论文标题
用晶格对称性的Abelian拓扑相量化的晶量规场和量化的离散几何响应
Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
论文作者
论文摘要
连续体中清洁的各向同性量子厅液体具有许多受对称保护的量化不变性物质，例如霍尔电导率，偏移和霍尔粘度。在这里，我们开发了一种在晶格上定义的拓扑阶段的对称性保护的量化不变性的理论，其中可能会出现无连续类似物的量化不变性。 We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders with symmetry group $G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of orientation-preserving space group symmetries on the lattice.我们展示了如何以离散的自旋矢量为特征的离散旋转和翻译对称性分数是如何表征的，它是一个离散的扭转矢量，在连续体中或没有晶格旋转对称性的情况下没有类似物，而区域矢量也没有类似物，在连续体中也没有类似物。离散的扭转矢量意味着一种水晶动量分数化，仅以$ 2 $，$ 3 $和$ 4 $倍的旋转对称性而不是平凡。量化的拓扑响应理论包括转移的离散版本，将分数电荷结合到脱节和角落，分数量化的脱节的角动量，旋转对称的分数电荷极化及其角动量对应物，其在电荷动量上的约束和每个单位电池的角度动量，以及量化的动量界限到区域的主体和偏置和独立的区域。分数量化的电荷极化，仅在$ 2 $，$ 3 $和$ 4 $倍的旋转对称性的晶格上，这是无处不在的，这意味着与晶格位错的分数充电和边界沿边界的单位长度的分数充电。一个重要的角色是在汉堡矢量上的有限群体分级扮演的，这取决于晶格的点组对称性。
Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected quantized invariants, such as the Hall conductivity, shift and Hall viscosity. Here we develop a theory of symmetry-protected quantized invariants for topological phases defined on a lattice, where quantized invariants with no continuum analog can arise. We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders with symmetry group $G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of orientation-preserving space group symmetries on the lattice. We show how discrete rotational and translational symmetry fractionalization can be characterized by a discrete spin vector, a discrete torsion vector which has no analog in the continuum or in the absence of lattice rotation symmetry, and an area vector, which also has no analog in the continuum. The discrete torsion vector implies a type of crystal momentum fractionalization that is only non-trivial for $2$, $3$, and $4$-fold rotation symmetry. The quantized topological response theory includes a discrete version of the shift, which binds fractional charge to disclinations and corners, a fractionally quantized angular momentum of disclinations, rotationally symmetric fractional charge polarization and its angular momentum counterpart, constraints on charge and angular momentum per unit cell, and quantized momentum bound to dislocations and units of area. The fractionally quantized charge polarization, which is non-trivial only on a lattice with $2$, $3$, and $4$-fold rotation symmetry, implies a fractional charge bound to lattice dislocations and a fractional charge per unit length along the boundary. An important role is played by a finite group grading on Burgers vectors, which depends on the point group symmetry of the lattice.
