{"id":"182cd800-aa3f-4863-a37e-ffe05b84226c","arxiv_id":"2412.01301","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In a quasiperiodic non-Hermitian lattice, wave packets spread diffusively with exponent 1/2 in the delocalized phase and subdiffusively with exponent 1/3 in the localized phase, in contrast to ballistic and halted Hermitian dynamics.","lead":"This paper derives universal wave-spreading exponents for a non-Hermitian version of the Aubry-André model, finding diffusive spreading (exponent 1/2) in the delocalized regime and subdiffusive spreading (exponent 1/3) in the localized regime. A smart generalist should care because it shows how non-Hermiticity can turn localization into slow transport, which is testable in photonic lattices and cold-atom experiments.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The iid extreme-value assumption in SM(II) is not satisfied by the quasiperiodic model; Eq. (12) needs a deterministic empirical-tail justification, and the exponents depend on it.","rationale":"The reader's weakest assumption names exactly the step I find most load-bearing: Eq. (12) is the bridge between spectral statistics and the spreading exponents, and its proof in SM(II) assumes iid imaginary eigenvalue parts, which the quasiperiodic potential does not provide. I therefore agree with the reader's identification. However, I do not regard the concern as fatal, because there is a plausible repair: for an equidistributed deterministic sequence with a square-root tail, the maximum of the first N sites still leaves a tail integral ∼ N^{-1}, so the exponents may survive. The direct finite-size order-statistics test I propose would settle this: it distinguishes the iid mechanism from the weaker empirical-tail mechanism and would expose any correction to δ. The numerical evidence in Fig. 3, the LE-based iDOS extraction, and the mobility-edge model in SM(IV) give independent support for the phenomenology, so a conditional verdict remains appropriate. The paper's generality claim for 'generic disordered non-Hermitian systems' is, however, not fully established until Eq. (12) is justified for deterministic quasiperiodic disorder; that is why the current CONDITIONAL verdict should stand unchanged rather than being upgraded to ACCEPT.","tokens_in":12003,"tokens_out":10431,"duration_ms":107333,"concrete_test":"For model (1) with |V|=5, ϕV=π/3, OBC, and Fibonacci system sizes L=610, 987, 1597, 2584, 4181, diagonalize H and record λ_max(L) = max_j Im E_j. Using the thermodynamic iDOS ρ_I(s) from Fig. 4(d), compute T(L) = ∫_{λ_max(L)}^{∞} ρ_I(s) ds. Eq. (12) predicts T(L) ∝ L^{-1}. If instead T(L) ∝ L^{-2} or shows a different power/log drift, the exponents in Eq. (13) do not follow. As a control, repeat with the on-site imaginary parts λ_j randomly permuted (same marginal distribution, no quasiperiodic correlations) and check that T(L) again scales as L^{-1}; this isolates whether the iid assumption itself or only the empirical-tail scaling is required. A nonzero result at this step would leave the numerics intact but would require rewriting the universal derivation around deterministic extremal statistics rather than iid extreme-value theory.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The scaling exponents in Eq. (13) follow from combining the maximization conditions (9)/(11) with Eq. (12), ∫_λ^∞ ρ_I(s) ds ∼ X^{-d}. Eq. (12) is proved in SM(II) by treating the imaginary parts λ_j of the O(X^d) local eigenvalues as iid random variables drawn from the thermodynamic iDOS ρ_I(s), yielding E[∫_{λ_max}^∞ ρ] = 1/(N+1). For the quasiperiodic model (1), this premise is not met: to leading order in the deep localized limit the λ_j are deterministic functions of the site, λ_j ≈ Im[|V| e^{iϕV} cos(2παj+ϕ)], so the extremes are order statistics of a low-discrepancy deterministic sequence, not of independent draws. The iid proof therefore does not apply as stated. What is actually needed is the weaker statement that the upper empirical tail of the first N sites has measure ∼ N^{-1}; this may follow from equidistribution plus the square-root Van Hove tail, but the paper neither states nor proves it. If the deterministic extreme gap instead scaled as N^{-α} with α≠1, the exponents would become δ=(β+1)/(α