Strict area law implies commuting parent Hamiltonian

• URL: http://arxiv.org/abs/2404.05867v1
• Date: Mon, 8 Apr 2024 21:01:49 GMT
• Title: Strict area law implies commuting parent Hamiltonian
• Authors: Isaac H. Kim, Ting-Chun Lin, Daniel Ranard, Bowen Shi,
• Abstract summary: We show that when a quantum state has entanglement entropy obeying a strict area law, 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.
• Score: 11.62855746863658
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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