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Short-range defect density controls Wigner solid melting

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2026-08-04 09:59 UTC pith:VT3FSKLT

load-bearing objection Strong experiment, partly circular HDD simulation: the two-regime story is plausible and worth refereeing, but the QMC 'validation' of the amorphous WS relies on defect assignments taken from the same images. the 4 major comments →

arxiv 2510.12009 v3 pith:VT3FSKLT submitted 2025-10-13 cond-mat.str-el cond-mat.mes-hall

Visualizing the Impact of Quenched Disorder on 2D Electron Wigner Solids

Zhehao Ge , Conor Smith , Zehao He , Yubo Yang , Qize Li , Ha-Leem Kim , Ziyu Xiang , Jianghan Xiao
show 12 more authors
Wenjie Zhou Salman Kahn Aining Hu Melike Erdi Rounak Banerjee Takashi Taniguchi Kenji Watanabe Seth Ariel Tongay Miguel A. Morales Shiwei Zhang Feng Wang Michael F. Crommie
This is my paper
classification cond-mat.str-el cond-mat.mes-hall PACS 73.20.Qt71.30.+h71.27.+a68.37.Ef
keywords Wigner solidquenched disordershort-range disorderscanning tunneling microscopyneural quantum statesquantum Monte Carlore-entrant meltingFriedel oscillations
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 reports that the fate of a two-dimensional electron Wigner solid under quenched disorder is determined by the density of short-range (isovalent) defects relative to the electron density, not by disorder strength alone. Scanning tunneling microscopy of gate-tunable bilayer MoSe2, combined with quantum Monte Carlo simulations that use the exact measured defect positions, reveals two regimes. When short-range defects are scarce, long-range charged defects dominate, producing mixed solid-liquid phases, a new re-entrant melting/crystallization, and Friedel oscillations. When short-range defects are abundant, they pin electrons into an amorphous Wigner solid that survives to higher electron densities and suppresses those features. The result provides a concrete criterion for when a disordered 2D electron solid behaves as a glassy amorphous solid rather than a locally crystalline one.

Core claim

The paper claims that the ratio of short-range isovalent defect density to electron density selects between two regimes of a disordered 2D electron Wigner solid. When short-range defects are rarer than electrons, long-range charged-defect disorder dominates, producing re-entrant melting/crystallization and Friedel oscillations. When short-range defects are much denser than electrons, they pin electrons into an amorphous Wigner solid that persists to higher electron densities and suppresses those features. STM imaging and neural quantum Monte Carlo simulations with the measured defect positions support this dichotomy.

What carries the argument

The multiple-plane-waves message-passing neural quantum state ((MP)^2-NQS) ansatz — a neural-network trial wavefunction that represents both Wigner solid and electron liquid in a single unbiased form — augmented with an electron-defect Jastrow factor to maintain the cusp condition. This allows variational quantum Monte Carlo to simulate melting with disorder potentials placed at the exact defect positions measured by STM. The in-gap tunneling technique provides minimally perturbative electron-density maps for direct comparison with the simulation.

Load-bearing premise

The central claim rests on the model of isovalent defects as exponentially screened point potentials with a fixed short-range length, with individual defects classified as attractive, repulsive, or negligible based on their visible influence on electron density in the liquid regime; if this defect representation is wrong, the simulated HDD amorphous Wigner solid could be an artifact of the model rather than a real material property.

What would settle it

Image a high-defect-density device at electron densities between about 8×10^11 and 2.5×10^12 cm^-2 with sub-nanometer resolution: the paper predicts a persistent amorphous Wigner solid with a diffuse ring in the structure factor and no Friedel oscillations around charged defects. Observing sharp crystalline Bragg peaks, re-entrant melting, or 2k_F Friedel oscillations in that window would contradict the central dichotomy.

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

If this is right

  • In the LDD regime, re-entrant melting/crystallization arises from commensuration-incommensuration: as electron density changes, the electron lattice locks and unlocks to structure-factor peaks of the long-range disorder distribution.
  • Friedel oscillations around charged defects in the LDD regime follow Wigner-crystal spacing (~1/√n_e) rather than non-interacting 2k_F spacing, showing electron correlations reshape single-defect response.
  • In the HDD regime, the amorphous Wigner solid persists to electron densities above the pristine crystallization threshold, indicating short-range disorder can stabilize an electron solid rather than only melt it.
  • The methodology of feeding STM-derived defect maps into a unified neural quantum Monte Carlo ansatz enables quantitative prediction of disorder-induced phases in other 2D materials.

