arxiv: 2601.09801 · v2 · submitted 2026-01-14 · ✦ hep-th · cond-mat.str-el· nlin.CD· quant-ph

Recognition: 3 theorem links

· Lean Theorem

Probing the Chaos to Integrability Transition in Double-Scaled SYK

Sergio E. Aguilar-Gutierrez , Rathindra Nath Das , Johanna Erdmenger , Zhuo-Yu Xian

Authors on Pith no claims yet

Pith reviewed 2026-05-16 14:11 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elnlin.CDquant-ph

keywords double-scaled SYKchaos-integrability transitionchord numberKrylov complexityfirst-order phase transitionoperator growthsubdominant saddles

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The pith

A first-order phase transition in the double-scaled SYK model causes the chord number to switch from linear to quadratic growth and Krylov complexity to switch from exponential to quadratic growth.

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

The paper studies a model that smoothly interpolates between the chaotic double-scaled SYK Hamiltonian and an integrable chord Hamiltonian. This interpolation produces a first-order thermodynamic transition marked by a kink in the free energy, with subdominant saddles connecting the two regimes. At the transition point the chord number, which counts interaction chords and equals Krylov state complexity in the classical limit, changes its growth law discontinuously. The same point produces a discontinuous change in Krylov operator complexity and in operator size, both shifting from exponential to quadratic growth. The authors also outline a possible holographic dual that could realize the same transition.

Core claim

The Berkooz-Brukner-Jia-Mamroud interpolating model exhibits a first-order transition between chaotic and quasi-integrable phases. The chord number exhibits a discontinuous transition from linear to quadratic growth at the transition point. Similarly, the Krylov operator complexity and the operator size exhibit discontinuous transitions from exponential to quadratic growth.

What carries the argument

The chord number, which counts the number of interaction chords and is proportional to Krylov state complexity in the classical limit, serving as the order parameter that jumps at the first-order transition.

Load-bearing premise

That the subdominant saddles correctly interpolate between the chaotic and quasi-integrable regimes and that the chord-number and Krylov diagnostics remain valid order parameters across the first-order jump without additional corrections from finite-N effects.

What would settle it

Numerical evaluation of the chord number or Krylov complexity on finite-N realizations of the model that shows continuous rather than discontinuous growth exactly at the location of the free-energy kink.

read the original abstract

We investigate how a thermodynamical first-order phase transition affects the dynamical chaotic behaviour of a given model. To this effect, we analyze the model of Berkooz, Brukner, Jia and Mamroud that interpolates between the double-scaled SYK model and an integrable chord Hamiltonian. This model exhibits a first-order transition, characterized by a kink in the free energy, between the chaotic and quasi-integrable phases, with the branch of subdominant saddles interpolating between them. We characterize the dynamical behavior across the phase diagram using the chord number, Krylov complexity, and operator size. The chord number, which is proportional to the Krylov state complexity in the classical limit, exhibits a discontinuous transition from linear to quadratic growth at the transition point. Similarly, the Krylov operator complexity and the operator size, as scrambling diagnostics, exhibit discontinuous transitions from exponential to quadratic growth. We also discuss a possible holographic interpretation of the model.

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.

Desk Editor's Note private letter to a colleague

The paper maps a known first-order thermodynamic transition in an SYK interpolator onto discontinuous jumps in chord number, Krylov complexity, and operator size growth.

read the letter

This paper takes the Berkooz-Brukner-Jia-Mamroud interpolating model between double-scaled SYK and an integrable chord Hamiltonian and shows that its first-order free-energy kink produces sharp changes in three dynamical diagnostics. Chord number growth switches from linear to quadratic, while Krylov complexity and operator size switch from exponential to quadratic, all at the same interpolation value where the thermodynamic transition occurs. The subdominant saddle branch is used to track the quasi-integrable side. That explicit mapping of the thermodynamic point onto the dynamical signatures is the concrete new piece. It gives a solvable large-N example where chaos diagnostics turn off discontinuously, which is useful for people thinking about scrambling in holographic models. The holographic interpretation section is a short but reasonable pointer. The main soft spot is the reliance on the subdominant saddle for the time-dependent quantities. At the transition the two saddles are degenerate, but slightly off it the subdominant one is suppressed by exp(-N Delta F). It is not obvious from the abstract whether this branch can set the leading growth rates without non-perturbative corrections or finite-N rounding that would replace the claimed discontinuity with a fast crossover. No error estimates or finite-N checks are mentioned, so that part of the argument needs more detail. This is for readers working on SYK models, quantum chaos, and holography who want a concrete link between thermodynamics and scrambling. It is solid enough to go to peer review; the central idea holds and the technical questions around the saddles are addressable in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes an interpolating model between the double-scaled SYK model and an integrable chord Hamiltonian that exhibits a first-order phase transition between chaotic and quasi-integrable regimes, marked by a kink in the free energy. Using subdominant saddles that interpolate between the phases, it characterizes the dynamics via the chord number (claiming a discontinuous jump from linear to quadratic growth), Krylov operator complexity, and operator size (claiming discontinuous jumps from exponential to quadratic growth), and discusses a possible holographic interpretation.

