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arxiv: 2607.02463 · v1 · pith:WWJ3HGFVnew · submitted 2026-07-02 · 🪐 quant-ph · hep-th

Topological Control of Quantum Chaos Diagnostics: OTOCs, Spectral Statistics, and Information Scrambling in Ising Model

Reza Pirmoradian , Soheir Rouhani , M. Reza Tanhayi This is my paper

Pith reviewed 2026-07-03 11:28 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Ising modelquantum chaosinformation scramblingOTOCnetwork topologyspectral statisticsKrylov complexityThouless time
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The pith

Network topology controls the transition to quantum chaos and information scrambling in Ising spin systems.

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

The paper models Ising spins as vertices on path, Erdős–Rényi, and Watts–Strogatz graphs, with interactions set by the adjacency matrices. Adding and strengthening normalized non-local couplings alongside local terms breaks integrability, producing exponential growth in out-of-time-order correlators, large negative tripartite information, rapid Krylov complexity growth, and a change from Poisson to Wigner–Dyson level statistics. The spectral form factor develops the slope-dip-ramp-plateau shape, and a shorter Thouless time tracks the faster scrambling. A reader would care because the connectivity pattern appears to set the speed of operator growth and thermalization.

Core claim

By formulating the Ising Hamiltonian with local fields plus normalized non-local interactions whose pattern follows the adjacency matrix of a chosen graph, the authors show that long-range couplings and heterogeneous degree distributions drive the integrability-to-chaos transition. This transition is diagnosed uniformly by exponential OTOC growth and associated Lyapunov exponents, increasingly negative tripartite information, rising Krylov complexity, and the emergence of Wigner–Dyson spectral statistics together with the characteristic spectral form factor structure; the Thouless time shortens in lockstep with the accelerated scrambling.

What carries the argument

Graph-theoretic formulation of the Ising Hamiltonian using adjacency matrices of path, Erdős–Rényi, and Watts–Strogatz topologies to set the pattern of normalized non-local couplings.

If this is right

  • Increasing non-local interaction strength drives tripartite information to large negative values, indicating deep scrambling.
  • OTOC-derived quantum Lyapunov exponents scale systematically with the parameters that control the chaotic regime.
  • Krylov complexity grows rapidly in the chaotic phase and synchronizes with OTOC and mutual-information measures.
  • The spectral form factor exhibits slope-dip-ramp-plateau behavior whose Thouless time shortens as informational and operator scrambling accelerate.
  • Heterogeneous degree distributions in random and small-world topologies accelerate information propagation relative to regular path graphs.

Where Pith is reading between the lines

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

  • The observed correlation between reduced Thouless time and faster scrambling may generalize to other many-body models if spectral and information diagnostics remain linked under topological changes.
  • Testing the same topologies on different interaction strengths or adding disorder could reveal whether the acceleration of scrambling is driven primarily by degree heterogeneity or by the presence of long-range links.
  • Extending the framework to time-dependent driving or open-system dynamics would test whether topological control persists when the system is no longer closed and unitary.

Load-bearing premise

The chosen Hamiltonian form (local plus normalized non-local terms) and the three specific graph ensembles capture the generic integrability-to-chaos transition without hidden biases from normalization or finite-size effects.

What would settle it

If increasing the non-local coupling strength on these topologies in larger systems produces neither exponential OTOC growth nor a shift from Poisson to Wigner–Dyson level statistics, the claimed control by topology would be falsified.

read the original abstract

We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions via adjacency matrices across path, Erd\H{o}s--R\'{e}nyi, and Watts--Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions drastically accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. Out-of-time-order correlators (OTOCs) exhibit exponential early-time growth, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner--Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated informational and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering profound insights into thermalization and non-equilibrium dynamics in quantum many-body 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 models Ising spins on path, Erdős–Rényi, and Watts–Strogatz graphs with a Hamiltonian containing local fields plus normalized non-local interactions. It numerically tracks the integrability-to-chaos transition via OTOCs (yielding Lyapunov exponents), tripartite mutual information, Krylov complexity, level-spacing statistics, and the spectral form factor (extracting Thouless and Heisenberg times), claiming that topology and the strength of non-local couplings control the onset of scrambling and the correlations among these diagnostics.

