{"id":"7fa658de-8c19-4ec4-a218-1b7abc0bde5d","arxiv_id":"2411.16862","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper defines fundamental length scales for Landau levels and fractional quantum Hall phases below which discrete lattice potentials stop breaking continuous magnetic translational symmetry.","lead":"This paper shows that a lattice of very closely spaced voltage spikes leaves the motion of electrons within a single Landau level almost unchanged, because the electrons cannot resolve features smaller than the magnetic length. The result suggests that the quantum particles in fractional quantum Hall states, including anyons, are largely immune to the short-wavelength part of disorder.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3) is not the spectral bandwidth: it is the diagonal/variational range of the LLL-projected delta lattice, giving e^{-2π^2/t^2}, while the exact bandwidth is set by e^{-π^2/t^2}; quantitative robustness claims and FQH lengths built on the 2.1e-8 cutoff are unsupported.","rationale":"Good-faith reading: the paper's qualitative claim is physically sensible: the LLL projection filters short wavelengths, so a sufficiently dense delta lattice is smoothed. That part is standard and survives. But the paper quantifies this by Eq. (3), and the supplementary derivation shows Eq. (3) is obtained from the real-space diagonal of H', i.e., from the range of or equivalently the coherent-state expectation. For a one-body periodic potential in the LLL, the spectrum is controlled by Fourier amplitudes v_G e^{-G^2/4}; the diagonal expectation squares the form factor. Hence Eq. (3) underestimates the exact bandwidth by ~e^{+π^2/t^2} — a factor ~3×10^4 at t=ℓ_B. The claim that this variational range 'converges' to the exact bandwidth contradicts the standard Harper-type spectrum of e^{iG·R}+e^{-iG·R}, whose range is 4 regardless of G; only the prefactor e^{-G^2/4} decays. Thus the quantitative benchmark 2.1e-8 and all fundamental lengths derived from it are not reliable. The FQH part compounds this: it uses real-space density profiles to compute only the diagonal energy landscape of a quasihole, again a lower bound on the projected spectrum; off-diagonal quasihole tunneling is uncontrolled. The density-profile input from Ref. [47] and the unspecified extrapolation add further uncertainty. The abstract's 'long wavelength' versus the main text's 'short wavelength' is a separate reporting inconsistency. If the exact spectral bandwidth is recomputed and found to follow e^{-π^2/t^2}, the qualitative robustness claim remains, but the specific numbers and the claim to 'quantitatively define' fundamental lengths must be revised. Hence REJECT in current form, not because the idea is wrong, but because the central quantitative derivation is of a different quantity than claimed.","tokens_in":14521,"tokens_out":32145,"duration_ms":320785,"concrete_test":"On a torus with N_Φ flux quanta, construct the LLL-projected delta lattice P V P for spacing t=ℓ_B and diagonalize it exactly (N_Φ≥36, with lattice commensurate). Compare the eigenvalue spread with Eq. (3). If the spread is ~10^{-4}, not ~10^{-8}, Eq. (3) is the diagonal range rather than the spectrum; then recompute Fig. 2 with the corrected bandwidth before accepting any FQH fundamental length.","verdict_should_be":"REJECT","load_bearing_attack":"Equation (3) is the load-bearing quantitative result. The supplementary derivation computes the range of the real-space diagonal of the projected delta lattice, not the spectrum of the projected one-body operator PVP. For a periodic potential in the LLL each Fourier component enters with one power of the LLL form factor: H_LLL = t^{-2} Σ_{n,m} e^{-π^2(n^2+m^2)/t^2} T_{nm}. The first harmonics therefore carry amplitude t^{-2} e^{-π^2/t^2}, and T_{10}+T_{-10} has spectral range 4, so the exact bandwidth is of order t^{-2} e^{-π^2/t^2} (≈10^{-4} at t=ℓ_B), not the t^{-2} e^{-2π^2/t^2} of Eq. (3). The exponent in Eq. (3) is the square of the form factor, which appears in the coherent-state expectation , not in the single-particle spectrum. The variational diagonal range is a lower bound on the spectral bandwidth, and it is not what is claimed to be numerically verified. Consequently the 2.1×10^{-8} benchmark, the LLL fundamental length ℓ_B, all higher-LL fundamental lengths in Fig. 2(a,e), and the FQH values a_ν and the Moore-Read value in Fig. 2(b,f) — all extracted with the same diagonal density-profile procedure — are not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that a periodic array of delta-function potentials, when projected into a single Landau level (LL), produces an effective Hamiltonian that is extremely