Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
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DOI：
10.1103/physrevresearch.3.013040
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发表时间：
2020-05
期刊：
影响因子：
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通讯作者：
N. Manjunath;M. Barkeshli
中科院分区：
文献类型：
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作者：
N. Manjunath;M. Barkeshli
连续介质中干净的各向同性量子霍尔流体具有许多对称保护的量子化不变量，如霍尔电导率、位移和霍尔粘度。在这里，我们发展了晶格上定义的拓扑相的对称保护的量子化不变量理论，在晶格上可以产生没有连续介质类比的量子化不变量。本文利用离散晶体规范场建立了拓扑场理论，充分刻画了对称群$G = U(1) \乘以G_{\text{space}}$的（2+1）D Abelian拓扑序的量子化不变量，其中$G_{\text{space}}$由晶格上保持方向的空间群对称组成。我们展示了如何用一个离散的自旋矢量、一个在连续介质中没有类似物或在没有晶格旋转对称的情况下没有类似物的离散扭转矢量和一个在连续介质中也没有类似物的面积矢量来表征离散旋转和平移对称分形化。离散扭转矢量意味着一种晶体动量分数化，这种分数化仅对$2$、$3$和$4$旋转对称是非平凡的。量子化拓扑响应理论包括位移的离散版本，它将分数电荷与斜位和角相结合，分数量子化的斜位角动量，旋转对称的分数电荷极化及其角动量对应物，对单位细胞的电荷和角动量的约束，以及与位错和面积单位相结合的量子化动量。分数量子化电荷极化仅在$2$,$3$和$4$-fold旋转对称的晶格上是非平凡的，这意味着与晶格位错绑定的分数电荷和沿边界的单位长度的分数电荷。Burgers向量上的有限群分级依赖于晶格的点群对称性，在此过程中起着重要的作用。
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.