Quantum metric and localization in a quasicrystal

Annica M. Black-Schaffer; Patric Holmvall; Quentin Marsal

REVIEW 3 major objections 3 minor 1 cited by

Quantum metric and localization in a quasicrystal

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The full quantum metric of the Fibonacci quasicrystal is bounded below by the Chern number, equal to the gap label, tying spatial localization to the fractal energy spectrum.

desk verdict A clever and mostly honest paper that introduces the quantum metric as a localization probe for the Fibonacci chain, but the advertised analytic bound linking the spatial metric to gap labels only holds for a cutoff-regularized 2D model, and the bridge back to 1D is a single weak-modulation numerics point. read the letter →

arxiv 2506.15575 v3 pith:PX7KGPXC submitted 2025-06-18 cond-mat.mes-hall

Quentin Marsal , Patric Holmvall , Annica M. Black-Schaffer This is my paper · ORCID

classification cond-mat.mes-hall

keywords quantummetricFibonaccichainquasicrystallocalizationphasonChernnumbergaplabelgeometry

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

This paper tries to establish the quantum metric as the right tool for describing electronic localization in quasicrystals, using the one-dimensional Fibonacci chain as a test case. The central claim is that the full quantum metric—the sum of a spatial component and a newly introduced phasonic component—is bounded below by the Chern number associated with the chain's hidden two-dimensional structure, and that this Chern number equals the integer label of each energy gap. If the claim is right, spatial localization and the fractal energy spectrum are two faces of the same quantum geometry: deeper gaps force the corresponding bulk states to spread over larger distances. The paper also shows that the quantum metric resolves structure that the inverse participation ratio cannot see, because it encodes how far apart the symmetry centers of the states are, not just how many sites they occupy. This matters because localization in quasicrystals controls superconductivity, topological phases, and many-body behavior, and the quantum metric is experimentally accessible.

What carries the argument

The load-bearing object is the two-component quantum metric built by treating the phason angle φ as a synthetic second dimension: Ωx = (1/2π)∫dφ Tr[P(φ) x Q(φ) x] and Ωφ = (1/2π)∫dφ Tr[∂φP(φ) Q(φ) ∂φP(φ)]. The spatial piece is the usual real-space quantum metric of the chain; the phasonic piece is new and measures the response of the projector to changes in the chain's termination mode. The proof of the bound adapts the known Chern-insulator inequality Ω > |C|/π to this mixed position–phason geometry, using a Chern number defined through a position-space formulation of the Berry curvature. Because the hopping function is a step function of φ, a Fourier cutoff L = N is introduced to regularize Ωφ, and the paper argues that this preserves the gap structure. The same machinery links the metric to the renormalization hierarchy of local symmetry centers (+/−/o labels) that generates the band structure.

What would settle it

A decisive check is to compute Ωφ for a fixed approximant while increasing the Fourier cutoff L: if Ωφ keeps growing with L even at weak modulation δ = 0.2, the bound Ω > |C|/π holds only for the smoothed model and not for the original step-function chain. A complementary check is to test, on long approximants at weak modulation, whether Ωx ever dips below |ν|/π for a resolved gap; any such violation would falsify the claimed link between spatial localization and gap labels.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the quantum metric of the Fibonacci chain has two components: Ωx, measuring the real-space spread of the occupied states, and Ωφ, measuring how strongly the states change as the phason angle φ is varied. Together they obey Ω = Ωx + Ωφ > |C|/π, where C is the Chern number of the synthetic two-dimensional system obtained by promoting φ to a second dimension. Since C is the gap label ν, the bound reads Ω > |ν|/π, and in the weak-modulation regime Ωφ is negligible so the spatial metric itself satisfies Ωx > |ν|/π. The paper further shows, through a real-space renormalization scheme, that each gap in the spectrum corresponds to hybridization of states living on mirror-related groups of sites, and that the quantum metric measures the squared distance between those groups. This is what makes the metric grow smoothly with gap label and ties the localization of bulk states to the self-similar hierarchy of the chain.

Load-bearing premise

The load-bearing premise is that the phasonic component Ωφ, defined after truncating the Fourier series of the step-function hopping at harmonic L = N, is a genuine part of the physical quantum metric of the Fibonacci chain rather than a regularization artifact; without the cutoff Ωφ diverges, and only in the weak-modulation case does the paper show it is numerically small.

Editorial extensions

If this is right

If the bound is correct, spatial localization and the fractal energy spectrum of a quasicrystal are not independent: each spectral gap imposes a minimal spatial spread on the states below it.
At weak modulation the spatial quantum metric tracks the bound, so the localization length of bulk states grows at least linearly with the gap label, a quantitative statement the inverse participation ratio cannot provide.
Because the same Chern number both labels the gaps and bounds the metric, localization in one dimension is inherited from the geometry of a two-dimensional parent crystal.
The quantum metric can tell apart bulk states that occupy the same number of sites but are centered at different distances apart, resolving structure invisible to the inverse participation ratio.
The construction generalizes to any quasicrystal built by cut-and-project, so the bound should hold for other irrational-slope chains with their own gap labels.