close to the identity for lattice spacings below the magnetic length, so that continuous magnetic translational invariance survives inside the LL Hilbert space even though the potential breaks it in the full Hilbert space. It defines a 'fundamental length' for each LL and for fractional quantum Hall (FQH) conformal Hilbert spaces by imposing a bandwidth cutoff, extracts values for Laughlin-1/2, 1/3, 1/4, 1/5 and Moore-Read quasiholes from real-space density profiles, and concludes that short-wavelength disorder has negligible effect on anyon dynamics below these length scales.","tokens_in":14909,"tokens_out":7338,"duration_ms":73623,"significance":"If correct, the paper would introduce a useful hierarchy of disorder length scales for topological quantum Hall systems and would have concrete implications for anyon braiding experiments. The manuscript includes an analytic supplementary derivation, explicit formulas for the diagonal of the projected delta lattice, and a separate check of Landau-level mixing effects; these are valuable elements. However, the central quantitative result is built on a variational/diagonal quantity rather than the spectral bandwidth of the projected operator, and the FQH fundamental lengths depend on an unspecified fitting and extrapolation procedure and on density profiles from a same-group reference. The qualitative idea is defensible and potentially interesting, but the quantitative claims in their present form are not established.","major_comments":[{"comment":"Equation (3) is not the spectral bandwidth of the LLL-projected delta-potential lattice. The supplementary derivation computes the range of the diagonal element , i.e. the variational range of the coherent-state expectation, and then asserts that this is the bandwidth. For a periodic potential the projected operator is PVP = t^{-2} Σ_{n,m} e^{-π^2(n^2+m^2)/t^2} T_{nm}, so the first harmonics have amplitude t^{-2}e^{-π^2/t^2}, and T_{10}+T_{-10} has spectral range 4. The exact bandwidth is therefore of order t^{-2}e^{-π^2/t^2}, not t^{-2}e^{-2π^2/t^2} as in Eq. (3). At t=ℓ_B the exact bandwidth is roughly 10^{-4}, six orders of magnitude larger than the value implied by Eq. (3); at t=1.7ℓ_B it is of order 0.05. Consequently the 2.1×10^{-8} benchmark, the statement a_LLL=ℓ_B, and every fundamental length in Fig. 2(a),(b),(e),(f) that is calibrated against this cutoff are not supported by the presented calculation.","section":"Main text, 'Lattice potential in the LLL'; Supplementary Eq. (S3)"},{"comment":"The extraction of the FQH fundamental lengths is not reproducible as reported. The text states that the bandwidth-spacing relation is fitted and then extrapolated to 2.1×10^{-8}, but it does not specify the fitting function, the number of data points, the fitting range, or any uncertainty estimate for the Laughlin-1/2, 1/3, 1/4, 1/5, or Moore-Read cases. Since the quoted values a_ν ≈ 1.5, 1.7, 2.0, 2.2 ℓ_B and the Moore-Read value 1.29 ℓ_B are central quantitative outputs, the absence of this information prevents assessment and verification.","section":"Main text, 'Fundamental length scales of conformal Hilbert spaces'"},{"comment":"All FQH quasihole fundamental lengths are derived from the real-space density profiles ρ(r) of Ref. [47], which is by the same group, and no independent check or error propagation is provided. Because the bandwidths in Fig. 2(b),(f) are computed directly from these density profiles, any systematic error in ρ(r) propagates linearly into a_{1/2}, a_{1/3}, a_{1/4}, a_{1/5}, and the Moore-Read value. The paper should include a sensitivity analysis or a comparison with an independent calculation of the quasihole density profile.","section":"Main text, 'Fundamental length scales of conformal Hilbert spaces'; Ref. [47]"},{"comment":"The claim that the variational bandwidth 'will converge to the exact bandwidth from exact diagonalization' and has been 'numerically verified both on the disk and torus geometry' is not backed by a direct comparison in the manuscript. The supplementary figure compares the disk and torus results of the same diagonal-range quantity; it does not compare that quantity with the spectrum of PVP obtained by exact diagonalization. Given the large difference between the diagonal range and the true spectral bandwidth noted in the first comment, this verification is essential rather than auxiliary.","section":"Main text, 'Lattice potential in the LLL'; Supplementary Fig. S1"}],"minor_comments":[{"comment":"The abstract says the dynamics of anyonic excitations are robust against the 'long wavelength' part of the disorder, while the main text and summary state that the 'short wavelength' part (large wavevector) has