Area law in two spatial dimensions

The area law says entanglement entropy of a region $A$ scales with the size of its boundary $|\partial A|$ rather than its volume. It underlies why gapped ground states are efficiently describable by tensor networks and is a cornerstone of Hamiltonian complexity.

What is known: The one-dimensional unique-gapped-ground-state area law is established, with subsequently improved quantitative gap and local-dimension dependence. In 2D there are important structured cases. The Anshu–Arad–Gosset frustration-free theorem assumes a local gap and a specified cut geometry; it is not a theorem for every frustration-free Hamiltonian. Suitable stabilizer constructions and controlled deformations give other positive examples.

Ground-space assumptions are essential: on-site projectors pinning one sublattice and leaving the other spins free produce a gap of one but allow volume-entangled ground states. Thus a gap alone cannot imply the claimed law for every vector of an unrestricted degenerate ground space. Topologically degenerate models require their own explicit local-ground-space hypotheses.

The unique-ground-state formulation above remains a central open target. A theorem about entanglement is not itself a total algorithm for deciding whether the gap promise holds; U1 does not automatically preclude such a conditional theorem. Representation and efficient computation are distinct further questions (B2/B3).

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.