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arxiv: 2508.00060 · v2 · submitted 2025-07-31 · ✦ hep-th · cond-mat.str-el

Entanglement spreading and emergent locality in Brownian SYK chains

Jatin Narde , Onkar Parrikar , Harshit Rajgadia , Sandip Trivedi This is my paper

Pith reviewed 2026-05-19 01:34 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-el
keywords Brownian SYK chainentanglement spreadingemergent localitybutterfly velocityFKPP equationquantum error correctioninformation light-cone
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The pith

In a Brownian SYK chain at strong coupling, information from an injected qudit spreads inside a sharp light-cone bounded by the butterfly velocity.

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

The paper studies the spread of quantum information in a solvable one-dimensional model of chaotic dynamics consisting of a chain of Brownian SYK sites. It injects a qudit into an infinite-temperature state at a single point and computes the fraction of that qudit's information that is recoverable from a centered interval of length 2ℓ after a fixed time T. At strong coupling this fraction remains near zero for small ℓ and jumps to near its maximum value once ℓ exceeds v_B T, where v_B is the butterfly velocity. The transition arises because the late-time operator growth obeys the FKPP equation, whose domain-wall solutions enforce an emergent light-cone structure.

Core claim

At strong coupling the amount of information of the injected qudit contained in an interval of length 2ℓ shows a sharp transition as a function of ℓ from near zero to near maximal correlation, with the transition located at ℓ ∼ v_B T. This sharp light-cone is produced by the domain-wall solutions of the FKPP equation that governs the late-time, strong-coupling dynamics of the model.

What carries the argument

The FKPP equation, a nonlinear generalization of the diffusion equation whose domain-wall solutions at late times and strong coupling produce the observed sharp light-cone.

If this is right

  • Quantum information spreads ballistically inside the butterfly-velocity light-cone rather than diffusively.
  • The sharp transition supplies an explicit realization of the emergent locality required by an RT-like formula for entanglement entropy.
  • Tools from quantum error correction can be used to track information content explicitly in this solvable chaotic model.
  • The same FKPP mechanism links operator growth in chaotic systems to the appearance of geometric bulk duals.

Where Pith is reading between the lines

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

  • The same sharp transition should appear in any chaotic many-body system whose scrambling dynamics reduce to an FKPP-like equation at late times.
  • Finite-size or finite-coupling corrections would round the step; their scaling with system size or coupling strength could be measured numerically.
  • Direct comparison of the information content with out-of-time-order correlators in the same model would test whether both quantities are controlled by the same domain-wall profile.

Load-bearing premise

The late-time strong-coupling dynamics of the Brownian SYK chain are governed by the FKPP equation whose domain-wall solutions produce the sharp light-cone, justified from general properties of operator growth in chaotic systems.

What would settle it

A direct computation or simulation of the recoverable information fraction versus interval length at late times and strong coupling, testing whether the rise from near-zero to near-maximal remains abrupt and centered at ℓ = v_B T.

read the original abstract

The Ryu-Takayanagi (RT) formula and its interpretation in terms of quantum error correction (QEC) implies an emergent locality for the spread of quantum information in holographic CFTs, where information injected at a point in the boundary theory spreads within a sharp light-cone corresponding to the butterfly velocity. This emergent locality is a necessary condition for the existence of a geometric bulk dual with an RT-like formula for entanglement entropy. In this paper, we use tools from QEC to study the spread of quantum information and the emergence of a sharp light-cone in an analytically tractable model of chaotic dynamics, namely a one-dimensional Brownian SYK chain. We start with an infinite temperature state in this model and inject a qudit at time $t=0$ at some point $p$ on the chain. We then explicitly calculate the amount of information of the qudit contained in an interval of length $2\ell$ (centered around $p$) at some later time $t=T$. We find that at strong coupling, this quantity shows a sharp transition as a function of $\ell$ from near zero to near maximal correlation. The transition occurs at $\ell \sim v_B T$, with $v_B$ being the butterfly velocity. Underlying the emergence of this sharp light-cone is a non-linear generalization of the diffusion equation called the FKPP equation, which admits sharp domain wall solutions at late times and strong coupling. These domain wall solutions can be understood on physical grounds from properties of operator growth in chaotic systems.

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 / 1 minor

Summary. The paper examines the spread of quantum information in a one-dimensional Brownian SYK chain using quantum error correction tools. A qudit is injected at a point p in an infinite-temperature state at t=0, and the information content of this qudit within a centered interval of length 2ℓ is computed at later time T. At strong coupling the quantity exhibits a sharp transition from near-zero to near-maximal correlation at ℓ ∼ v_B T, with the sharpness attributed to domain-wall solutions of the FKPP equation that emerge from the model's operator-growth dynamics.

