论文标题
在不均匀电场中的固有异常霍尔电导率
Intrinsic anomalous Hall conductivity in a nonuniform electric field
论文作者
论文摘要
我们研究了在二维晶体中如何在二维晶体中改变了固有的异常电导率，由于施加的电场的不均匀性而导致时间反转对称性损坏。专注于无带交叉的干净的非交互式两波段系统，我们在小小限度的波动矢量$ q $上得出了霍尔电导率的一般表达，以订购$ q^2 $，该$ q^2 $，该$ q^2 $，该$ q^2 $，该响应响应了对电场的第二梯度的响应。我们使用Kubo公式表明，答案可以通过浆果曲率，fubini-study量子度量标准和与量子几何连接相关的等级3对称张量表示，并且物理上对应于位置操作员第三累积累积的一部分。我们进一步将结果与半经典方法中的预测进行了比较。通过得出半经典运动方程，我们在某些限制下重现了从久保公式获得的结果。但是，我们还发现，从电子的明确位置和动力来看，传统的半经典描述是由于源自海森伯格的不确定性原理的奇异术语，因此并不完全一致。因此，我们提供了一个案例的明确例子，即半经典方法固有地遭受不确定性原理的侵害，这意味着应将其应用于额外谨慎的非均匀领域的系统。
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.
