We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb–Schultz–Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

中文翻译：

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量子电荷传输的多体指数

我们提出了多体量子系统上的幺正映射和聚类状态对的索引。我们要求地图保存整数值电荷并保持状态不变，例如有间隙的基态。该指数是整数值且在扰动下稳定。通常，该指数测量跨基准线的电荷传输。我们表明，它简化为（i）非相互作用费米子情况下的投影指数，（ii）平移不变系统的电荷密度，以及（iii）二维设置中的量子霍尔电导，无需任何额外对称。示例 (ii) 恢复了 Lieb-Schultz-Mattis 定理，并且 (iii) 提供了在相互作用的多体系统中霍尔电导量化的新的简短证明。