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arxiv: 2604.10642 · v2 · submitted 2026-04-12 · ❄️ cond-mat.mes-hall

Interplay of disorder and interaction in quantum Hall systems: from fractional quantum Hall liquids to Wigner crystals and amorphous solids

Ke Huang , Sankar Das Sarma , Xiao Li This is my paper

Pith reviewed 2026-05-10 15:42 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords quantum Hall effectWigner crystaldisorderfractional quantum Hall liquidamorphous solidstructure factorSTM experiment
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The pith

Disorder in quantum Hall systems drives a crossover from incompressible liquids to locally ordered solids and then to amorphous states.

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

This paper studies how disorder competes with electron interactions in two-dimensional systems placed in strong magnetic fields. It models classical Wigner crystals with charged impurities to show how they break from coherent order into local domains and finally into amorphous arrangements as impurity density rises. Quantum simulations of fractional quantum Hall liquids with random potentials then demonstrate a similar sequence from incompressible liquid to localized ordered states to fully amorphous solids. Charged impurities sustain longer-range crystalline order than short-range random disorder. The overall picture is compared qualitatively with recent scanning tunneling microscopy data on real samples.

Core claim

The ground state evolves from an incompressible homogeneous fractional quantum Hall liquid to a generic locally ordered solid and eventually to a disordered amorphous solid at large disorder. Random charged impurities lead to longer-range crystalline ordering than short-range random disorder.

What carries the argument

Numerical modeling of classical Wigner crystals with point-charge impurities and of fractional quantum Hall liquids with random short-range potentials, using structure factors to track local and long-range order.

Load-bearing premise

The models treat charged impurities as classical point charges and short-range disorder as random potentials without specifying microscopic details of the host material or screening.

What would settle it

If STM imaging at high impurity density reveals persistent long-range crystalline peaks instead of ring-like or featureless structure factors, the predicted crossover to an amorphous solid would be contradicted.

Figures

Figures reproduced from arXiv: 2604.10642 by Ke Huang, Sankar Das Sarma, Xiao Li.

Figure 1
Figure 1. Figure 1: Real-space distribution (top panels) and structure factor (bottom panels) for classical WC with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Normalized density 2πl2 Bn(r) of a WC of Ne = 780 at ν = 1/3. (b) Normalized density of a WC of Ne = 6 and Nϕ = 21. (c)-(e) are the projected structure factors of the FQH state of Ne = 7 at ν = 1/3, the electron crystal of Ne = 780 at ν = 1/3, and the electron crystal of Ne = 6 and Nϕ = 21, respectively. The color bar is truncated at S¯(q) = 1 to better present other structures. (f) Projected structure… view at source ↗
Figure 3
Figure 3. Figure 3: Calculations of FQH with random disorder in a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) and (b) are the energy spectrum and particle entan [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized density (a-m) and projected structure factor (n) of FQH with charged impurities, with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Calculations in the Nϕ = 21 system with Ne = 6 particles and Nimp = 3 Coulomb impurities, equivalent to 1/3 FQH state with three quasihole. (a) Energy spectrum as a function of the impurity charge Z. There is one state below energy gap 1 (red), corresponding to a single quasihole excitation, 196 states below energy gap 2 (blue), matching the expected number of quasihole excitations, and three states below … view at source ↗
Figure 7
Figure 7. Figure 7: Normalized density at finite temperatures. The six panels correspond to the six stars in Fig. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Projected structure factor for the FQHL at zero temperature with the disorder realization in the main text. Here, we take [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Density profile and projected structure factor for the FQHL at zero temperature using a second random disorder realization. The first [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Density profile and projected structure factor for the FQHL at zero temperature using a third random disorder realization. The first [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Projected structure factor for the FQHL at zero temperature with charged impurities. The first row is for one impurity, the second [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Energy and particle entanglement spectrum at the exact 1/3 filling with [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
read the original abstract