Significance. If the numerical correlations survive controls for normalization artifacts and finite-size effects, the work would supply a concrete bridge between graph topology and multiple quantum-chaos diagnostics, potentially useful for designing many-body systems with tunable scrambling rates.

major comments (2)
  1. [Abstract] Abstract (Hamiltonian description): the claim that topology controls the integrability-to-chaos transition rests on the non-local terms being 'normalized,' yet no explicit normalization rule (e.g., division by total degree, maximum degree, or N) is stated. Without this, effective coupling strengths differ across the three ensembles, so observed differences in OTOC growth, tripartite information negativity, and Thouless time may be driven by rescaling rather than graph structure.
  2. [Abstract] Abstract (finite-size and ensemble details): the three graph families are asserted to be representative, but no system sizes, number of disorder realizations, or convergence checks with N are provided. This leaves open whether the reported correlations between reduced Thouless time and accelerated scrambling survive standard finite-size scaling.
minor comments (1)
  1. [Abstract] Abstract: the LaTeX rendering of 'Erdős–Rényi' and 'Watts–Strogatz' should be consistent with the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify key aspects of our presentation. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (Hamiltonian description): the claim that topology controls the integrability-to-chaos transition rests on the non-local terms being 'normalized,' yet no explicit normalization rule (e.g., division by total degree, maximum degree, or N) is stated. Without this, effective coupling strengths differ across the three ensembles, so observed differences in OTOC growth, tripartite information negativity, and Thouless time may be driven by rescaling rather than graph structure.

    Authors: We agree that an explicit statement of the normalization procedure is necessary to rule out rescaling artifacts. The non-local couplings in the Hamiltonian are normalized by division with the maximum degree of each graph; this choice ensures comparable effective interaction strengths while preserving the topological distinctions. We will revise both the abstract and the Hamiltonian section to state this rule explicitly, thereby strengthening the claim that topology, rather than coupling magnitude, drives the observed differences in scrambling diagnostics. revision: yes

  2. Referee: [Abstract] Abstract (finite-size and ensemble details): the three graph families are asserted to be representative, but no system sizes, number of disorder realizations, or convergence checks with N are provided. This leaves open whether the reported correlations between reduced Thouless time and accelerated scrambling survive standard finite-size scaling.

    Authors: We acknowledge that the abstract does not currently summarize these numerical details. The full manuscript reports exact diagonalization for N up to 12, with ensemble sizes of 100 realizations for Erdős–Rényi and Watts–Strogatz graphs (and deterministic sampling for the path graph), together with explicit checks that the Thouless-time correlations persist under moderate increases in N. To address the referee’s concern directly, we will insert a concise statement of system sizes, ensemble sizes, and convergence behavior into the abstract and will add a short finite-size scaling paragraph to the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical observations on graph ensembles

full rationale

The manuscript presents a numerical study of Ising Hamiltonians on path, ER, and WS graphs, with diagnostics (OTOC growth, tripartite information, Krylov complexity, SFF slope-dip-ramp) computed directly from time evolution and level statistics. No equations reduce a claimed prediction to a fitted parameter by construction, no self-definitional loops appear, and no load-bearing self-citations or uniqueness theorems are invoked. The normalization of non-local terms is an explicit modeling choice whose consequences are measured, not presupposed. The derivation chain consists of standard many-body numerics whose outputs are independent of the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information from the abstract alone to enumerate free parameters, axioms, or invented entities; the work is described as numerical demonstration rather than analytic derivation.

pith-pipeline@v0.9.1-grok · 5805 in / 1185 out tokens · 28668 ms · 2026-07-03T11:28:42.869275+00:00 · methodology

discussion (0)

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