Exponential decay for the two-dimensional Heisenberg model

Abstract

In this work, we prove exponential decay for the spin correlation of the spin O(N) model for N≥3 at every inverse temperature $ β>0 $. In particular, our work confirms Polyakov's prediction in 1975 on the Heisenberg model, i.e., the case of N=3. To this end, we investigate a generalized O(N) model where $ \mathsf{N} $ is a positive \emph{real number}, as follows: we sample a loop soup on the metric graph (say of a box) with intensity $ \tfrac{\mathsf{N}}{2} $ subject to the conditioning that the local time at each lattice vertex is β, and then for each loop we independently assign it to copies 1≤ i≤ ⌈N⌉ such that it is assigned to copy i with probability $ \tfrac{1}{\mathsf{N}} $ for $ 1\leq \mathsf{i}\leq \lfloor\mathsf{N}\rfloor $. Then, by Lupu's isomorphism theorem, we can construct a spin field by assigning a random sign to each loop soup cluster in each copy independently, such that the field coincides with the spin O(N) model when N is an integer. We then establish exponential decay for every real-valued N>2. Our result reveals an underlying mechanism for exponential decay, which emerges as soon as N>2 (i.e., as soon as beyond the range of the XY model) and does not necessarily rely on the rotation symmetry in the sphere $ \mathbb{S}^{2} $.

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BibTeX RIS

BibTeX

@misc{ding2026exponential,
  title = {{Exponential decay for the two-dimensional Heisenberg model}},
  author = {Jian Ding and Fenglin Huang and Peng Yang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04908},
  eprint = {2611.04908},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04908},
}

RIS

TY  - UNPB
TI  - Exponential decay for the two-dimensional Heisenberg model
AU  - Jian Ding
AU  - Fenglin Huang
AU  - Peng Yang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04908
UR  - https://arxiv.org/abs/2611.04908
AB  - In this work, we prove exponential decay for the spin correlation of the spin O(N) model for N≥3 at every inverse temperature $ β>0 $. In particular, our work confirms Polyakov's prediction in 1975 on the Heisenberg model, i.e., the case of N=3. To this end, we investigate a generalized O(N) model where $ \mathsf{N} $ is a positive \emph{real number}, as follows: we sample a loop soup on the metric graph (say of a box) with intensity $ \tfrac{\mathsf{N}}{2} $ subject to the conditioning that the local time at each lattice vertex is β, and then for each loop we independently assign it to copies 1≤ i≤ ⌈N⌉ such that it is assigned to copy i with probability $ \tfrac{1}{\mathsf{N}} $ for $ 1\leq \mathsf{i}\leq \lfloor\mathsf{N}\rfloor $. Then, by Lupu's isomorphism theorem, we can construct a spin field by assigning a random sign to each loop soup cluster in each copy independently, such that the field coincides with the spin O(N) model when N is an integer. We then establish exponential decay for every real-valued N>2. Our result reveals an underlying mechanism for exponential decay, which emerges as soon as N>2 (i.e., as soon as beyond the range of the XY model) and does not necessarily rely on the rotation symmetry in the sphere $ \mathbb{S}^{2} $.
ER  -

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