Electrically Reversible Layer-Tensor Responses in Bilayers

Abstract

The layer Hall effect provides a prototypical example of a hidden layer response in van der Waals bilayers: opposite layers carry compensating Hall currents at zero out-of-plane electric field, while an applied out-of-plane electric field $E_z$ selects one layer over the other and switches the net Hall response. Here we show that this field-switching mechanism is not restricted to Hall transport, but follows from a general symmetry principle for response tensors. We formulate a selection rule for the layer tensor effect (LTE) using Jahn symbols. For a tensor with Jahn symbol $J$ and rank $r$, the stacking operation that exchanges the two layers acts as a coset representative relating the $+E_z$ and $-E_z$ states. Tensor components that are odd under this operation are forced to cancel between layers at $E_z=0$, and acquire a macroscopic response satisfying $T_{\alpha }(+E_z)=-T_{\alpha }(-E_z)$. This criterion allows us to enumerate the symmetry-allowed electrically reversible tensor components associated with different Jahn symbols and to identify their physical realizations in common response tensors. We apply the theory to representative anomalous Hall, shift-current, nonlinear Drude, piezoelectric, spin Hall, and spin-photocurrent channels. Model calculations and first-principles simulations further confirm the predicted field-odd responses. Our results establish an electrically switchable LTE framework, providing a unified route to multifunctional layertronic responses in bilayer materials.