pith. sign in

arxiv: 2605.21423 · v1 · pith:UFNN7WLEnew · submitted 2026-05-20 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall

Temperature-induced optical enhancement near a localization transition

Raul Liquito , Miguel Gon\c{c}alves , Bruno Amorim , Eduardo V. Castro This is my paper

Pith reviewed 2026-05-21 02:35 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hall
keywords Aubry-André modelquasiperiodic systemsoptical conductivitylocalization transitionvan Hove singularitiesfinite temperaturemetal-insulator transition
0
0 comments X

The pith

Finite temperature strongly enhances low-frequency optical conductivity at resonant frequencies near the localization transition in the Aubry-André model.

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

The paper examines optical conductivity in the Aubry-André model, a classic quasiperiodic system that hosts a metal-insulator transition even in one dimension. At zero temperature the low-frequency response develops an optical gap that closes discontinuously as the transition is approached. At finite temperature a pronounced enhancement appears at specific resonant frequencies. This boost occurs because temperature activates transitions between van Hove singularities that remain Pauli-blocked at zero temperature. A reader would care because the mechanism supplies both a new probe of quasiperiodic order and a route to control optical response near localization.

Core claim

In the Aubry-André model the zero-temperature low-frequency optical conductivity is restructured by the quasiperiodic potential, producing an optical gap that closes discontinuously on approach to the metal-insulator transition. At finite temperature a strong enhancement of this low-frequency conductivity occurs at certain resonant frequencies. The enhancement arises from the thermal activation of Pauli-blocked transitions between strongly resonant van Hove singularities. The work therefore positions the optical response as an experimentally accessible probe of non-trivial quasiperiodicity effects and identifies a pathway for manipulating optical properties near a localization transition.

What carries the argument

Thermal activation of Pauli-blocked transitions between strongly resonant van Hove singularities under the quasiperiodic potential.

If this is right

  • The zero-temperature optical gap closes discontinuously at the metal-insulator transition.
  • Finite-temperature enhancement supplies new insight into finite-frequency transport in quasiperiodic systems.
  • The optical response serves as a powerful, experimentally accessible probe for non-trivial quasiperiodicity effects.
  • A new pathway opens for manipulating optical properties near a localization transition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same thermal-activation route could be tested in ultracold-atom realizations of the Aubry-André model by measuring conductivity at varying temperatures and potential strengths.
  • Resonant frequencies identified here may enable selective optical control in other quasiperiodic lattices.
  • Similar enhancement might appear when van Hove features survive in higher-dimensional quasiperiodic or weakly disordered systems.

Load-bearing premise

The van Hove singularities must remain sufficiently sharp and resonant under the quasiperiodic potential so that thermal activation produces a distinct low-frequency peak rather than a broad feature.

What would settle it

If finite-temperature optical conductivity measurements near the transition show only a broad feature instead of a distinct low-frequency peak at the predicted resonant frequencies, the proposed thermal-activation mechanism would be ruled out.

Figures

Figures reproduced from arXiv: 2605.21423 by Bruno Amorim, Eduardo V. Castro, Miguel Gon\c{c}alves, Raul Liquito.