d + β +1) and δ=(β+1)/(α d), changing the headline values. The numerical agreement in Fig. 3 and the mobility-edge test in SM(IV) corroborate the phenomenology but do not validate the iid step in the derivation; SM(III) only checks stability of β=−1/2, not the order-statistics relation. This is the most load-bearing gap: the framework's universality claim is exactly the derivation of Eq. (12) for generic disorder, and here its proof uses an assumption the model violates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies wave-packet spreading in the one-dimensional non-Hermitian Aubry-André model with complex on-site potential V cos(2παj + ϕ). The authors report three numerical facts: in the delocalized phase |V| < 2 the normalized second-moment width grows as X(t) ∼ t^{1/2}, in the localized phase |V| > 2 it grows as t^{1/3}, and at the transition the fitted exponent is approximately 0.57. They propose a scaling theory in which the largest imaginary part among O(X^d) local eigenvalues controls the propagation, leading to Eqs. (9), (11), and (12) and to the general relations δ = (β+1)/(d+β+1) (localized) and δ = (β+1)/d (delocalized), where β is the Van Hove tail exponent of the imaginary density of states. Using the generalized Thouless relation, they extract ρ_I(λ) from the Lyapunov exponent and find β = −1/2 in both regimes, reproducing δ = 1/3 and δ = 1/2. A supplemental model with a mobility edge yields a fitted β ≈ −0.435 and a predicted δ ≈ 0.36, confirmed by direct dynamics.","tokens_in":109,"tokens_out":6295,"duration_ms":118674,"significance":"If the derivation were made rigorous, the result would establish a useful universality: non-Hermitian dynamics converts the Hermitian halted/ballistic dichotomy into subdiffusion/diffusion controlled by the tail of the imaginary density of states. The main numerical observation (δ ≈ 1/3 and δ ≈ 1/2 across the entire phase diagram) is clean and internally consistent, and the Lyapunov-exponent method is a practical alternative to exact diagonalization. The mobility-edge test in SM Section IV is a genuine dynamical confirmation of the scaling relation once β is known. The main reservation is that the key relation Eq. (12) is derived in SM Section II under an iid assumption that the quasiperiodic model does not satisfy; the universality claim therefore currently rests on an unproven empirical-tail statement.","major_comments":[{"comment":"The derivation of Eq. (12) is load-bearing and is not justified for the model studied. SM Section II explicitly says \"By treating λ_j as random variables drawn from the distribution ρ_I(s),\" and the calculation E[∫_{λ_max}^∞ ρ_I(s) ds] = 1/(N+1) uses the product structure P(λ_max ≤ λ) = [F(λ)]^N, which is exact only for independent draws. For the quasiperiodic potential in model (1), the imaginary parts of eigenvalues are deterministic functions of the site index in the deep localized limit (SM Section III gives λ(k) = |V| sin ϕ_V cos k [1 + O(t^2)]), so the extremes are order statistics of a low-discrepancy deterministic sequence, not of independent samples. The authors need to replace the iid step with a proof or explicit statement that the upper empirical tail of the first N sites has measure ∼ N^{−1}; if the empirical tail exponent were α ≠ 1, Eq. (13) would become δ = (β+1)/(α d + β + 1) and δ = (β+1)/(α d). The numerical agreement in Fig. 3 and the mobility-edge test in SM Section IV corroborate the phenomenology but do not validate this step; SM Section III only checks the stability of β = −1/2, not the order-statistics relation.","section":"SM Section II and Eq. (12)"},{"comment":"The saddle-point/maximization conditions (9) and (11) treat λ(X) as a smooth function of the localization center X and replace the discrete set of eigenstates by a continuum. In the quasiperiodic model, λ(X) is a deterministic pseudo-random sequence in X, and the derivative ∂λ/∂X is not defined in the usual sense; the propagator ansatz (8) also assumes a purely exponential spatial profile with a single localization length ξ. These are reasonable heuristic scaling arguments, but the paper does not state the conditions under which the saddle point is valid. Since Eqs. (9) and (11) are used together with Eq. (12) to obtain the exponents, the