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REVIEW 3 major objections 5 minor 76 references

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

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that long-time averaged spread complexity kinks exactly when a quantum quench crosses a mobility edge, making it a dynamical order parameter for localization transitions in Aubry-André-Harper models.

desk verdict The kink in time-averaged spread complexity at mobility-edge crossings is a plausible and probably correct observation, but the analytic Lanczos-coefficient formulas are wrong and the numerical case for 'nonanalytic' is under-supported. read the letter →

arxiv 2608.02451 v1 pith:IDXBWU2W submitted 2026-08-03 quant-ph cond-mat.dis-nncond-mat.stat-mech

Triyas Sapui , Tanoy Kanti Konar , Subinay Dasgupta , Aditi Sen De This is my paper · ORCID
classification quant-phcond-mat.dis-nncond-mat.stat-mech
keywords spreadcomplexityKrylovAubry-André-HarpermodelmobilityedgelocalizationtransitionLanczoscoefficientsquantumquenchlong-rangehopping
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 tries to establish that a single dynamical quantity—the time-averaged spread complexity of a state—can act as an order parameter for localization transitions in one-dimensional quasiperiodic systems, including cases with energy-dependent mobility edges. It argues that when the final quasiperiodic potential in a quench crosses the mobility edge tied to the initial state's energy, the long-time averaged complexity develops a sharp kink, and this kink matches the phase boundary obtained from the inverse participation ratio. The mechanism is the broadening of the local density of states across the transition, which directly feeds the first Lanczos coefficient through its variance. The paper also derives analytic moments and Lanczos coefficients for extreme quenches and shows that their shape (constant versus plateau-then-decay) encodes whether a mobility edge is present. If correct, this gives a dynamical probe of localization that requires only time evolution and survival-probability data.

What carries the argument

Spread complexity is the average position of the time-evolved state on the Krylov basis built from the Hamiltonian, C(t)=Σ_n n |⟨K_n|ψ(t)⟩|². The Lanczos coefficients a_n, b_n from the Lanczos recursion define this basis, and the identity b₁² = variance of the local density of states connects spectral broadening directly to complexity growth. The diagnostic used throughout is the long-time averaged value C̄. In the generalized AAH model the mobility edge is the energy E0 satisfying βE0 = 2t − λ; in the long-range model the paper uses the energy-dependent boundary E = λ cosh(α ln 2) − t.

What would settle it

Compute the first Lanczos coefficient directly for β=0.3, λ_f=100: Eq. (16) yields a₀ ≈ 683, while a direct integration of the survival-amplitude moment gives a₀ ≈ 16 and the Hamiltonian's operator norm is about 143; if a₀ exceeds the norm bound, the analytic expressions in Eqs. (15)–(17) cannot be correct.

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Extended reading notes

Core claim

For quenches between eigenstates of the Aubry-André-Harper Hamiltonian at different potential strengths, the long-time averaged spread complexity shows a nonanalytic kink exactly when the final potential λ_f crosses the mobility edge associated with the initial state's energy. In the standard AAH model, which has no mobility edge, the kink sits at λ_f=2; in the generalized β≠0 model, the kink position shifts with the initial state's energy according to βE0 = 2t − λ, matching IPR phase boundaries. The broadening of the local density of states across the transition is identified as the cause, via b₁² = σ²_LDOS. In the long-range hopping model, the averaged spread complexity still finds the mob

Load-bearing premise

The load-bearing assumption is that in the infinite-potential limit the post-quench eigenstates are exactly single-site localized, so the survival amplitude reduces to the averaged phase in Eq. (14); if that idealization is wrong or the derived moments are inconsistent, the analytic support for the Lanczos-coefficient claims collapses even though the numerical kink detection may still stand.