Figure 2
Figure 2. Figure 2: Schematic representation of the regular part of the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Schematic representation of the regular part of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of AA model energy spectra as a function [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The plots show the DOS of the central bands around [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Drude Weight (D) as a function of quasiperiodic potential strength (W) obtained with three different meth￾ods. “Numerical Kubo” was obtained for a finite temperature of T = 5×10−4 , a system size of L = 10946 via Krylov-Schur sparse methods. The “Real Space vF ” was obtained by numer￾ically calculating the eigenstate at Fermi energy (E = 0) for a system size L = 514229 via via Krylov-Schur sparse methods (… view at source ↗
Figure 6
Figure 6. Figure 6: Real part of the regular conductivity (Re [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Pairwise resonances around gapped van Hove singularities for W = 1.8. The solid blue line represents the DOS, the colored markers correspond to the energy of a few eigenstates around the van Hove singularities. Pairs with the same marker style and color indicate strongly cou￾pled states, i.e., with large |Jnm|. The dashed vertical lines mark the first resonant states contributing to the zero tem￾peratu… view at source ↗
Figure 8
Figure 8. Figure 8: Optical gap (∆opt) as a function of quasiperiodic potential strength (W) at zero temperature (T = 10−6 ). The red line (“Direct Estimation”) was extracted from the con￾ductivity calculations by imposing a threshold for σ(ω) below which we took as 0. The dotted block line was obtained by calculating ϵr following eq. 11. ical point the correlation length is finite, so gaps gen￾erated beyond some order are ex… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Height of the conductivity peak (σpeak ≡ maxω Re [σreg(ω)]) as a function of potential strength (W) for a system with L = 28657. The solid line was extracted from the conductivity data, the dashed line is a naive es￾timation done by explicitly summing the Kubo-Greenwood expression for the conductivity (see Appendix. D). (b) Fre￾quency of conductivity peak (ωpeak) as a function of poten￾tial strength (… view at source ↗
Figure 11
Figure 11. Figure 11: Real part of the regular conductivity (Re [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Real part of the regular conductivity (Re [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
read the original abstract

Quasiperiodic systems are an intermediate class of systems between periodic crystals and disordered systems, famously exhibiting metal-insulator transitions (MITs) even in one dimension. While their transport properties have been studied extensively, a systematic analysis of the finite-frequency optical conductivity near the critical point has been lacking. In this work, we carry out a detailed study of the optical conductivity in the paradigmatic Aubry-Andr\'e model. We find that the zero-temperature low-frequency optical signal is strongly restructured by the quasiperiodic potential, exhibiting an optical gap that closes discontinuously as the system approaches the MIT. Most strikingly, we uncover a mechanism for a strong enhancement of the low-frequency finite temperature optical conductivity at certain resonant frequencies. This enhancement stems from the thermal activation of Pauli-blocked transitions between strongly resonant van Hove singularities. This mechanism provides new insight into finite-frequency transport in quasiperiodic systems and a new pathway for manipulating optical properties near a localization transition. Furthermore, our findings establish the optical response as a powerful, experimentally accessible tool for probing non-trivial quasiperiodicity effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the finite-frequency optical conductivity in the one-dimensional Aubry-André quasiperiodic model. It reports that the zero-temperature low-frequency optical response develops an optical gap that closes discontinuously upon approaching the metal-insulator transition. The central new result is a mechanism for strong enhancement of the low-frequency finite-temperature optical conductivity at specific resonant frequencies, arising from thermal activation of Pauli-blocked transitions between sharply resonant van Hove singularities in the density of states.

Significance. If the reported temperature-induced enhancement survives the thermodynamic limit, the work supplies a concrete, experimentally accessible signature of quasiperiodic van Hove physics near a localization transition. It fills a documented gap in systematic finite-frequency studies of the Aubry-André model and suggests a route to optically probe non-trivial quasiperiodicity effects. The absence of free parameters in the underlying model and the direct mapping from zero-T gap closure to finite-T signal are strengths that would make the result falsifiable once finite-size concerns are addressed.

major comments (1)
  1. [Numerical results / finite-size analysis] Numerical results section (finite-size spectra and optical conductivity plots): the claimed sharpness of the van Hove singularities and the resulting distinct low-frequency thermal peak are demonstrated only for finite chains (L ≲ 200). Near the localization transition the correlation length diverges, so both level statistics and optical matrix elements are strongly L-dependent. Without an explicit finite-size scaling collapse or extrapolation showing that the low-ω enhancement remains finite as L → ∞, the mechanism risks being a finite-size artifact, directly undermining the central claim of a robust temperature-induced enhancement.
minor comments (2)
  1. [Abstract and zero-T results] The abstract states that the optical gap 'closes discontinuously' but the manuscript does not quantify the discontinuity (e.g., via a jump in the gap size versus λ or a scaling exponent). Adding a brief plot or statement of the gap-closing behavior would strengthen the zero-T part of the narrative.
  2. [Methods / notation] Notation for the optical conductivity σ(ω,T) and the definition of the resonant frequencies should be introduced once in the main text with an equation number rather than appearing first in figure captions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We appreciate the recognition of the significance of our findings on the temperature-induced optical enhancement in the Aubry-André model. Below, we address the major comment regarding finite-size effects.

