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arxiv: 2606.05303 · v1 · pith:JTBH3632new · submitted 2026-06-03 · ✦ hep-th · cond-mat.str-el· quant-ph

Krylov Complexity: Flat bands and Carroll breaking deformations

Aritra Banerjee , Arpan Bhattacharyya , Rudranil Basu , Sayan Das This is my paper

Pith reviewed 2026-06-28 04:56 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords Krylov complexityflat bandsCarrollian symmetriessupertranslationsfermionic ladderscompact localised statesquenchesUV/IR mixing
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The pith

Krylov complexity growth distinguishes phase-dependent resilience of Carrollian sectors against delocalising perturbations in flat-band systems.

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

Systems with flat band structures, expressed via Compact Localised States, are invariant under supertranslation symmetries and thus carry Carrollian symmetries. The paper examines state dynamics under quenches induced by Carroll-breaking perturbations, using Krylov complexity as the probe in all-bands-flat fermionic ladder Hamiltonians that include a supertranslation-preserving interaction. Complexity growth across critical lines is shown to resolve how different phases resist or yield to delocalisation. A parallel continuum Carroll scalar field with gradient deformation supplies a complementary case exhibiting strong ultraviolet sensitivity through UV/IR mixing.

Core claim

In fermionic ladder Hamiltonians with all bands flat, augmented by supertranslation-preserving interactions, the growth of Krylov state complexity for quenches across critical lines sharply resolves the phase-dependent resilience of Carrollian sectors against delocalising perturbations. This resolution is augmented by analysis of a continuum Carroll scalar field theory with a gradient deformation, which exhibits strong ultraviolet sensitivity arising from UV/IR mixing.

What carries the argument

Krylov (spread) complexity as a probe of state dynamics under quenches induced by Carroll-breaking perturbations in all-bands-flat models.

If this is right

  • Quenches across critical lines produce distinct Krylov complexity growth signatures tied to Carrollian phase resilience.
  • Carrollian sectors exhibit phase-dependent protection against delocalisation that complexity growth detects.
  • Gradient deformations in the continuum Carroll scalar theory produce UV-sensitive complexity growth with UV/IR mixing.

Where Pith is reading between the lines

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

  • Krylov complexity may function as a general diagnostic for symmetry-protected flat-band dynamics beyond the ladder models studied.
  • The UV/IR mixing observed in the continuum case suggests similar sensitivity could appear in discrete models at strong deformations.
  • Quench protocols that preserve or break supertranslations selectively could isolate Carrollian contributions in other many-body systems.

Load-bearing premise

Flat band structures written in the language of Compact Localised States are explicitly invariant under supertranslation symmetries.

What would settle it

Observation that Krylov complexity growth rates remain identical across phases in the all-bands-flat fermionic ladder models under Carroll-breaking quenches would falsify the claim that growth resolves phase-dependent resilience.

Figures

Figures reproduced from arXiv: 2606.05303 by Aritra Banerjee, Arpan Bhattacharyya, Rudranil Basu, Sayan Das.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
read the original abstract

Systems with flat band structures, when written in the language of Compact Localised States (CLS), have been shown to be explicitly invariant under supertranslation symmetries, making Carrollian symmetries inherently important for such systems. In this work, we explore the state dynamics of these systems, focusing on quenches induced by Carroll breaking perturbations, through the probe of Krylov (spread) Complexity. We specialise to Fermionic ladder Hamiltonians with all bands flat (ABF) scenario, augmented by a supertranslation preserving interaction, and discuss Krylov state complexity for quenches across critical lines. We further discuss how the growth of Krylov complexity sharply resolves the phase-dependent resilience of Carrollian sectors against delocalising perturbations. This is augmented by a complementary mechanism for Krylov growth in a continuum Carroll scalar field theory with a gradient deformation, which exhibits strong ultraviolet sensitivity (UV/IR mixing).

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

3 major / 2 minor

Summary. The manuscript claims that flat-band Hamiltonians expressed in the Compact Localised States (CLS) basis are invariant under supertranslation symmetries, rendering Carrollian symmetries relevant; it then uses Krylov (spread) complexity to diagnose the dynamics of quenches induced by Carroll-breaking perturbations in an all-bands-flat (ABF) fermionic ladder augmented by a supertranslation-preserving interaction, asserting that the growth rate sharply distinguishes phase-dependent resilience of the Carrollian sector, and supplements this with a continuum Carroll scalar field theory exhibiting UV/IR mixing under gradient deformations.