Reading between the lines

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

Editorial inference: if the inequality survives the thermodynamic limit, it predicts a quantitative scaling law—bulk-state spread growing at least as fast as the gap label—that could be measured through the quantum metric's experimental signatures, such as nonlinear transport or optical response.
Editorial inference: the cutoff dependence of Ωφ suggests that the phason direction behaves like a synthetic dimension with effective long-range hopping; this predicts phason-driven delocalization that could be probed in photonic or cold-atom realizations where the phason is swept in time.
Editorial inference: the same two-component geometry should apply to other cut-and-project quasicrystals, where the parent lattice's Chern numbers would bound the full quantum metric through the appropriate gap labels, extending the result beyond the golden-mean slope.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know about this one for two reasons: it is the first real attempt I have seen to make the quantum metric a practical localization diagnostic for quasicrystals, and it contains a clean statement of where the argument gets soft. The paper is worth reading, but the headline claim should be read carefully.

What is actually new: they define a position-space quantum metric for the Fibonacci chain, show it resolves bulk bands that the IPR cannot tell apart, and connect the smooth growth of Ωx with gap label to the renormalization-group structure of local symmetry centers. That Section V story—the metric as a measure of squared distance between symmetry-related groups of sites—is genuinely nice and I think it will survive. They also introduce a phasonic component Ωφ and a mixed Chern number C, and the numerics for C=ν in the large gaps is convincing. Code and data are on Zenodo; the numerical claims are reproducible.

The soft spot is exactly where the stress-test note lands. Eq. (27) bounds Ωx+Ωφ in a 2D model obtained by promoting the phason to a coordinate and cutting the Fourier series at L=N. Without the cutoff, ∂φP is ill-defined and Ωφ diverges. With it, you have long-range phasonic hoppings, and the bound constrains that smoothed parent model, not the original chain. The authors openly say the phasonic part alone gives no useful 1D information and that Ωφ∝L² for strong modulation. So the 1D claim—that Ωx alone is bounded by |C|/π for weak modulation—rests on a single numerical observation at δ=0.2 on F9, with no small-δ estimate and no N-dependence check. That is a real gap between the abstract and the proof. I do not think it is fatal: the numerical tracking of Ωx with gap label is there in Fig. 3(b) for larger chains, and for weak modulation the phasonic contribution is visibly small. But the advertised 'analytic lower bound on spatial localization in 1D' is, strictly, a bound on the regularized 2D model plus a plausibility argument.

I also want to give credit for honesty: the manuscript flags the divergence, the cutoff, and the cutoff-dependence. That is more than many papers do. The citation pattern looks fine; the Peotta-Törmä inequality is imported, but they say so, and the Chern number identification is well anchored in the earlier quasicrystal topology literature.

Verdict: send it out. A serious referee should ask for a perturbative estimate or a thermodynamic check on Ωφ at fixed small δ, and the authors should either prove the weak-modulation bound or state clearly that Eq. (27) applies to the regularized 2D model. With that revision, the numerical and conceptual content is solid. I would take it to reading group; I would not cite it this year, but I would watch the follow-up.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the one-dimensional Fibonacci hopping chain and proposes the position-space quantum metric of Eq. (12) as a localization probe that is more sensitive than the inverse participation ratio. The authors show numerically that the quantum metric distinguishes bulk bands with different local symmetry environments and that it increases with the gap label. They then promote the phason angle to a synthetic dimension, introduce a mixed position-phason Chern number C in Eq. (23) and a two-component quantum metric Ω=Ω_x+Ω_φ in Eqs. (24)-(26), and invoke the 2D Chern-insulator inequality Ω>|C|/π to argue that the full quantum metric is bounded below by the gap label, with the spatial component alone inheriting the bound in the weak-modulation limit. The paper also gives a renormalization-scheme interpretation of why the quantum metric grows with gap label.

Significance. If the main claims were fully established, the paper would be a valuable conceptual advance: it would connect the spatial localization of quasicrystalline eigenstates to the fractal gap structure through quantum geometry, and it would offer a practical probe that goes beyond IPR-type measures. The position-space formulation, the numerical comparison with the IPR, and the symmetry-center interpretation in Sec. V are appealing and well supported. The availability of code and data (Ref. [88]) is a strength. However, the key analytical bridge from the 2D parent model to the original 1D chain is incomplete, because the phasonic quantum-metric component is cutoff-dependent and its smallness is only asserted numerically; the advertised 1D bound therefore needs additional support.

major comments (3)

