Strict Area Law Entanglement versus Chirality

Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through nonzero thermal or electrical Hall conductance. In this Letter, we prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies. We also show that the finite dimensional local Hilbert space condition can be relaxed to the condition that the state has finite local entanglement entropies. As a crucial ingredient in the proofs, we introduce a new quantum information-theoretic primitive called instantaneous modular flow, which has many other potential applications.