  Motivated by recent studies of odd-parity multipole order in condensed matter
physics, we theoretically study magnetoelectric responses in an electric
octupole state. Investigating the Edelstein effect and spin Hall effect in a
locally noncentrosymmetric bilayer Rashba model, we clarify characteristic
properties due to parity violation in the electric octupole state. Furthermore,
a possible realization of electric octupole order in bilayer high-Tc cuprate
superconductors is proposed. Our calculation of magnetic torque is consistent
with recent experimental observation of a kink above the superconducting
transition temperature. We also show significant enhancement of the in-plane
anisotropy in spin susceptibility due to the superconductivity, and propose an
experimental test by means of the nuclear magnetic resonance in the
superconducting state. A spin-orbit coupled metal state in Cd2Re2O7 is also
discussed.
