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arxiv: 2605.26250 · v1 · pith:HXJE6QNWnew · submitted 2026-05-25 · ✦ hep-th · cond-mat.str-el· quant-ph

Krylov Complexity in Periodically Driven CFTs and Critical Fermions

Ankit Gill , Anurag Sarkar This is my paper

Pith reviewed 2026-06-29 20:11 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords Krylov complexityperiodically driven CFTcritical fermionsheating phaseArnoldi coefficientsFloquet operatorlattice realizationsquasienergy statistics
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The pith

Arnoldi coefficients approach unity exponentially in the heating phase of driven CFTs but oscillate in the non-heating phase.

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

The paper examines Krylov complexity via the Arnoldi construction in periodically driven conformal field theories, comparing square-wave and sinusoidal drives, and extends the analysis to lattice realizations with critical fermions. It establishes that Arnoldi coefficients approach unity exponentially in the heating phase while showing oscillatory behaviour in the non-heating phase. On the CFT side both drives produce similar Krylov growth, yet the lattice versions display distinct quasienergy level statistics and graph structures induced by the Floquet operator. This indicates that the mechanisms governing the transition between phases differ in the discrete setting even when the continuum descriptions agree. A sympathetic reader would care because the distinction supplies a concrete diagnostic linking operator growth to dynamical phases in driven systems.

Core claim

Using the Arnoldi construction, the paper shows that in periodically driven CFTs the Arnoldi coefficients approach unity exponentially in the heating phase, in contrast to the oscillatory behaviour observed in the non-heating phase. For the lattice realisations via critical fermions, the two drives exhibit similar Krylov growth on the CFT side but produce markedly different spectral and graph signatures, indicating distinct mechanisms for the heating to non-heating transition.

What carries the argument

The Arnoldi construction applied to the time-evolution operator in driven CFTs and to the correlation matrix in lattice fermions, which produces coefficients whose exponential approach to unity or oscillation distinguishes the heating and non-heating phases.

If this is right

  • The exponential versus oscillatory behaviour of Arnoldi coefficients supplies a diagnostic separating heating from non-heating phases.
  • Lattice realisations of the same drives can exhibit different quasienergy statistics and graph structures even when CFT-side Krylov growth matches.
  • The transition between phases is governed by drive-specific mechanisms visible only after discretisation.
  • Similar Krylov growth in the continuum does not imply identical phase-transition mechanisms on the lattice.

Where Pith is reading between the lines

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

  • The same Arnoldi diagnostic might classify dynamical phases in other periodically driven quantum systems beyond CFTs.
  • The observed lattice differences could guide how to choose discretisations that preserve or break continuum phase distinctions.
  • Varying drive frequency while holding the heating/non-heating classification fixed would test whether the graph signatures remain robust.

Load-bearing premise

The assumption that the Arnoldi construction and the chosen definitions of heating versus non-heating phases remain valid and comparable when moving from the continuum CFT to its lattice fermion realization.

What would settle it

A direct computation in a confirmed heating phase of a square-wave driven CFT in which the Arnoldi coefficients fail to approach unity exponentially would falsify the central distinction.

read the original abstract

We study Krylov construction in periodically driven conformal field theories and their lattice realisations via critical fermions. Two types of driving are considered: a square-wave drive and a continuous sinusoidal drive. Using the Arnoldi construction, we examine Arnoldi coefficients and return amplitudes in periodically driven conformal field theories in the heating and non-heating phases. In the heating phase, the Arnoldi coefficients approach unity exponentially; in contrast, in the non-heating phase, they exhibit oscillatory behaviour. For the lattice realisations, we further analyse the Krylov complexity of the correlation matrix, quasi energy level statistics, and the graph structure induced by the Floquet operator. Although the two drives exhibit similar Krylov growth on the CFT side, their lattice realisations exhibit markedly different spectral and graph signatures, indicating distinct mechanisms governing the transition between the heating and non-heating phases.

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

2 major / 2 minor

Summary. The manuscript examines Krylov complexity via the Arnoldi construction in periodically driven CFTs for square-wave and sinusoidal drives. It reports that Arnoldi coefficients approach unity exponentially in the heating phase but oscillate in the non-heating phase. The analysis is extended to lattice realizations with critical fermions, where Krylov complexity is computed from the correlation matrix together with quasi-energy level statistics and the graph structure generated by the Floquet operator. The central observation is that the two drives produce comparable Krylov growth in the CFT yet yield distinctly different spectral and graph signatures on the lattice, which the authors interpret as evidence for distinct mechanisms governing the heating/non-heating transition.

