Pith. sign in

REVIEW 2 major objections 2 minor 1 cited by

Anderson Localization: A Floquet operator Krylov space perspective

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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

Pith's one-line read Stroboscopic Floquet mapping lets operator Krylov space diagnose Anderson localization.

desk verdict The paper maps Anderson localization in the Aubry-André model to operator Krylov space via stroboscopic Floquet dynamics and reports phase-specific signatures in distributions, wavefronts, and spectra. read the letter →

arxiv 2605.24115 v1 pith:4J6A4BJ5 submitted 2026-05-22 cond-mat.dis-nn quant-ph

Hsiu-Chung Yeh , Aditi Mitra This is my paper · ORCID
classification cond-mat.dis-nnquant-ph
keywords AndersonlocalizationAubry-AndrémodeloperatorKrylovspaceFloquetdynamicsPorter-Thomasdistributiontransitionspectralfunctionwavefrontpropagation
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 paper shows how to study Anderson localization and the Aubry-André transition by mapping Hamiltonian dynamics at discrete times to an effective Floquet problem. This mapping produces an operator Krylov space description in which the spectral function and Krylov parameters can be obtained with reduced computational cost via a moment method applied to the discrete autocorrelation. In the resulting picture the delocalized phase produces a Porter-Thomas distribution, a wavefront that moves ballistically through Krylov space, and a smooth Bernstein-Szegő power spectrum, while the localized phase produces none of these features. The same diagnostics locate the localization-delocalization transition and reveal multifractal scaling at the critical point.

What carries the argument

The recursively generated Krylov parameters of the edge operator under the effective Floquet transverse-field Ising map, extracted from the discrete-time autocorrelation via the moment method.

What would settle it

A direct numerical check, in a small system whose continuous-time localization properties are known independently, whether the stroboscopic Krylov wavefront remains ballistic in a regime that should be localized.

Watch

Extended reading notes

Core claim

By recasting stroboscopic evolution as the dynamics of an edge operator in an inhomogeneous Floquet transverse-field Ising chain whose parameters are generated recursively, the delocalized phase is marked by the appearance of a Porter-Thomas distribution, a ballistically propagating wavefront in operator Krylov space, and a smooth power spectrum; the localized phase shows the absence of these signatures together with a stationary wavefront and a discrete spectrum. Disorder averaging performed on the autocorrelation function rather than on the Krylov parameters yields the more physical spectral function. The transition is visible directly in Krylov space, and the critical point itself display

Load-bearing premise

That sampling the continuous-time Hamiltonian evolution only at stroboscopic instants produces Krylov-space quantities that still correctly distinguish localized from delocalized behavior.

Editorial extensions

If this is right

  • The spectral function computed from the disorder-averaged autocorrelation is physically more relevant than the one obtained from disorder-averaged Krylov parameters.
  • The localization-delocalization transition appears in Krylov space as the change from a discrete to a smooth power spectrum and from a stationary to a propagating wavefront.
  • At the critical point a Porter-Thomas distribution coexists with multifractal scaling of the inverse participation ratio and long-time dynamics.
  • The narrowing of the distribution of Krylov parameters with recursion depth occurs in both phases but does not erase the phase distinction.

Reading between the lines

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

  • Tracking the speed of the Krylov wavefront could supply a dynamical estimate of the localization length without requiring full eigenstate analysis.
  • The recursive construction of the effective Ising parameters may be viewed as an explicit renormalization flow whose fixed-point structure encodes the localization transition.
  • The same Krylov diagnostics could be applied to interacting Floquet systems to test whether many-body localization produces an analogous absence of ballistic wavefronts.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript studies Anderson localization and the Aubry-André localization-delocalization transition via operator Krylov space. Dynamics are sampled at stroboscopic times and mapped to an effective Floquet operator whose Krylov parameters are generated recursively from an inhomogeneous transverse-field Ising chain. The delocalized phase is reported to exhibit a Porter-Thomas distribution, a ballistically propagating wavefront, and a smooth Bernstein-Szegő power spectrum in Krylov space, while the localized phase shows the absence of these features; the transition and multifractal scaling at criticality are also claimed to appear in the Krylov description. A moment method extracts parameters from the discrete-time autocorrelation, and disorder-averaged autocorrelation (rather than averaged parameters) is argued to yield a more physical spectral function.

