Faster-than-power-law decay of order parameter correlations in the two-dimensional RP2 model: a rigorous result by vibe-proving
Abstract
We consider the RP2 model on the two-dimensional square lattice with nearest-neighbor interactions. It is obtained by replacing the spin in the standard classical Heisenberg model with a director, which does not distinguish two opposite directions. Unlike the standard classical Heisenberg model or the O(3) nonlinear sigma model, the RP2 (and its cousin RP3) model allows point vortex defects classified by π1(RP2)= Z2 in two dimensions. It is tempting to expect the Z2 vortices to induce a phase transition analogous to the well-established Berezinskii-Kosterlitz-Thouless transition in the O(2) model (XY model) in two dimensions. Indeed, the possibilities of a critical phase or a critical point with power-law correlations of the order parameter in the RP2 and RP3 models have been discussed in the literature. However, the challenging nature of the problem has prevented a definitive conclusion on the phase diagram so far. In this paper, we report a rigorous mathematical proof by Artificial Intelligence (ChatGPT 6 Astra) that the absolute value of the two-point correlation function of order parameters at distance r in the RP2 and RP3 models on the two-dimensional square lattice is bounded above by Cβexp(−cβ[log(2+r)]2) for any finite inverse temperature β≥0, with positive constants Cβ and cβ depending only on β. Namely, the correlation function decays faster than any power of r. The proof is formalized and verified in Lean; the formalization is publicly available at doi:10.5281/zenodo.23548847 . A heuristic argument based on a one-parameter effective theory ansatz at large distances combined with the rigorous correlation bound suggests (but does not prove) that the Z2 vortices are always unbound at any nonzero temperature.
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BibTeX
@misc{oshikawa2026fasterthanpowerlaw,
title = {{Faster-than-power-law decay of order parameter correlations in the two-dimensional RP$^2$ model: a rigorous result by vibe-proving}},
author = {Masaki Oshikawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41256},
eprint = {2610.41256},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41256},
}
RIS
TY - UNPB
TI - Faster-than-power-law decay of order parameter correlations in the two-dimensional RP^2 model: a rigorous result by vibe-proving
AU - Masaki Oshikawa
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41256
UR - https://arxiv.org/abs/2610.41256
AB - We consider the RP^2 model on the two-dimensional square lattice with nearest-neighbor interactions. It is obtained by replacing the spin in the standard classical Heisenberg model with a director, which does not distinguish two opposite directions. Unlike the standard classical Heisenberg model or the O(3) nonlinear sigma model, the RP^2 (and its cousin RP^3) model allows point vortex defects classified by π_1(RP^2)= Z_2 in two dimensions. It is tempting to expect the Z_2 vortices to induce a phase transition analogous to the well-established Berezinskii-Kosterlitz-Thouless transition in the O(2) model (XY model) in two dimensions. Indeed, the possibilities of a critical phase or a critical point with power-law correlations of the order parameter in the RP^2 and RP^3 models have been discussed in the literature. However, the challenging nature of the problem has prevented a definitive conclusion on the phase diagram so far. In this paper, we report a rigorous mathematical proof by Artificial Intelligence (ChatGPT 6 Astra) that the absolute value of the two-point correlation function of order parameters at distance r in the RP^2 and RP^3 models on the two-dimensional square lattice is bounded above by C_βexp(−c_β[log(2+r)]^2) for any finite inverse temperature β≥0, with positive constants C_β and c_β depending only on β. Namely, the correlation function decays faster than any power of r. The proof is formalized and verified in Lean; the formalization is publicly available at doi:10.5281/zenodo.23548847 . A heuristic argument based on a one-parameter effective theory ansatz at large distances combined with the rigorous correlation bound suggests (but does not prove) that the Z_2 vortices are always unbound at any nonzero temperature.
ER -