Moire Control of Alterelectric Quadrupolar Order

Alterelectricity is a compensated ferroic state in which quadrupolar electronic order reshapes low-energy electronic structure without producing a net polarization. Here we show that a moir\'e superlattice can turn such order into a controllable phase. Within a Bloch-periodic two-orbital theory, the slowly varying interlayer registry is coarse-grained into an effective moir\'e field acting on a self-consistent two-component alterelectric quadrupole. The resulting phase develops above a strongly filling-dependent instability threshold and crosses over from a weakly selected regime into a robust axial-dominated ground state, while the diagonal-dominated branch remains only a weak competitor. A registry-phase sweep supplies an explicit continuous path through internal quadrupole space, demonstrating that the moir\'e superlattice does more than stabilize alterelectricity: it steers its internal orientation. This orientational selection is encoded directly in the redistribution of low-energy spectral weight across the moir\'e Brillouin zone. These results identify moir\'e superlattices as a generic route to controllable alterelectric order and to programmable anisotropic electronic functionality.