negligible effects. This is a physically important inconsistency and should be corrected.","section":"Abstract"},{"comment":"The empirical tanh fit for higher-LL fundamental lengths is reported with R²=0.9997, but no parameter uncertainties or residual information are given; with only a few Landau-level data points, standard errors are needed to support the fitted curve.","section":"Main text, 'Fundamental length scales of conformal Hilbert spaces'"},{"comment":"The statement that the cutoff 2.1×10^{-8} is chosen 'so that the fundamental length of the LLL is the magnetic length' should be presented as a calibration convention, not as an independent determination, and the main text should carry the same transparency.","section":"Supplementary Sec. II"},{"comment":"The equality H_vNL = Σ_X |X>, which is a variational bound, not the spectrum. For the projected delta lattice in the LLL, each reciprocal vector q enters with one power of the form factor e^{-|q|^2/4}. The first harmonics at |q|=2π/t give an amplitude ~ t^{-2} e^{-π^2/t^2}, and T(q)+T(-q) has spectrum spanning [-2,2], so the true bandwidth is ~ (4/t^2)e^{-π^2/t^2}. At t=ℓ_B that's ~10^{-4}, about 10^4 times larger than the 2.1×10^{-8} claimed. So the cutoff benchmark and the resulting fundamental lengths (a_LLL=ℓ_B, the higher-LL tanh fit, and the Laughlin/Moore-Read values) are not established. The density-profile input comes from a same-group reference, the fits are unspecified, and there are no error bars, so the FQH lengths are doubly soft.\n\nThat said, the qualitative story is credible and the paper has good bones. The idea of using the von Neumann lattice to define a Hamiltonian for a conformal Hilbert space is nice, and the contrast with the quantum metric (which increases with LL index while the fundamental length decreases) is genuinely interesting. The supplementary's all-orders LL-mixing analysis is clean and convincing: the dressed-LL bandwidth stays very close to the within-LL result. Also, the abstract/main text wavelength inconsistency that worried the reader doesn't appear in the manuscript I read; both say the short-wavelength part of disorder is suppressed.\n\nThe bottom line: the central quantitative result needs to be recomputed. The authors should acknowledge that Eq. (3) is a Husimi-function range and redo the analysis with exact diagonalization or a proper Bloch-state calculation. Once that's done, the qualitative claims about robustness and the existence of a fundamental length might still hold, but the numbers will shift substantially.\n\nThis is worth sending to referees—the conceptual idea is good enough that a careful referee could help fix the form-factor mistake. But I wouldn't cite the quantitative values until a corrected version appears, and I wouldn't present this as a quantitative theory of anyon protection yet.","headline":"The central quantitative claim is undercut by a form-factor exponent error: Eq. (3) is the Husimi range, not the spectral bandwidth, so the fundamental lengths are unsupported, though the qualitative robustness idea may survive.","tokens_in":15394,"tokens_out":9073,"would_cite":false,"duration_ms":88999,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.43.-f","73.43.Cd"],"model":"deepseek-v4-flash","headline":"Continuous magnetic translational symmetry survives a lattice of point potentials inside a single Landau level, because the projected bandwidth collapses faster than exponentially as the lattice spacing drops below the magnetic length.","keywords":["magnetic translational invariance","Landau levels","von Neumann lattice","fractional quantum Hall effect","anyon dynamics","disorder robustness","conformal Hilbert space","quasihole fundamental length"],"falsifier":"Exact diagonalization of a single Laughlin-1/3 quasihole on a torus with a periodic array of delta potentials of spacing $a=1.7\\ell_B$ should give a bandwidth below $10^{-7}$; if the computed bandwidth is orders of magnitude larger, the extrapolation from the density profiles is wrong. A high-resolution STM measurement of the local density of states of a fractional quantum Hall droplet under a fine tip-potential array would directly test the predicted robustness.","tokens_in":14331,"feed_emoji":"🧲","tokens_out":5930,"duration_ms":54627,"temperature":0.7,"pith_summary":"This paper argues that the continuous magnetic translational symmetry of a single Landau level—and of the Hilbert spaces of fractional quantum Hall states—can survive a lattice of point-like potentials, even though the potential breaks the symmetry in the full two-dimensional Hilbert space. The reason is a strong suppression of the projected bandwidth: once the lattice spacing drops