Editorial extensions

If this is right

  • C̄ can locate mobility edges without computing eigenstate localization measures like IPR; a single time-averaged quantity suffices.
  • The first Lanczos coefficient b₁ inherits the LDOS broadening, so Lanczos-coefficient data carry the same transition signal as C̄.
  • Lanczos coefficients distinguish models with and without mobility edges: nearly constant in the AAH model, plateau-then-decay in the generalized AAH model, and decaying in the long-range model for localized-to-extended quenches.
  • For quenches from extended to localized phases in the long-range model, the moments coincide with those of the short-range AAH model; the reverse direction does not.
  • The kinks sharpen with system size, consistent with a genuine dynamical phase transition in the thermodynamic limit.

Reading between the lines

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

  • Extension not pursued in the paper: if LDOS broadening is the mechanism, then initial states tuned exactly to the mobility-edge energy should show the sharpest kink in C̄, which could be tested by scanning C̄ continuously over initial-state energy rather than only three energy sectors.
  • The same reasoning suggests C̄ should also detect mobility edges in disordered Anderson models, where no quasiperiodic structure exists, as long as a mobility edge separates localized and delocalized eigenstates.
  • The plateau-then-decay shape of Lanczos coefficients may offer a finite-size-independent fingerprint of mobility edges extractable from short-time survival data, avoiding the need for long-time averaging.
  • Because C̄ depends only on survival-amplitude moments, it could be measurable in cold-atom or photonic waveguide experiments through interference or intensity readouts, although the paper does not address experimental implementation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the long-time averaged spread complexity of quantum quenches in generalized and long-range Aubry-André-Harper models. The central claim is that, in the generalized AAH model with an energy-dependent mobility edge, the long-time averaged spread complexity exhibits nonanalytic behavior when the post-quench quasiperiodic potential crosses the mobility edge associated with the initial eigenstate energy, thereby serving as an accurate dynamical probe of the mobility-edge transition. The authors support this with numerical kink detection in C(λ_f) for three initial states, an LDOS-broadening argument tied to the first Lanczos coefficient, and an analytic derivation of the moments and Lanczos coefficients for quenches between the limits λ_in=0 and λ_f→∞. They also analyze the long-range hopping extension, reporting qualitatively different Lanczos-coefficient behavior for forward versus backward quenches.

Significance. If the central claim is correct, time-averaged spread complexity provides a simple and experimentally relevant dynamical order parameter for mobility-edge transitions in single-particle quasiperiodic systems. The proposed mechanism—LDOS broadening feeding the first Lanczos coefficient—is physically plausible and connects Krylov-space diagnostics to established localization concepts. The analytic moment computation for extreme quenches, once corrected, could be a useful benchmark for the community. However, the numerical evidence for nonanalyticity is currently incomplete, and the analytic formulas contain internal inconsistencies that undermine the claimed analytical verification. The qualitative distinction in Lanczos-coefficient shapes (plateau-then-decay versus constant) is interesting but rests on the same numerical and analytical foundations.

