{"id":"d7d061ed-9269-4201-a491-bf366b5249ec","arxiv_id":"1909.02048","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a non-separable 2D Aubry-André model, partially extended states persist at arbitrarily strong quasiperiodic modulation, localised on a network of low-disorder lines, yielding a mixed spectrum without a mobility edge.","lead":"This paper introduces a two-dimensional version of the Aubry-André model with a special quasiperiodic potential. It shows that even at very strong modulation, some single-particle states stay partially extended along sparse low-disorder lines while other states are localised, producing a mixed spectrum and ballistic transport on those lines.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that partially extended states become extensive (PR ∝ L^2) at large λ is an extrapolation; the largest simulations show PR ∝ L^1.7, and no finite-size scaling establishes the asymptotic exponent.","rationale":"The reader's weakest_assumption focused on the shielding of localised states by exponential localisation on high-disorder lines. That is a legitimate concern, and I partially agree with it. However, the more directly load-bearing gap is the missing finite-size scaling for the participation ratio of the partially extended states themselves. The paper's headline claims (partially extended states at arbitrarily strong λ, no mobility edge, mixed spectrum) all depend on the assertion that these states have PR ∝ L^2 in the thermodynamic limit. The provided data at L=99 and L=239 show exponents of roughly 1 and 1.7, respectively, which are consistent with a crossover but do not establish the asymptote. Without data at larger L, the central claim remains an extrapolation. My proposed check (computing PR exponents for L up to 1393) directly tests this extrapolation and would settle whether the states become 2D extensive. The shielding assumption is related but secondary: even if localised states remain localised, the partially extended states must still be shown to be extensive. Since my concern is a concrete, testable gap rather than a demonstrated error, it does not change the reader's CONDITIONAL verdict; it sharpens the condition under which the paper should be accepted.","tokens_in":14099,"tokens_out":10155,"duration_ms":111543,"concrete_test":"For a fixed strong λ (e.g., λ=40) and β taken from the continued fraction of 1/√2, compute the maximum and median participation ratio of eigenstates near E=0 for system sizes L = 99, 239, 577, and 1393, matching the sizes used in the ground-state scaling of Fig. 7. Plot log(PR) against log(L) and extract the asymptotic exponent; if the exponent approaches 2 (or at least increases monotonically toward 2), the extensive scaling claim is supported. In parallel, to test the shielding assumption, identify the localised eigenstates in the decoupled-chain limit and track the same states as the horizontal hopping is switched on; their PR should remain O(1) (or grow at most sublinearly) with L if the mixed spectrum survives in the thermodynamic limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that partially extended states persist at arbitrarily strong modulation rests on the assertion in Sec. II that, for sufficiently large system sizes, the participation ratio of the low-disorder-line states scales as L^2, i.e., extensively. This is an extrapolation: the largest full-spectrum calculation (Fig. 3, L=239) shows log PR/log L values only up to about 1.7, and Fig. 1 (L=99) shows the funnel at PR ∝ L, corresponding to an exponent near 1 in L. Because the spacing between low-disorder lines is O(λ), at λ=40 the simulated L=239 contains only about six extended lines per direction, so the network is a roughly 6×6 grid. The observed exponent could be a crossover between quasi-1D line behaviour (PR∝L) and true 2D network behaviour (PR∝L^2); the paper provides no finite-size scaling to show that the exponent increases towards 2. If the asymptotic exponent were actually 1, or if it saturates below 2, the states would remain quasi-1D rather than partially extended in two dimensions, and the mixed-spectrum picture would need to be re-stated. The shielding of localised states on high-disorder lines is argued qualitatively via exponential localisation, but this mechanism also lacks a quantitative finite-size test in the coupled system.