We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of $1/p$, where $p$ is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb–Schultz–Mattis (LSM) theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron–Dana–Zak relation between the Hall conductance and the filling factor.

中文翻译：

---

量子基态扇区的合理指数

我们考虑电荷传输，用于与具有有限退化和拓扑有序的带隙基态子空间相互作用的多体系统。对于保留基态空间的任何保留位置，保持电荷的单一元素，我们将一个索引作为以下各项的整数：$\mathrm{1个}/p$，在哪里 $p$是基态简并。我们证明该指数在under的组成下是可加的。这种形式主义引起了多种应用：分数量子霍尔电导，分数里布-舒尔茨-马蒂斯（LSM）定理，将标准LSM推广到平移不变性被打破的系统，以及Avron-Dana-Zak的相互作用推广霍尔电导与填充因子之间的关系。