[Section VI, Eqs. (24)-(27)] The central inequality Eq. (27) is derived for the sum Ω_x+Ω_φ in a smoothed 2D model, not for the spatial quantum metric of the original 1D Fibonacci chain. The manuscript states that ∂_φP is ill-defined for the step-function hopping tn(φ) in Eq. (10), that Ω_φ diverges without the Fourier cutoff L in Eq. (20), and that for strong modulation Ω_φ∝L². With L=N, the truncation introduces long-range hoppings along the phasonic direction and changes the model. The advertised conclusion Ω_x>|C|/π for the 1D chain therefore requires a controlled statement that Ω_φ is negligible for weak modulation in the thermodynamic limit. The only evidence given is the comparison of Figs. 5(c) and 5(d) for δ=0.2 and one chain size (three concatenated F9 blocks); no N-dependence, no small-δ expansion, and no gap-label dependence of Ω_φ are provided. The paper itself concedes near the end of Section VI that we cannot really deduce useful information for the 1D Fibonacci from the phasonic part alone. Without a proof or a thermodynamic scaling analysis of Ω_φ, Eq. (27) constrains a regularized parent model and does not establish the claimed link between spatial localization and gap labels in the Fibonacci chain.
[Section VI, Fig. 5(a) and Eq. (23)] The bound Ω>|C|/π presupposes a well-defined, quantized Chern number for the gapped 2D system. Figure 5(a) shows C≈ν only in the largest gaps, with clear deviations for smaller gaps attributed to finite-size effects. Since the paper states the result for every gap label, the identification C=ν needs a controlled thermodynamic check, such as showing C(N)-ν→0 at fixed gap label; the inequality should be applied only to gaps where quantization is established. If the finite-size C is smaller than the integer ν, then Ω>|C|/π does not imply Ω>|ν|/π. In addition, the transfer of the Ref. [54] inequality requires that the projectors P(φ) define smooth bands over the full φ torus after the cutoff; the manuscript does not demonstrate that the cutoff L=N leaves the relevant gap open for all φ.
[Section VI, paragraph after Eq. (26)] The paper borrows the known 2D Chern-insulator inequality rather than deriving the bound directly for the quasicrystal. This is a legitimate strategy, but it makes the mapping exactness load-bearing. Because the phason derivative is distributional for the original step-function hopping, the mapping is exact only after a cutoff, and the physical meaning of the cutoff is not settled. The manuscript needs either an explicit small-δ argument showing Ω_φ is parametrically suppressed relative to Ω_x as N→∞, or a demonstration that the cutoff can be removed after a suitable renormalization. As written, the conclusion that the spatial localization of the Fibonacci chain inherits a gap-label lower bound is a numerical observation rather than an analytical consequence of Eq. (27).

minor comments (3)

[Section IV, Eqs. (13)-(14)] The interpretation of Tr[P x Q x P] as ⟨x²⟩−⟨x⟩² is exact for a single band; for the many-band projector used below a gap, the trace is a multi-band spread functional and should be described as such to avoid a misleading variance identity.
[Section III and Fig. 3] The comparison between the quantum metric computed for F13 under OBC and the IPR computed for F12 under PBC rests on the assertion that these approximants have the same gap structure; this matching deserves a more explicit justification, since the central numerical comparison depends on it.
[Section VI, Eq. (23)] The choice of the central third of the chain for the bulk trace in the Chern number is arbitrary; the sensitivity of C to the bulk-window size should be reported, especially for smaller gaps where finite-size effects are visible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bound Ω ≥ |C|/π is an application of an external 2D inequality to an explicitly defined regularized model, with C = ν cited from independent prior work.

full rationale

The paper's central chain is: define a 2D mixed position–phason model via Eqs. (19)–(22), define the mixed Chern number C in Eq. (23) and the two-component metric Ω = Ωx + Ωϕ in Eqs. (24)–(26), then invoke the known 2D Chern-insulator inequality Ω > |C|/π from Ref. [54] to obtain Eq. (27). Each of these steps is either a definition, a numerical evaluation, or an application of an external theorem; no quantity is defined in terms of the result it is supposed to predict. The identification C = ν is cited to Refs. [34–36], whose authors do not overlap with the present paper, so the gap-label bound is not imported from a self-citation chain. The only self-citation in the narrative, Ref. [53], is background motivation and is not load-bearing. The paper explicitly acknowledges the regularization issue: 'In principle, the derivatives with respect to ϕ are thus ill-defined for the Fibonacci chain... This results in Ωϕ diverging in the thermodynamic limit', and later says 'we cannot really deduce useful information for the 1D Fibonacci from the phasonic part alone as it is strongly influenced by the cutoff L.' These are genuine limitations: Eq. (27) strictly bounds the cutoff model, and the step to Ωx > |C|/π for the original 1D chain rests on the numerical observation Ωϕ ≪ Ωx at δ = 0.2. That step is an unproven robustness claim, not a fitted parameter renamed as a prediction, and it does not make the derivation circular. The numerical plots confirm the inequality but are not the source of it. Hence the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 3 invented entities

The central derivation rests on prior results for the quantum metric, the renormalization/symmetry labeling of the Fibonacci spectrum, the identification of gap labels with Chern numbers, and the 2D Chern-insulator inequality. The genuinely new constructions, the phasonic metric component and the mixed Chern number, both require the Fourier cutoff L; the divergence of Ωϕ and its L² growth mean that the 'full quantum metric' bound is not by itself a statement about the original 1D chain.

free parameters (1)

Fourier cutoff L for the phasonic component = L = N (55 in Fig. 5; 377 in Fig. 3)
Introduced in Section VI after Eq. (20) to smooth the ϕ-dependence and make Ωϕ finite. The result depends strongly on this hand-chosen scale, since Ωϕ grows as L² for strong modulation.

