The massive phase of the two-dimensional Heisenberg model is stable under small anisotropy
Abstract
Polyakov predicted that the classical Heisenberg model on ℤ2 has exponentially decaying correlations at every temperature, and a proof of this has recently been posted. We show that the massive phase survives small anisotropy. For every inverse temperature β there is δ(β)>0 such that the S2-valued model with couplings ∑iJiσxiσyi has exponentially decaying spin two-point functions, uniformly in finite subgraphs with free boundary conditions and in edge strengths up to β, whenever |Ji−1|≤δ(β) for i=1,2,3. For the easy-plane model with couplings σxσx+σyσy+λσzσz this shows that the onset of the Berezinskii-Kosterlitz-Thouless phase, proved by Garban and de Montgolfier for every λ∈(−1,1), moves to zero temperature as λ↑1: in free-boundary limits, correlations decay exponentially for every β≤ c |log(1−λ)|1/2. The physical prediction is of order |log(1−λ)|. The same holds for aligned single-ion anisotropy, which includes a confining potential ε∑(σz)2, and for nearly isotropic O(n) models with n≥3. The proof keeps the sign-cluster argument of the isotropic proof, using Ginibre's correlation inequality for anisotropic planar energies, and uses the rotation symmetry only on one finite annulus, where the anisotropy is a bounded perturbation of the Hamiltonian.
Related
- Stable Rigidity in the Heisenberg Groups
- The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
- The Bernstein problem in higher dimensional Heisenberg groups
- Classification of finite energy positive solutions to Yamabe-type equation on the fifteen dimensional octonionic Heisenberg group
- Finite- and long-range spin-wave free-energy asymptotics for the two-dimensional quantum Heisenberg ferromagnet
BibTeX
@misc{pei2026massive,
title = {{The massive phase of the two-dimensional Heisenberg model is stable under small anisotropy}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01532},
eprint = {2611.01532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The massive phase of the two-dimensional Heisenberg model is stable under small anisotropy
AU - Yan Ru Pei
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01532
UR - https://arxiv.org/abs/2611.01532
AB - Polyakov predicted that the classical Heisenberg model on ℤ^2 has exponentially decaying correlations at every temperature, and a proof of this has recently been posted. We show that the massive phase survives small anisotropy. For every inverse temperature β there is δ(β)>0 such that the S^2-valued model with couplings ∑_iJ^iσ_x^iσ_y^i has exponentially decaying spin two-point functions, uniformly in finite subgraphs with free boundary conditions and in edge strengths up to β, whenever |J^i−1|≤δ(β) for i=1,2,3. For the easy-plane model with couplings σ^xσ^x+σ^yσ^y+λσ^zσ^z this shows that the onset of the Berezinskii-Kosterlitz-Thouless phase, proved by Garban and de Montgolfier for every λ∈(−1,1), moves to zero temperature as λ↑1: in free-boundary limits, correlations decay exponentially for every β≤ c |log(1−λ)|^1/2. The physical prediction is of order |log(1−λ)|. The same holds for aligned single-ion anisotropy, which includes a confining potential ε∑(σ^z)^2, and for nearly isotropic O(n) models with n≥3. The proof keeps the sign-cluster argument of the isotropic proof, using Ginibre's correlation inequality for anisotropic planar energies, and uses the rotation symmetry only on one finite annulus, where the anisotropy is a bounded perturbation of the Hamiltonian.
ER -