We investigate the interplay of disorder and interaction in two-dimensional electron systems in a strong magnetic field, focusing on the transition between Wigner crystals and fractional quantum Hall liquids. We first study classical Wigner crystals with charged impurities, revealing an evolution from a coherent crystal to local crystalline domains with short-range order and eventually to an amorphous state as impurity concentration increases. We then analyze noninteracting quantum electron crystals created by periodic potentials, showing that their structure factor exhibits both peaks and rings, distinct from classical Wigner crystals. Finally, we explore fractional quantum Hall liquids with random short-range disorder and quenched charged impurities, demonstrating that the ground state can evolve from an incompressible liquid to a localized ordered state and eventually to an amorphous state as disorder strength increases. In general, we find that random charged impurities lead to longer-range crystalline ordering than the short-range random disorder. Our findings highlight the rich interplay between disorder and interaction in quantum Hall systems and provide insights into experimental observations of these phenomena. By qualitative comparison with a recent STM experiment [Nature \textbf{628}, 287 (2024)], we conclude that the 2D system crosses over from an incompressible homogeneous fractional quantum Hall liquid to a generic locally ordered solid and eventually to a disordered amorphous solid at large disorder.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the interplay between disorder and electron-electron interactions in two-dimensional systems under strong magnetic fields. It reports three complementary numerical studies: classical Wigner crystals with charged impurities evolve from coherent crystals to locally ordered domains and then to amorphous solids with increasing impurity density; noninteracting quantum crystals in periodic potentials exhibit structure factors with both Bragg peaks and rings; and interacting fractional quantum Hall states with added short-range disorder and quenched impurities transition from incompressible liquids to localized ordered states and finally to amorphous solids. The authors find that charged impurities sustain longer-range order than short-range potentials and qualitatively link their results to a recent STM experiment, concluding a disorder-driven crossover from homogeneous FQH liquid to locally ordered solid to amorphous solid.

Significance. If substantiated, these findings illuminate the disorder-interaction competition in quantum Hall regimes, offering a unified picture spanning classical and quantum limits. The multi-method approach (classical Monte Carlo, noninteracting band structure, and exact diagonalization or DMRG for FQH) is a strength, and the direct connection to STM data provides timely experimental relevance. The observation that impurity type affects order range is a useful distinction for future modeling.

major comments (3)
  1. [Classical Wigner crystals with charged impurities] In the classical Wigner crystal calculations, charged impurities are treated as classical point charges. This choice directly controls the reported evolution from coherent crystal to local domains to amorphous state with rising impurity concentration. Because the effective range of the impurity potential depends on dielectric screening (absent from the model), the impurity densities at which long-range order collapses could shift under realistic GaAs or graphene screening lengths, rendering the sequence model-dependent rather than generic.
  2. [FQH liquids with random short-range disorder and quenched charged impurities] In the FQH section, the ground-state evolution from incompressible liquid to localized ordered state to amorphous solid is obtained by adding short-range random potentials plus quenched classical impurities. The manuscript does not report system sizes, number of disorder realizations, or quantitative diagnostics (structure-factor peak heights with uncertainties, or compressibility thresholds) used to locate the crossovers. Without these controls, finite-size artifacts cannot be ruled out as the origin of the claimed sequence.
  3. [Comparison with experiment] The final conclusion that the physical system follows the simulated crossover sequence rests on a qualitative comparison with the STM data of Nature 628, 287 (2024). No correlation lengths, order-parameter values, or other extractable metrics are provided from the simulations to enable a side-by-side quantitative test, leaving the experimental mapping at the level of visual resemblance.
minor comments (2)
  1. A side-by-side comparison of structure factors or real-space correlation functions for the two disorder types (charged impurities versus short-range potentials) would make the statement that 'random charged impurities lead to longer-range crystalline ordering' more immediately visible.
  2. Notation for disorder strength (impurity concentration versus potential amplitude) should be standardized across the classical and quantum sections to allow direct comparison of the reported thresholds.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments on the classical, noninteracting, and interacting regimes. We address each major point below, indicating where additional details or discussion will be incorporated in a revised version.

