Intrinsic Anomalous Hall Conductivity in a Nonuniform Electric Field
Intrinsic Anomalous Hall Conductivity in a Nonuniform Electric Field
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DOI：
10.1103/physrevlett.126.156602
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发表时间：
2021-04-12
影响因子：
8.6
通讯作者：
Moore, Joel E.
中科院分区：
文献类型：
--
作者：
Kozii, Vladyslav;Avdoshkin, Alexander;Moore, Joel E.
本文研究了由于外加电场的弱非均匀性而破坏时间反转对称性的二维晶体的本征反常霍尔电导率是如何被修正的。以一个干净的无交叉带的非相互作用双带系统为研究对象，我们导出了小有限波矢量q到q(2)阶的霍尔电导率的一般表达式，它决定了电场第二次梯度下的霍尔响应。利用Kubo公式，我们证明了答案可以通过Berry曲率、Fubini-Study量子度量和与量子几何连接相关的秩3对称张量来表示，该张量在物理上对应于位置算子的第三累积量的规范不变部分。我们进一步将我们的结果与半经典方法所作的预测进行比较。通过对半经典运动方程的推导，在一定极限下得到了由Kubo公式得到的结果。然而，我们也发现，根据电子的确定位置和动量的传统半经典描述并不完全一致，因为源自海森堡测不准原理的奇异项。因此，我们提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，这意味着它应该特别小心地应用于非均匀领域的系统。
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.