Significance. If the reduction to the FKPP equation and the resulting sharp fronts are rigorously established, the work supplies a concrete, analytically tractable demonstration of emergent locality in a chaotic quantum system. This directly supports the link between microscopic operator growth, sharp light-cones, and the Ryu-Takayanagi formula in holographic settings. The use of QEC diagnostics to quantify information spread is a methodological strength.

major comments (2)
  1. [Abstract / FKPP section] Abstract and subsequent discussion of the FKPP reduction: the claim that an explicit calculation yields a sharp transition at ℓ ∼ v_B T is not accompanied by the derivation steps that map the Brownian SYK Lindblad (or Schwinger-Dyson) equations onto the FKPP form, nor by error estimates or finite-size checks. Because this mapping is load-bearing for the quantitative location of the transition, the support for the central claim remains only moderate.
  2. [Discussion of late-time dynamics] The justification for the FKPP domain-wall sharpness invokes general properties of chaotic operator growth, but the gradient expansion and truncation of higher-order nonlinearities are not shown to be controlled near the front. If the neglected terms remain O(1), they can produce a finite-width smoothing that would eliminate the claimed sharp transition in the information content.
minor comments (1)
  1. [Introduction] Notation for the interval length 2ℓ and the time T should be introduced with a clear figure or equation reference to avoid ambiguity when comparing to the butterfly velocity v_B.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. The feedback highlights the need for greater detail on the FKPP reduction and the control of approximations near the domain wall. We have revised the manuscript to address these points and respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract / FKPP section] Abstract and subsequent discussion of the FKPP reduction: the claim that an explicit calculation yields a sharp transition at ℓ ∼ v_B T is not accompanied by the derivation steps that map the Brownian SYK Lindblad (or Schwinger-Dyson) equations onto the FKPP form, nor by error estimates or finite-size checks. Because this mapping is load-bearing for the quantitative location of the transition, the support for the central claim remains only moderate.

    Authors: We agree that the mapping from the microscopic equations to the FKPP form was insufficiently detailed in the original submission. In the revised manuscript we have added a new subsection that explicitly derives the reduction from the Schwinger-Dyson equations of the Brownian SYK chain to the FKPP equation, including the strong-coupling limit and the truncation steps. We also include error estimates that show corrections are suppressed by inverse powers of the coupling and present finite-size numerical checks on small chains that confirm the location and sharpness of the transition at ℓ ∼ v_B T. These additions directly strengthen the quantitative support for the central claim. revision: yes

  2. Referee: [Discussion of late-time dynamics] The justification for the FKPP domain-wall sharpness invokes general properties of chaotic operator growth, but the gradient expansion and truncation of higher-order nonlinearities are not shown to be controlled near the front. If the neglected terms remain O(1), they can produce a finite-width smoothing that would eliminate the claimed sharp transition in the information content.

    Authors: We thank the referee for raising this important point about the validity of the truncation near the front. In the revision we have expanded the discussion of late-time dynamics to include a scaling argument showing that, in the strong-coupling regime, the front is controlled by the linear instability of the operator growth, with nonlinear corrections entering only perturbatively. We argue that this keeps the domain wall sharp on the scales relevant to the information measure. A fully non-perturbative demonstration that all higher-order terms remain negligible would require additional techniques and lies beyond the scope of the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit QEC calculation and general chaotic operator growth

full rationale

The paper explicitly computes the information content of an injected qudit inside a finite interval using quantum error correction tools applied to the Brownian SYK chain dynamics. The observed sharp transition is then identified with the butterfly velocity v_B extracted from the model's independent operator-growth (OTOC) dynamics. The FKPP equation is introduced as a general nonlinear diffusion description whose domain-wall solutions are justified by broad properties of chaotic operator spreading rather than by any self-referential definition, fitted parameter, or self-citation chain internal to the present work. No equation or result in the provided derivation reduces to an input by construction; the central claim therefore remains self-contained against the model's microscopic rules.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of the FKPP equation to the late-time operator spreading in this model and on the existence of sharp domain-wall solutions at strong coupling; no new particles or forces are introduced.

free parameters (1)
  • strong-coupling regime
    The sharp transition is derived under the assumption that the system is deep in the strong-coupling limit where the FKPP description holds.
axioms (1)
  • domain assumption The effective dynamics of information spreading reduce to the FKPP equation at late times and strong coupling
    Invoked to explain the emergence of sharp domain walls from operator growth properties.

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Reference graph

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