Where Pith is reading between the lines

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

  • If the n_SR/n_e ratio is the true control parameter, gating a single sample across the LDD/HDD crossover should continuously transform re-entrant melting into a stable amorphous solid; this is a direct experimental check.
  • The quenching of Friedel oscillations by dense short-range disorder implies suppression of long-range electronic response around charged impurities, which may alter magnetic exchange couplings in such electron solids — a consequence not explored in the paper.
  • The effective isovalent defect density n0≈7.6×10^12 cm^-2 used in HDD simulations depends on excluding 'undetectable' defects; a systematic variation of the detection threshold would map the phase boundary quantitatively in the (n_SR, n_e) plane.
  • The LDD/HDD dichotomy may extend to other 2D materials with native short-range defects, such as chalcogen vacancies in TMDs; a comparative study would test whether the two regimes are universal.

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

4 major / 5 minor

Summary. The paper reports STM imaging of electron Wigner solids in gate-tunable bilayer MoSe2 devices with two markedly different densities of short-range (isovalent) disorder. In the low-defect-density (LDD) regime (n_SR ≲ n_e), the authors observe mixed solid-liquid phases, local re-entrant melting/crystallization, and Friedel oscillations around charged defects. In the high-defect-density (HDD) regime (n_SR ≫ n_e), they observe a robust amorphous Wigner solid with suppressed re-entrant melting and Friedel oscillations. Neural-network QMC simulations using disorder positions extracted from STM reproduce the LDD melting/oscillation behavior and, when isovalent defects are modeled as short-range screened potentials, produce a crossover with increasing n_SR. The central claim is that the type and density of quenched disorder, not just its overall strength, controls the phase behavior of disordered 2D Wigner solids.

Significance. If the conclusions hold, this is a significant advance: it provides single-defect-resolved experimental evidence that short-range disorder qualitatively changes the melting of 2D Wigner solids, a regime not cleanly addressed by transport or optical measurements. The experimental technique (in-gap tunneling STM) and the unified (MP)2-NQS QMC approach are genuine strengths; the QMC reproduces several non-trivial features (local melting, Friedel oscillations, re-entrant crystallization) without a phase-biased trial wavefunction. However, the simulation support is weakened by the fact that the HDD disorder model's sign assignments, defect-activity selection, and the only fitting parameter (ε, with two values) are calibrated against the same STM density maps used for validation. The two-regime picture is plausible and supported by direct imaging, but the quantitative 'confirmation' is partially circular and the HDD conclusion is more model-dependent than the text suggests.

major comments (4)
  1. [SI §S9, Fig. 5f–i] The HDD-regime QMC model is not an independent test of the short-range-disorder hypothesis. The sign of each isovalent defect and the effective defect density n0≈7.63×10^12 cm^-2 are determined by 'experimentally observing its influence on electron probability density in the liquid regime' and excluding 'undetectable' defects. Because the same electron-density maps are then used to validate the HDD simulations (Fig. 5d vs 5i), the apparent one-to-one correspondence between defects and electron localization is partly built in. Please provide an out-of-sample test (e.g., predict a field of view not used for input extraction), a sensitivity analysis over the exclusion threshold, or an independent first-principles assignment of defect sign/activity.
  2. [SI §S9] The short-range disorder potential is an exponentially screened charge with range set to 0.15 a0* (~0.05 nm) and half the defects attractive/half repulsive. This range is two orders of magnitude smaller than the mean electron spacing, so the effective disorder strength is extremely sensitive to this choice; no sensitivity study is given. Since the central HDD conclusion (suppressed Friedel oscillations, robust amorphous WS) relies on this potential, please show that the qualitative phase behavior is robust to varying the range (e.g., 0.1–1 a0*) and the attractive/repulsive ratio.
  3. [SI §S9, Figs. 2–4] The QMC 'confirmation' uses the dielectric constant ε as a fitting parameter, with ε=2.58 at low n_e and ε=3.73 at intermediate/high n_e, chosen by visual comparison to experiment. Thus the agreement is not an ab initio prediction, and the statement that ε is 'the only fitting parameter' is misleading because two values are used for different density windows. Please either justify a physical mechanism for the ε switch or treat the qualitative LDD results as the primary evidence and rephrase the validation claim accordingly.
  4. [§5, Fig. 5] The LDD and HDD devices are fabricated from different MoSe2 crystals (HQ Graphene vs self-grown). Although n_LR is stated to be comparable, the charged-defect configurations differ between the two devices; the experimental HDD observations could in principle be influenced by the specific arrangement of charged defects or by uncontrolled strain/doping differences between crystals. To attribute the crossover to n_SR, please show that the charged-defect structure factor and local environment statistics are similar, or analyze multiple HDD regions with different charged-defect patterns.
minor comments (5)
  1. [Main text, Fig. 4 caption] The non-interacting Friedel oscillation expression is referred to SI Section S5 in the main text but SI Section S6 in the Fig. 4 caption. Please reconcile the cross-reference.
  2. [SI §S13, Fig. S16] The text states n_e ≈ 2.54×10^12 cm^-2 corresponds to r_s ≈ 14.0, while the Fig. S16 caption gives r_s ≈ 9.72. One of these is inconsistent; please correct.
  3. [Abstract/Methods] The abstract uses 'neural-quantum-state quantum Monte Carlo (NQS-QMC)' but the Methods and SI use '(MP)2-NQSs'. Define the acronym at first use and keep terminology consistent throughout.
  4. [Fig. 5 caption] The caption gives n0 ≈ 7.63×10^12 cm^-2 but does not explicitly state that panels g and h use n0/4 and n0/2, respectively. The notation n_SR ≈ n0/4, n0/2 is clear from the figure axes, but a brief statement would improve readability.
  5. [SI §S10] Typo: 'shor-range' should be 'short-range' in the first sentence.