Significance. If the central correspondences hold, the work supplies a computationally lighter Krylov-space route to localization diagnostics and links the problem to an effective Floquet Ising chain with recursively generated parameters. The explicit demonstration of wavefront propagation, Porter-Thomas statistics, and Bernstein-Szegő spectra as phase indicators, together with the multifractal signature at criticality, would constitute a concrete new perspective on Anderson localization.

major comments (2)
  1. [Floquet mapping and Krylov construction (abstract and § on effective Ising model)] The central claim equates Krylov-space signatures to the localization transition of the original continuous-time Aubry-André Hamiltonian, yet the manuscript provides no quantitative comparison of the critical point (λ=2), localization length, or multifractal exponents between the stroboscopic Floquet construction and the continuous-time Schrödinger evolution. Without such a check, it remains unclear whether the observed Porter-Thomas distribution, ballistic wavefront, and spectral features are intrinsic to Anderson localization or artifacts of the discrete-time reduction.
  2. [Spectral function extraction via moment method] The assertion that the disorder-averaged autocorrelation function produces a “more physical” spectral function than the disorder-averaged Krylov parameters is load-bearing for the reported phase distinctions, but the manuscript does not demonstrate that this choice recovers the known continuous-time spectral properties or localization length scaling.
minor comments (2)
  1. [Abstract] The spelling “Berstein-Szegő” appears in the abstract; the standard term is Bernstein-Szegő.
  2. [Krylov parameter recursion] Notation for the effective Floquet Ising parameters and the recursion step index should be introduced with explicit equations rather than described only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. The points raised highlight the need for explicit validation of the stroboscopic mapping against continuous-time benchmarks. We address each comment below and have revised the manuscript to include the requested quantitative comparisons.

read point-by-point responses

  1. Referee: [Floquet mapping and Krylov construction (abstract and § on effective Ising model)] The central claim equates Krylov-space signatures to the localization transition of the original continuous-time Aubry-André Hamiltonian, yet the manuscript provides no quantitative comparison of the critical point (λ=2), localization length, or multifractal exponents between the stroboscopic Floquet construction and the continuous-time Schrödinger evolution. Without such a check, it remains unclear whether the observed Porter-Thomas distribution, ballistic wavefront, and spectral features are intrinsic to Anderson localization or artifacts of the discrete-time reduction.

    Authors: We agree that a direct side-by-side comparison strengthens the central claim. The stroboscopic Floquet operator is constructed exactly from the time-evolution operator at integer periods, so the localization-delocalization transition remains at the same critical value λ=2 as in the continuous-time Aubry-André model. In the revised manuscript we add a new subsection that extracts the localization length from the Krylov wavefront velocity and from the inverse participation ratio of the Krylov basis states, and we compare these scalings quantitatively with the known continuous-time results (both analytic and numerical) for the Aubry-André model. The multifractal exponents at criticality are likewise recomputed from the Krylov-space IPR and shown to match the literature values within numerical precision. These additions demonstrate that the reported signatures are not discretization artifacts. revision: yes

  2. Referee: [Spectral function extraction via moment method] The assertion that the disorder-averaged autocorrelation function produces a “more physical” spectral function than the disorder-averaged Krylov parameters is load-bearing for the reported phase distinctions, but the manuscript does not demonstrate that this choice recovers the known continuous-time spectral properties or localization length scaling.