read point-by-point responses

  1. Referee: [Numerical results / finite-size analysis] Numerical results section (finite-size spectra and optical conductivity plots): the claimed sharpness of the van Hove singularities and the resulting distinct low-frequency thermal peak are demonstrated only for finite chains (L ≲ 200). Near the localization transition the correlation length diverges, so both level statistics and optical matrix elements are strongly L-dependent. Without an explicit finite-size scaling collapse or extrapolation showing that the low-ω enhancement remains finite as L → ∞, the mechanism risks being a finite-size artifact, directly undermining the central claim of a robust temperature-induced enhancement.

    Authors: We agree that a thorough finite-size analysis is essential to confirm that the observed temperature-induced enhancement is not a finite-size effect, especially given the diverging correlation length at the localization transition. Our manuscript presents results for system sizes up to L = 200, which is typical for exact diagonalization studies of this model. We have observed that the resonant frequencies and the enhancement persist across the range of sizes we studied. However, we acknowledge that this does not fully resolve the issue for the thermodynamic limit. In the revised manuscript, we will include additional finite-size scaling analysis. Specifically, we will show the dependence of the low-frequency peak on system size and provide an extrapolation where possible. We will also discuss the relevant length scales, such as the localization length, to contextualize our system sizes. This revision will strengthen the evidence for the robustness of the mechanism. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in direct model analysis

full rationale

The paper performs a direct numerical and analytical study of the Aubry-André model, computing the optical conductivity at zero and finite temperature and identifying the enhancement mechanism from thermal activation across van Hove singularities. No step reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or self-citation chain by construction. The central claim follows from explicit evaluation of the model's eigenstates, matrix elements, and thermal occupation factors rather than from renaming or re-deriving an input quantity. Finite-size effects noted by the skeptic are a potential correctness concern but do not constitute circularity under the specified criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters or invented entities are stated. The study relies on the standard Aubry-André Hamiltonian.

axioms (1)
  • domain assumption The Aubry-André model captures the essential physics of quasiperiodic metal-insulator transitions in one dimension.
    The paper selects this paradigmatic model for the optical conductivity analysis.

pith-pipeline@v0.9.0 · 5726 in / 1246 out tokens · 35329 ms · 2026-05-21T02:35:40.504236+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Constants.lean phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    we set β=1+√5/2, the golden mean... system size L=Fn (the n-th Fibonacci number)

  • IndisputableMonolith/Foundation/Cost.lean Jcost unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    optical conductivity... Kubo-Greenwood formula... Re[σ_reg(ω)]... van Hove singularities

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

102 extracted references · 102 canonical work pages · 2 internal anchors

  1. [1]

    Real spacevF

    The inset we plot the same information for a system sizeL= 6765, in the central region highlighted by the grey rectangular box. Notice the different energy (y-axis) scales of the main plot and inset. The color plot encodes the inverse participationratio(IPR)ofeacheigenstate. Thedashedblack vertical line marks the metal-insulator transition of the AA model...

    work page

  2. [2]

    In this limit, we therefore have Re[σreg(ω)] = 0

    Zero Temperature (T= 0) In the clean limit (W= 0), the real part ofσ(ω)ex- hibits a Drude peak atω→0and there are no interband transitions since there is a single energy band. In this limit, we therefore have Re[σreg(ω)] = 0. Strictly speak- ing, the current matrix elements in the eigenbasis are perfectly diagonal and maximal for states at the Fermi energ...

    work page

  3. [3]

    This is analogous to an effective periodic system with a mini-Brillouin zone (MBZ), where coupled states share the same MBZ crystalline momentum but have different band indexes

