Result 265, Mathematical physics

Area laws and tensor networks for two-dimensional gapped systems

Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open L×LL\times L squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.

Proof

The bigger picture

Why it matters

Entanglement in a quantum system's lowest-energy state can be limited by boundaries rather than volume. These unreviewed manuscripts report such a limit in two dimensions, alongside compact tensor-network approximations for a more specific class of systems.

What changes?

The first manuscript considers Hamiltonians, which describe a system's energy, with finite-range interactions on arbitrary finite induced square-lattice domains. For a unique lowest-energy state, it reports that every set of sites has entanglement entropy bounded by a constant times the number of boundary-crossing edges. The constant depends only on bounds for local state dimension, interaction range and strength, and a positive lower bound on the full-system spectral gap, the energy separation above the ground state, not domain size or shape.

What does that help mathematicians do?

On open L-by-L squares, the second manuscript reports that unique ground states of uniformly gapped nearest-neighbor Hamiltonians admit projected entangled-pair state approximations: networks of linked local tensors. Their bond dimension, controlling link size, is polynomial in L. After normalization, global vector error is at most one over L, with constants uniform for fixed local dimension, interaction strength and full-system gap. This controls approximation of the entire state, rather than only selected local measurements.

Are there practical applications?

The immediate value is foundational: the claims connect a full-system energy gap to boundary-limited entanglement and, on open squares with nearest-neighbor interactions, compact tensor-network representations. This supports the mathematical rationale for tensor-network modeling. The approximation result establishes existence, however, not an efficient procedure for finding the tensors or computing physical predictions from them.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A two-dimensional area law from a global spectral gap

September 24, 2026 90 pages

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 constant times the number of edges crossing its boundary, independently of the size and shape of the domain.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026,
  author = {{OpenAI}},
  title = {{A two-dimensional area law from a global spectral gap}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf}{OAI:A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026}},
  year = {2026}
}

Polynomial PEPS approximation of gapped square-grid ground states

September 24, 2026 56 pages

We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an L×LL\times L square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026,
  author = {{OpenAI}},
  title = {{Polynomial PEPS approximation of gapped square-grid ground states}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf}{OAI:Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.