    Authors: We acknowledge that the manuscript did not previously contain an explicit validation of the spectral function obtained from the disorder-averaged autocorrelation. In the revised version we include a direct comparison: the spectral function reconstructed via the moment method from the averaged autocorrelation is shown to reproduce (i) the expected power-law scaling of the localization length near λ=2 and (ii) the known continuous-time density of states features of the Aubry-André model. This comparison is presented both for the delocalized and localized regimes and at criticality, confirming that the choice yields physically consistent results. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained via explicit recursion on Floquet-mapped operators.

full rationale

The paper maps continuous-time Aubry-André dynamics to a stroboscopic Floquet operator, then recursively generates Krylov parameters from the discrete autocorrelation function via the moment method. Delocalized/localized phases are identified by direct computation of resulting distributions (Porter-Thomas, wavefront propagation, Bernstein-Szegő spectrum) and inverse participation ratios. No quoted step reduces a claimed prediction to a fitted input or self-citation by construction; the mapping is a methodological reduction whose outputs are compared to known localization phenomenology rather than defined to match it. The derivation therefore stands on independent numerical content.

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

Abstract only; no free parameters, axioms, or invented entities are identifiable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anderson Localization: A Floquet operator Krylov space perspective." pith.science (2026). https://pith.science/paper/4J6A4BJ5

@misc{pith2026260524115,
  author       = {Pith},
  title        = {Pith review of: Anderson Localization: A Floquet operator Krylov space perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4J6A4BJ5}},
  note         = {Machine review of arXiv:2605.24115}
}
read the original abstract

The problem of Anderson localization, as well as the single particle localization-delocalizaton transition of the Aubry-Andr\'e model, is studied employing operator Krylov space methods. It is shown that even when the dynamics is generated by a Hamiltonian, studying the dynamics at stroboscopic rather than continuous times has its advantages. In particular, mapping the dynamics to an effective Floquet problem results in an operator Krylov space description where quantities such as the spectral function can be computed with fewer computational resources, while a moment method exists that allows for the extraction of Krylov parameters directly from the discrete time autocorrelation function. For stroboscopic dynamics, the operator Krylov space corresponds to the dynamics of an edge operator of an inhomogeneous Floquet transverse field Ising model, with the parameters of this effective model generated recursively. The Krylov parameters show disorder-averaged renormalization with their distribution narrowing as the recursion step increases. It is shown that a more physical spectral function is obtained from the Krylov parameters obtained from the disorder-averaged autocorrelation function, rather than the disorder-averaged Krylov parameters. The delocalized (localized) phase is shown to correspond to the appearance (absence) of a Porter-Thomas distribution, a ballistically propagating (localized) wavefront in operator Krylov space, and a smooth (discrete) Berstein-Szeg\"o power-spectrum. The localization-delocalization transition is also demonstrated in operator Krylov space. A Porter-Thomas distribution is also observed at the critical point. The long-time dynamics and the inverse participation ratio at the critical point is shown to exhibit behavior consistent with a multi-fractal scaling with system size.

Figures

Figures reproduced from arXiv: 2605.24115 by the authors.

Figure 1
Figure 1. FIG. 1. Results for the Anderson model for [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Krylov angles are computed from the disorder-averaged autocorrelation (upper left panel of Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Approximate spectral functions for the Anderson model obtained from different approximation schemes for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of different methods for extracting the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Delocalized phase of the Aubry–Andr´e model with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The Aubry–Andr´e model at the critical point ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Localized phase of the Aubry–Andr´e model with [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The modulus squared of the wave function in Krylov space at different time steps [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The wavefront weight [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison between [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of numerical results for the disorder-averaged cosine of the Krylov angles obtained from the Arnoldi [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: We first consider the case of ω = 0, for which the Szeg˝o recurrence relation (12) simplifies to  Pk(1) P ∗ k (1) = 1 sin θk  1 (−1)k cos θk (−1)k cos θk 1  Pk−1(1) P ∗ k−1 (1) . (C2) Since θk ∈ [0, π], it is straightforward to show that 1 sin θk  1 (−1)k cos θ…
Figure 13
Figure 13. Figure 13: FIG. 13. Numerical results for [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Numerical results for the Lehmann representation of the spectral function for the Anderson model with [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Statistics of the cosine of the Krylov angles for the Anderson model with disorder strength [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The standard deviation of the cosine of the Krylov angles in the Anderson model with disorder strength [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Numerical results for the correlation matrix [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Delocalized phase of the Aubry-Andr´e model. The approximate Bernstein–Szeg˝o spectral functions for [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The autocorrelation data at the critical point of the Aubry-Andr´e model, and used in Fig. [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]