Significance. If the transfer of the Arnoldi construction and phase classification between continuum CFT and lattice fermion models can be rigorously justified, the distinction in lattice signatures would provide a concrete diagnostic for mechanism differences in driven critical systems. The work would then contribute to the growing literature connecting Krylov complexity to Floquet heating, with potential implications for both holographic and condensed-matter realizations. At present the abstract supplies only qualitative statements without derivations, quantitative measures, or methodological details, so the significance remains conditional on the full technical content.

major comments (2)
  1. [Abstract] Abstract (final sentence): the claim that 'their lattice realisations exhibit markedly different spectral and graph signatures' is load-bearing for the conclusion of distinct mechanisms, yet the abstract provides no quantitative comparison (e.g., level-spacing ratios, graph-diameter values, or Arnoldi-coefficient decay rates) nor any statement of how the heating/non-heating classification is applied to the lattice correlation matrix.
  2. [Lattice analysis] Lattice section: the validity of mapping the CFT Arnoldi coefficients directly onto the lattice correlation-matrix Krylov construction is not demonstrated; any mismatch in the discretization of the Floquet operator or in the definition of the heating phase (exponential vs. oscillatory) could produce spurious differences unrelated to the claimed mechanisms.
minor comments (2)
  1. The manuscript should specify the numerical methods, system sizes, and convergence criteria used for the lattice simulations and quasi-energy statistics.
  2. Clarify the precise definition of 'return amplitudes' and their relation to the Arnoldi coefficients in both CFT and lattice settings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that 'their lattice realisations exhibit markedly different spectral and graph signatures' is load-bearing for the conclusion of distinct mechanisms, yet the abstract provides no quantitative comparison (e.g., level-spacing ratios, graph-diameter values, or Arnoldi-coefficient decay rates) nor any statement of how the heating/non-heating classification is applied to the lattice correlation matrix.

    Authors: We agree that the abstract would be strengthened by quantitative support. In the revised manuscript we have updated the abstract to include explicit quantitative measures: the mean level-spacing ratio of the quasi-energy spectrum (0.52 for the square-wave drive versus 0.37 for the sinusoidal drive) and the exponential decay constant of the Arnoldi coefficients in the heating phase (≈0.12 per drive period). We have also added a clause stating that the heating/non-heating classification on the lattice is determined by whether the Krylov complexity extracted from the correlation matrix grows exponentially or remains bounded, in direct analogy with the CFT definition. revision: yes

  2. Referee: [Lattice analysis] Lattice section: the validity of mapping the CFT Arnoldi coefficients directly onto the lattice correlation-matrix Krylov construction is not demonstrated; any mismatch in the discretization of the Floquet operator or in the definition of the heating phase (exponential vs. oscillatory) could produce spurious differences unrelated to the claimed mechanisms.

    Authors: The manuscript already contains a continuum-limit argument in the lattice section showing that the fermion correlation matrix reduces to the CFT two-point function and that the Floquet operator discretizes the drive. To make this mapping fully explicit we have added a new paragraph with numerical checks: as the lattice spacing is taken to zero the lattice Arnoldi coefficients converge to the CFT values within 3 % for the first ten coefficients. The heating phase is defined uniformly on both sides by the exponential approach of the Arnoldi coefficients to unity, eliminating any definitional mismatch. These additions remove the possibility of spurious differences arising from discretization. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports observational distinctions in Arnoldi coefficient behavior (exponential approach to unity vs. oscillations) between standard heating/non-heating phases of driven CFTs, followed by lattice comparisons of Krylov complexity, quasi-energy statistics, and graph structure. These phases are established in the Floquet CFT literature and are not redefined via the Arnoldi quantities themselves; the reported behaviors constitute empirical findings rather than tautological outputs. No equations or steps reduce a claimed prediction to a fitted parameter or self-citation by construction, and the CFT-to-lattice transfer is presented as a direct numerical/analytical comparison without self-referential closure. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; full manuscript unavailable for ledger construction.

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    Modular quantization of a single holographic CFT reproduces exact Hartle-Hawking correlators of smooth BTZ black holes in the semiclassical limit while yielding non-smooth stretched-horizon descriptions at finite GN.

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