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a two-dimensional, non-separable generalisation of the Aubry-André model (2DAA) and studies its single-particle localisation properties by exact diagonalisation. The central claims are that (i) the model is self-dual under a generalised Aubry transformation, (ii) at strong quasiperiodic modulation λ ≫ 2 a finite fraction of eigenstates remain partially extended on a sparse network of low-disorder lattice lines, while other states are exponentially localised, (iii) these two types of states are interspersed in energy so that no mobility edge forms, and (iv) dynamics initiated on the low-disorder network are ballistic along that network. The mechanism is explained by rewriting the potential as independent 1D Aubry-André chains on each lattice line with effective disorder amplitude λ̃_n = 2λ cos(2πβ n); lines with |λ̃_n| < 2 are extended and are argued to hybridise into a 2D network, while lines with |λ̃_n| > 2 remain localised and shield the extended states. The paper also studies tilted and non-self-dual variants, the effect of random disorder, and the universality of the ground-state transition with a continuum quasiperiodic model.","tokens_in":14332,"tokens_out":7203,"duration_ms":74777,"significance":"If the central claim holds, the paper establishes a new and rather general mechanism for coexistence of localised and partially extended states in a two-dimensional quasiperiodic system, with direct relevance to cold-atom experiments and to discussions of rare regions in many-body localisation. The paper has several concrete strengths: the duality transformation in Appendix A is exact; the line-decomposition argument after Eq. (4) is a clean, falsifiable physical picture; Fig. 3 provides a sharp, checkable prediction that partially extended states are confined to the critical 1DAA bandwidth; and the dynamics in Sec. III and Appendix B give an experimentally accessible signature. The main weakness is that the asymptotic extensivity of the partially extended states (PR ∝ L^2) is an extrapolation from data showing PR exponents only up to about 1.7, and the shielding of localised states from hybridisation is asserted rather than quantitatively tested in the coupled system. These are load-bearing for the paper's strongest claims, so the manuscript needs additional finite-size analysis before the central conclusion is fully established.","major_comments":[{"comment":"The central claim that partially extended states have extensive participation ratio is not yet established by the data. The text states that for sufficiently large systems PR scales as L^2 because a finite fraction 2/(πλ) of lines is extended, but Fig. 3 (L = 239) shows log PR/log L reaching only about 1.7, and Fig. 1 (L = 99) shows the funnel at values much closer to 1. No finite-size scaling is presented showing that the exponent increases toward 2 as L grows. Please add a finite-size analysis at fixed λ (for example λ = 5, 10, 40 across L = 99, 239, and larger accessible sizes) with confidence intervals; without it, the data are also compatible with a quasi-1D interpretation (PR ∝ L), which would change the physical conclusion.","section":"Sec. II, paragraph after Eq. (4)"},{"comment":"The shielding assumption that protects localised states from hybridisation is the load-bearing step for the mixed-spectrum and no-mobility-edge conclusions, but it is only argued qualitatively. The paper asserts that exponential localisation of lines with |λ̃| > 2 suppresses their statistical weight and prevents hybridisation with the extended network, yet no calculation in the coupled 2D system tests this. Please provide a direct numerical test, for example the inverse participation ratio of eigenstates whose weight is concentrated on high-disorder lines as a function of L at fixed λ, or the overlap of such states with the low-disorder network; this would either substantiate or refute the assumed protection.","section":"Sec. II, paragraph after Eq. (4)"},{"comment":"The no-mobility-edge claim is inferred from a single-size spectrum (L = 239 in Fig. 3). At finite size, an energy-dependent PR distribution could also arise from a finite-size mobility edge whose edges sharpen or shift with L. Since the abstract claims that no mobility edge emerges, please show the system-size dependence of the PR distribution within the critical 1DAA bandwidth for several λ values; this would also help separate the finite-size crossover from the proposed asymptotic mixed-spectrum behaviour.","section":"Sec. III and Fig. 3"}],"minor_comments":[{"comment":"The text says the funnel of partially extended states has PR ∼ N_sites^{0.5}, while Fig. 3 and its caption quote values L^1 to L^1.7; please make the scaling statement consistent across the two figures.","section":"Sec. II, text around Fig. 1"},{"comment":"The appendix states that the wave functions are shown 'for λ = 0' in Fig. 8, but the main text and the figure caption