below the magnetic length, or below a phase-specific fundamental length, the band becomes exponentially flat. This makes disorder with short wavelengths nearly invisible to anyonic excitations in the quantum Hall regime. The paper defines these fundamental lengths quantitatively and shows they behave oppositely to the quantum metric length as the Landau level index increases.","feed_headline":"Lattice potentials barely break a Landau level's hidden symmetry","feed_subtitle":"Short-wavelength disorder leaves anyon dynamics untouched once its spacing drops below the magnetic length.","key_machinery":"The von Neumann lattice (vNL) Hamiltonian, defined as $H_{\\rm vNL}=\\sum_X |X\\rangle\\langle X|$, is a lattice of projection operators onto coherent states that equals the projection of a delta-potential lattice into the Landau level; its analytic bandwidth is given in terms of elliptic $\\theta$ functions, $\\Delta E = \\frac{2}{\\pi t^2}\\,\\theta[2,0,e^{-8\\pi^2/t^2}]\n\\theta[3,0,e^{-8\\pi^2/t^2}]$, which controls the exponential suppression. The generalization to interacting phases uses the conformal Hilbert space (CHS) of a topological phase, spanned by the ground state and quasihole states, with the quasihole real-space density profile $\\rho(r)$ as the input that determines how the bandwidth depends on lattice spacing.","core_discovery":"Within the Hilbert subspace of a single Landau level, a superlattice of delta-function potentials yields a projected Hamiltonian whose bandwidth decreases faster than exponentially with the ratio $t/\\ell_B$, so for $t \\lesssim \\ell_B$ the spectrum is flat to better than $2.1\\times10^{-8}$ in units of the potential strength. Consequently the continuous magnetic translational invariance of the truncated space is restored to excellent approximation even though the potential breaks it in real space. For fractional quantum Hall phases, the same argument applies inside the conformal Hilbert space of a phase: each phase has a fundamental length $a_\\nu$ below which quasihole dynamics are insensitive to the potential. For Laughlin states these lengths track the composite-fermion scale, approximately $\\sqrt{m}\\,\\ell_B$ for filling $1/m$, while the Moore-Read $e/2$ quasihole subspace at $\\nu=1/2$ has a smaller fundamental length than the Laughlin-1/2 subspace, indicating greater sensitivity to disorder. The paper also shows that short-wavelength components of a random disorder potential have negligible effect on anyon dynamics, because their wavelengths fall below the relevant fundamental length.","pith_inferences":["If the mechanism is general, flat Chern bands with non-commutative geometry should show the same blindness to fine-grained one-body potentials, since the suppression is a property of the projected Hilbert space rather than of the specific Landau-level wavefunctions.","Because the absolute fundamental lengths depend on the chosen bandwidth cutoff, the physically robust statements are the ratios between phases; experiments and numerics should focus on measuring these ratios rather than absolute values.","A direct testable extension would place a periodic array of tip-induced potentials with spacing just below the predicted fundamental length over a $\\nu=1/3$ sample and measure quasihole mobility or interferometric visibility; it should remain essentially unchanged compared to a clean sample.","The smaller fundamental length of the Moore-Read quasihole subspace suggests that non-Abelian anyons may be more susceptible to disorder pinning than Abelian ones at the same filling, which could be probed by intentionally introducing short-wavelength disorder in a $\\nu=5/2$ sample."],"forward_implications":["Short-wavelength disorder potentials in realistic samples have negligible effect on the dynamics of anyons in fractional quantum Hall states, provided the disorder is weaker than the incompressibility gap and its wavelength is below the phase's fundamental length.","Anyonic braiding and interferometry can be performed in the presence of fine-grained impurities without compromising the exchange statistics, as long as the spatial separations between braiding paths exceed the relevant fundamental length.","The fundamental length provides a quantitative measure of quasihole size that is distinct from the quantum metric length; low-filling Laughlin states have larger fundamental lengths and are therefore more robust to disorder than integer quantum Hall states.","The von Neumann Hamiltonian gives a nearly flat many-body band within each conformal Hilbert space, enabling selective superlattice engineering that perturbs different topological phases differently based on their fundamental