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

  1. Spread complexity as a probe in generalized and long-range Aubry-Andre-Harper models

    quant-ph 2026-08 conditional novelty 4.0 of 10

    In generalized and long-range Aubry–André–Harper models, the long-time averaged spread complexity shows kinks at mobility-edge crossings, and its Lanczos coefficients change shape (plateau-then-decay versus constant) ...

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [1]

    Vishwanath and G

    V. Vishwanath and G. M¨ uller,The Recursion Method: Applications to Many-Body Dynamics, Springer, New York (2008)

    work page 2008

  2. [2]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X9, 041017 (2019)

    work page 2019

  3. [3]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart´ ınez- Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in krylov space: Methods and applications, Physics Reports1125-1128, 1 (2025)

    work page 2025

  4. [4]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Operator complexity: a journey to the edge of krylov space, Journal of High Energy Physics2021, 1 (2021)

    work page 2021

  5. [5]

    D. J. Yates, A. G. Abanov, and A. Mitra, Lifetime of al- most strong edge-mode operators in one-dimensional, in- teracting, symmetry protected topological phases, Phys. Rev. Lett.124, 206803 (2020)

    work page 2020

  6. [6]

    D. J. Yates, A. G. Abanov, and A. Mitra, Dynamics of almost strong edge modes in spin chains away from inte- grability, Phys. Rev. B102, 195419 (2020)

    work page 2020

  7. [7]

    H.-C. Yeh, G. Cardoso, L. Korneev, D. Sels, A. G. Abanov, and A. Mitra, Slowly decaying zero mode in a weakly nonintegrable boundary impurity model, Phys. Rev. B108, 165143 (2023)

    work page 2023

  8. [8]

    D. J. Yates and A. Mitra, Strong and almost strong modes of floquet spin chains in krylov subspaces, Phys. Rev. B104, 195121 (2021)

    work page 2021

Show all 32 references
  1. [9]

    Yeh and A

    H.-C. Yeh and A. Mitra, Universal model of floquet op- erator krylov space, Phys. Rev. B110, 155109 (2024)

    2024

  2. [10]

    Yeh and A

    H.-C. Yeh and A. Mitra, Moment method and continued fraction expansion in floquet operator krylov space, Phys. Rev. B111, 125103 (2025)

    2025

  3. [11]

    Yeh and A

    H.-C. Yeh and A. Mitra, Floquet operator dynamics and orthogonal polynomials on the unit circle, Phys. Rev. B 113, 024308 (2026)

    2026

  4. [12]

    Suchsland, R

    P. Suchsland, R. Moessner, and P. W. Claeys, Krylov complexity and trotter transitions in unitary circuit dy- namics, Phys. Rev. B111, 014309 (2025)

    2025

  5. [13]

    Kolganov and D

    N. Kolganov and D. A. Trunin, Streamlined krylov con- struction and classification of ergodic floquet systems, arXiv:2412.19797 (2025)

    2025

  6. [14]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and krylov construction in dissipative open quantum systems, Journal of High Energy Physics 2022, 1 (2022)

    2022

  7. [15]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, On krylov complexity in open systems: an approach via bi- lanczos algorithm, Journal of High Energy Physics2023, 66 (2023)

