Although the odd-parity multipole order barely occurs in crystals with high symmetry, it can be formed in locally noncentrosymmetric crystals. We illustrate the odd-parity electric octupole order generated in a bilayer structure. When the local electric quadrupole is alternatively stacked between layers, it is regarded as an electric octupole order from the viewpoint of symmetry. We show that the p y s x + p x s y spin nematic order is induced by the spin-orbit coupling in the electric octupole state, and it results from the spontaneous inversion symmetry breaking leading to the D 2d point group symmetry. We investigate the possible realization of the electric octupole order in the bilayer ruthenate Sr 3 Ru 2 O 7 , assuming a local electric quadrupole arising from the Pomeranchuk instability and/or the orbital order. Effects of the lattice distortion and magnetic field are also clarified, and the nature of the "electronic nematic state" of Sr 3 Ru 2 O 7 is discussed. It is proposed that the asymmetric band structure is a signature of the electric octupole order in Sr 3 Ru 2 O 7 . The odd-parity multipole order in other strongly correlated electron systems is discussed.

The odd-parity multipole is an emergent degree of freedom, leading to spontaneous inversion symmetry breaking. The odd-parity multipole order may occur by forming staggered even-parity multipoles in a unit cell. We focus on a locally noncentrosymmetric bilayer Rashba system, and study an odd-parity electric octupole order caused by the antiferro stacking of local electric quadrupoles. Analyzing the forward scattering model, we show that the electric octupole order is stabilized by a layer-dependent Rashba spin-orbit coupling. The roles of the spin-orbit coupling are clarified on the basis of the analytic formula of multipole susceptibility. The spin texture allowed in the D 2d point group symmetry and its magnetic response are revealed. Furthermore, we show that the parity-breaking quantum critical point appears in the magnetic field. The possible realization of the electric octupole order in bilayer high-T c cuprate superconductors is discussed.

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-T c 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 Cd 2 Re 2 O 7 is also discussed.