authors should either justify the continuum approximation for this model or clearly label it as an assumption on the same footing as Eq. (12).","section":"Eqs. (8), (9), and (11)"},{"comment":"The claim that the framework applies to \"generic disordered non-Hermitian systems, whether the disorder is correlated or uncorrelated\" is broader than the derivation supports. Equation (12) requires knowledge of the empirical tail of local eigenvalues; for uncorrelated random disorder the iid argument is plausible, but for correlated or quasiperiodic disorder it is not automatic. The authors should either prove a deterministic analogue of Eq. (12), state it as an explicit assumption, or restrict the universality claim to the cases where the empirical-tail relation can be verified.","section":"Conclusion, final paragraph"}],"minor_comments":[{"comment":"In the description of Fig. 2(b), the text says the phase boundary separates \"the delocalized regime (|V| < 2) from the localized regime (|V| < 2)\"; the second inequality should be |V| > 2.","section":"Fig. 2 caption and text"},{"comment":"The supplement has two sections labeled \"(III)\": \"Perturbative analysis of the spectral structure\" and \"Dynamical spreading in the presence of mobility edge.\" The latter should be numbered (IV).","section":"Supplemental Material headings"},{"comment":"Equation (12) is attributed to reference [44] (the supplemental material), but the derivation is in SM Section II; the citation should be made explicit at first use so that readers know where the proof is located.","section":"Main text, Eq. (12)"},{"comment":"The caption for Fig. 3 refers to colored dots and dashed fitting lines but does not specify the color coding or the time window used for the steady-evolution fit; a legend or colorbar would help the reader assess how robust the extracted slopes are.","section":"Fig. 3"},{"comment":"The sentence \"δ > 1/2, δ = 1/2, and δ < 1/2 corresponds to superdiffusive, diffusive, and subdiffusive transport\" has a subject-verb agreement error; \"corresponds\" should be \"correspond.\"","section":"Text after Eq. (7)"}],"recommendation":"major_revision","confidential_remarks":"The iid step in SM Section II is the main risk to the paper's central claim. The numerics are convincing and the mobility-edge test is a good consistency check, but without a deterministic-order-statistics argument the universality claim is under-supported. If the authors can supply such an argument, or clearly state it as an assumption, the paper would be suitable for publication. I would not reject outright because the reported exponents are likely correct and the framework is valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper reports something real and worth knowing: in the non-Hermitian Aubry-André model, normalized wave-packet spreading goes as t^{1/3} in the localized regime and t^{1/2} in the delocalized regime. The numerics are clean enough to believe the phenomenology, and the mobility-edge supplement in the SM is the strongest part—they fit the iDOS tail exponent β, plug it into their scaling relation, get δ=0.36, and then confirm δ=0.36 from direct dynamics. That is a genuine parameter-free check, and it works.\n\nThe genuinely new pieces are the delocalized-regime diffusive law, the unified localized/delocalized scaling treatment, and the extraction of δ from Lyapunov exponents via the generalized Thouless relation. The localized-regime δ=1/3 and the extreme-value machinery overlap with Ref. [33], which they cite; this is a solid extension, not a new subfield.\n\nNow the soft spots. The derivation of Eq. (12), the key scaling relation, rests on treating the imaginary parts of local eigenvalues as iid draws from the thermodynamic iDOS. That is fine for genuinely random disorder, but the quasiperiodic model's λ_j are deterministic, strongly correlated functions of the site index. The SM proof does not apply to that case as written. If the upper tail of the deterministic sequence behaved differently from iid, the exponents would change. I suspect the statement is still true for this model—the sequence is equidistributed, and the empirical tail above the maximum is 1/N by definition, so with a discrepancy bound you likely get Eq. (12) without any randomness—but the paper does not make that argument. As written, the universality framework has a rigor gap at its base, even though the numerics support the conclusions.