lengths.","The exponential suppression of bandwidth implies that LL mixing and higher-order perturbation effects remain negligible even when the cyclotron gap is comparable to the potential strength, protecting the robustness of the translational invariance."],"supporting_citations":[{"why":"Supplies the completeness of the von Neumann lattice of coherent states, which underlies the vNL Hamiltonian as a projection of the delta-potential lattice.","marker":"[38]"},{"why":"Provides the real-space quasihole density profiles used to compute bandwidths and extract fundamental lengths for Laughlin and Moore-Read phases.","marker":"[47]"},{"why":"Defines the conformal Hilbert space of a topological phase as the subspace spanned by ground and quasihole states, the framework for generalizing the argument beyond a single Landau level.","marker":"[49]"},{"why":"Contains the analytic bandwidth derivation, disk/torus numerics, and the LL-mixing perturbation calculation that support the robustness claim.","marker":"[54]"},{"why":"Provides the short-range two-body model Hamiltonian whose null space is the Laughlin conformal Hilbert space, used to define the quasihole Hilbert spaces.","marker":"[57]"}],"fun_headline_variants":["Landau level symmetry survives lattice potentials below magnetic length","Short-wavelength disorder leaves anyon dynamics intact","Hidden symmetry in quantum Hall fluids persists under potentials","Lattice potentials can't break magnetic translation invariance in LL subspace","Fundamental length decides robustness of anyon dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantitative length scales claimed for fractional quantum Hall phases depend on an earlier computation of quasihole charge distributions and on a fitted extrapolation to a very small bandwidth cutoff; if that input is inaccurate, the specific numbers shift, though the qualitative robustness below the magnetic length is an analytic result.","fun_headline_variants_meta":{"raw":{"variants":["Landau level symmetry survives lattice potentials below magnetic length","Short-wavelength disorder leaves anyon dynamics intact","Hidden symmetry in quantum Hall fluids persists under potentials","Lattice potentials can't break magnetic translation invariance in LL subspace","Fundamental length decides robustness of anyon dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1802,"prompt_tokens":929,"completion_tokens":873,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":798}},"tokens_in":545,"tokens_out":873,"duration_ms":7429,"temperature":1.0,"reasoning_tokens":798,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:47:32.240059+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exact diagonalization of a single Laughlin-1/3 quasihole on a torus with a periodic array of delta potentials of spacing $a=1.7\\ell_B$ should give a bandwidth below $10^{-7}$; if the computed bandwidth is orders of magnitude larger, the extrapolation from the density profiles is wrong. A high-resolution STM measurement of the local density of states of a fractional quantum Hall droplet under a fine tip-potential array would directly test the predicted robustness.","supporting_citations":[{"cited_title":"On the completeness of a system of coherent states,","cited_arxiv_id":null,"evidence_quote":"Supplies the completeness of the von Neumann lattice of coherent states, which underlies the vNL Hamiltonian as a projection of the delta-potential lattice."},{"cited_title":"Universal Mod- elling of Emergent Oscillations in Fractional Quantum Hall Fluids,","cited_arxiv_id":null,"evidence_quote":"Provides the real-space quasihole density profiles used to compute bandwidths and extract fundamental lengths for Laughlin and Moore-Read phases."},{"cited_title":"Anyons in conformal Hilbert spaces: Statis- tics and dynamics of gapless excitations in fractional quantum Hall systems,","cited_arxiv_id":null,"evidence_quote":"Defines the conformal Hilbert space of a topological phase as the subspace spanned by ground and quasihole states, the framework for generalizing the argument beyond a single Landau level."},{"cited_title":"Robust translational invariance in topological bands against lattice potentials and disorders","cited_arxiv_id":null,"evidence_quote":"Contains the analytic bandwidth derivation, disk/torus numerics, and the LL-mixing perturbation calculation that support the robustness claim."},{"cited_title":"Landau level mix- ing and levitation of extended states in two dimensions,","cited_arxiv_id":null,"evidence_quote":"Provides the short-range two-body model Hamiltonian whose null space is the Laughlin conformal Hilbert space, used to define the quasihole Hilbert spaces."}],"review_version":1}