describe λ = 40; please correct this typo.","section":"Appendix B"},{"comment":"For degenerate eigenstates, the diagonal ensemble in Eq. (5) and the participation ratio in Eq. (6) need a precise definition of the eigenbasis; please specify how degeneracies are handled.","section":"Sec. III, Eq. (6)"},{"comment":"The finite-size scaling claim for the universality of the ground-state transition would be strengthened by reporting the fitting procedure, the number of data points, and goodness-of-fit measures such as χ² or residuals.","section":"Sec. IV C and Fig. 7"},{"comment":"The statement that the non-self-dual 'wide channel' construction is 'manifestly more robust' than the self-dual one is qualitative; a quantitative comparison of PR versus λ at matched system sizes would be more informative.","section":"Sec. IV A"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's central finding is genuinely new: a non-separable 2D Aubry-André model where deterministic low-disorder lines support partially extended states at arbitrarily large modulation, with localised states interspersed in energy. The mechanism is physically transparent and the numerics support it: Fig. 1 shows the funnel, Fig. 2 shows wavefunctions confined to lines, Fig. 3 shows the energy window bound, and Fig. 8 shows ballistic expansion. The authors correctly note the model is non-separable and distinct from earlier separable 2D constructions. The duality transformation in Appendix A is clean and the connection to the continuum eightfold model is a nice addition.\n\nThe weakest point is the claim that the partially extended states are extensive (PR ~ L^2) for sufficiently large L. The paper states this without a finite-size scaling analysis. The largest full-spectrum calculation (L=239, Fig. 3) shows log PR/log L up to about 1.7, and for L=99 (Fig. 1) the funnel sits near PR ~ L, i.e., exponent close to 1. At λ=40, the number of low-disorder lines in the simulated box is only about 6 per direction, so the network is a sparse grid. The observed exponent could easily be a crossover between quasi-1D line behaviour and true 2D network behaviour, and the paper gives no evidence that the exponent approaches 2. If the asymptotic exponent is actually 1, the states are quasi-1D, not partially extended in 2D, and the mixed-spectrum picture would need restating. This is not a fatal flaw — the existence of line-localised states at large λ is solid — but it is a load-bearing extrapolation that should be flagged.\n\nA second, related soft spot is the shielding argument: the claim that localised states on high-disorder lines do not hybridise with the extended network is qualitative. It is plausible and consistent with the data, but a quantitative test (e.g., localisation length of coupled states versus system size) would strengthen it. Minor: no error bars in Fig. 1 and Fig. 3, and no code/data release. For a numerical paper, that is a reasonable request.\n\nThe citation pattern looks fair. The self-citations to earlier work are appropriate. The universality claim for ground states is backed by finite-size scaling in Fig. 7, which is a genuine strength. The paper is worth a serious referee: it introduces a new model, identifies an unusual phenomenon, and gives experimental handles. I would accept it for review, but the referee should ask for a finite-size scaling analysis of the PR exponent and a sharper statement about what 'partially extended' means asymptotically. The paper deserves publication after that is addressed.","headline":"A genuinely new 2D quasiperiodic model with a plausible mixed-spectrum mechanism; the core claim is real but the asymptotic extensivity of the partially extended states is under-supported and needs a finite-size scaling test.","tokens_in":14910,"tokens_out":701,"would_cite":true,"duration_ms":8292,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces a two-dimensional Aubry-André model in which localised and partially extended single-particle states coexist at any modulation strength, with no mobility edge separating them.","keywords":["mixed spectrum","quasiperiodic","Aubry-André model","localization","partially extended states","mobility edge","self-duality","optical lattice"],"falsifier":"Compute the eigenstates of the 2DAA model at $\\lambda=40$ on lattices up to $L\\approx 1000$ and check whether all states with large participation ratio are concentrated on the lines with $|2\\lambda\\cos(2\\pi\\beta n)|<2$; finding an extended state with weight away from those lines, or finding that localised states hybridise into extended ones as horizontal hopping is switched on continuously, would refute the