major comments (3)
  1. [Section III.B, Fig. 4] The central claim of 'nonanalytic behavior' and 'accurately identifying the mobility edge' is not established. The observed kinks at λ_f≈3, 2, 1 for β=0.3 deviate from the predictions of Eq. (9), which give λ* = 2.6, 2.0, 1.4 for the ground, middle, and highest excited initial states. The deviations for the ground and highest excited states are about 0.4, i.e., 15–30%, and no finite-size scaling or phase averaging is provided to show these kinks converge to the predicted values in the thermodynamic limit. Footnote 75 asserts that the second derivative exhibits sharp kinks, but no second-derivative data or scaling analysis is shown. A kink in a finite-size system is necessarily smooth; demonstrating a genuine nonanalyticity requires evidence of a developing singularity (e.g., a diverging derivative or a scaling collapse) as N→∞. Without this, the term 'nonanalytic' is an overstatement, an
  2. [Section III.C, Eqs. (15)–(17)] The analytic moment formulas are internally inconsistent and violate basic spectral constraints. From Eq. (15) with n=1, one obtains a0 = (λ_f/β)(√(1−β²)+1), whereas Eq. (16) states a0 = (λ_f/β)(1/√(1−β²)+1). Direct integration of the survival amplitude in Eq. (14) gives a0 ≈ 16.1 for β=0.3, λ_f=100, while Eq. (16) gives 683, which exceeds the operator norm ||H|| ≈ 143 for this parameter set. Additionally, Eq. (17) diverges as β→0, whereas it should reduce to b1=λ_f/√2, as derived later in the same section. These errors mean the 'analytical verification' in Fig. 6 is not reliable; the claimed agreement between numerics and analytics for β≠0 needs to be re-examined with corrected formulas.
  3. [Section IV, Eq. (26) and Fig. 8] For the long-range hopping model, the mobility-edge prediction Eq. (26) is not quantitatively compared with the kink positions in Fig. 8. The text states that C exhibits a kink whenever the mobility edge is crossed, but no numerical values of the kink locations are given, and no finite-size scaling or phase averaging is provided. Given that the IPR in the localized phase does not approach unity for small α, the interpretation of the kinks as marking the mobility edge requires a quantitative check against Eq. (26). The same concerns as in Fig. 4 apply here, and the claim that spread complexity 'accurately identifies' the mobility edge in the long-range model is not substantiated.
minor comments (5)
  1. [Eq. (4)] The moment definition μ_n = ⟨K0|(iH)^n|K0⟩ is inconsistent with the survival amplitude S(t)=⟨e^{−iHt}⟩ in Eq. (14). The derivative in Eq. (4) should yield (−iH)^n, or equivalently μ_n should be defined as ⟨H^n⟩. This sign convention affects Eq. (15); please clarify and make the formulas consistent.
  2. [Fig. 4 caption] The white vertical lines are not explained in the caption. Please state that they mark the predicted mobility-edge crossings from Eq. (9) and list the corresponding λ_f values. Also specify the system size N and the quasiperiodic phase φ used for all figures.
  3. [Footnote 75] The claim that the second derivative of C exhibits sharp kinks is central to the nonanalyticity argument, but no plot or quantitative analysis is provided. Either include the second-derivative data or temper the claim.
  4. [Eq. (25)] The binomial sum expression for the NNN-hopping moments is difficult to parse, and the parity condition 'p even' appears after the sum over p without clear enforcement. Please check the formula and clarify the summation limits.
  5. [Section II] Minor typos include 'the the Sachdev-Ye-Kitaev model' in the introduction. Also, the symbols λ and λ_f are used interchangeably at times (e.g., in the AAH transition description); please standardize the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: spread complexity is benchmarked against independent IPR and external exact mobility-edge formulas, and the analytic-moment section is a limiting calculation rather than a fit to the target claim.

full rationale

The paper's central claim is that the long-time averaged spread complexity C shows a kink when the post-quench potential crosses the mobility edge of the final Hamiltonian. C is computed directly from the Krylov/Lanczos construction (Eqs. 3-6), while the mobility-edge location is taken from IPR of exact eigenstates and from the independent exact formula βE0=2t−λ (Eq. 9, Ref. [49], no author overlap). No parameter of C is fitted to the IPR or to the mobility-edge formula, so the comparison is an external benchmark, not a constructed prediction. The analytical moments in Sec. III.C are derived from a stated λ_f→∞/site-localized-eigenstate assumption plus Weyl equidistribution (Eqs. 13-14), and the Lanczos coefficients obtained from those moments are checked against direct Lanczos recursion (Fig. 6); this is an independent limiting calculation and does not presuppose the kink claim. The apparent internal inconsistency in Eqs. (15)-(17) noted by the reader is a correctness/validation risk, not a circular reduction, because those equations are not used to define the mobility edge or to generate the numerical C data. No load-bearing self-citations or imported uniqueness theorems were found. Finite-size limitations (no system-size scaling or phase averaging, approximate kink locations) weaken the evidence for nonanalyticity but do not amount to circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the exact mobility-edge formulas from Refs [49, 51], the λ→∞ site-localized eigenstate assumption, and Weyl equidistribution. No new degrees of freedom are postulated; the hand-chosen protocol parameters (λ_in=100, β=0.3, α=0.5/1.5) do not enter the mobility-edge predictions themselves. The fragile entry is the moment/Lanczos derivation in Sec. III.C, whose printed formulas are internally inconsistent.

