Area laws and tensor networks for two-dimensional gapped systems

What OpenAI claims

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× 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.

Papers (2)

1. A two-dimensional area law from a global spectral gap

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. Polynomial PEPS approximation of gapped square-grid ground states

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an L× 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.