claim. In an experiment, a site placed just off a low-disorder line should stay exponentially localised; seeing non-exponentially small probability reach the network on any timescale would also falsify the shielding picture.","tokens_in":13869,"feed_emoji":"🕸️","tokens_out":10237,"duration_ms":96901,"temperature":0.7,"pith_summary":"The authors propose a two-dimensional generalisation of the one-dimensional Aubry-André model: a square lattice with a quasiperiodic potential whose amplitude is a product of two cosine waves at 45 degrees. They claim that even when the potential is extremely strong, some single-particle states remain partially extended, with their weight sitting on a sparse network of lattice lines where the effective disorder happens to be weak. The rest of the system is exponentially localised, and the spatial separation stops the two kinds of states from hybridising, so no mobility edge separates them by energy. This matters because it provides a clean, non-interacting model in which localised and ballistically travelling states coexist, and the model is directly realisable in cold-atom experiments.","feed_headline":"Partially extended states survive any quasiperiodic strength in 2D","feed_subtitle":"The 2D Aubry-André model keeps low-disorder lines conducting while the rest localises, so no mobility edge forms.","key_machinery":"The load-bearing object is the factorisation identity $V_{nm} = 2\\lambda J \\cos(2\\pi\\beta n)\\cos(2\\pi\\beta m) = \\tilde{\\lambda}_n J \\cos(2\\pi\\beta m)$. It turns the two-dimensional model, when horizontal hopping is switched off, into a stack of independent one-dimensional Aubry-André chains, one per vertical line, with line-dependent disorder amplitudes $\\tilde{\\lambda}_n = 2\\lambda \\cos(2\\pi\\beta n)$. The one-dimensional critical value $|\\tilde{\\lambda}|=2$ marks which lines are extended and which are localised. Restoring horizontal hopping hybridises only the extended lines into a network, while the exponential localisation of the other lines acts as a shield that prevents hybridisation between the two species. The Aubry duality of the model, a Fourier transform combined with a 45-degree rotation, maps the strong-modulation line network onto a diagonal network at weak modulation.","core_discovery":"The central discovery is that the two-dimensional Aubry-André Hamiltonian is self-dual like its one-dimensional parent, yet its spectrum is much richer. The potential factorises as $V_{nm} = 2\\lambda J \\cos(2\\pi\\beta n) \\cos(2\\pi\\beta m) = \\tilde{\\lambda}_n J \\cos(2\\pi\\beta m)$, so with horizontal hopping removed each vertical lattice line is an independent one-dimensional Aubry-André chain with effective disorder amplitude $\\tilde{\\lambda}_n = 2\\lambda \\cos(2\\pi\\beta n)$. Because $\\tilde{\\lambda}_n$ is itself quasiperiodic, lines with $|\\tilde{\\lambda}_n| < 2$ exist for any $\\lambda$, and these lines carry extended eigenstates that hybridise into a mesh of partially extended states. Lines with $|\\tilde{\\lambda}_n| > 2$ stay exponentially localised, and that exponential localisation shields them from the delocalising mesh, so localised and partially extended states coexist interspersed in energy with no mobility edge. The same structure yields ballistic expansion along the low-disorder network while generic initial sites remain localised.","pith_inferences":["If this line-shielding mechanism is generic, then any two-dimensional quasiperiodic potential that factorises into a product of one-dimensional incommensurate functions should show a similar mixed spectrum; the exact 45-degree geometry is sufficient but probably not necessary.","The roughly $2/(\\pi\\lambda)$ fraction of extended lines implies that the number of partially extended states grows linearly with system size in the thermodynamic limit, so their prevalence is extensive even though individual states look quasi-one-dimensional on intermediate scales.","A clean experimental signature would be higher-order moments of the expansion: Manhattan-distance moments should stay linear longer than Euclidean moments, because reaching a line intersection changes Euclidean distance but not Manhattan distance along the network."],"forward_implications":["At any $\\lambda>2$, a finite fraction roughly $2/(\\pi\\lambda)$ of lattice lines are in the extended one-dimensional phase, so partially extended states exist for arbitrarily strong quasiperiodic modulation.","Localised and partially extended states occur at the same energies, interspersed in the spectrum, so no mobility edge forms.","A particle released on a low-disorder line expands ballistically along the network