free parameters (4)
  • initial potential strength λ_in = 100
    Hand-chosen as 'deep in the localized phase' for backward quenches (Sec. III.A–B); the kink positions are not strongly sensitive to this value as long as it is deep in the localized regime, but the choice is unquantified.
  • deformation parameter β = 0.3
    Representative value used for all gAAH numerics (Figs. 4–6); no β-scan of the kink positions is shown.
  • LR hopping exponent α = 0.5, 1.5
    Representative values for the long-range model (Fig. 8); no systematic α-dependence of the kink positions is shown.
  • quasiperiodic phase φ = unspecified (implicitly 0)
    The phase φ in Eq. (7) is never stated; for finite N the observables depend on φ, so this is an unstated numerical convention.
assumptions (6)
  • domain assumption Weyl equidistribution: as N→∞, averages over the quasiperiodic phases θ_j = 2πqj + φ equal uniform integrals over θ
    Used in Eqs. (14) and (19) to convert survival-amplitude sums into (1/2π)∫dθ integrals; standard for the irrational-rotation AAH model.
  • domain assumption For λ_f→∞, eigenstates of H(λ_f) are exactly site-localized (|ψ_m⟩ = c†_m|0⟩) with energies λ_f cosθ_m/(1−βcosθ_m)
    Invoked in Sec. III.C to evaluate the forward-quench survival amplitude; holds only asymptotically and ignores the hopping remnant at any finite λ_f.
  • domain assumption The mobility-edge formulas βE₀ = 2t − λ (Eq. 9) and E = λcosh(α ln2) − t (Eq. 26) are exact and valid at the system sizes used
    Taken from Refs [49] and [51]; the kink positions are compared against these formulas as ground truth.
  • standard math Lanczos recursion preserves survival-amplitude moments (μ_n ↔ {a_n, b_n} via the Motzkin/unwrapped Markov-chain representation)
    Used in Sec. III.C to obtain Lanczos coefficients from moments; established in Ref [22].
  • domain assumption The single-particle tight-binding Hilbert space is sufficient; interactions and spin are neglected
    The entire analysis uses the quadratic Hamiltonian (Eq. 7) with fermionic creation/annihilation operators and no interaction term.
  • domain assumption The long-time average in Eq. (6) converges at the integration times used
    No convergence checks on the C(t) integral are reported.

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Pith. "Pith review of Spread complexity as a probe in generalized and long-range Aubry-Andre-Harper models." pith.science (2026). https://pith.science/paper/IDXBWU2W

@misc{pith2026260802451,
  author       = {Pith},
  title        = {Pith review of: Spread complexity as a probe in generalized and long-range Aubry-Andre-Harper models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDXBWU2W}},
  note         = {Machine review of arXiv:2608.02451}
}
read the original abstract

We investigate the spread complexity of quantum quenches in generalized and long-range Aubry-Andre-Harper (AAH) models, encompassing regimes with and without mobility edges. In particular, in the generalized AAH models supporting energy-dependent mobility edges, we demonstrate that the long-time averaged spread complexity exhibits nonanalytic behavior when the post-quench quasiperiodic potential crosses the mobility edge associated with the energy of the initial eigenstate, thereby accurately identifying the mobility-edge transition. Such a behavior is supported by the spreading of local density of states. We further derive analytical expressions for the moments and the corresponding Lanczos coefficients for quenches between the limits of vanishing and strong quasiperiodic potentials. The Lanczos coefficients display qualitatively distinct behavior depending on the presence of mobility edges - they exhibit an initial plateau followed by a decay with the Krylov basis index, in contrast to the nearly constant behavior of the conventional AAH model without mobility edges. For LR hopping, the coefficients decay with the Krylov basis index for quenches from the localized to the extended phase, while they coincide with the AAH results for quenches in the opposite direction.

Figures

Figures reproduced from arXiv: 2608.02451 by the authors.