\n\nOther issues are minor: Fig. 3 has no error bars, “dozens of phase samples” is vague, and the reported δ=0.57 at the transition sits awkwardly with their statement that the critical exponent is an open question.\n\nBottom line: the paper is worth engaging. The phenomenology looks right, the LE method is practical, and the mobility-edge prediction is a good sign. The iid gap is addressable and should be fixed in revision. Send it to a referee; it should not be desk-rejected.","headline":"Solid, clean numerics and a useful LE-based method; the central scaling derivation leans on an iid assumption that does not literally apply to the quasiperiodic model, but the gap looks repairable and the results are probably right.","tokens_in":12903,"tokens_out":5074,"would_cite":true,"duration_ms":46429,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B44","81Q12"],"pacs":[],"model":"deepseek-v4-flash","headline":"In the non-Hermitian Aubry-André model, wave packets spread as t^{1/3} in the localized phase and t^{1/2} in the delocalized phase, instead of halting or moving ballistically.","keywords":["non-Hermitian Aubry-André model","quasiperiodic disorder","wave-packet spreading","dynamical delocalization","imaginary density of states","Van Hove singularity","Lyapunov exponent","subdiffusion"],"falsifier":"For the non-Hermitian Aubry-André model at $|V|=5$, compute the largest imaginary eigenvalue among blocks of length $X$ and check whether the ensemble-averaged tail integral $\\mathbb{E}[\\int_{\\lambda_{\\max}}^{\\infty}\\rho_I(s)\\,ds]$ decays as $1/X$; if it does not, Eq. (12) fails and the claimed $\\delta=1/3$ need not follow. One can also simulate the wave-packet spreading for much longer times and larger systems and check whether the fitted exponent stays at $1/3$ in the localized regime and $1/2$ in the delocalized regime.","tokens_in":11825,"feed_emoji":"🌊","tokens_out":12513,"duration_ms":93525,"temperature":0.7,"pith_summary":"Non-Hermitian systems can keep a wave packet spreading even when every eigenstate is localized, because eigenstates with larger imaginary energies dominate the time evolution. This paper studies the non-Hermitian Aubry-André model, a one-dimensional lattice with a complex quasiperiodic potential, and claims that normalized wave packets always spread algebraically: with exponent $\\delta=1/3$ in the localized regime and $\\delta=1/2$ in the delocalized regime, in place of the Hermitian behaviors of halted and ballistic motion. The authors connect both exponents to a single spectral feature, a Van Hove singularity with exponent $\\beta=-1/2$ at the tail of the imaginary density of states, through scaling relations $\\delta=(\\beta+1)/(d+\\beta+1)$ and $\\delta=(\\beta+1)/d$. They also give a Lyapunov-exponent method for extracting the spreading exponent without diagonalizing large non-Hermitian matrices. If the claim holds, slow algebraic spreading is a generic dynamical signature of non-Hermitian disorder, not a special property of one model.","feed_headline":"Non-Hermitian quasicrystals spread at t^{1/3} and t^{1/2} rates","feed_subtitle":"Non-Hermitian disorder turns halted or ballistic motion into slow algebraic spreading set by a spectral singularity.","key_machinery":"The load-bearing object is the imaginary density of states, $\\rho_I(s)$, the normalized distribution of $\\operatorname{Im} E$ over the spectrum. The engine of the argument is an extreme-value identity for that distribution: if the imaginary parts of the eigenvalues in a local region of size $N=X^d$ are drawn independently from $\\rho_I(s)$, then the expected tail integral beyond the largest one, $\\mathbb{E}[\\int_{\\lambda_{\\max}}^{\\infty}\\rho_I(s)\\,ds]$, equals $1/(N+1)$ and therefore decays as $X^{-d}$. Plugging that into the steepest-descent conditions $\\partial\\lambda/\\partial X\\sim t^{-1}$ and $X\\partial\\lambda/\\partial X\\sim t^{-1}$ converts a spectral tail exponent $\\beta$ into the spreading exponents $\\delta=(\\beta+1)/(d+\\beta+1)$ and $\\delta=(\\beta+1)/d$. The paper's practical tool is the generalized Thouless relation $\\rho(E)=\\frac{1}{2\\pi}\\nabla^2\\gamma(E)$, which gives the density of states from the Laplacian of the Lyapunov exponent in the complex plane and lets the authors read off $\\rho_I$ without exact diagonalization.","core_discovery":"The paper's central claim is that in the non-Hermitian Aubry-André model with complex on-site potential $V=|V|e^{i\\phi_V}$, the ensemble-averaged spreading distance obeys $X(t)\\sim t^{1/3}$ for $|V|>2t$ (localized regime) and $X(t)\\sim t^{1/2}$ for $|V|<2t$ (delocalized regime), with an intermediate exponent near $0.57$ at the transition. To explain this, the authors propose a general random-variable mechanism: a wave packet that has spread over a volume $X^d$ is controlled by the eigenstate with the largest imaginary energy $\\lambda_{\\max}$ in that region, and the tail integral of the imaginary density of states satisfies $\\int_{\\lambda_{\\max}}^{\\infty}\\rho_I(s)\\,ds\\sim X^{-d}$. Combining this with the maximization conditions $\\partial\\lambda/\\partial X\\sim t^{-1}$ (localized) and $X\\partial\\lambda/\\partial X\\sim t^{-1}$ (delocalized) yields the universal scaling relations $\\delta=(\\beta+1)/(d+\\beta+1)$ and $\\delta=(\\beta+1)/d$ when the tail is $\\rho_I(s)\\sim(s_0-s)^{\\beta}$. For this model the tail is a Van Hove singularity with $\\beta=-1/2$, which produces $\\delta=1/3$ and $\\delta=1/2$; the authors show perturbatively in the deep localized regime that this singularity is stable.","pith_inferences":["The most fragile step is the independent-samples treatment of eigenvalue imaginary parts: in a quasiperiodic system these values are deterministic and strongly correlated, so a finite-size scaling study of $\\lambda_{\\max}$ within blocks of length $X$ would directly test whether the $1/X$ tail decay really holds in the thermodynamic limit.","If the scaling relations are right, a two-dimensional quasiperiodic non-Hermitian lattice with the same $\\beta=-1/2$ tail should spread with $\\delta=1/5$ (localized) and $\\delta=1/4$ (delocalized); those exponents are distinct enough to be checked in photonic or ultracold-atom platforms.","The same random-variable argument would predict different spreading behavior for disorder whose imaginary-spectrum tail is not algebraic, for example Gaussian, since the extreme-value statistics would change; this makes the quasiperiodic model's deterministic algebraic tail the special feature that produces clean power laws."],"forward_implications":["In any $d$-dimensional non-Hermitian disordered system whose imaginary density of states has an algebraic tail with exponent $\\beta$, normalized wave-packet spreading should follow these scaling laws; the quasiperiodic model is the $\\beta=-1/2$ case of that general statement.","The Hermitian dichotomy of halted versus ballistic transport is replaced, whenever such a tail singularity exists, by a non-Hermitian dichotomy of subdiffusion versus diffusion governed by the imaginary spectrum.","Spreading exponents can be computed from Lyapunov exponents in the complex plane through $\\rho(E)=\\frac{1}{2\\pi}\\nabla^2\\gamma(E)$, a route that avoids large-scale exact diagonalization of non-Hermitian matrices.","In the deep localized regime, second-order perturbation theory shows the $\\beta=-1/2$ Van Hove singularity is stable, so the $1/3$ exponent is not a transition artifact but persists throughout the localized phase."],"supporting_citations":[{"why":"Reports the experimental observation of dynamical delocalization despite spectrally localized eigenstates, the phenomenon this paper explains quantitatively.","marker":"[32]"},{"why":"Supplies the earlier maximization-condition treatment of non-Hermitian transport that the authors generalize into scaling relations.","marker":"[33]"},{"why":"Global theory of one-frequency Schrödinger operators fixes the exact transition at |V|=2t and the in-spectrum Lyapunov exponent.","marker":"[34]"},{"why":"Defines the original Hermitian Aubry-André model whose self-duality and localization transition are the reference point.","marker":"[35]"},{"why":"Supplemental material contains the derivation of Eq. (12), the extreme-value tail identity, and the perturbative proof of the beta=-1/2 tail's stability.","marker":"[44]"},{"why":"Establishes the