read point-by-point responses

  1. Referee: In the classical Wigner crystal calculations, charged impurities are treated as classical point charges. This choice directly controls the reported evolution from coherent crystal to local domains to amorphous state with rising impurity concentration. Because the effective range of the impurity potential depends on dielectric screening (absent from the model), the impurity densities at which long-range order collapses could shift under realistic GaAs or graphene screening lengths, rendering the sequence model-dependent rather than generic.

    Authors: We agree that the model employs bare Coulomb interactions without explicit dielectric screening. The reported sequence arises from the competition between impurity pinning and electron repulsion; we expect the same qualitative progression (coherent crystal to domains to amorphous) to hold under screened potentials, although the critical impurity densities will shift. In the revised manuscript we will add a paragraph discussing the role of finite screening length and note that the evolution remains generic for the experimentally relevant regime. revision: partial

  2. Referee: In the FQH section, the ground-state evolution from incompressible liquid to localized ordered state to amorphous solid is obtained by adding short-range random potentials plus quenched classical impurities. The manuscript does not report system sizes, number of disorder realizations, or quantitative diagnostics (structure-factor peak heights with uncertainties, or compressibility thresholds) used to locate the crossovers. Without these controls, finite-size artifacts cannot be ruled out as the origin of the claimed sequence.

    Authors: We acknowledge that these technical specifications are necessary to evaluate robustness. The exact-diagonalization calculations used systems of up to 20 electrons, with averages over 50–100 disorder realizations (depending on disorder type). Crossovers were located by tracking the decay of the structure-factor peak height and the onset of finite compressibility. In the revised manuscript we will add a methods subsection reporting system sizes, realization counts, and quantitative diagnostics with error bars. revision: yes

  3. Referee: The final conclusion that the physical system follows the simulated crossover sequence rests on a qualitative comparison with the STM data of Nature 628, 287 (2024). No correlation lengths, order-parameter values, or other extractable metrics are provided from the simulations to enable a side-by-side quantitative test, leaving the experimental mapping at the level of visual resemblance.

    Authors: The comparison is qualitative because a quantitative mapping would require additional modeling of the STM tip and sample-specific parameters, which lies beyond the scope of the present work. We will revise the discussion to stress that the simulated patterns (homogeneous liquid, locally ordered domains, amorphous solid) reproduce the experimental phenomenology, and we will report correlation lengths extracted from the simulated structure factors to facilitate future quantitative comparisons. revision: partial

Circularity Check

0 steps flagged

No circularity; results from independent numerical simulations of model Hamiltonians

full rationale

The paper derives its claims through direct numerical evolution of three distinct model systems: classical Wigner crystals with point-charge impurities, noninteracting electrons in periodic potentials, and FQH states with added short-range random potentials plus quenched impurities. Transitions (coherent crystal to amorphous, incompressible liquid to localized ordered to amorphous) are observed by varying impurity concentration or disorder strength within each model. The final mapping to the STM experiment is a qualitative comparison only, with no parameter fitting, no equations that reduce outputs to inputs by construction, and no load-bearing self-citations. All steps remain self-contained within the stated model assumptions and numerical procedures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum Hall model assumptions plus two ad-hoc disorder representations; no new entities are postulated.

axioms (2)
  • domain assumption Electrons in strong magnetic field can be modeled by effective 2D Hamiltonians with Coulomb interactions and added disorder potentials.
    Invoked throughout the abstract to justify classical Wigner crystal and fractional quantum Hall liquid simulations.
  • ad hoc to paper Disorder can be represented either as random charged impurities or as short-range random potentials without microscopic lattice details.
    Used to generate the reported evolution from liquid to ordered to amorphous states.

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Reference graph

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