Figure 1
Figure 1. (a)). Detection of criticality through forward and backward quenches. When λ < 2, IPR vanishes, while a finite IPR is obtained when λ > 2 (see map plot of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Works this paper leans on

76 extracted references · 2 linked inside Pith

  1. [1]

    o .(2) The vectors inKare generally not orthonormal

    Time-averaged spread complexity Consider an initial pure state|ψ(0)⟩evolving under a time-independent Hamiltonian ˆH, leading to the time-evolved state, |ψ(t)⟩=e −i ˆHt |ψ(0)⟩= ∞X n=0 (−it ˆH) n n! |ψ(0)⟩.(1) Since the evolved state is generated through repeated action of ˆHon the initial state, it can be expressed as a linear com- bination of the vectors...

  2. [2]

    Technology Vertical - Quantum Communication

    Long-range and generalized Aubry–Andr ´e–Harper (AAH) Hamiltonian To investigate the behavior of spread complexity across the extended-to-localized quantum phase transition, we consider a generalized Aubry–Andr ´e–Harper model with long-range hopping [47, 50]. The Hamiltonian reads as H=−t NX j<k c† jck |j−k| α + h.c. ! + λ′ cos(2πqj+ϕ)c † jcj 1−βcos(2πqj...

    2000

  3. [3]

    Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Information and Computation6, 213–262 (2006)

    M. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Information and Computation6, 213–262 (2006)

    2006

  4. [4]

    Dowling and M

    M. Dowling and M. Nielsen, The geometry of quantum com- putation, Quantum Information and Computation8, 861–899 (2008)

    2008

  5. [5]

    Kolmogorov, On tables of random numbers, Theoretical Computer Science207, 387–395 (1998)

    A. Kolmogorov, On tables of random numbers, Theoretical Computer Science207, 387–395 (1998)

    1998

  6. [6]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer r ´enyi entropy, Phys. Rev. Lett.128, 050402 (2022)

    2022

  7. [7]

    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, 1–82 (2025)

    2025

  8. [8]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Physical Review X9, 10.1103/physrevx.9.041017 (2019)

    doi:10.1103/physrevx.9.041017 2019

Show all 76 references
  1. [9]

    Avdoshkin and A

    A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Physical Review Research2, 10.1103/phys- revresearch.2.043234 (2020)

    2020 doi

  2. [10]

    Barb ´on, E

    J. Barb ´on, E. Rabinovici, R. Shir, and R. Sinha, On the evolu- tion of operator complexity beyond scrambling, Journal of High Energy Physics2019, 10.1007/jhep10(2019)264 (2019)

    2019 doi

  3. [11]

    Rabinovici, A

    E. Rabinovici, A. S ´anchez-Garrido, R. Shir, and J. Sonner, Op- erator complexity: a journey to the edge of krylov space, Jour- nal of High Energy Physics2021, 10.1007/jhep06(2021)062 (2021)

    2021 doi

  4. [12]

    Rabinovici, A

    E. Rabinovici, A. S ´anchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, Journal of High Energy Physics2022, 10.1007/jhep07(2022)151 (2022)

    2022 doi

  5. [13]

    Rabinovici, A

    E. Rabinovici, A. S ´anchez-Garrido, R. Shir, and J. Sonner, Krylov localization and suppression of complexity, Journal of High Energy Physics2022, 10.1007/jhep03(2022)211 (2022)

    2022 doi

  6. [14]

    S.-K. Jian, B. Swingle, and Z.-Y . Xian, Complexity growth of operators in the syk model and in jt gravity, Journal of High Energy Physics2021, 10.1007/jhep03(2021)014 (2021)

    2021 doi

  7. [16]

    Dymarsky and M

    A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D104, L081702 (2021)

    2021

  8. [17]

    A. A. Nizami and A. W. Shrestha, Spread complexity and quan- tum chaos for periodically driven spin chains, Phys. Rev. E110, 034201 (2024)

    2024

  9. [18]