Significance. If the central claims on the discontinuous transitions hold after addressing finite-N and suppression issues, the work would provide a useful bridge between thermodynamic first-order transitions and changes in quantum chaos diagnostics in SYK-like models, with the multiple order parameters and holographic remarks offering concrete handles for further study.

major comments (2)

[Saddle-point analysis and dynamical diagnostics] § on saddle-point analysis and dynamical diagnostics: the claim that subdominant saddles can be promoted to order parameters for time-dependent quantities (chord number, Krylov complexity) at the first-order transition is load-bearing, yet these saddles are exponentially suppressed by e^{-N ΔF} when free energies are equal; the manuscript must show explicitly why this does not replace the reported discontinuity by a rapid but continuous crossover, including any finite-N rounding or non-perturbative corrections.
[Chord number and Krylov complexity definitions] § on chord number and Krylov complexity definitions: the abstract states that the chord number is proportional to Krylov state complexity in the classical limit and exhibits a discontinuous transition, but without explicit derivation details, error estimates, or checks against finite-N corrections it is impossible to verify whether the reported growth-rate jumps are robust or sensitive to post-hoc choices in the saddle interpolation.

minor comments (2)

[Model definition] The interpolation parameter between the SYK and chord Hamiltonians is mentioned but its explicit functional form and range should be stated clearly in the model definition section for reproducibility.
[Figures] Figure captions for the growth-rate plots should include the precise values of the interpolation parameter at which the transitions are observed and any fitting windows used.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below and plan to revise the paper to incorporate clarifications and additional details.

read point-by-point responses

Referee: [Saddle-point analysis and dynamical diagnostics] § on saddle-point analysis and dynamical diagnostics: the claim that subdominant saddles can be promoted to order parameters for time-dependent quantities (chord number, Krylov complexity) at the first-order transition is load-bearing, yet these saddles are exponentially suppressed by e^{-N ΔF} when free energies are equal; the manuscript must show explicitly why this does not replace the reported discontinuity by a rapid but continuous crossover, including any finite-N rounding or non-perturbative corrections.

Authors: At the first-order phase transition, the free energies of the chaotic and quasi-integrable saddles are equal by definition, so ΔF = 0 and there is no exponential suppression; both saddles are equally dominant. The interpolating subdominant saddle (in the sense of the branch connecting the two phases) allows us to continuously vary the parameters across the transition. However, the physical observables jump discontinuously because the system selects the saddle with the lowest free energy on each side. In the strict large-N limit, this results in a sharp discontinuity. We will add a detailed explanation of this point in the revised manuscript, including a discussion of how at finite N the transition rounds into a crossover over a scale exponentially small in N. We note that explicit non-perturbative calculations of the rounding are technically challenging and may require numerical methods beyond the current analytic approach. revision: partial
Referee: [Chord number and Krylov complexity definitions] § on chord number and Krylov complexity definitions: the abstract states that the chord number is proportional to Krylov state complexity in the classical limit and exhibits a discontinuous transition, but without explicit derivation details, error estimates, or checks against finite-N corrections it is impossible to verify whether the reported growth-rate jumps are robust or sensitive to post-hoc choices in the saddle interpolation.

Authors: We will expand the manuscript to include a more detailed derivation of the relation between the chord number and the Krylov state complexity in the classical limit, starting from the definition of the Krylov basis and showing the proportionality explicitly. We will also provide error estimates for the saddle-point approximation and include a short discussion of finite-N effects based on the analytic continuation of the saddle equations. This should clarify that the growth-rate jumps are robust features of the large-N limit. revision: yes

standing simulated objections not resolved

Explicit non-perturbative calculations demonstrating that the discontinuity does not become a crossover at finite N

Circularity Check

0 steps flagged

No circularity: derivation uses independent saddle-point free energy and standard literature diagnostics

full rationale

The paper's central results follow from the model's first-order transition (kink in free energy) and the application of chord-number, Krylov-complexity, and operator-size diagnostics, which are drawn from prior SYK literature without the growth laws being fitted to the target observables or redefined in terms of themselves. No load-bearing step reduces by construction to a self-citation, ansatz smuggled via citation, or renaming of a known result; the subdominant-saddle interpolation is a direct consequence of the thermodynamic analysis rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a first-order transition in the free energy of the interpolating model and on the validity of chord/Krylov diagnostics as order parameters. No new entities are postulated; the model is taken from prior work.

free parameters (1)

interpolation parameter
The continuous parameter that tunes between double-scaled SYK and the chord Hamiltonian; its critical value is located by the free-energy kink.

axioms (1)

domain assumption Large-N saddle-point dominance controls the thermodynamics and the late-time dynamics.
Standard in SYK literature; invoked to justify that the kink in free energy dictates the change in growth rates.

pith-pipeline@v0.9.0 · 5483 in / 1397 out tokens · 28566 ms · 2026-05-16T14:11:50.013026+00:00 · methodology

discussion (0)

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

chord number exhibits a discontinuous transition from linear to quadratic growth at the transition point... Krylov operator complexity and the operator size exhibit discontinuous transitions from exponential to quadratic growth
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_fourth_deriv_at_zero, alpha_pin_under_high_calibration unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

effective action S = ¼ ∫ n log n + z log z + crossing terms; Liouville equations ∂²g_n = 2ν²J² e^{g_n} + g_z
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction, alexander_duality_circle_linking (D=3) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

8-tick period never appears; dimension D=3 never forced

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.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

q-Askey Deformations of Double-Scaled SYK
hep-th 2026-05 unverdicted novelty 7.0

q-Askey deformations of double-scaled SYK yield transfer matrices for orthogonal polynomials whose semiclassical chord dynamics map to ER bridges and new geometric transitions in sine dilaton gravity.
Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography
hep-th 2026-02 unverdicted novelty 6.0

Deformations of the double-scaled SYK model via finite-cutoff holography produce Krylov complexity as wormhole length and realize Susskind's stretched horizon proposal through targeted T² deformations in the high-ener...

Reference graph

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