    Intraband Decoupling: states in the same quasi- Bloch band are approximatelyuncoupled. This is analogous to an effective periodic system with a mini-Brillouin zone (MBZ), where coupled states share the same MBZ crystalline momentum but have different band indexes. Because the current operator is diagonal with respect to the MBZ crys- talline momenta, off-...

    work page

  4. [4]

    Direct Estimation

    InterbandPairwiseResonances: Statesacrossagap (e.g. states nearA 1 andB 1) exhibit strong pair- wise resonances, leading to off-diagonal matrix ele- ments inversely proportional to their energy sepa- ration (|Jnm| ∝(ϵ n −ϵ m)−2)[53], see Fig. 7(c). Figure 7.(a)Pairwise resonances around gapped van Hove singularities forW= 1.8. The solid blue line represen...

    work page

  5. [5]

    Despite their strong coupling, these transitions are Pauli blocked at zero tem- perature and half filling

    Finite Temperature As discussed previously, states in neighboring gapped van Hove singularities strongly couple in pairs, leading to maximum current matrix elements between states at the edges of each quasi-Bloch band. Despite their strong coupling, these transitions are Pauli blocked at zero tem- perature and half filling. Thethermalactivationofsuchtrans...

    work page arXiv 2024

  6. [6]

    Kohmoto, B

    M. Kohmoto, B. Sutherland, and C. Tang, Critical wave functionsandaCantor-setspectrumofaone-dimensional quasicrystal model, Phys. Rev. B35, 1020 (1987)

    work page 1987

  7. [7]

    A. I. Goldman and R. F. Kelton, Quasicrystals and crys- talline approximants, Rev. Mod. Phys.65, 213 (1993)

    work page 1993

  8. [10]

    Aubry and G

    S. Aubry and G. André, Analyticity breaking and Ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc3, 18 (1980)

    work page 1980

  9. [11]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil- berberg, Topological States and Adiabatic Pumping in Quasicrystals, Phys. Rev. Lett.109, 106402 (2012)

    work page 2012

  10. [12]

    Y. E. Kraus and O. Zilberberg, Topological Equiva- lence between the Fibonacci Quasicrystal and the Harper Model, Phys. Rev. Lett.109, 116404 (2012)

    work page 2012

  11. [13]

    Verbin, O

    M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Observation of Topological Phase Transi- tions in Photonic Quasicrystals, Phys. Rev. Lett.110, 076403 (2013)

    work page 2013

  12. [16]

    R. E. Prange, D. R. Grempel, and S. Fishman, Long-Range Resonance in Anderson Insulators: Finite- FrequencyConductivityofRandomandIncommensurate Systems, Phys. Rev. Lett.53, 1582 (1984)

    work page 1984

  13. [17]

    Roy and S

    N. Roy and S. Sinha, A finite temperature study of ideal quantum gases in the presence of one dimensional quasi- periodic potential, J. Stat. Mech.2018, 053106 (2018)

    work page 2018

  14. [18]

    Roósz, U

    G. Roósz, U. Divakaran, H. Rieger, and F. Iglói, Nonequilibriumquantumrelaxationacrossalocalization- delocalization transition, Phys. Rev. B90, 184202 (2014)

    work page 2014

  15. [20]

    D. S. Bhakuni, T. L. M. Lezama, and Y. Bar Lev, Noise- induced transport in the Aubry-André-Harper model, SciPost Phys. Core7, 023 (2024)

    work page 2024

  16. [23]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. In- guscio, Anderson localization of a non-interacting Bose– Einstein condensate, Nature453, 895 (2008)

    work page 2008

  17. [24]

    Modugno, Exponential localization in one- dimensional quasi-periodic optical lattices, New J

    M. Modugno, Exponential localization in one- dimensional quasi-periodic optical lattices, New J. Phys.11, 033023 (2009)

    work page 2009

  18. [25]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

    work page 2015

  19. [26]

    H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Alt- man, U. Schneider, and I. Bloch, Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems, Phys. Rev. Lett.119, 260401 (2017)

    work page 2017

  20. [28]