assumptions (6)

domain assumption The position-space quantum metric formula Eq. (12), averaged over phason modes with OBCs, correctly measures electronic localization in the 1D Fibonacci chain.
Introduced in Section IV based on Refs. [52,53]; the extension to a phason-averaged 1D quasicrystal is not derived in the paper.
domain assumption The band structure of every approximant is obtained from the recursion H_n = -H_{n-2} ⊕ o H_{n-3} ⊕ + H_{n-2} (Eq. 17), and each band carries local symmetry eigenvalues.
Taken from Refs. [49,66]; used in Section V to explain the gap-label dependence of the quantum metric.
domain assumption Gap labels ν equal the winding number of edge states and equal the Chern number C of the 2D cut-and-project parent system.
Cited to Refs. [34-36,64]; used to set C=ν in Fig. 5 and Eq. (27).
standard math For a 2D Chern insulator, the trace of the quantum metric satisfies Ω ≥ |C|/π (Ref. [54]).
Quoted, not re-derived; the paper then applies it to the constructed (x,ϕ) 2D model.
ad hoc to paper The Fourier series Eq. (20) with cutoff L=N represents the phasonic degree of freedom without closing gaps with label up to N.
The paper states that without the cutoff Ωϕ diverges and gaps with |ν|>L do not open; L=N is chosen by hand to preserve the gap structure.
domain assumption Symmetric and antisymmetric mirror-symmetric pairs |±⟩ give a quantum metric contribution (x_n - x_{-n})² as in Eq. (18).
Two-state illustrative argument in Section V; not extended to the full many-band occupied projector.

invented entities (3)

Phasonic quantum metric Ωϕ
purpose: Momentum-like component of the quantum metric along the phason angle; added to Ωx to restore the 2D Chern-insulator inequality.
Diverges in the thermodynamic limit without a cutoff and depends on L as Ωϕ∝L² for strong modulation, so it has no independent falsifiable handle.
Mixed phason-position Chern number C (Eq. 23)
purpose: Topological invariant mixing position x and phason ϕ to label gaps and to enter the lower bound.
The quantization C=ν is cited from earlier work, and the construction is verified on the same eigenstates used to compute the quantum metric.
Conjugate position m along the phasonic direction
purpose: Position-space picture in which the Fourier harmonics of the hopping become long-range hoppings along the synthetic phasonic dimension.
An auxiliary synthetic dimension; it is not directly observable in the 1D chain and is regulated by the cutoff L.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum metric and localization in a quasicrystal." pith.science (2026). https://pith.science/paper/PX7KGPXC

@misc{pith2026250615575,
  author       = {Pith},
  title        = {Pith review of: Quantum metric and localization in a quasicrystal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PX7KGPXC}},
  note         = {Machine review of arXiv:2506.15575}
}

read the original abstract

We use the quantum metric to understand the properties of quasicrystals, represented by the one-dimensional (1D) Fibonacci chain. We show that the quantum metric can relate the localization properties of the eigenstates to the self-similarity of both the chain and its energy spectrum. In particular, the quantum metric incorporates information about distances between the local symmetry centers of each eigenstate, making it much more sensitive to the localization properties of quasicrystals than other measures of localization, such as the inverse participation ratio. Importantly, we further find that a complete description of localization requires us to, in addition, introduce a new phasonic component to the quantum metric, along with a similarly mixed phason-position Chern number. Using this, we show that the sum of both position and phasonic components of the quantum metric is lower-bounded by the gap labels associated with each energy gap of the Fibonacci chain, which stem from the Chern number. This establishes a direct link through the quantum geometry between spatial localization and fractal energy spectrum of quasicrystals. Taken together, quantum geometry provides a unifying, yet accessible, understanding of quasicrystals, rooted in their self-similarity and with intriguing consequences also for many-body physics.

Figures

Figures reproduced from arXiv: 2506.15575 by the authors.

**Figure 2.** (a) Local density of states map of the eigenstates of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a) plots the IPR of the states lying at the edge of the conduction and valence bands, immediately below and above each gap. Compared to the data shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Schematic band structure of the six smallest Fibonacci [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Chern number C and quantum metric Ω = Ωx + Ωϕ as a function of the gap label ν with two colors corresponding to different modulation strengths δ = 0.2, or δ = 0.43, using the 55- site approximant F9 and OBCs. In order to limit finite-size effects, we consider a chain m…

**Figure 6.** Figure 6: Same figure as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Hierarchical Structures of Quantum Geometric Spectrum in Quasicrystals: A Renormalization-Group Study
cond-mat.mes-hall 2025-07 conditional novelty 6.0 of 10

Quantum metric in the Fibonacci chain scales as an inverse power of the spectral gap, with exponents from renormalization-group recursion, and the scaling marks criticality in the Aubry-André-Harper model.

Reference graph

Works this paper leans on

89 extracted references · 69 canonical work pages · cited by 1 Pith paper

[88]

J. D. Hunter, Matplotlib: A 2d graphics environment, Comput- ing in Science & Engineering9, 90 (2007)

2007
[54]

Q.MarsalandA.M.Black-Schaffer,Enhancedquantummetric due to vacancies in graphene, Phys. Rev. Lett.133, 026002 (2024)

work page 2024
[1]

Shechtman, I

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett.53, 1951 (1984)

1984
[2]

N. Macé, A. Jagannathan, P. Kalugin, R. Mosseri, and F. Pié- chon, Critical eigenstates and their properties in one- and two- dimensional quasicrystals, Phys. Rev. B96, 045138 (2017)

2017
[3]

Y.E.Kraus,Y.Lahini,Z.Ringel,M.Verbin,andO.Zilberberg, Topologicalstatesandadiabaticpumpinginquasicrystals,Phys. Rev. Lett.109, 106402 (2012)

2012
[4]

D.Varjas,A.Lau,K.Pöyhönen,A.R.Akhmerov,D.I.Pikulin, andI.C.Fulga,Topologicalphaseswithoutcrystallinecounter- parts, Phys. Rev. Lett.123, 196401 (2019)

2019
[5]

G. Rai, H. Schlömer, C. Matsumura, S. Haas, and A. Jagan- nathan,Bulktopologicalsignaturesofaquasicrystal,Phys.Rev. B 104, 184202 (2021)

2021
[6]

Sites with conumbers−1, 0, and1 sit in-between long bonds

All these sites form pairs around a strongtS bond and can therefore be called molecular sites. Sites with conumbers−1, 0, and1 sit in-between long bonds. Those sites thus become isolated in the strongly modulated limit,tL≪ tS, and can therefore be called atomic sites [13]. The sequence of hopping terms obtained by the ’cut-and- project’ method in fact dep...

