模交换子作为鲁棒拓扑不变量与近似马尔可夫性
Modular commutator as a robust topological invariant and approximate Markovianity
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中文总结 AI 辅助
本文证明在近似局域量子马尔可夫条件下，模交换子作为带隙基态的手征中心荷探针在形变下渐近不变，并给出非零模交换子所需条件互信息的指数下界，同时推广至霍尔电导估计器。
中文摘要 AI 辅助
模交换子为带隙基态的手征中心荷 $c_-$ 提供了体态、局域、单波函数探针。其形变不变性先前仅在局域量子马尔可夫条件下确立——即对于将圆盘分割为三个连续条带 A、B 和 C 的所有三分割，条件互信息 $I(A:C|B)$ 为零。然而，局域量子马尔可夫条件也迫使该探针消失，从而留下问题：在马尔可夫性质仅近似成立的物理相关态中，模交换子是否仍保持鲁棒。本文针对有限维希尔伯特空间解决了这一矛盾：近似局域量子马尔可夫性质意味着在保持拓扑的形变下模交换子的变化渐近消失。我们接着证明了模交换子的迹范数连续性。将形变不变性与迹范数连续性相结合，我们确立了在由保持近似局域量子马尔可夫条件的准局域酉路径连接的带隙量子相内，模交换子保持渐近不变。反之，通过分析由模哈密顿量生成的有限时间动力学，我们推导出非零模交换子所需条件互信息的定量下界：在态满足纠缠面积定律的前提下，此类条带三分割上的 $I(A:C|B)$ 不能比 B 宽度的指数衰减更快，将严格无-go 定理推广为定量有限界。最后，我们证明这些结论同样适用于霍尔电导估计器。
英文摘要
The modular commutator provides a bulk, local, single-wave-function probe of the chiral central charge $c_-$ for gapped ground states. Its invariance under deformations was previously established under a local quantum Markov condition---namely, the conditional mutual information $I(A:C|B)$ is zero for all tripartitions of a disk into three consecutive strips A, B, and C \cite{Modular-commutator-Gapped}. However, the local quantum Markov condition also forces the probe to vanish, leaving open whether modular commutator remains robust in physically relevant states where the Markov property holds only approximately. In this paper, we resolve this tension for finite-dimensional Hilbert spaces: the approximate local quantum Markov property implies the change of the modular commutator under topology-preserving deformations vanishes asymptotically. We next prove trace-norm continuity of the modular commutator. Combining deformation invariance with trace-norm continuity, we establish that the modular commutator remains asymptotically invariant within gapped quantum phases connected by quasi-local unitary paths that preserve the approximate local quantum Markov condition. Conversely, by analyzing finite-time dynamics generated by modular Hamiltonians, we derive a quantitative lower bound on the conditional mutual information required for a non-zero modular commutator: $I(A:C|B)$ across such strip tripartitions cannot decay faster than exponentially with the width of B given that the state satisfies entanglement area law, extending the exact no-go theorem of \cite{strict-J-2024} to a quantitative finite bound . Finally, we demonstrate that those conclusions apply equally to the Hall conductance estimator \cite{FanSahayVishwanath2023}.
发表机构
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
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