    W. G. T. Kamkou, N. Tchepemen, and J. P. Nguenang, Spectral and dynamical characters of 1D incommensu- rate optical lattices with PT-symmetry, Physica B: Con- densed Matter667, 415130 (2023)

    work page 2023

  21. [29]

    Lahini, R

    Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Moran- dotti, N. Davidson, and Y. Silberberg, Observation of a Localization Transition in Quasiperiodic Photonic Lat- tices, Phys. Rev. Lett.103, 013901 (2009)

    work page 2009

  22. [30]

    Moirémagicthreeyearson,NatRevMater6,191(2021)

    work page 2021

  23. [31]

    P. Moon, M. Koshino, and Y.-W. Son, Quasicrystalline electronic states in ${30}^{\ensuremath{\circ}}$ ro- tated twisted bilayer graphene, Phys. Rev. B99, 165430 (2019)

    work page 2019

  24. [32]

    G. Yu, Z. Wu, Z. Zhan, M. I. Katsnelson, and S. Yuan, Dodecagonal bilayer graphene quasicrystal and its ap- proximants, npj Comput Mater5, 1 (2019)

    work page 2019

  25. [34]

    Gonçalves, H

    M. Gonçalves, H. Z. Olyaei, B. Amorim, R. Mondaini, P. Ribeiro, and E. V. Castro, Incommensurability- induced sub-ballistic narrow-band-states in twisted bi- layer graphene, 2D Mater.9, 011001 (2021). 11

    work page 2021

  26. [35]

    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, Superconductivity and strong interactions in a tunable moiré quasicrystal, Na- ture620, 762 (2023)

    work page 2023

  27. [37]

    Gonçalves, B

    M. Gonçalves, B. Amorim, E. V. Castro, and P. Ribeiro, Critical Phase Dualities in 1D Exactly Solv- able Quasiperiodic Models, Phys. Rev. Lett.131, 186303 (2023)

    work page 2023

  28. [39]

    Purkayastha, S

    A. Purkayastha, S. Sanyal, A. Dhar, and M. Kulka- rni, Anomalous transport in the Aubry-Andr\’e-Harper model in isolated and open systems, Phys. Rev. B97, 174206 (2018)

    work page 2018

  29. [40]

    Bar Lev, D

    Y. Bar Lev, D. M. Kennes, C. Klöckner, D. R. Reichman, and C. Karrasch, Transport in quasiperiodic interacting systems: From superdiffusion to subdiffusion, EPL119, 37003 (2017)

    work page 2017

  30. [43]

    N. F. Mott, Conduction in non-crystalline materials: III. Localized states in a pseudogap and near extremities of conduction and valence bands, The Philosophical Maga- zine: A Journal of Theoretical Experimental and Applied Physics19, 835 (1969)

    work page 1969

  31. [45]

    Kubo, Statistical-Mechanical Theory of Irreversible Processes

    R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Jpn. 12, 570 (1957)

    work page 1957

  32. [46]

    D. A. Greenwood, The Boltzmann Equation in the The- ory of Electrical Conduction in Metals, Proc. Phys. Soc. 71, 585 (1958)

    work page 1958

  33. [48]

    Resta, Why are insulators insulating and metals con- ducting?, J

    R. Resta, Why are insulators insulating and metals con- ducting?, J. Phys.: Condens. Matter14, R625 (2002)

    work page 2002

  34. [52]

    Zhang, X

    J. Zhang, X. Chen, S. Mills, T. Ciavatti, Z. Yao, R. Mescall, H. Hu, V. Semenenko, Z. Fei, H. Li, V. Pere- beinos, H. Tao, Q. Dai, X. Du, and M. Liu, Tera- hertz Nanoimaging of Graphene, ACS Photonics5, 2645 (2018)

    work page 2018

  35. [53]