Show all 89 references

[7]

D. V. Else, S.-J. Huang, A. Prem, and A. Gromov, Quantum many-body topology of quasicrystals, Phys. Rev. X11, 041051 (2021)

2021
[8]

Fan and H

J. Fan and H. Huang, Topological states in quasicrystals, Front. Phys. 17, 13203 (2022)

2022
[9]

N. Macé, N. Laflorencie, and F. Alet, Many-body localization inaquasiperiodicFibonaccichain,SciPostPhys. 6,050(2019)

2019
[10]

Chiaracane, F

C. Chiaracane, F. Pietracaprina, A. Purkayastha, and J. Goold, Quantumdynamicsintheinteractingfibonaccichain,Phys.Rev. B 103, 184205 (2021)

2021
[11]

S.Sakai,N.Takemori,A.Koga,andR.Arita,Superconductivity on a quasiperiodic lattice: Extended-to-localized crossover of cooper pairs, Phys. Rev. B95, 024509 (2017)

2017
[12]

Kamiya, T

K. Kamiya, T. Takeuchi, N. Kabeya, N. Wada, T. Ishimasa, A. Ochiai, K. Deguchi, K. Imura, and N. K. Sato, Discovery of superconductivityinquasicrystal,Nat.Commun. 9,154(2018)

2018
[13]

Zhang and M

X. Zhang and M. S. Foster, Enhanced amplitude for super- conductivity due to spectrum-wide wave function criticality in quasiperiodic and power-law random hopping models, Phys. Rev. B106, L180503 (2022)

2022
[14]

Y. Wang, G. Rai, C. Matsumura, A. Jagannathan, and S. Haas, Superconductivity in the fibonacci chain, Phys. Rev. B109, 214507 (2024)

2024
[15]

Tokumoto, K

Y. Tokumoto, K. Hamano, S. Nakagawa, Y. Kamimura, S. Suzuki, R. Tamura, and K. Edagawa, Superconductivity in a vanderwaalslayeredquasicrystal,NatureCommunications 15, 1529 (2024)

2024
[16]

G. Rai, S. Haas, and A. Jagannathan, Proximity effect in a superconductor-quasicrystal hybrid ring, Phys. Rev. B100, 165121 (2019)

2019
[17]

G. Rai, S. Haas, and A. Jagannathan, Superconducting prox- imity effect and order parameter fluctuations in disordered and quasiperiodic systems, Phys. Rev. B102, 134211 (2020)

2020
[18]

A.Sandberg,O.A.Awoga,A.M.Black-Schaffer,andP.Holm- vall, Josephson effect in a fibonacci quasicrystal, Phys. Rev. B 110, 104513 (2024)

2024
[19]

I.C.Fulga,D.I.Pikulin,andT.A.Loring,Aperiodicweaktopo- logical superconductors, Phys. Rev. Lett.116, 257002 (2016)

2016
[20]

Ghadimi, T

R. Ghadimi, T. Sugimoto, K. Tanaka, and T. Tohyama, Topo- logical superconductivity in quasicrystals, Phys. Rev. B104, 144511 (2021)

2021
[21]

Kobiałka, O

A. Kobiałka, O. A. Awoga, M. Leĳnse, T. Domański, P. Holm- vall, and A. M. Black-Schaffer, Topological superconductivity in fibonacci quasicrystals, Phys. Rev. B110, 134508 (2024)

2024
[22]

M. Hori, T. Sugimoto, T. Tohyama, and K. Tanaka, Self- consistent study of topological superconductivity in two- dimensional quasicrystals, Phys. Rev. B110, 144512 (2024)

2024
[23]

Tamura, T

R. Tamura, T. Abe, S. Yoshida, Y. Shimozaki, S. Suzuki, A. Ishikawa, F. Labib, M. Avdeev, K. Kinjo, K. Nawa, and T. J. Sato, Observation of antiferromagnetic order in a quasicrystal, Nat. Phys. 10.1038/s41567-025-02858-0 (2025)

2025 doi
[24]

Odom, High-rotational symmetry lattices fabricated by moiré nanolithography, Nano Lett.12, 4948 (2012)

S.M.Lubin,W.Zhou,A.J.Hryn,M.D.Huntington,andT.W. Odom, High-rotational symmetry lattices fabricated by moiré nanolithography, Nano Lett.12, 4948 (2012)

2012
[25]

S. J. Ahn, P. Moon, T.-H. Kim, H.-W. Kim, H.-C. Shin, E. H. Kim, H. W. Cha, S.-J. Kahng, P. Kim, M. Koshino, Y.-W. Son, C.-W. Yang, and J. R. Ahn, Dirac electrons in a dodecagonal graphene quasicrystal, Science361, 782 (2018)

