Kozii Vladyslav, Avdoshkin Alexander, Zhong Shudan, Moore Joel E
Department of Physics, University of California, Berkeley, California 94720, USA.
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.
Phys Rev Lett. 2021 Apr 16;126(15):156602. doi: 10.1103/PhysRevLett.126.156602.
We study how the intrinsic anomalous Hall conductivity is modified in two-dimensional crystals with broken time-reversal symmetry due to weak inhomogeneity of the applied electric field. Focusing on a clean noninteracting two-band system without band crossings, we derive the general expression for the Hall conductivity at small finite wave vector q to order q^{2}, which governs the Hall response to the second gradient of the electric field. Using the Kubo formula, we show that the answer can be expressed through the Berry curvature, Fubini-Study quantum metric, and the rank-3 symmetric tensor which is related to the quantum geometric connection and physically corresponds to the gauge-invariant part of the third cumulant of the position operator. We further compare our results with the predictions made within the semiclassical approach. By deriving the semiclassical equations of motion, we reproduce the result obtained from the Kubo formula in some limits. We also find, however, that the conventional semiclassical description in terms of the definite position and momentum of the electron is not fully consistent because of singular terms originating from the Heisenberg uncertainty principle. We thus present a clear example of a case when the semiclassical approach inherently suffers from the uncertainty principle, implying that it should be applied to systems in nonuniform fields with extra care.
我们研究了由于外加电场的弱不均匀性，在具有破缺时间反演对称性的二维晶体中，本征反常霍尔电导率是如何被修正的。聚焦于一个没有能带交叉的干净无相互作用双能带系统，我们推导了在小有限波矢(q)到(q^{2})阶的霍尔电导率的一般表达式，它决定了对电场二阶梯度的霍尔响应。利用久保公式，我们表明答案可以通过贝里曲率、富比尼 - 斯图迪量子度量以及与量子几何联络相关且物理上对应于位置算符三阶累积量的规范不变部分的三阶对称张量来表示。我们进一步将我们的结果与半经典方法中的预测进行比较。通过推导半经典运动方程，我们在某些极限情况下重现了从久保公式得到的结果。然而，我们也发现，由于源于海森堡不确定性原理的奇异项，传统的关于电子确定位置和动量的半经典描述并不完全一致。因此，我们给出了一个明确的例子，说明半经典方法固有地受到不确定性原理的影响，这意味着在非均匀场中的系统应用它时应格外小心。