Significance. If the central claims are substantiated, the work would supply a concrete dynamical probe (Krylov complexity) that distinguishes Carrollian-protected phases in flat-band models, thereby linking Krylov-space techniques from quantum information with Carrollian symmetry structures; the continuum example further suggests a mechanism for UV sensitivity that could be testable in lattice regularizations.

major comments (3)
  1. [Model construction / ABF fermionic ladder] The load-bearing premise that the CLS basis remains invariant under supertranslation generators once the supertranslation-preserving interaction is added is stated as having been shown in prior work, yet the model-construction section provides no explicit commutation check or re-derivation for the interacting ABF ladder Hamiltonian; without this verification the subsequent interpretation of complexity growth as resolving Carrollian resilience lacks an anchor in the specific Hamiltonian.
  2. [Results on Krylov complexity growth] The abstract and results sections assert that Krylov-complexity growth 'sharply resolves' phase-dependent resilience, but no explicit Hamiltonians, quench protocols, numerical data, or error analysis are supplied in the provided text; the central claim therefore cannot be verified from the manuscript alone.
  3. [Continuum Carroll scalar field theory] In the continuum Carroll scalar field theory section, the claimed UV/IR mixing under the gradient deformation is presented qualitatively; the manuscript does not derive the precise form of the Krylov operator or the resulting complexity growth rate, leaving the UV sensitivity statement without quantitative support.
minor comments (2)
  1. [Introduction / Preliminaries] Notation for the supertranslation generators and their action on the CLS basis should be introduced explicitly before the quench analysis.
  2. [Numerical results] Figure captions for any complexity-vs-time plots should state the precise quench parameters, system size, and averaging procedure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the manuscript can be strengthened. We address each major comment below and have revised the text accordingly to include explicit verifications, numerical details, and quantitative derivations.

read point-by-point responses

  1. Referee: [Model construction / ABF fermionic ladder] The load-bearing premise that the CLS basis remains invariant under supertranslation generators once the supertranslation-preserving interaction is added is stated as having been shown in prior work, yet the model-construction section provides no explicit commutation check or re-derivation for the interacting ABF ladder Hamiltonian; without this verification the subsequent interpretation of complexity growth as resolving Carrollian resilience lacks an anchor in the specific Hamiltonian.

    Authors: We agree that an explicit commutation check strengthens the presentation. Although the invariance follows from the construction in our prior work on the non-interacting ABF ladder, the revised manuscript now includes a direct verification that the supertranslation generators still commute with the full interacting Hamiltonian (including the added supertranslation-preserving term). This anchors the subsequent dynamical analysis. revision: yes

  2. Referee: [Results on Krylov complexity growth] The abstract and results sections assert that Krylov-complexity growth 'sharply resolves' phase-dependent resilience, but no explicit Hamiltonians, quench protocols, numerical data, or error analysis are supplied in the provided text; the central claim therefore cannot be verified from the manuscript alone.

    Authors: The Hamiltonians and quench protocols are defined in Section 3, but we acknowledge that the numerical implementation details, raw data tables, and error analysis were insufficiently explicit. The revised version adds the full quench protocols, the precise Krylov basis truncation used, representative complexity curves with statistical error bars, and a brief discussion of numerical convergence. revision: yes

  3. Referee: [Continuum Carroll scalar field theory] In the continuum Carroll scalar field theory section, the claimed UV/IR mixing under the gradient deformation is presented qualitatively; the manuscript does not derive the precise form of the Krylov operator or the resulting complexity growth rate, leaving the UV sensitivity statement without quantitative support.

    Authors: We agree the continuum discussion was qualitative. The revised manuscript now derives the explicit form of the Krylov operator for the deformed Carroll scalar, computes the leading growth rate of the complexity, and shows how the UV cutoff enters the late-time slope, thereby quantifying the UV/IR mixing. revision: yes

Circularity Check

1 steps flagged

CLS supertranslation invariance imported via citation as load-bearing premise for Carrollian interpretation of Krylov growth.

specific steps
  1. self citation load bearing [Abstract]
    "Systems with flat band structures, when written in the language of Compact Localised States (CLS), have been shown to be explicitly invariant under supertranslation symmetries, making Carrollian symmetries inherently important for such systems."

    The claim that Krylov complexity growth resolves phase-dependent resilience of Carrollian sectors requires the existence of those sectors, which rests solely on the cited invariance. No equations or verification are supplied for the specific model with the added supertranslation-preserving interaction, so the interpretive framework depends on the prior result rather than being re-established here.

full rationale

The paper's setup states that flat-band CLS structures are explicitly invariant under supertranslation symmetries (making Carrollian symmetries important), then uses this to frame the Krylov complexity analysis of Carroll-breaking quenches and phase-dependent resilience. This premise is justified only by the phrase 'have been shown' without re-derivation or verification inside the present Hamiltonian (ABF fermionic ladder plus interaction). The complexity computations themselves appear independent of any fitted parameters or self-defined quantities, so the central claim retains independent content beyond the citation, but the Carrollian anchor reduces to the imported result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the central narrative rests on the stated invariance of CLS flat bands under supertranslations and on the utility of Krylov complexity as a probe.

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