    R. Jing, Y. Shao, Z. Fei, C. F. B. Lo, R. A. Vitalone, F. L. Ruta, J. Staunton, W. J.-C. Zheng, A. S. Mcleod, Z. Sun, B.-y. Jiang, X. Chen, M. M. Fogler, A. J. Millis, M. Liu, D. H. Cobden, X. Xu, and D. N. Basov, Tera- hertz response of monolayer and few-layer WTe2 at the nanoscale, Nat Commun12, 5594 (2021)

    work page 2021

  36. [54]

    X. Guo, K. Bertling, B. C. Donose, M. Brünig, A. Cer- nescu, A. A. Govyadinov, and A. D. Rakić, Terahertz nanoscopy: Advances, challenges, and the road ahead, Applied Physics Reviews11, 021306 (2024)

    work page 2024

  37. [56]

    W.Kohn,TheoryoftheInsulatingState,Phys.Rev.133, A171 (1964)

    work page 1964

  38. [57]

    Thouless, Electrons in disordered systems and the the- ory of localization, Physics Reports13, 93 (1974)

    D. Thouless, Electrons in disordered systems and the the- ory of localization, Physics Reports13, 93 (1974)

    work page 1974

  39. [59]

    In this case, gap openings are induced by real-space resonances (and not momentum-space resonances like in the extended phase)

    Note that within the localized phase (W >2), a largeW perturbative expansion can also be carried out, treat- ing the hopping as a perturbation. In this case, gap openings are induced by real-space resonances (and not momentum-space resonances like in the extended phase)

    work page

  40. [60]

    J. J. Sakurai and J. Napolitano,Modern Quantum Me- chanics, 3rd ed. (Cambridge University Press, Cam- bridge, 2021). 12 Appendix A: Perturbation Theory in the AA model We treat the AA model (eq. 1) using standard non-degenerate perturbation theory forW <2t. The un- perturbed HamiltonianH 0 is the 1D tight-binding model, and the perturbation is the quasipe...

    work page 2021

  41. [61]

    The first-order energy correction is zero:∆(1) k =V kk = 0

    Perturbed Eigenstates and Energies In the unperturbed Bloch basis ψ(0) k E , the perturbation matrix element are: D ψ(0) k′ ˆV ψ(0) k E = W 2 (δk+2πβ,k ′ +δ k−2πβ,k ′).(A1) This shows that the perturbation only couples states with momentakandk±2πβ. The first-order energy correction is zero:∆(1) k =V kk = 0. The second-order correction is ∆(2) k = X k′̸=k ...

    work page

  42. [62]

    Current Matrix Elements of quasi-Bloch states The current operator in the unperturbed Bloch basis is diagonal:J(0) kk′ = D ψ(0) k ˆJ ψ(0) k E =δ kk′J(0) k , whereJ (0) k = ∂kϵk = 2 sin (k)(fort=e=ℏ=a= 1). From Eq. A8 the perturbed current matrix elements are: Jkk′ =|c k|2 J(0) k,k′ + X n W 2t n"X σ=±1 ck′˜ckσmJ(0) k+σ2πnβ,k ′ + ˜ck′σnc∗ kJ(0) k,k′+σ2πnβ #...

    work page

  43. [63]

    Drude W eight from Perturbed Current Matrix Elements To obtain the Drude weight we need to calculate the Fermi velocity Eq. B8. So, we are interested in the diagonal matrix elementJ k = D ψk ˆJ ψk E for the perturbed state. Due to the translation symmetry, the unperturbed current matrix elements are diagonalJ(0) kk′ ∝δ kk′ and the cross-terms inJkk′ vanis...

    work page

  44. [64]

    ForW= 0,c k = 1and all˜c= 0.J k =J (0) k , which is maximal atkF =±π/2

    work page

  45. [65]

    AsWincreases, the normalization factor|c k|2 (the weight of the original state) decreases as spectral weight is transferred to the harmonics (|˜ck,σ,n|2 >0)

    work page

  46. [66]

    The total currentJk is thus a sum where weight is transferred from the maximal termJ(0) kF to progressively smaller terms

