. Tsaloukidis and P. Sur´ owka, Elastic li´ enard-wiechert potentials of dynamical dislocations from tensor gauge theory in 2+1 dimensions, Physical Review B109, 10.1103/PhysRevB.109.104118 (2024)

    doi:10.1103/physrevb.109.104118 2024

  54. [55]

    and references therein. III. DEFECT FIELDS AND INTERACTIONS In this section, we compute the fields produced by iso- lated, static defects. For reference, the equations of mo- tion derived from the action (17) read: A− tanh(kd) 2ε k ρvac −en 0C −1 b τii = 0,(20a) ∂jτij −en 0∂iA= 0,(20b) 1 2 ϵijϵkl∂j∂l ˜Cikmnτmn − 1 2 ϵij∂iρdisl j = 0,(20c) where we have se...

  55. [56]

    Matus, Defects in wigner crystals: Fracton-elasticity duality and vacancy proliferation, Phys

    P. Matus, Defects in wigner crystals: Fracton-elasticity duality and vacancy proliferation, Phys. Rev. B113, 075138 (2026)

    2026

  56. [57]

    Zhang, L

    L. Zhang, L. Gu, H. S. Adlong, A. Christianen, E. Dizer, R. Ni, R. Ma, S. Park, H. Jang, T. Taniguchi, K. Watan- abe, I. Esterlis, R. Schmidt, A. Imamoglu, and Y. Zhou, Wigner polarons reveal wigner crystal dynamics in a monolayer semiconductor (2025), arXiv:2512.16631 [cond-mat.mes-hall]

    arXiv 2025

  57. [58]

    G l´ odkowski, P

    A. G l´ odkowski, P. Matus, F. Pe˜ na-Ben ´ ıtez, and L. Tsa- loukidis, Quadrupole gauge theory: Anti-higgs mech- anism and elastic dual, Physical Review D112, 10.1103/wnvt-gmwv (2025)

    doi:10.1103/wnvt-gmwv 2025

  58. [59]

    H. S. Adlong, E. Dizer, R. Schmidt, A. Imamoglu, and A. Christianen, Theory of exciton polarons in 2d wigner crystals (2025), arXiv:2512.16651 [cond-mat.str-el]

    arXiv 2025

  59. [60]

    J. H. Nyhegn, E. R. Christensen, and G. M. Bruun, An exciton interacting with the phonons of an electronic wigner crystal (2025), arXiv:2512.16888 [cond-mat.str- el]

    Pith/arXiv arXiv 2025

  60. [61]

    L. Wang, F. Menzel, F. Pichler, P. Kn¨ uppel, K. Watan- abe, T. Taniguchi, M. Knap, and T. Smole´ nski, Spec- troscopy of wigner crystal polarons in an atomically thin semiconductor (2025), arXiv:2512.16552 [cond-mat.mes- hall]

    arXiv 2025

  61. [62]

    Tozzini and M

    V. Tozzini and M. P. Tosi, Lattice vibrations and elastic constants of three- and two-dimensional quantal wigner crystals near melting, Journal of Physics: Condensed Matter8, 8121 (1996)

    1996

  62. [63]

    Bonsall and A

    L. Bonsall and A. A. Maradudin, Some static and dy- namical properties of a two-dimensional wigner crystal, Phys. Rev. B15, 1959 (1977)

    1959

  63. [64]

    Y. Dong, E. Demler, and Z. Sun, Optical detection and manipulation of pseudospin orders in wigner crystals (2026), arXiv:2512.21097 [cond-mat.str-el]

    arXiv 2026

  64. [65]

    with the exception that the bulk stress is strongly suppressed in the absence of gates, and the energy of the defect grows asE dscl ∝L 2. D. Interaction energies Defect charges No gates With gates Q1-Q2 Q1Q2 4πϵ l2 scr−3 3Q1Q2 4πεd q π 2k∗r exp(−k∗r) b-Q− Cs 2πen0 Q(b×r) r2 − d d+lsc Cs 2πen0 Q(b×r) r2 b1-b2 − Cs 2π (b1 ·b 2) lnr− Cs(1−α) 2π (b1 ·b 2) lnr...

  65. [66]

    Grimes and G

    C. Grimes and G. Adams, Crystallization of electrons on the surface of liquid helium, Surface Science98, 1 (1980)

    1980

  66. [67]

    With that in mind, let us explore the consequences of the proliferation of charged defects on the properties of the crystal

    rather than a secondary crystal. With that in mind, let us explore the consequences of the proliferation of charged defects on the properties of the crystal. In App. S1 we use the semiclassical Boltz- mann equation approach based on the wavepacket con- struction [68] to derive the hydrodynamic equations for the ideal Fermi liquid: the continuity equation ...

  67. [68]

    L. D. Landau and E. M. Lifshitz,Theory of Elasticity, 3rd ed., Course of Theoretical Physics, Vol. 7 (Butterworth- Heinemann, 1986)

    1986

  68. [69]

    deWit, Theory of disclinations: Iv

    R. deWit, Theory of disclinations: Iv. straight discli- nations, Journal of Research of the National Bureau of Standards. Section A: Physics and Chemistry77A, 607 (1973)

    1973

  69. [70]

    Lu and G

    Z.-K. Lu and G. V. Shlyapnikov, Fermi liquid of two- dimensional polar molecules, Phys. Rev. A85, 023614 (2012)

    2012

  70. [71]

    Sundaram and Q

    G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and berry-phase effects, Phys. Rev. B59, 14915 (1999)

    1999

  71. [72]

    Holstein, Theory of ultrasonic absorption in metals: the collision-drag effect, Phys

    T. Holstein, Theory of ultrasonic absorption in metals: the collision-drag effect, Phys. Rev.113, 479 (1959)

    1959

  72. [73]

    F. S. Khan and P. B. Allen, Deformation potentials and electron-phonon scattering: Two new theorems, Phys. Rev. B29, 3341 (1984)

    1984

  73. [74]

    E. M. Lifshitz and L. P. Pitaevskii,Physical Kinetics (Pergamon, 1981)

    1981

  74. [75]

    Zeng and A

    Y. Zeng and A. J. Millis, Berry phase dynamics of sliding electron crystals, Phys. Rev. X15, 031059 (2025). S1 Supplemental Material: Theory of two-dimensional Wigner crystals with defects: Interactions, collective modes and melting transition S1. EFFECTIVE ACTION FOR THE MIXED FERMI LIQUID-WIGNER CR YST AL SYSTEM To describe the dynamics of a Fermi liqui...

    2025