Correlation decay in two-dimensional projective spin models
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In two-dimensional nearest-neighbor RP² and RP³ spin models, do quadrupolar correlations decay faster than every power at each finite temperature? The observables are (nₓ·nᵧ)²−1/3 and (qₓ·qᵧ)²−1/4.
Why it matters and what counts as progress
Why it matters: The answer constrains disputed algebraic-order scenarios for these projective order parameters.
What counts as progress: A bound Cβ exp[−cβ(log(2+r))²] uniform over finite square-lattice subgraphs and signed couplings |bₑ|≤β, or evidence rigorously contradicting it.
How the problem developed
- 1982 · Numerical evidence [1]
Fukugita, Kobayashi, Okawa, Oyanagi and Ukawa Reported numerical evidence for a phase transition in the two-dimensional RP² spin model.
- 2023 · Numerical evidence [1]
Lander Burgelman and coauthors Reported pseudocritical scaling consistent with a zero-temperature transition and finite-temperature crossover in the classical RP² model.
- 22 Oct 2026 · Reported claim [1]
Masaki Oshikawa and ChatGPT 6 Astra Reported a rigorous faster-than-power-law bound on quadrupolar correlations in the two-dimensional nearest-neighbor RP² and RP³ models at every finite coupling.
AI contribution
Faster-than-power-law decay of order parameter correlations in the two-dimensional RP2 model: a rigorous result by vibe-proving
In the two-dimensional nearest-neighbor RP² and RP³ projective spin models, the quadrupolar two-point correlation is bounded by Cβ exp[−cβ(log(2+r))²] for every finite coupling bound β. The positive constants are uniform over finite square-lattice subgraphs, signed couplings |bₑ|≤β and observation sites; r is their lattice ℓ∞ distance. The spherical observables are (nₓ·nᵧ)²−1/3 and (qₓ·qᵧ)²−1/4.
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