Néel order in the square-lattice spin-one-half antiferromagnet
For even $L \ge 4$ let $\Lambda_L = (\mathbb{Z}/L\mathbb{Z})^2$, and let $$H_L = \sum_{\{x, y\}} \mathbf{S}_x \cdot \mathbf{S}_y$$ be the spin-one-half Heisenberg antiferromagnet on $\bigotimes_{x \in \Lambda_L} \mathbb{C}^2$, the sum running over nearest-neighbour pairs of $\Lambda_L$, each counted once. Here $\mathbf{S}_x = (S^1_x, S^2_x, S^3_x)$ with $S^a_x = \sigma^a/2$ acting on the factor at $x$, and $\sigma^1, \sigma^2, \sigma^3$ are the Pauli matrices. The ground state $\psi_L$ of $H_L$ is unique (Lieb and Mattis); take it normalized. Let $\eta_x = (-1)^{x_1 + x_2}$. In two dimensions the Mermin–Wagner–Hohenberg theorem excludes Néel order at positive temperature. Jordão Neves and Fernando Perez, with a numerical correction by Affleck, Kennedy, Lieb and Tasaki, proved Néel order in the two-dimensional ground state for every spin $S \ge 1$ by the method of Dyson, Lieb and Simon, and Kennedy, Lieb and Shastry proved it for spin one-half on the cubic lattice.
Problem. Is it true that $$\liminf_{\substack{L \to \infty \\ L \text{ even}}} \frac{1}{L^4} \sum_{x, y \in \Lambda_L} \eta_x \eta_y \, \langle \psi_L, \mathbf{S}_x \cdot \mathbf{S}_y \, \psi_L \rangle > 0\,?$$
Source: Kennedy, Lieb and Shastry (1988); Lieb (1999), Problem B of his contribution to the IAMP list of open problems in mathematical physics. In this quantitative form, AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, long-range order, quantum Heisenberg antiferromagnets
References
@article{kennedy1988,
author = {Kennedy, Tom and Lieb, Elliott H. and Shastry, B. Sriram},
title = {Existence of {N}{\'e}el order in some spin-1/2 {H}eisenberg antiferromagnets},
journal = {J. Stat. Phys.},
volume = {53},
number = {5-6},
year = {1988},
pages = {1019--1030},
doi = {10.1007/BF01023854},
}
@misc{lieb1999,
author = {Lieb, Elliott H.},
title = {Long range order for the quantum {H}eisenberg model},
howpublished = {Open Problems in Mathematical Physics, International Association of Mathematical Physics, ed. M. Aizenman},
year = {1999},
url = {https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9901.HeisenbergFerr.html},
note = {Problem B},
}
@article{lieb1962,
author = {Lieb, Elliott and Mattis, Daniel},
title = {Ordering energy levels of interacting spin systems},
journal = {J. Math. Phys.},
volume = {3},
number = {4},
year = {1962},
pages = {749--751},
doi = {10.1063/1.1724276},
}
@article{dyson1978,
author = {Dyson, Freeman J. and Lieb, Elliott H. and Simon, Barry},
title = {Phase transitions in quantum spin systems with isotropic and nonisotropic interactions},
journal = {J. Stat. Phys.},
volume = {18},
number = {4},
year = {1978},
pages = {335--383},
doi = {10.1007/BF01106729},
}
@article{neves1986,
author = {Jord{\~a}o Neves, E. and Fernando Perez, J.},
title = {Long range order in the ground state of two-dimensional antiferromagnets},
journal = {Phys. Lett. A},
volume = {114},
number = {6},
year = {1986},
pages = {331--333},
doi = {10.1016/0375-9601(86)90571-2},
}
@article{affleck1988,
author = {Affleck, Ian and Kennedy, Tom and Lieb, Elliott H. and Tasaki, Hal},
title = {Valence bond ground states in isotropic quantum antiferromagnets},
journal = {Comm. Math. Phys.},
volume = {115},
number = {3},
year = {1988},
pages = {477--528},
doi = {10.1007/BF01218021},
}
@article{mermin1966,
author = {Mermin, N. D. and Wagner, H.},
title = {Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic {H}eisenberg models},
journal = {Phys. Rev. Lett.},
volume = {17},
number = {22},
year = {1966},
pages = {1133--1136},
doi = {10.1103/PhysRevLett.17.1133},
}
@misc{aim-square-lattice-neel-order,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Néel order in the square-lattice spin-one-half antiferromagnet}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/203-square-lattice-neel-order.md},
note = {Entry 203 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00475, "Néel order in the square-lattice spin-one-half antiferromagnet", clopen.solutions.