    Suchsland, R

    P. Suchsland, R. Moessner, and P. W. Claeys, Krylov complex- ity and trotter transitions in unitary circuit dynamics, Phys. Rev. B111, 014309 (2025)

    2025

  10. [19]

    Kolganov and D

    N. Kolganov and D. A. Trunin, Streamlined krylov construction and classification of ergodic floquet systems, Phys. Rev. E111, L052202 (2025)

    2025

  11. [20]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Op- erator growth and krylov construction in dissipative open quantum systems, Journal of High Energy Physics2022, 10.1007/jhep12(2022)081 (2022)

    2022 doi

  12. [21]

    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, 10.1007/jhep12(2023)066 (2023)

    2023 doi

  13. [22]

    H. G. Menzler and R. Jha, Krylov delocalization/localization across ergodicity breaking, Phys. Rev. B110, 125137 (2024)

    2024

  14. [23]

    G. R. Malik, J. Sharma, R. K. Shukla, S. Aravinda, and S. K. Mishra, Krylov complexity in the ergodically constrained nonintegrable transverse-field ising model, Phys. Rev. B113, 184207 (2026)

    2026

  15. [24]

    Balasubramanian, P

    V . Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quan- tum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022)

    2022

  16. [25]

    Seetharaman, C

    S. Seetharaman, C. Singh, and R. Nath, Properties of krylov state complexity in qubit dynamics, Phys. Rev. D111, 076014 (2025)

    2025

  17. [26]

    P. H. S. Bento, A. del Campo, and L. C. C ´eleri, Krylov com- plexity and dynamical phase transition in the quenched lipkin- meshkov-glick model, Phys. Rev. B109, 224304 (2024)

    2024

  18. [27]

    Baggioli, K.-B

    M. Baggioli, K.-B. Huh, H.-S. Jeong, K.-Y . Kim, and J. F. Pedraza, Krylov complexity as an order parameter for quan- tum chaotic-integrable transitions, Phys. Rev. Res.7, 023028 (2025)

    2025

  19. [28]

    Balasubramanian, J

    V . Balasubramanian, J. M. Magan, and Q. Wu, Quantum chaos, integrability, and late times in the krylov basis, Phys. Rev. E 111, 014218 (2025)

    2025

  20. [29]

    Bhattacharjee and P

    B. Bhattacharjee and P. Nandy, Krylov fractality and complex- ity in generic random matrix ensembles, Phys. Rev. B111, L060202 (2025)

    2025

  21. [31]

    Caputa and S

    P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B106, 195125 (2022)

    2022

  22. [32]

    Bhattacharjee, S

    B. Bhattacharjee, S. Sur, and P. Nandy, Probing quantum scars and weak ergodicity breaking through quantum complexity, Phys. Rev. B106, 205150 (2022)

    2022

  23. [33]

    Takahashi, Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models, Phys

    K. Takahashi, Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models, Phys. Rev. B112, 054312 (2025)

    2025

  24. [34]

    Caputa, N

    P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan, and H. J. R. Van Zyl, Spread complexity and topological transi- tions in the kitaev chain, Journal of High Energy Physics2023, 10.1007/jhep01(2023)120 (2023)

    2023 doi

  25. [35]

    Pirmoradian, E

    R. Pirmoradian, E. Sadoogh, M. Teymouri, N. Abolqasemi- Azad, M. R. Lahooti, and Z. Mohammad-Ali, Investigation 10 of quantum chaos in local and non-local ising models (2025), arXiv:2512.21713 [quant-ph]

    2025

  26. [36]

    Balasubramanian, J

    V . Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D107, 126001 (2023)

    2023

  27. [37]

    Gautam, K

    M. Gautam, K. Pal, K. Pal, A. Gill, N. Jaiswal, and T. Sarkar, Spread complexity evolution in quenched interacting quantum systems, Phys. Rev. B109, 014312 (2024)

    2024

  28. [38]

    Y . Zhou, W. Xia, L. Li, and W. Li, Diagnosing quantum many- body chaos in non-hermitian quantum spin chain via krylov complexity, Phys. Rev. Res.7, 033281 (2025)

