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Zero-Velocity Lieb–Robinson Bounds for Disordered Spin Systems

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
A quantum spin model on $\mathbb Z^d$ has finite-dimensional local Hilbert spaces and a Hamiltonian built from finite-range interaction terms. Its Heisenberg evolution is $\tau_t(A)=e^{itH}Ae^{-itH}$. A zero-velocity Lieb--Robinson estimate bounds the commutator of distant local observables by a decaying function of their separation uniformly in time, rather than with a light cone widening linearly in $|t|$. The disorder averaging or probability quantifiers are part of the model.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Can zero-velocity Lieb–Robinson bounds be proved for general disordered finite-range quantum spin models on $\mathbb Z^d$, thereby establishing a universal many-body-localized phase?
Source scope and qualification.
The word general cannot be interpreted as every disorder distribution at every disorder strength: those classes include delocalized systems. A proposed localized regime and quantitative disorder hypotheses require source-specific specification; no unconditional universal theorem is created here.

Short English statement

Can time-uniform locality bounds establish a many-body localized regime for general disordered finite-range spin systems?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-304-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-304

Zero-Velocity Lieb–Robinson Bounds for Disordered Spin Systems

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
A quantum spin model on $\mathbb Z^d$ has finite-dimensional local Hilbert spaces and a Hamiltonian built from finite-range interaction terms. Its Heisenberg evolution is $\tau_t(A)=e^{itH}Ae^{-itH}$. A zero-velocity Lieb--Robinson estimate bounds the commutator of distant local observables by a decaying function of their separation uniformly in time, rather than with a light cone widening linearly in $|t|$. The disorder averaging or probability quantifiers are part of the model.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Can zero-velocity Lieb–Robinson bounds be proved for general disordered finite-range quantum spin models on $\mathbb Z^d$, thereby establishing a universal many-body-localized phase?
Source scope and qualification.
The word general cannot be interpreted as every disorder distribution at every disorder strength: those classes include delocalized systems. A proposed localized regime and quantitative disorder hypotheses require source-specific specification; no unconditional universal theorem is created here.

Short English statement

Can time-uniform locality bounds establish a many-body localized regime for general disordered finite-range spin systems?

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proves a deterministic zero-velocity bound for commuting finite-range spin Hamiltonians and shows why a direct norm perturbation does not yield a stable general localized regime.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
An exactly localized commuting class.
In a finite spin volume let $H_0=\sum_X h_X$, where all self-adjoint interaction terms commute, every $X$ has diameter at most $R$, and $h_X$ acts only on $X$. Their sizes and coefficients may be random. If $A$ is supported in a finite set $S$, split \[ H_0=H_S+H_{S^c},\qquad H_S=\sum_{X:X\cap S\ne\varnothing}h_X. \] Every term of $H_{S^c}$ commutes with $A$, and by hypothesis $[H_S,H_{S^c}]=0$. Hence \[ e^{itH_0}Ae^{-itH_0}=e^{itH_S}Ae^{-itH_S}. \] The right side is supported in $S_R=\{x:d(x,S)\leq R\}$ for every real $t$. If $B$ is supported in $T$ with $d(S,T)>R$, the commutator is exactly zero. Together with the universal bound $2\|A\|\|B\|$, this gives, for any $\mu>0$, \[ \|[\tau_t^{H_0}(A),B]\| \leq 2e^{\mu R}\|A\|\|B\|e^{-\mu d(S,T)}, \qquad t\in\mathbb R. \] For fixed local supports the proof is volume independent. Diagonal finite-range Ising Hamiltonians with arbitrary random coefficients supply examples in every dimension. Commutativity, rather than a probabilistic localization estimate, is doing the work.
Testing perturbative stability.
Put $H=H_0+\varepsilon V$ with $V$ bounded and self-adjoint in a finite volume. Duhamel's formula and unitary invariance of the norm imply \[ \|\tau_t^H(A)-\tau_t^{H_0}(A)\| \leq 2|\varepsilon|\,|t|\,\|V\|\,\|A\|. \] For distant $B$, this gives only a commutator bound $4|\varepsilon|\,|t|\,\|V\|\,\|A\|\,\|B\|$ in addition to the trivial norm bound. It neither decays with separation nor is useful uniformly in time; moreover an extensive $V$ has volume-growing norm. Thus ordinary small-perturbation continuity cannot extend the exact commuting calculation to the requested regime.
Remaining scope.
One must replace this time-growing expansion by a localization mechanism with volume-uniform, time-uniform spatial estimates, together with stated disorder probability or averaging quantifiers. No such estimate is supplied for noncommuting many-body interactions here. As the definition emphasizes, the target cannot mean every disorder distribution at every strength. This calculation gives a restricted exact model and identifies the failed perturbative step, not a universal many-body-localized phase.
Verification and adversarial audit.
Checked that disjoint terms can be discarded only because all interaction terms commute, the support enlargement R, the exponential bound at distance R, and Duhamel constants and volume dependence. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked that disjoint terms can be discarded only because all interaction terms commute, the support enlargement R, the exponential bound at distance R, and Duhamel constants and volume dependence. Retained unresolved status and zero original-target progress; Obtain volume-uniform and time-uniform decay for noncommuting interactions under explicit disorder assumptions; the direct Duhamel estimate grows linearly in time.

Coordinator checked the support localization of commuting finite-range interactions and the volume-sensitive time-growing Duhamel perturbation bound. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Obtain volume-uniform and time-uniform decay for noncommuting interactions under explicit disorder assumptions; the direct Duhamel estimate grows linearly in time.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. Obtain volume-uniform and time-uniform decay for noncommuting interactions under explicit disorder assumptions; the direct Duhamel estimate grows linearly in time.

Sources

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