  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 WTe$_2$ where nonlinear anomalous Hall current switches in
odd-layer WTe$_2$ while remaining invariant in even-layer WTe$_2$ 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 WTe$_2$. 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, therefore,
suggest that ferroelectric metals and Weyl semimetals may offer unprecedented
opportunities for the development of nonlinear quantum electronics.