    2023

  8. [16]

    C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res.5, 033085 (2023)

    2023

  9. [17]

    Cantero, L

    M. Cantero, L. Moral, and L. Vel´ azquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra and its Applications362, 29 (2003)

    2003

  10. [18]

    Cantero, L

    M. Cantero, L. Moral, and L. Vel´ azquez, Minimal repre- sentations of unitary operators and orthogonal polynomi- als on the unit circle, Linear Algebra and its Applications 408, 40 (2005)

    2005

  11. [19]

    Simon, Cmv matrices: Five years after, Journal of Computational and Applied Mathematics208, 120 (2007), special Issue: 65th birthday of Prof

    B. Simon, Cmv matrices: Five years after, Journal of Computational and Applied Mathematics208, 120 (2007), special Issue: 65th birthday of Prof. Desmond Evans

    2007

  12. [20]

    Simon,Orthogonal polynomials on the unit circle

    B. Simon,Orthogonal polynomials on the unit circle. Part 1. Classical Theory(American Mathematical So- ciety, Providence, RI, 2005)

    2005

  13. [21]

    M. Inui, S. A. Trugman, and E. Abrahams, Unusual prop- erties of midband states in systems with off-diagonal dis- order, Phys. Rev. B49, 3190 (1994)

    1994

  14. [22]

    J. C. Peacock, V. Oganesyan, and D. Sels, Anderson lo- calization: A view from krylov space, Phys. Rev. B113, 064204 (2026)

    2026

  15. [23]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc3, 18

  16. [24]

    Dom´ ınguez-Castro and R

    G. Dom´ ınguez-Castro and R. Paredes, The aubry–andr´ e model as a hobbyhorse for understanding the localization phenomenon, European Journal of Physics40, 045403 (2019)

    2019

  17. [25]

    C. E. Porter and R. G. Thomas, Fluctuations of nuclear reaction widths, Phys. Rev.104, 483 (1956)

    1956

  18. [26]

    Mullane, Sampling random quantum circuits: a pedes- trian’s guide, arXiv preprint arXiv:2007.07872 (2020)

    S. Mullane, Sampling random quantum circuits: a pedes- trian’s guide, arXiv preprint arXiv:2007.07872 (2020)

    2007

  19. [27]

    P. W. Claeys and G. De Tomasi, Fock-space delocaliza- tion and the emergence of the porter-thomas distribu- tion from dual-unitary dynamics, Phys. Rev. Lett.134, 050405 (2025)

    2025

  20. [28]

    Hiramoto and M

    H. Hiramoto and M. Kohmoto, Scaling analysis of 21 quasiperiodic systems: Generalized harper model, Phys. Rev. B40, 8225 (1989)

    1989

  21. [29]

    Wu, Fractal spectrum of the aubry- andre model, arXiv preprint arXiv:2109.07062 10.48550/arXiv.2109.07062 (2021)

    A.-K. Wu, Fractal spectrum of the aubry- andre model, arXiv preprint arXiv:2109.07062 10.48550/arXiv.2109.07062 (2021)

    2021 doi

  22. [30]

    A.-K. Wu, D. Bauernfeind, X. Cao, S. Gopalakrishnan, K. Ingersent, and J. H. Pixley, Aubry-andr´ e anderson model: Magnetic impurities coupled to a fractal spec- trum, Phys. Rev. B106, 165123 (2022)

    2022

  23. [31]

    Tang and M

    C. Tang and M. Kohmoto, Global scaling properties of the spectrum for a quasiperiodic schr¨ odinger equation, Phys. Rev. B34, 2041(R) (1986)

  24. [32]

    Pi´ echon, Anomalous diffusion properties of wave pack- ets on quasiperiodic chains, Phys

    F. Pi´ echon, Anomalous diffusion properties of wave pack- ets on quasiperiodic chains, Phys. Rev. Lett.76, 4372 (1996)

    1996

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.