Magnetic Toroidal Moment under Partial Magnetic Order in Hexagonal Zigzag-Chain Compound Ce$_3$TiBi$_5$

We theoretically investigate an antiferromagnetic structure in a hexagonal zigzag-chain compound Ce$_3$TiBi$_5$, which shows a linear magnetoelectric effect in metals and an unfamiliar temperature dependence of magnetic susceptibility. We find a partial magnetic order as a coexistence of a nonmagnetic zigzag chain and staggered antiferromagnetic zigzag chains. The latter accompany in-plane magnetic toroidal dipole moments which are responsible for the observed behaviors. We show that magnetic frustration arising from competing exchange interactions plays an important role to realize such a partial magnetic order. Furthermore, we present the important spin-orbit coupling parameters to induce the physical phenomena driven by the magnetic toroidal dipole moment, such as the linear magnetoelectric effect and nonlinear transport.