relation between the density of states and the Laplacian of the Lyapunov exponent, the basis for extracting the imaginary density of states.","marker":"[49]"},{"why":"Extends the Lyapunov-exponent/density-of-states formalism to one-dimensional non-Hermitian Schrödinger equations, used here for the Lyapunov-exponent analysis.","marker":"[50]"},{"why":"Used in the supplemental spectral analysis to label quasiperiodic eigenenergies by a real wave number, needed for the tail calculation.","marker":"[51]"}],"fun_headline_variants":["Non-Hermitian quasicrystals: subdiffusive t^{1/3} and diffusive t^{1/2} spreading","Non-Hermitian disorder sets universal t^{1/3} and t^{1/2} spreading","Quasiperiodic non-Hermitian systems spread anomalously: t^{1/3} and t^{1/2}","Van Hove singularities dictate spreading in non-Hermitian quasicrystals","Universal spreading laws from Van Hove singularities in non-Hermitian systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scaling law assumes that the imaginary parts of eigenvalues inside a local region behave like independent random samples from the thermodynamic imaginary density of states, whereas the quasiperiodic model's spectrum is deterministic and correlated.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian quasicrystals: subdiffusive t^{1/3} and diffusive t^{1/2} spreading","Non-Hermitian disorder sets universal t^{1/3} and t^{1/2} spreading","Quasiperiodic non-Hermitian systems spread anomalously: t^{1/3} and t^{1/2}","Van Hove singularities dictate spreading in non-Hermitian quasicrystals","Universal spreading laws from Van Hove singularities in non-Hermitian systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001153,"raw_usage":{"total_tokens":4830,"prompt_tokens":1047,"completion_tokens":3783,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":3653}},"tokens_in":663,"tokens_out":3783,"duration_ms":24775,"temperature":1.0,"reasoning_tokens":3653,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:31:11.362772+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the non-Hermitian Aubry-André model at $|V|=5$, compute the largest imaginary eigenvalue among blocks of length $X$ and check whether the ensemble-averaged tail integral $\\mathbb{E}[\\int_{\\lambda_{\\max}}^{\\infty}\\rho_I(s)\\,ds]$ decays as $1/X$; if it does not, Eq. (12) fails and the claimed $\\delta=1/3$ need not follow. One can also simulate the wave-packet spreading for much longer times and larger systems and check whether the fitted exponent stays at $1/3$ in the localized regime and $1/2$ in the delocalized regime.","supporting_citations":[{"cited_title":"Weidemann, M","cited_arxiv_id":null,"evidence_quote":"Reports the experimental observation of dynamical delocalization despite spectrally localized eigenstates, the phenomenon this paper explains quantitatively."},{"cited_title":"Avila, Global theory of one-frequency Schr¨ odinger op- erators, Acta Math","cited_arxiv_id":null,"evidence_quote":"Global theory of one-frequency Schrödinger operators fixes the exact transition at |V|=2t and the in-spectrum Lyapunov exponent."},{"cited_title":"Aubry and G","cited_arxiv_id":null,"evidence_quote":"Defines the original Hermitian Aubry-André model whose self-duality and localization transition are the reference point."},{"cited_title":"(12) in the main text; (III) Perturbative analysis of the spectral structure; (IV) Dynamical spreading in the presence of mobility edge","cited_arxiv_id":null,"evidence_quote":"Supplemental material contains the derivation of Eq. (12), the extreme-value tail identity, and the perturbative proof of the beta=-1/2 tail's stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the relation between the density of states and the Laplacian of the Lyapunov exponent, the basis for extracting the imaginary density of states."},{"cited_title":"Derrida, J.L","cited_arxiv_id":null,"evidence_quote":"Extends the Lyapunov-exponent/density-of-states formalism to one-dimensional non-Hermitian Schrödinger equations, used here for the Lyapunov-exponent analysis."},{"cited_title":"Yang and Y","cited_arxiv_id":null,"evidence_quote":"Used in the supplemental spectral analysis to label quasiperiodic eigenenergies by a real wave number, needed for the tail calculation."}],"review_version":1}