Circularity Check

2 steps flagged

QMC 'validation' partly re-uses the experimental density maps it is compared against, but the central two-regime claim rests on direct STM observation and is not circular by construction.

specific steps
  1. fitted input called prediction [SI Section S9, QMC simulation method; Figs. 2-5]
    "The value of the dielectric constant ε used in our QMC simulations is a fitting parameter and is selected based on comparing the similarity between experimental and QMC electron density maps (it is the only fitting parameter in our simulations)."

    The QMC maps are presented as reproducing/validating the experimental density maps (for example, 'the simulations reproduce key experimental features'), but a Hamiltonian parameter is chosen by maximizing similarity to those same density maps. Hence the close match of wavepacket width, localization, and melting pattern is partly a consequence of the fit rather than an independent parameter-free prediction. The experimental LDD/HDD classification itself remains direct STM evidence, so the circularity is partial.

  2. fitted input called prediction [SI Section S9, effective isovalent defect density n0; Fig. 5f-i caption]
    "We determined whether an isovalent defect is attractive or repulsive by experimentally observing its influence on electron probability density in the liquid regime. Such analysis shows that some isovalent defects have an undetectable perturbation to the electron probability density and so we exclude those isovalent defects from our QMC simulations. This results in an effective isovalent defect density of n0≈7.63×10^12 cm^-2 as described in the caption of Fig. 5."

    The HDD simulations are populated by assigning each isovalent defect an attractive/repulsive sign and defining the 'effective' defect density n0 from the same experimental electron-probability-density maps that are later used for validation. The striking one-to-one correspondence between electron localization and defect positions in Fig. 5i is therefore partly built into the simulation input, and the QMC demonstration that short-range disorder produces the robust amorphous WS is not fully independent of the data it claims to reproduce. The direct experimental observation of distinct LDD/HDD regimes is not itself circular.

full rationale

The paper's central claim—that the regime changes when short-range disorder density crosses the electron density—is anchored in independent STM measurements: defects are counted from topographs, n_e is estimated from WS wavevectors and capacitance, and the LDD versus HDD behavior is directly observed. That classification is not defined in terms of a fitted parameter. The circularity concern is limited to the QMC 'confirmation' loop. In SI S9 the authors state that the dielectric constant is a fitting parameter selected by comparing simulated and experimental density maps, and that each isovalent defect's sign as well as the effective defect density are inferred from experimental electron probability density in the liquid regime. Consequently, the impressive agreement between the simulations and the same STM-derived maps (especially the HDD pinning pattern in Fig. 5i) is partially a reconstruction of the experimental input rather than an independent prediction. This does not collapse the whole paper, but it means the theoretical support for the HDD amorphous-WS mechanism is not as clean as the main text implies. The self-citation to the group's (MP)2-NQSs ansatz is a normal methodological citation and is not the load-bearing evidence for the regime dichotomy, so I do not count it as circularity. Overall, the central claim has independent experimental content, but the QMC validation is partially fitted, giving a moderate circularity score of 4.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