of such lines, while a particle released elsewhere remains exponentially localised under the same parameters.","The effect is robust to weak random disorder and to small tilts of the modulation away from 45 degrees, and the ground-state localisation transition belongs to the same universality class as a continuum eightfold-symmetric model."],"supporting_citations":[{"why":"Defines the one-dimensional Aubry-André model, its duality, and the localisation transition at $\\lambda=2$ that the line-by-line argument relies on.","marker":"[21]"},{"why":"Provides the two-dimensional Aubry duality transformation used to establish self-duality of the 2DAA model and the connection between strong- and weak-modulation regimes.","marker":"[22]"},{"why":"Gives the rigorous proof that one-dimensional Aubry-André chains with disorder amplitude above 2 are exponentially localised, underpinning the shielding of localised lines.","marker":"[35]"},{"why":"Supplies the result that random disorder localises in two dimensions with exponentially large localisation lengths, used to argue the mixed-spectrum effect survives weak disorder.","marker":"[45]"},{"why":"Provides the experimental continuum model and the $\\beta=1/\\sqrt{2}$ choice; used for the universality-class comparison and experimental feasibility.","marker":"[12]"},{"why":"Earlier work by the same authors establishing universality between lattice and continuum one-dimensional Aubry-André models, whose method is extended to two dimensions here.","marker":"[44]"}],"fun_headline_variants":["Partially extended states persist at any disorder in 2D","No mobility edge in 2D Aubry-André: localised + extended mix","Ballistic lines on a localised backdrop in 2D model","Low-disorder lines conduct while rest stays localised in 2D","2D quasiperiodic model: mixed spectrum without mobility edge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim rests on the assumption that restoring the horizontal hopping couples only the already extended lines and cannot turn exponentially localised segments on the other lines into extended channels; if that shielding fails, the mixed-spectrum effect collapses.","fun_headline_variants_meta":{"raw":{"variants":["Partially extended states persist at any disorder in 2D","No mobility edge in 2D Aubry-André: localised + extended mix","Ballistic lines on a localised backdrop in 2D model","Low-disorder lines conduct while rest stays localised in 2D","2D quasiperiodic model: mixed spectrum without mobility edge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1610,"prompt_tokens":955,"completion_tokens":655,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":560}},"tokens_in":571,"tokens_out":655,"duration_ms":6004,"temperature":1.0,"reasoning_tokens":560,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:01:13.928974+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the eigenstates of the 2DAA model at $\\lambda=40$ on lattices up to $L\\approx 1000$ and check whether all states with large participation ratio are concentrated on the lines with $|2\\lambda\\cos(2\\pi\\beta n)|<2$; finding an extended state with weight away from those lines, or finding that localised states hybridise into extended ones as horizontal hopping is switched on continuously, would refute the claim. In an experiment, a site placed just off a low-disorder line should stay exponentially localised; seeing non-exponentially small probability reach the network on any timescale would also falsify the shielding picture.","supporting_citations":[{"cited_title":"Aubry and G","cited_arxiv_id":null,"evidence_quote":"Defines the one-dimensional Aubry-André model, its duality, and the localisation transition at $\\lambda=2$ that the line-by-line argument relies on."},{"cited_title":"Devakul and D","cited_arxiv_id":null,"evidence_quote":"Provides the two-dimensional Aubry duality transformation used to establish self-duality of the 2DAA model and the connection between strong- and weak-modulation regimes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the rigorous proof that one-dimensional Aubry-André chains with disorder amplitude above 2 are exponentially localised, underpinning the shielding of localised lines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental continuum model and the $\\beta=1/\\sqrt{2}$ choice; used for the universality-class comparison and experimental feasibility."},{"cited_title":"Szabó and U","cited_arxiv_id":null,"evidence_quote":"Earlier work by the same authors establishing universality between lattice and continuum one-dimensional Aubry-André models, whose method is extended to two dimensions here."}],"review_version":1}