Under broken time reversal symmetry such as in the presence of external magnetic field or internal magnetization, a transverse voltage can be established in materials perpendicular to both longitudinal current and applied magnetic field, known as classical Hall effect. However, this symmetry constraint can be relaxed in the nonlinear regime, thereby enabling nonlinear anomalous Hall current in time-reversal invariant materials – an underexplored realm with exciting new opportunities beyond classical linear Hall effect. Here, using group theory and first-principles theory, we demonstrate a remarkable ferroelectric nonlinear anomalous Hall effect in time-reversal invariant few-layer WTe₂ where nonlinear anomalous Hall current switches in odd-layer WTe₂ except 1T′ monolayer while remaining invariant in even-layer WTe₂ upon ferroelectric transition. This even-odd oscillation of ferroelectric nonlinear anomalous Hall effect was found to originate from the absence and presence of Berry curvature dipole reversal and shift dipole reversal due to distinct ferroelectric transformation in even and odd-layer WTe₂. Our work not only treats Berry curvature dipole and shift dipole on an equal footing to account for intraband and interband contributions to nonlinear anomalous Hall effect, but also establishes Berry curvature dipole and shift dipole as new order parameters for noncentrosymmetric materials. The present findings suggest that ferroelectric metals and Weyl semimetals may offer unprecedented opportunities for the development of nonlinear quantum electronics.

在中断时间反向对称的情况下，例如在存在外部磁场或内部磁化的情况下，可以在垂直于纵向电流和施加磁场的材料中建立横向电压，这就是经典的霍尔效应。但是，这种对称性约束可以在非线性范围内得到放松，从而在时变不变材料中实现非线性异常的霍尔电流，这是一个尚未开发的领域，除了经典的线性霍尔效应之外，它还具有令人兴奋的新机遇。在这里，使用群论和第一性原理，我们证明了在时间反转不变的几层WTe ₂中非凡的铁电非线性异常霍尔效应，其中非线性异常霍尔电流在奇数层WTe _2中切换除了1T'单层外，铁电跃迁在偶数层WTe _2中保持不变。铁电非线性异常霍尔效应的这种奇数振荡是由于在偶数和奇数层WTe _2中存在明显的铁电相变而引起的贝里曲率偶极子反转和位移偶极子反转的存在和不存在而引起的。我们的工作不仅在相等的基础上对待Berry曲率偶极子和频移偶极子，以说明带内和带间对非线性异常霍尔效应的贡献，而且还将Berry曲率偶极子和频移偶极子确立为非中心对称材料的新阶参数。目前的发现表明，铁电金属和魏尔半金属可能为非线性量子电子学的发展提供前所未有的机会。