论文标题
与空间对称性的分数量子厅状态的分类
Classification of fractional quantum Hall states with spatial symmetries
论文作者
论文摘要
分数量子大厅（FQH）状态是对称式拓扑状态（集合）的示例：除了内在的拓扑顺序外，它们对对称性破裂具有鲁棒性外，它们还具有对称性保护的拓扑不变性，例如任何人的分数电荷和分数霍尔电导率。在本文中，我们通过使用$ g $ g $ ossed的编织的张量类别（$ g \ times $ btcs）将具有空间对称性的FQH状态开发了具有空间对称性的FQH状态的全面理论，该理论适用于Abelian和非阿贝尔拓扑状态。我们在连续体中使用$ u（1）$电荷保护，磁翻译和空间旋转对称性的系统，并以二维为单位单元格中的任意每个单位电池中的任意理性磁通量，并假设符号不适合Anyons。在晶体环境中，适用于分数切绝缘子和旋转液体，对称分数的完全特征是对电荷，自旋，离散扭转和面积矢量的非 - 阿布尔状态的概括，这些载体，旋转，离散的扭转和面积向量，这些载体指定分数电荷，角度动量，线性动量，线性动量和分数为每种构成任何人的翻译。拓扑响应理论包含$ 9 $项，将电荷，线性动量和角动量附加到磁通量，晶格位错，脱节，角落和区域单位。使用$ g \ times $ btc形式主义，我们得出了与霍尔电导率和每个单位电池通量填充的公式相关的公式；在连续体中，这与填充分数和霍尔电导率有关，而无需假设伽利亚不变。我们为$ g \ times $ btc框架内的拓扑不变性提供系统的公式；例如，这给出了霍尔电导率的新分类定义。
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs) for SETs. We consider systems with $U(1)$ charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all $5$ orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains $9$ terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the $G\times$BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the $G\times$BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.
