Antiferromagnetic order in the half-filled square-lattice Hubbard model
For even $L \ge 4$ and $U > 0$ let $\Lambda_L = (\mathbb{Z}/L\mathbb{Z})^2$ and $$H_{L,U} = -\sum_{\{x, y\}} \sum_{s = \uparrow, \downarrow} \bigl(c_{xs}^\dagger c_{ys} + c_{ys}^\dagger c_{xs}\bigr) + U \sum_{x \in \Lambda_L} n_{x\uparrow} n_{x\downarrow},$$ the Hubbard model on $\Lambda_L$, the first sum running over nearest-neighbour pairs, each counted once. The fermionic operators satisfy $\{c_{xs}, c_{yt}^\dagger\} = \delta_{xy}\delta_{st}$ and $\{c_{xs}, c_{yt}\} = 0$, and $n_{xs} = c_{xs}^\dagger c_{xs}$. Restrict $H_{L,U}$ to exactly $L^2$ electrons (half filling), let $\psi_{L,U}$ be its normalized ground state, and let $S^3_x = (n_{x\uparrow} - n_{x\downarrow})/2$. By Lieb's theorem this ground state is unique and has total spin zero, and Shen, Qiu and Tian showed that $\langle \psi_{L,U}, \mathbf{S}_x \cdot \mathbf{S}_y \, \psi_{L,U} \rangle$ is positive when $x$ and $y$ lie in the same sublattice and negative otherwise. For large $U$, second-order perturbation theory in the hopping gives the spin-one-half Heisenberg antiferromagnet as the effective Hamiltonian (see Tasaki). In Hartree–Fock theory, Langmann and Lenells proved Hirsch's prediction that the antiferromagnetic gap vanishes like $e^{-2\pi/\sqrt{U}}$ as $U \to 0$.
Problem. Does every fixed $U > 0$ satisfy $$\liminf_{\substack{L \to \infty \\ L \text{ even}}} \frac{1}{L^4} \Bigl\langle \psi_{L,U}, \Bigl(\sum_{x \in \Lambda_L} (-1)^{x_1 + x_2} S^3_x\Bigr)^2 \psi_{L,U} \Bigr\rangle > 0\,?$$
Source: Lieb (1995; arXiv 1993), Problem 1, for the half-filled repulsive Hubbard model on the hypercubic lattice with any positive U, in the ground state when D = 2; see also Tasaki (1998), §5.2. 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, antiferromagnetism, Hubbard models, long-range order
References
@incollection{lieb1995,
author = {Lieb, Elliott H.},
title = {The {H}ubbard model: some rigorous results and open problems},
booktitle = {Advances in Dynamical Systems and Quantum Physics (Capri, 1993)},
publisher = {World Scientific},
address = {River Edge, NJ},
year = {1995},
pages = {173--193},
eprint = {cond-mat/9311033},
archivePrefix = {arXiv},
}
@article{tasaki1998,
author = {Tasaki, Hal},
title = {The {H}ubbard model -- an introduction and selected rigorous results},
journal = {J. Phys.: Condens. Matter},
volume = {10},
number = {20},
year = {1998},
pages = {4353--4378},
doi = {10.1088/0953-8984/10/20/004},
eprint = {cond-mat/9512169},
archivePrefix = {arXiv},
}
@article{lieb1989,
author = {Lieb, Elliott H.},
title = {Two theorems on the {H}ubbard model},
journal = {Phys. Rev. Lett.},
volume = {62},
number = {10},
year = {1989},
pages = {1201--1204},
doi = {10.1103/PhysRevLett.62.1201},
}
@article{shen1994,
author = {Shen, Shun-Qing and Qiu, Zhao-Ming and Tian, Guang-Shan},
title = {Ferrimagnetic long-range order of the {H}ubbard model},
journal = {Phys. Rev. Lett.},
volume = {72},
number = {8},
year = {1994},
pages = {1280--1282},
doi = {10.1103/PhysRevLett.72.1280},
}
@article{langmann2026,
author = {Langmann, Edwin and Lenells, Jonatan},
title = {On the mean-field antiferromagnetic gap for the half-filled {2D} {H}ubbard model at zero temperature},
journal = {Lett. Math. Phys.},
volume = {116},
number = {2},
year = {2026},
pages = {50},
doi = {10.1007/s11005-026-02085-5},
eprint = {2501.18141},
archivePrefix = {arXiv},
}
@misc{aim-hubbard-antiferromagnetism,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Antiferromagnetic order in the half-filled square-lattice Hubbard model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/204-hubbard-antiferromagnetism.md},
note = {Entry 204 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-00441, "Antiferromagnetic order in the half-filled square-lattice Hubbard model", clopen.solutions.