    2025

  29. [39]

    Bhattacharya, R

    A. Bhattacharya, R. N. Das, B. Dey, and J. Erdmenger, Spread complexity and localization inPT-symmetric systems, Phys. Rev. B110, 064320 (2024)

    2024

  30. [40]

    Medina-Guerra, I

    E. Medina-Guerra, I. V . Gornyi, and Y . Gefen, Correlations and krylov spread for a non-hermitian hamiltonian: Ising chain with a complex-valued transverse magnetic field, Phys. Rev. B111, 174207 (2025)

    2025

  31. [41]

    Nandy, T

    P. Nandy, T. Pathak, Z.-Y . Xian, and J. Erdmenger, Krylov space approach to singular value decomposition in non-hermitian sys- tems, Phys. Rev. B111, 064203 (2025)

    2025

  32. [42]

    H. Sahu, A. Bhattacharya, and P. P. Nath, Quantum complexity and localization in random and time-periodic unitary circuits, Phys. Rev. B113, 214312 (2026)

    2026

  33. [43]

    Chaki, A

    P. Chaki, A. Sil, P. Ghosh, U. Sen, and S. S. Roy, Resource gen- eration and dynamical complexities in open random quantum circuits (2026), arXiv:2605.20350 [quant-ph]

    2026 arXiv

  34. [44]

    S. PG, J. B. Kannan, R. Modak, and S. Aravinda, Dependence of krylov complexity saturation on the initial operator and state, Phys. Rev. E112, L032203 (2025)

    2025

  35. [45]

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

    2026

  36. [46]

    Yeh and A

    H.-C. Yeh and A. Mitra, Anderson localization: A floquet oper- ator krylov space perspective (2026), arXiv:2605.24115 [cond- mat.dis-nn]

    2026 arXiv

  37. [47]

    Aubry and G

    S. Aubry and G. Andre, Analyticity breaking and anderson lo- calization, Ann. Israel Phys. Soc. 3, 133 (1980)

    1980

  38. [48]

    P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proceedings of the Physical Society. Section A68, 874–878 (1955)

    1955

  39. [49]

    X. Deng, S. Ray, S. Sinha, G. V . Shlyapnikov, and L. Santos, One-dimensional quasicrystals with power-law hopping, Phys. Rev. Lett.123, 025301 (2019)

    2019

  40. [50]

    Hiramoto and M

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

    1989

  41. [51]

    Ganeshan, J

    S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimen- sion, Phys. Rev. Lett.114, 146601 (2015)

    2015

  42. [52]

    Roy and A

    N. Roy and A. Sharma, Fraction of delocalized eigenstates in the long-range aubry-andr ´e-harper model, Phys. Rev. B103, 075124 (2021)

    2021

  43. [53]

    Biddle, D

    J. Biddle, D. J. Priour, B. Wang, and S. Das Sarma, Localization in one-dimensional lattices with non-nearest-neighbor hopping: Generalized anderson and aubry-andr ´e models, Phys. Rev. B 83, 075105 (2011)

    2011

  44. [54]

    Biddle, B

    J. Biddle, B. Wang, D. J. Priour, and S. Das Sarma, Localization in one-dimensional incommensurate lattices beyond the aubry- andr´e model, Phys. Rev. A80, 021603(R) (2009)

    2009

  45. [55]

    Biddle and S

    J. Biddle and S. Das Sarma, Predicted mobility edges in one- dimensional incommensurate optical lattices: An exactly solv- able model of anderson localization, Phys. Rev. Lett.104, 070601 (2010)

    2010

  46. [56]

    C. Yang, Y . Wang, P. Wang, X. Gao, and S. Chen, Dynami- cal signature of localization-delocalization transition in a one- dimensional incommensurate lattice, Phys. Rev. B95, 184201 (2017)

    2017

  47. [57]

    Qing, Y .-Q

    Y . Qing, Y .-Q. Chen, and S.-X. Zhang, Entanglement growth and information capacity in a quasiperiodic system with a single-particle mobility edge, Phys. Rev. B113, 064308 (2026)