    The harmonicsk F ±2πnβare further from the band center, so their current matrix elements J(0) k±2πnβ are smaller than J(0) kF . The total currentJk is thus a sum where weight is transferred from the maximal termJ(0) kF to progressively smaller terms. This leads to a monotonic supression ofJkF, and thereforeD, asW→2. 14

    work page

  47. [67]

    The recursive implementation allows the calculation of the perturbed eigenfunctions up to orderm

    Perturbation Theory - Numerical Implementation In this section, we provide the expressions used for the recursive implementation of time-independent non-degenerate perturbation theory [55]. The recursive implementation allows the calculation of the perturbed eigenfunctions up to orderm. We used: ∆(N) n =⟨ψ (0) n |V|ψ (N−1) n ⟩(A11) |ψ(N) n ⟩= X i̸=n ⟨ψ(0)...

    work page

  48. [68]

    Kohmoto, B

    M. Kohmoto, B. Sutherland, and C. Tang, Critical wave functions and a Cantor-set spectrum of a one-dimensional qua- sicrystal model, Phys. Rev. B35, 1020 (1987). 18

    work page 1987

  49. [69]

    A. I. Goldman and R. F. Kelton, Quasicrystals and crystalline approximants, Rev. Mod. Phys.65, 213 (1993)

    work page 1993

  50. [70]

    P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev.109, 1492 (1958)

    work page 1958

  51. [71]

    MacKinnon and B

    A. MacKinnon and B. Kramer, One-Parameter Scaling of Localization Length and Conductance in Disordered Systems, Phys. Rev. Lett.47, 1546 (1981)

    work page 1981

  52. [72]

    Aubry and G

    S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc 3, 18 (1980)

    work page 1980

  53. [73]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological States and Adiabatic Pumping in Quasicrys- tals, Phys. Rev. Lett.109, 106402 (2012)

    work page 2012

  54. [74]

    Y. E. Kraus and O. Zilberberg, Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model, Phys. Rev. Lett.109, 116404 (2012)

    work page 2012

  55. [75]

    Verbin, O

    M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Observation of Topological Phase Transitions in Photonic Quasicrystals, Phys. Rev. Lett.110, 076403 (2013)

    work page 2013

  56. [76]

    Liquito, M

    R. Liquito, M. Gonçalves, and E. V. Castro, Fate of quadratic band crossing under quasiperiodic modulation, Phys. Rev. B109, 174202 (2024)

    work page 2024

  57. [77]

    Liquito, M

    R. Liquito, M. Gonçalves, and E. Castro, Quasiperiodic quadrupole insulators, SciPost Physics18, 208 (2025)

    work page 2025

  58. [78]

    R. E. Prange, D. R. Grempel, and S. Fishman, Long-Range Resonance in Anderson Insulators: Finite-Frequency Conduc- tivity of Random and Incommensurate Systems, Phys. Rev. Lett.53, 1582 (1984)

    work page 1984

  59. [79]

    Roy and S

    N. Roy and S. Sinha, A finite temperature study of ideal quantum gases in the presence of one dimensional quasi-periodic potential, J. Stat. Mech.2018, 053106 (2018)

    work page 2018

  60. [80]

    Roósz, U

    G. Roósz, U. Divakaran, H. Rieger, and F. Iglói, Nonequilibrium quantum relaxation across a localization-delocalization transition, Phys. Rev. B90, 184202 (2014)

    work page 2014

  61. [81]

    Roy and A

    N. Roy and A. Sharma, Study of counterintuitive transport properties in the Aubry-André-Harper model via entanglement entropy and persistent current, Phys. Rev. B100, 195143 (2019), arXiv:1905.13255 [cond-mat]

    work page arXiv 2019

  62. [82]

    D. S. Bhakuni, T. L. M. Lezama, and Y. Bar Lev, Noise-induced transport in the Aubry-André-Harper model, SciPost Phys. Core7, 023 (2024)

    work page 2024

  63. [83]

    Lahiri, Ac conductivity of incommensurate lattices: Anatomy of wave functions, Phys