2018
[26]

Mahmood, A

R. Mahmood, A. V. Ramirez, and A. C. Hillier, Creating two- dimensionalquasicrystal,supercell,andmoirélatticeswithlaser interferencelithography: Implicationsforphotonicbandgapma- terials, ACS Appl. Nano Mater.4, 8851 (2021)

2021
[27]

A. Uri, S. C. de la Barrera, M. T. Randeria, D. Rodan-Legrain, T.Devakul,P.J.D.Crowley,N.Paul,K.Watanabe,T.Taniguchi, R. Lifshitz, L. Fu, R. C. Ashoori, and P. Jarillo-Herrero, Super- conductivity and strong interactions in a tunable moiré qua- sicrystal, Nature620, 762 (2023)

2023
[28]

Jagannathan, The fibonacci quasicrystal: Case study of hid- dendimensionsandmultifractality,Rev.Mod.Phys

A. Jagannathan, The fibonacci quasicrystal: Case study of hid- dendimensionsandmultifractality,Rev.Mod.Phys. 93,045001 (2021)

2021
[29]

M.Kohmoto,L.P.Kadanoff,andC.Tang,Localizationproblem in one dimension: Mapping and escape, Phys. Rev. Lett.50, 1870 (1983)

1983
[30]

Ostlund, R

S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia,One-DimensionalSchrödingerEquationwithanAlmost Periodic Potential, Phys. Rev. Lett.50, 1873 (1983)

1983
[31]

Sire,ClémentandMosseri,Rémy,Spectrumof1dquasicrystals near the periodic chain, J. Phys. France50, 3447 (1989)

1989
[32]

Piéchon, M

F. Piéchon, M. Benakli, and A. Jagannathan, Analytical results for scaling properties of the spectrum of the fibonacci chain, Phys. Rev. Lett.74, 5248 (1995)

1995
[33]

Moustaj, M

A. Moustaj, M. Röntgen, C. V. Morfonios, P. Schmelcher, and C. Morais Smith, Spectral properties of two coupled fibonacci chains, New Journal of Physics25, 093019 (2023)

2023
[34]

M.Reisner,Y.Tahmi,F.Piéchon,U.Kuhl,andF.Mortessagne, Experimentalobservationofmultifractalityinfibonaccichains, Phys. Rev. B108, 064210 (2023)

2023
[35]

109, 116404 (2012)

Y.E.KrausandO.Zilberberg,Topologicalequivalencebetween thefibonacciquasicrystalandtheharpermodel,Phys.Rev.Lett. 109, 116404 (2012)

2012
[36]

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, Topological equivalence of crystal and quasicrystal band structures, Phys. Rev. B88, 125118 (2013)

2013
[37]

Jagannathan, Missing link between the 2D Quantum Hall problemand1Dquasicrystals(2025),arXiv:2506.01122[cond- mat.str-el]

A. Jagannathan, Missing link between the 2D Quantum Hall problemand1Dquasicrystals(2025),arXiv:2506.01122[cond- mat.str-el]

2025
[38]

Dal Negro, C

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, 11 A.Lagendĳk,R.Righini,M.Colocci,andD.S.Wiersma,Light transportthroughtheband-edgestatesoffibonacciquasicrystals, Phys. Rev. Lett.90, 055501 (2003)

2003
[39]

Steurer and D

W. Steurer and D. Sutter-Widmer, Photonic and phononic qua- sicrystals, J. Phys. D: Appl. Phys.40, R229 (2007)

2007
[40]

Verbin, O

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silber- berg, Observation of topological phase transitions in photonic quasicrystals, Phys. Rev. Lett.110, 076403 (2013)

2013
[41]

Tanese, E

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akker- mans, Fractal energy spectrum of a polariton gas in a fibonacci quasiperiodic potential, Phys. Rev. Lett.112, 146404 (2014)

2014
[42]

Verbin, O

M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silber- berg,Topologicalpumpingoveraphotonicfibonacciquasicrys- tal, Phys. Rev. B91, 064201 (2015)

2015
[43]

Baboux, E

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, Measuring topological invariants from generalized edge states inpolaritonicquasicrystals,Phys.Rev.B 95,161114(R)(2017)

2017
[44]

Zilberberg, S

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y.E.Kraus,andM.C.Rechtsman,Photonictopologicalbound- arypumpingasaprobeof4dquantumhallphysics,Nature 553, 59 (2018)

2018
[45]

Lisiecki, J

F. Lisiecki, J. Rychły, P. Kuświk, H. Głowiński, J. W. Kłos, F. Groß, N. Träger, I. Bykova, M. Weigand, M. Zelent, E. J. Goering, G. Schütz, M. Krawczyk, F. Stobiecki, J. Dubowik, andJ.Gräfe,Magnonsinaquasicrystal: Propagation,extinction, andlocalizationofspinwavesinfibonaccis...

