Strict area law entanglement versus chirality

TL;DR

This work investigates whether a representative state in a finite-dimensional local Hilbert space can realize chirality in a 2+1D gapped phase, as signaled by nonzero $c_{-}$ or $\sigma_{xy}$. It develops instantaneous modular flow as a new quantum-information primitive and uses it to derive two no-go theorems: bulk A1-compatible states with strict area-law entanglement cannot realize nonzero $c_{-}$ via $J(A,B,C)$ nor nonzero $\sigma_{xy}$ via $\Sigma(A,B,C)$ under on-site $U(1)$ symmetry. The proofs show that modular-flow dynamics would force unbounded growth of $S_{BC}(t)$ or $\langle Q_{BC}^2\rangle(t)$, contradicting finite local dimensions or bounded spectra. The results constrain how chirality can be represented in quantum states and suggest exploring subleading area-law corrections, connections to CFT and Virasoro symmetry, and further applications of instantaneous modular flow.

Abstract

Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through non-zero thermal or electrical Hall conductance. In this paper, 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. 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.

Figures (4)

• Figure 1: Tripartite regions on the lattice. (Left) Regions for bulk A1. (Right) Regions for computing $c_{-}$ and $\sigma_{xy}$.
• Figure 2: Regions for $J(A,B,C')_{\left|\Psi'(t)\right\rangle}$. $C = MC'$. Note that $E = \overline{ABCD}$ does not touch $C$.
• Figure 3: Various region deformations of $\Sigma(A,B,C)$ away from (left) and near (right) a triple intersection point among $A,B,C,\overline{ABC}$.
• Figure 4: Region partitions $B = B_1B_2,C' = C'_1C'_2$, $C = C'M$.

Theorems & Definitions (17)

• Theorem 3.1
• Theorem 3.2
• Definition 4.1
• Lemma 4.2: Markov decomposition move
• proof : Proof of Theorem \ref{['thm:no-go-thermal-intro']}
• Lemma 5.1
• proof
• proof
• Lemma 3.4
• proof