No new particles, forces, conserved quantities, or dimensions are postulated. The short-range isovalent-defect potential and the effective defect density are parameterized models, not new physical entities. The main free parameters are the QMC dielectric constant (fitted to agreement with experiment), the screening range for isovalent defects, and the effective isovalent-defect density after excluding 'undetectable' defects.

free parameters (3)
  • Dielectric constant ε in QMC simulations = 2.58 ε0 (low n_e) and 3.73 ε0 (intermediate/high n_e)
    SI S9 states: 'The value of the dielectric constant ε used in our QMC simulations is a fitting parameter and is selected based on comparing the similarity between experimental and QMC electron density maps (it is the only fitting parameter in our simulations).' It takes two values for different density ranges.
  • Isovalent-defect screening gate distance = 0.15 a0* (~0.05 nm)
    SI S9: isovalent defects are treated as dual-gate screened charges with exponentially decaying interactions 'at the scale of an input gate distance parameter, which we set to 0.15 a0*'; this range is an input choice, not independently measured.
  • Effective isovalent defect density n0 = 7.63 × 10^12 cm^-2
    SI S9: defects whose perturbation to electron density is deemed undetectable are excluded from QMC simulations, yielding n0≈7.63×10^12 cm^-2. This effective density sets the HDD simulation scale used in Fig. 5f-i.
axioms (6)
  • domain assumption In-gap STM tunnel current is a faithful, minimally perturbing proxy for the local 2D electron density
    Main text and SI S2. The central experimental observable is the in-gap current map; if tip-induced perturbation or tunneling-matrix-element effects dominated, the images would not reflect the ground-state WS density.
  • domain assumption Charged defects act as fixed point charges with charge −e in the QMC Hamiltonian
    Methods and SI S9 state charged defects are treated as fixed point charges with net charge −e. No independent charge-state calculation is provided.
  • ad hoc to paper Isovalent defects can be represented by exponentially screened potentials with range 0.15 a0* and half attractive/half repulsive character
    SI S9-S10: the potential form, screening scale, and sign assignment are chosen for the simulations rather than derived from first principles or independent measurements.
  • domain assumption The (MP)^2-NQS variational ansatz accurately represents both Wigner-solid and liquid phases in the presence of disorder
    Methods/SI S9 rely on ref 17 for the clean 2DEG; the extension to disordered systems with an added electron-defect Jastrow factor is assumed to remain unbiased and accurate.
  • domain assumption Effective mass m* = 0.54 m_e and the stated dielectric constants describe BL-MoSe2 for density-to-r_s conversion
    Used in the main text and SI S3 to estimate r_s; taken from band-structure calculations (ref 28) and assumed dielectric screening values.
  • domain assumption Electron density n_e at intermediate gate voltages follows linear interpolation or geometric-capacitance scaling
    SI S3: n_e for Figs. 2b-c is estimated by assuming linear dependence on V_G; for V_G≥3.3 V it is estimated from capacitance using a reference point. Systematic uncertainties are not given.

pith-pipeline@v1.3.0-alltime-deepseek · 19289 in / 11269 out tokens · 99447 ms · 2026-08-04T09:59:48.840240+00:00 · methodology

0 comments
read the original abstract

Electron Wigner solids (WSs)1-12 provide an ideal system for understanding the competing effects of electron-electron and electron-disorder interactions, a central unsolved problem in condensed matter physics. Progress in this topic has been limited by a lack of single-defect-resolved experimental measurements as well as accurate theoretical tools to enable realistic experiment/theory comparison. Here we overcome these limitations by combining atomically-resolved scanning tunneling microscopy (STM) with neural-quantum-state quantum Monte Carlo (NQS-QMC) simulation of disordered 2D electron WSs to discover new disorder-induced physical regimes of correlated electron behavior. STM was used to image the electron density ($n_e$) dependent evolution of electron WSs in gate-tunable bilayer MoSe2 devices with varying long-range ($n_\mathrm{LR}$) and short-range ($n_\mathrm{SR}$) disorder densities. These images were compared to NQS-QMC simulations using realistic disorder maps extracted from experiment, thus allowing the roles of different disorder types to be disentangled. We identify two distinct physical regimes for disordered electron WSs that depend on $n_\mathrm{SR}$. For $n_\mathrm{SR} \lesssim n_e$ the WS behavior is dominated by long-range disorder and features extensive mixed solid-liquid phases, a new type of local re-entrant melting/crystallization, and prominent Friedel oscillations. In contrast, when $n_\mathrm{SR} \gg n_e$ these features are suppressed and a more robust amorphous WS phase emerges that persists to higher ne, highlighting the importance of short-range disorder in this regime. Our work establishes a powerful framework for studying disordered quantum solids via a combined experimental-theoretical approach.