2019
[46]

Goblot, A

V. Goblot, A. Štrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaître, L. Le Gratiet, A. Harouri, I. Sagnes, S. Ravets, A. Amo, J. Bloch, and O. Zilberberg, Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains, Nat. Phys.16, 832 (2020)

2020
[47]

Franca, T

S. Franca, T. Seidemann, F. Hassler, J. van den Brink, and I. C. Fulga, Impedance spectroscopy of chiral symmetric topoelec- trical circuits, Phys. Rev. B109, L241103 (2024)

2024
[48]

H. Yao, A. Khoudli, L. Bresque, and L. Sanchez-Palencia, Critical behavior and fractality in shallow one-dimensional quasiperiodic potentials, Phys. Rev. Lett.123, 070405 (2019)

2019
[49]

Lahini, R

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009)

2009
[50]

N. Macé, A. Jagannathan, and F. Piéchon, Fractal dimensions of wave functions and local spectral measures on the fibonacci chain, Phys. Rev. B93, 205153 (2016)

2016
[51]

Ahmed, A

A. Ahmed, A. Ramachandran, I. M. Khaymovich, and A. Sharma, Flat band based multifractality in the all-band-flat diamond chain, Phys. Rev. B106, 205119 (2022)

2022
[52]

Provost and G

J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics76, 289 (1980)

1980
[53]

Marzari and D

N. Marzari and D. Vanderbilt, Maximally localized generalized wannierfunctionsforcompositeenergybands,Phys.Rev.B 56, 12847 (1997)

1997
[55]

Commun.6, 8944 (2015)

S.PeottaandP.Törmä,Superfluidityintopologicallynontrivial flat bands, Nat. Commun.6, 8944 (2015)

2015
[56]

Taniguchi, M

H.Tian,X.Gao,Y.Zhang,S.Che,T.Xu,P.Cheung,K.Watan- abe, T. Taniguchi, M. Randeria, F. Zhang,et al., Evidence for diracflatbandsuperconductivityenabledbyquantumgeometry, Nature 614, 440 (2023)

2023
[57]

E. J. BERGHOLTZ and Z. LIU, Topological flat band models and fractional chern insulators, Interna- tional Journal of Modern Physics B 27, 1330017 (2013), https://doi.org/10.1142/S021797921330017X

2013 doi
[58]

P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractionalcherninsulatorstatesintwistedbilayergraphene: An analytical approach, Phys. Rev. Res.2, 023237 (2020)

2020
[59]

Varjas, A

D. Varjas, A. Abouelkomsan, K. Yang, and E. J. Bergholtz, Topological lattice models with constant Berry curvature, Sci- Post Phys.12, 118 (2022)

2022
[60]

T.Neupert,C.Chamon,andC.Mudry,Measuringthequantum geometry of bloch bands with current noise, Phys. Rev. B87, 245103 (2013)

2013
[61]

Ozawa and N

T. Ozawa and N. Goldman, Probing localization and quantum geometry by spectroscopy, Phys. Rev. Res.1, 032019 (2019)

2019
[62]

Mazzola, S

M.Kang,S.Kim,Y.Qian,P.M.Neves,L.Ye,J.Jung,D.Puntel, F. Mazzola, S. Fang, C. Jozwiak, A. Bostwick, E. Rotenberg, J. Fuji, I. Vobornik, J.-H. Park, J. G. Checkelsky, B.-J. Yang, and R. Comin, Measurements of the quantum geometric tensor in solids, Nature Physics21, 110 (2025)

2025
[63]

N.Verma,P.J.W.Moll,T.Holder,andR.Queiroz,Quantumge- ometry: Revisiting electronic scales in quantum matter (2025), arXiv:2504.07173 [cond-mat.mtrl-sci]

2025
[64]

J. Q. You and T. B. Hu, A global cut-and-project method to construct generalized fibonacci lattices and quasilattices, physica status solidi (b) 147, 471 (1988), https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.2221470203

1988 doi
[65]

Röntgen, C

M. Röntgen, C. V. Morfonios, R. Wang, L. Dal Negro, and P. Schmelcher, Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains, Phys. Rev. B99, 214201 (2019)

2019
[66]

T. O. Wehling, S. Yuan, A. I. Lichtenstein, A. K. Geim, and M. I. Katsnelson, Resonant scattering by realistic impurities in graphene, Phys. Rev. Lett.105, 056802 (2010)

2010
[67]

N. Macé, A. Jagannathan, and F. Piéchon, Gap structure of 1d cut and project hamiltonians, Journal of Physics: Conference Series 809, 012023 (2017)

2017
[68]

Moustaj, S

A. Moustaj, S. Kempkes, and C. M. Smith, Effects of disorder in the fibonacci quasicrystal, Phys. Rev. B104, 144201 (2021)

2021
[69]

Agarwala and V

A. Agarwala and V. B. Shenoy, Topological insulators in amor- phous systems, Phys. Rev. Lett.118, 236402 (2017)

2017
[70]

Thiem and M

S. Thiem and M. Schreiber, Wavefunctions, quantum diffusion, and scaling exponents in golden-mean quasiperiodic tilings, Journal of Physics: Condensed Matter25, 075503 (2013)

2013
[71]

V. K. Varma, S. Pilati, and V. E. Kravtsov, Conduction in quasiperiodic and quasirandom lattices: Fibonacci, riemann, and anderson models, Phys. Rev. B94, 214204 (2016)

2016
[72]

Morfonios, P

C. Morfonios, P. Schmelcher, P. Kalozoumis, and F. Diakonos, Local symmetry dynamics in one-dimensional aperiodic lat- tices: a numerical study, Nonlinear Dynamics78, 71 (2014)

2014
[73]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in poly- acetylene, Phys. Rev. Lett.42, 1698 (1979)

1979
[74]

