Area laws and tensor networks for two-dimensional gapped systems

2 papers

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 \(L\times L\) 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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

A two-dimensional area law from a global spectral gap

OpenAI · September 24, 2026

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.

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@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

OpenAI · September 24, 2026

We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an \(L\times L\) 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.

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Cite

@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.