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Mixed spectra and partially extended states in a two-dimensional quasiperiodic model

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1909.02048 v3 pith:IGZZESCJ submitted 2019-09-04 cond-mat.dis-nn cond-mat.quant-gascond-mat.stat-mechphysics.atom-phquant-ph

Attila Szabó , Ulrich Schneider This is my paper · ORCID
classification cond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechphysics.atom-phquant-ph
keywords mixedspectrumquasiperiodicAubry-Andrémodellocalizationpartiallyextendedstatesmobilityedgeself-dualityopticallattice
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Sec. II, paragraph after Eq. (4)] 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.
  2. [Sec. II, paragraph after Eq. (4)] 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.
  3. [Sec. III and Fig. 3] 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.
minor comments (5)
  1. [Sec. II, text around Fig. 1] 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.
  2. [Appendix B] 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.
  3. [Sec. III, Eq. (6)] 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.
  4. [Sec. IV C and Fig. 7] 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.
  5. [Sec. IV A] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the 2DAA mechanism is derived from an independent 1DAA benchmark and checked against exact diagonalisation; self-citations are motivational, not load-bearing.

full rationale

The central derivation is self-contained. The partially extended states are identified directly from exact-diagonalisation participation ratios (Eq. 3, Figs. 1-3), and the explanatory mechanism in Sec. II (Eq. 4) is an independent rewriting of the potential plus the textbook 1DAA localisation transition [21], not an input fitted to the 2D spectra. The energy-window test (Fig. 3) is not circular: the critical 1DAA bandwidth [38] is computed independently, while the states plotted are all eigenstates near E=0 and are labelled by their PR, not by their energy being inside the window. The L^2-extensivity statement for asymptotically large systems is an extrapolation from the observed log PR/log L values around 1.7 at L=239; that is a correctness/robustness concern, not a definitional reduction. The universality section invokes the authors' earlier 1D result [44] and their experiment [12] as motivation, but the same-universality-class conclusion rests on the finite-size scaling collapse of the ground-state curvature computed for both models in this paper, with exponents fitted to those data; the self-citations are not load-bearing. No uniqueness theorem, fitted parameter renamed as prediction, or ansatz smuggled via citation was found. The low score reflects only the presence of minor non-load-bearing self-citations and an asymptotic extrapolation, neither of which constitutes circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced; the low-disorder lines are deterministic features of the potential (Eq. 4), not postulated objects. No free parameters are fitted to produce the main result; lambda, beta, system sizes, and disorder strengths are simulation choices. The scaling exponents in Sec. IV.C are fit outputs for demonstrating universality, not inputs.

assumptions (4)
  • domain assumption The one-dimensional Aubry-André model has a localisation transition at lambda = 2: extended for lambda < 2, localised for lambda > 2.
    Invoked in Sec. II to identify lines with |lambda_tilde_n| < 2 as extended 1DAA models; based on Refs. 21, 33, 34, 35.
  • domain assumption In two dimensions, any amount of random disorder leads to exponential localisation with exponentially large localisation lengths.
    Invoked in Sec. IV.B to interpret the power-law decays in Fig. 6 as finite-size precursors; based on Ref. 45.
  • domain assumption The lattice model (2) and the continuum Hamiltonian (8) are continuously connected so that their ground-state localisation transitions share the same universality class.
    Assumed in Sec. IV.C from the parameter hierarchy V0 = V2 >> 8Er >> V1 = V3; used to justify comparing scaling exponents.
  • standard math Aubry duality for the 2D model maps lambda to 4/lambda via a Fourier transform on the square lattice.
    Derived in Appendix A; relies on standard Fourier analysis of tight-binding Hamiltonians.

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Pith. "Pith review of Mixed spectra and partially extended states in a two-dimensional quasiperiodic model." pith.science (2026). https://pith.science/paper/IGZZESCJ

@misc{pith2026190902048,
  author       = {Pith},
  title        = {Pith review of: Mixed spectra and partially extended states in a two-dimensional quasiperiodic model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGZZESCJ}},
  note         = {Machine review of arXiv:1909.02048}
}
read the original abstract

We introduce a two-dimensional generalisation of the quasiperiodic Aubry-Andr\'e model. Even though this model exhibits the same duality relation as the one-dimensional version, its localisation properties are found to be substantially more complex. In particular, partially extended single-particle states appear for arbitrarily strong quasiperiodic modulation. They are concentrated on a network of low-disorder lattice lines, while the rest of the lattice hosts localised states. This spatial separation protects the localised states from delocalisation, so no mobility edge emerges in the spectrum. Instead, localised and partially extended states are interspersed, giving rise to an unusual type of mixed spectrum and enabling complex dynamics even in the absence of interactions. A striking example is ballistic transport across the low-disorder lines while the rest of the system remains localised. This behaviour is robust against disorder and other weak perturbations. Our model is thus directly amenable to experimental studies and promises fascinating many-body localisation properties.

Figures

Figures reproduced from arXiv: 1909.02048 by the authors.

Figure 1
Figure 1. FIG. 1. Participation ratios of all eigenstates of the 2DAA model [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Wave function weight [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Participation ratios of the eigenstates of the 2DAA model [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Finite size scaling of the ground state curvature [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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    and atλ = 0.1 (linear scale to the right). The former is strongly localised with a very small localisation length (ξ≈ 0.1); the latter is fully extended. (c) Median energy state (E≈ 0) at λ = 40. The wave function is concentrated on a few lattice lines, along which the quasipe...

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Reviewed August 14, 2026 · model on record in the stance chip above.