Area law for gapped Hamiltonians beyond one dimension

Establish an area law for the bipartite entanglement entropy of ground states of local gapped Hamiltonians in two or higher spatial dimensions, thereby rigorously justifying PEPS as a variational class for such systems.

Background

An area law for one-dimensional gapped systems was proven by Hastings, underpinning MPS approximations. In higher dimensions, while PEPS naturally satisfy an area law, a general proof that ground states of local gapped Hamiltonians obey an area law remains a longstanding conjecture.

A rigorous resolution would solidify the theoretical foundation for using PEPS to approximate ground states of higher-dimensional gapped models.

References

Therefore PEPS are expected to form a good variational class for low-energy states of local gapped Hamiltonians, which have since long been conjectured to satisfy an area law.

— Les Houches Lectures Notes on Tensor Networks  (2512.24390 - Cuiper et al., 30 Dec 2025) in Section 4.3