Figures

Figures reproduced from arXiv: 2510.12009 by Aining Hu, Conor Smith, Feng Wang, Ha-Leem Kim, Jianghan Xiao, Kenji Watanabe, Melike Erdi, Michael F. Crommie, Miguel A. Morales, Qize Li, Rounak Banerjee, Salman Kahn, Seth Ariel Tongay, Shiwei Zhang, Takashi Taniguchi, Wenjie Zhou, Yubo Yang, Zehao He, Zhehao Ge, Ziyu Xiang.

Figure 1
Figure 1. Figure 1: Experiment setup. a, Sketch of STM setup for gate-tunable BL-MoSe2 devices. A graphite nanoribbon contact/BL-MoSe2/hBN heterostructure is placed on a 285 nm SiO2/Si substrate. A d.c. bias voltage 𝑉ௌ (with added a.c. modulation 𝑉௔௖ for STS) is applied between the STM tip and BL-MoSe2 sample. A backgate voltage 𝑉ீ is applied between the p-type silicon substrate and the BL-MoSe2. b, STM topograph of LDD BL-Mo… view at source ↗
Figure 2
Figure 2. Figure 2: Wigner solid behavior in LDD regime (low [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Wigner solid behavior in LDD regime (intermediate [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Electron solid-liquid transition in LDD regime (high [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 2D electronic behavior in HDD regime. a, [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

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Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Self-doped Crystal from Preempted Band-inversion Transitions

    cond-mat.str-el 2026-04 unverdicted novelty 7.0

    Self-doped Wigner crystals arise from preempted band-inversion transitions between commensurate crystals, as established by non-perturbative arguments and Hartree-Fock calculations in jellium and rhombohedral graphene models.

  2. Shape of Wigner Crystals and Hole Self-Doping in a Mexican-Hat Dispersion

    cond-mat.str-el 2026-07 conditional novelty 6.0

    A polaron-dressed variational Monte Carlo study finds that a Wigner crystal with hole self-doping near k=0 can be energetically favorable near the crystallization transition in a Mexican-hat band.

  3. 2D Transport in an in-plane magnetic field

    cond-mat.mes-hall 2026-07 conditional novelty 6.0

    An in-plane magnetic field shifts the 2D MIT critical density for Anderson localization (up or down depending on disorder type) but leaves Wigner crystallization unaffected, providing a distinguishing experimental test.

  4. 2D Transport in an in-plane magnetic field

    cond-mat.mes-hall 2026-07 conditional novelty 6.0

    If the 2D metal-insulator transition is Anderson localization, spin-polarizing with an in-plane field should double the critical density for Coulomb disorder and halve it for short-range disorder; a pinned Wigner crys...

  5. 2D Transport in an in-plane magnetic field

    cond-mat.mes-hall 2026-07 conditional novelty 5.5

    In-plane B-field spin polarization raises (lowers) the Anderson-localization critical density when long-range (short-range) disorder dominates, while leaving a Wigner-crystal critical density unchanged.

  6. Disorder-induced strong-field strong-localization in 2D systems

    cond-mat.mes-hall 2026-01 unverdicted novelty 3.0

    The spatially random localized phase at low filling factors in bilayer graphene is the disorder-induced Anderson solid phase.

Reference graph

Works this paper leans on

2 extracted references · cited by 4 Pith papers

  1. [18]

    short-range

    Quantum Monte Carlo (QMC) simulations predict that this transition occurs when the Wigner–Seitz radius (𝑟௦, defined as half the average inter-electron distance in units of the effective Bohr radius) exceeds ~3714,17. In real materials disorder is inevitable and significantly alters correlated electron behavior1-12. Transport and optics experiments19-22, f...

  2. [32]

    BN-sliding

    These two defect types show markedly different electronic signatures in scanning tunneling spectroscopy (STS). For charged defects (Fig. 1d) the BL-MoSe2 conduction band edge shifts upward in bias voltage (𝑉௦) over a large distance from the defect, indicating long-range interaction arising from a negative defect charge center31,32. In contrast, isovalent ...

    arXiv 1995