An area law for general gapped two-dimensional spin systems
Fix an integer $q \ge 2$ and $\Delta > 0$. On $\Lambda = \{1, \dots, L\} \times \{1, \dots, M\}$ put a copy of $\mathbb{C}^q$ at each vertex, and let $$H = \sum_{x} h_x + \sum_{\{x,y\}} h_{xy}$$ be self-adjoint, the second sum running over nearest-neighbour pairs, with each term acting on its indicated sites and of operator norm at most $1$. Suppose that $H$ has a unique normalized ground state $\psi$ and spectral gap at least $\Delta$. For $1 \le \ell < L$ let $A = \{1, \dots, \ell\} \times \{1, \dots, M\}$ and $\rho_A = \operatorname{Tr}_{\Lambda \setminus A} |\psi\rangle\langle\psi|$. Hastings proved an area law for the entanglement entropy of gapped one-dimensional systems. Anshu, Arad and Gosset proved that, for a locally gapped, frustration-free Hamiltonian with geometrically local interactions in two dimensions and a unique ground state, the ground-state entanglement entropy across a vertical cut of length $n$ is at most $n^{1+o(1)}$.
Problem. Is there a constant $C(q, \Delta)$, independent of $L$, $M$, $\ell$ and the interaction, such that always $$-\operatorname{Tr}(\rho_A \log \rho_A) \le C(q, \Delta)\, M\,?$$
Source: The area law conjecture for gapped ground states in two dimensions, as discussed by Anshu, Arad and Gosset (2022), §1, who ask whether their local gap can be replaced by a gap of the full Hamiltonian. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, area law, entanglement entropy, gapped ground states
References
@article{hastings2007,
author = {Hastings, M. B.},
title = {An area law for one-dimensional quantum systems},
journal = {J. Stat. Mech. Theory Exp.},
volume = {2007},
number = {8},
year = {2007},
pages = {P08024},
doi = {10.1088/1742-5468/2007/08/P08024},
eprint = {0705.2024},
archivePrefix = {arXiv},
}
@inproceedings{anshu2022,
author = {Anshu, Anurag and Arad, Itai and Gosset, David},
title = {An area law for 2d frustration-free spin systems},
booktitle = {Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022)},
publisher = {ACM},
year = {2022},
pages = {12--18},
doi = {10.1145/3519935.3519962},
eprint = {2103.02492},
archivePrefix = {arXiv},
}
@misc{aim-two-dimensional-area-law,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An area law for general gapped two-dimensional spin systems}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/199-two-dimensional-area-law.md},
note = {Entry 199 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00731, "An area law for general gapped two-dimensional spin systems", clopen.solutions.