Multipole properties of one-dimensional $f$-electron systems

By exploiting a density matrix renormalization group method, we investigate the ground-state properties of a one-dimensional three-orbital Hubbard model on the basis of a $j$-$j$ coupling scheme. Here we focus on the case where the $f$-electron number per site is one ($f^1$). When three orbitals are degenerate, we observe a peak at $q$=0 in $\Gamma_{3g}$ quadrupole correlation, indicating a ferro-orbital state. Namely, $f$ electron occupies an itinerant $\Gamma_8^b$ orbital to gain kinetic energy, while localized $\Gamma_8^a$ and $\Gamma_7$ orbitals are found to be almost empty. Furthermore, we find a peak at $q$=$\pi$ in $\Gamma_{4u}$ dipole correlation, suggesting an antiferromagnetic state. On the other hand, when we take account of the level splitting between $\Gamma_8$ and $\Gamma_7$ orbitals, due to the competition between itinerant and localized orbitals, we observe a characteristic change of $\Gamma_{3g}$ quadrupole correlation into an incommensurate structure in accordance with the change of the orbital structure. We will also discuss a key role of multipole degrees of freedom in $f^2$- and $f^3$-electron systems.