    2026

  48. [58]

    S. Ye, Z. Zhou, N. A. Khan, and G. Xianlong, Energy- dependent dynamical quantum phase transitions in quasicrys- tals, Phys. Rev. A109, 043319 (2024)

    2024

  49. [59]

    Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    S. Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    2011

  50. [60]

    J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, and M. Inguscio, Bose-einstein condensate in a random potential, Phys. Rev. Lett.95, 070401 (2005)

    2005

  51. [61]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zac- canti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting bose–einstein condensate, Na- ture453, 895–898 (2008)

    2008

  52. [62]

    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

  53. [63]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L ¨uschen, M. H. Fischer, R. V osk, E. Altman, U. Schneider, and I. Bloch, Ob- servation of many-body localization of interacting fermions in a quasirandom optical lattice, Science349, 842–845 (2015)

    2015

  54. [64]

    Kim, M.-S

    K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Quan- tum simulation of frustrated ising spins with trapped ions, Na- ture465, 590–593 (2010)

    2010

  55. [65]

    Richerme, Z.-X

    P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss- Feig, S. Michalakis, A. V . Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions, Nature511, 198–201 (2014)

    2014

  56. [66]

    Labuhn, D

    H. Labuhn, D. Barredo, S. Ravets, S. de L ´es´eleuc, T. Macr `ı, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single rydberg atoms for realizing quantum ising models, Na- ture534, 667–670 (2016)

    2016

  57. [67]

    Sahoo, U

    A. Sahoo, U. Mishra, and D. Rakshit, Localization-driven quan- tum sensing, Phys. Rev. A109, L030601 (2024)

    2024

  58. [68]

    Sahoo, A

    A. Sahoo, A. Saha, and D. Rakshit, Stark localization near aubry-andr´e criticality, Phys. Rev. B111, 024205 (2025)

    2025

  59. [69]

    Ganguli and A

    M. Ganguli and A. Jana, State dependent spread complexity dy- namics in many-body localization transition (2024)

    2024

  60. [70]

    G. F. Scialchi, A. J. Roncaglia, and D. A. Wisniacki, Integrability-to-chaos transition through the krylov approach for state evolution, Phys. Rev. E109, 054209 (2024)

    2024

  61. [71]

    H. A. Camargo, K.-B. Huh, V . Jahnke, H.-S. Jeong, K.-Y . Kim, and M. Nishida, Spread and spectral complexity in quantum spin chains: from integrability to chaos, Journal of High Energy Physics2024, 10.1007/jhep08(2024)241 (2024)

    2024 doi

  62. [72]

    Grabarits and A

    A. Grabarits and A. del Campo, Universal growth of krylov complexity across a quantum phase transition (2025)

    2025

  63. [73]

    Teh and T

    H.-H. Teh and T. Orito, Krylov complexity and mixed-state phase transition (2025)

    2025

  64. [74]

    Heyl, Dynamical quantum phase transitions: a review, Re- ports on Progress in Physics81, 054001 (2018)

    M. Heyl, Dynamical quantum phase transitions: a review, Re- ports on Progress in Physics81, 054001 (2018)

    2018

  65. [75]

    Shimasaki, H

    T. Shimasaki, H. E. Kondakci, M. Prichard, J. Pagett, Y . Bai, P. Dotti, A. Cao, T.-C. Lu, T. Grover, and D. M. Weld, Exper- imental realization of the kicked aubry-andre-harper hamilto- nian, inConference on Lasers and Electro-Optics, CLEO QELS (Optica Publishing Group, 2022)...

    2022

  66. [76]

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

    2019

  67. [77]

    We also notice that the double derivative ofCwith respect to λf exhibits the sharp kinks, thereby capable of capturing these mobility edges in the gAAH model

  68. [78]

    Bhattacharya, P

    A. Bhattacharya, P. P. Nath, and H. Sahu, Krylov complexity for nonlocal spin chains, Phys. Rev. D109, 066010 (2024)

    2024

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.