Strict area law implies commuting parent Hamiltonian

TL;DR

This paper proves that in two dimensions, quantum states with a strict area law for entanglement entropy can be described by commuting local Hamiltonians, linking entanglement properties to physical Hamiltonian structures.

Contribution

It establishes that strict area law states in 2D admit commuting parent Hamiltonians, extending to states with gapped domain walls and implications for chiral systems.

Findings

States with strict area law have commuting local Hamiltonians.

Entanglement bootstrap axioms imply existence of stable, gapped, local Hamiltonians.

Chiral gapless edge modes cannot obey strict area law with finite local Hilbert space.

Abstract

We show that in two spatial dimensions, when a quantum state has entanglement entropy obeying a strict area law, meaning $S(A)=\alpha |\partial A| - \gamma$ for constants $\alpha, \gamma$ independent of lattice region $A$, then it admits a commuting parent Hamiltonian. More generally, we prove that the entanglement bootstrap axioms in 2D imply the existence of a commuting, local parent Hamiltonian with a stable spectral gap. We also extend our proof to states that describe gapped domain walls. Physically, these results imply that the states studied in the entanglement bootstrap program correspond to ground states of some local Hamiltonian, describing a stable phase of matter. Our result also suggests that systems with chiral gapless edge modes cannot obey a strict area law provided they have finite local Hilbert space.