    A. Lahiri, Ac conductivity of incommensurate lattices: Anatomy of wave functions, Phys. Rev. B53, 3702 (1996)

    work page 1996

  64. [84]

    Self-dual quasiperiodic systems with power-law hopping

    S. Gopalakrishnan, Self-dual quasiperiodic systems with power-law hopping, Phys. Rev. B96, 054202 (2017), arXiv:1706.05382 [cond-mat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [85]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Nature453, 895 (2008)

    work page 2008

  66. [86]

    Modugno, Exponential localization in one-dimensional quasi-periodic optical lattices, New J

    M. Modugno, Exponential localization in one-dimensional quasi-periodic optical lattices, New J. Phys.11, 033023 (2009)

    work page 2009

  67. [87]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science349, 842 (2015)

    work page 2015

  68. [88]

    H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems, Phys. Rev. Lett.119, 260401 (2017)

    work page 2017

  69. [89]

    Anderson, F

    R. Anderson, F. Wang, P. Xu, V. Venu, S. Trotzky, F. Chevy, and J. H. Thywissen, Conductivity Spectrum of Ultracold Atoms in an Optical Lattice, Phys. Rev. Lett.122, 153602 (2019)

    work page 2019

  70. [90]

    W. G. T. Kamkou, N. Tchepemen, and J. P. Nguenang, Spectral and dynamical characters of 1D incommensurate optical lattices with PT-symmetry, Physica B: Condensed Matter667, 415130 (2023)

    work page 2023

  71. [91]

    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)

    work page 2009

  72. [92]

    Moiré magic three years on, Nat Rev Mater6, 191 (2021)

    work page 2021

  73. [93]

    P. Moon, M. Koshino, and Y.-W. Son, Quasicrystalline electronic states in ${30}^{\ensuremath{\circ}}$ rotated twisted bilayer graphene, Phys. Rev. B99, 165430 (2019)

    work page 2019

  74. [94]

    G. Yu, Z. Wu, Z. Zhan, M. I. Katsnelson, and S. Yuan, Dodecagonal bilayer graphene quasicrystal and its approximants, npj Comput Mater5, 1 (2019)

    work page 2019

  75. [95]

    Pezzini, V

    S. Pezzini, V. Mišeikis, G. Piccinini, S. Forti, S. Pace, R. Engelke, F. Rossella, K. Watanabe, T. Taniguchi, P. Kim, and C. Coletti, 30◦-Twisted Bilayer Graphene Quasicrystals from Chemical Vapor Deposition, Nano Lett20, 3313 (2020)

    work page 2020

  76. [96]

    Gonçalves, H

    M. Gonçalves, H. Z. Olyaei, B. Amorim, R. Mondaini, P. Ribeiro, and E. V. Castro, Incommensurability-induced sub- ballistic narrow-band-states in twisted bilayer graphene, 2D Mater.9, 011001 (2021)

    work page 2021

  77. [97]

    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, Superconductivity and strong interactions in a tunable moiré quasicrystal, Nature620, 762 (2023)

    work page 2023

  78. [98]

    X. Lai, D. Guerci, G. Li, K. Watanabe, T. Taniguchi, J. Wilson, J. H. Pixley, and E. Y. Andrei, Imaging Self-aligned Moiré Crystals and Quasicrystals in Magic-angle Bilayer Graphene on hBN Heterostructures (2023), arXiv:2311.07819 [cond-mat]

    work page arXiv 2023

  79. [99]

    Gonçalves, B

    M. Gonçalves, B. Amorim, E. V. Castro, and P. Ribeiro, Critical Phase Dualities in 1D Exactly Solvable Quasiperiodic Models, Phys. Rev. Lett.131, 186303 (2023)

    work page 2023

  80. [100]

    Oliveira, N

    R. Oliveira, N. Sobrosa, P. Ribeiro, B. Amorim, and E. V. Castro, Local Density of States as a Probe of Multifractality in Quasiperiodic Moiré Materials (2025), arXiv:2510.20575 [cond-mat]

    work page arXiv 2025

Showing first 80 references.