Electric Octupole Order in Bilayer Ruthenate Sr$_3$Ru$_2$O$_7$

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.