J. Wang, Y. Chen, and H. Huang, Quantum metric enhance- mentandhierarchicalscalinginone-dimensionalquasiperiodic systems (2025), arXiv:2507.04213 [cond-mat.mes-hall]

2025 arXiv
[75]

C.-K.Chiu,J.C.Y.Teo,A.P.Schnyder,andS.Ryu,Classifica- tionoftopologicalquantummatterwithsymmetries,Rev.Mod. Phys. 88, 035005 (2016)

2016
[76]

M. V. Berry, Quantal phase factors accompanying adiabatic changes,ProceedingsoftheRoyalSocietyofLondon.A.Math- ematical and Physical Sciences392, 45 (1984)

1984
[77]

R.BiancoandR.Resta,Mappingtopologicalorderincoordinate space, Phys. Rev. B84, 241106 (2011). 12

2011
[78]

D.SticletandF.Piéchon,Distant-neighborhoppingingraphene and haldane models, Phys. Rev. B87, 115402 (2013)

2013
[79]

H. R. Carrasco, J. Schirmann, A. Mordret, and A. G. Grushin, A family of aperiodic tilings with tunable quantum geometric tensor (2025), arXiv:2505.13304 [cond-mat.mes-hall]

2025
[80]

and Duneau, M., Quasiperiodic patterns and icosahe- dral symmetry, J

Katz, A. and Duneau, M., Quasiperiodic patterns and icosahe- dral symmetry, J. Phys. France47, 181 (1986)

1986
[81]

54, 2688 (1985)

M.DuneauandA.Katz,Quasiperiodicpatterns,Phys.Rev.Lett. 54, 2688 (1985)

1985
[82]

Elser, The diffraction pattern of projected structures, Acta Crystallographica Section A42, 36 (1986)

V. Elser, The diffraction pattern of projected structures, Acta Crystallographica Section A42, 36 (1986)

1986
[83]

Y. E. Kraus and O. Zilberberg, Quasiperiodicity and topology transcend dimensions, 12, 624
[84]

N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watan- abe,T.Tanigushi,B.Yan,andW.Gao,Quantum-metric-induced nonlineartransportinatopologicalantiferromagnet,Nature 621, 487 (2023)

2023
[85]

Z. Li, S. Zhang, T. Tohyama, X. Song, Y. Gu, T. Iitaka, H. Su, and H. Zeng, Optical detection of quantum geometric tensor in intrinsic semiconductors, Science China Physics, Mechanics & Astronomy 64, 107211 (2021)

2021
[86]

Julku, S

A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. Törmä, Geometric origin of superfluidity in the lieb-lattice flat band, Phys. Rev. Lett.117, 045303 (2016)

2016
[87]

C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: asoftwarepackageforquantumtransport,NewJournal of Physics16, 063065 (2014)

2014
[89]

13 0 -3 2 -1 4-4 1 -2 3 0 0 50 100 Site conumber 0 1filling fraction 0 -3 2 -1 4-4 1 -2 3 0 gap label 0.0 0.5 IPR 0 1 0.0 0.1 0.2 Density of states(a) (b) Figure 6

Q.Marsal,P.Holmvall,andA.Black-Schaffer,Quantummetric and localization in a quasicrystal, 10.5281/zenodo.15609220 (2025). 13 0 -3 2 -1 4-4 1 -2 3 0 0 50 100 Site conumber 0 1filling fraction 0 -3 2 -1 4-4 1 -2 3 0 gap label 0.0 0.5 IPR 0 1 0.0 0.1 0.2 Density of states(a) (b) ...

2025 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [88]

J. D. Hunter, Matplotlib: A 2d graphics environment, Comput- ing in Science & Engineering9, 90 (2007)

2007

[2] [54]

Q.MarsalandA.M.Black-Schaffer,Enhancedquantummetric due to vacancies in graphene, Phys. Rev. Lett.133, 026002 (2024)

work page 2024

[3] [1]

Shechtman, I

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett.53, 1951 (1984)

1984

[4] [2]

N. Macé, A. Jagannathan, P. Kalugin, R. Mosseri, and F. Pié- chon, Critical eigenstates and their properties in one- and two- dimensional quasicrystals, Phys. Rev. B96, 045138 (2017)

2017

[5] [3]

Y.E.Kraus,Y.Lahini,Z.Ringel,M.Verbin,andO.Zilberberg, Topologicalstatesandadiabaticpumpinginquasicrystals,Phys. Rev. Lett.109, 106402 (2012)

2012

[6] [4]

D.Varjas,A.Lau,K.Pöyhönen,A.R.Akhmerov,D.I.Pikulin, andI.C.Fulga,Topologicalphaseswithoutcrystallinecounter- parts, Phys. Rev. Lett.123, 196401 (2019)

2019

[7] [5]

G. Rai, H. Schlömer, C. Matsumura, S. Haas, and A. Jagan- nathan,Bulktopologicalsignaturesofaquasicrystal,Phys.Rev. B 104, 184202 (2021)

2021

[8] [6]

Sites with conumbers−1, 0, and1 sit in-between long bonds

All these sites form pairs around a strongtS bond and can therefore be called molecular sites. Sites with conumbers−1, 0, and1 sit in-between long bonds. Those sites thus become isolated in the strongly modulated limit,tL≪ tS, and can therefore be called atomic sites [13]. The sequence of hopping terms obtained by the ’cut-and- project’ method in fact dep...