Analysis of $f\text{−}p$ model for octupole ordering in $\mathrm{Np}{\mathrm{O}}_2$

In order to examine the origin of octupole ordering in $\mathrm{Np}{\mathrm{O}}_2$, we propose a microscopic model constituted of neptunium $5f$ and oxygen $2p$ orbitals. To study multipole ordering, we derive effective multipole interactions from the $f\text{−}p$ model by using the fourth-order perturbation theory in terms of $p\text{−}f$ hopping integrals. Analyzing the effective model numerically, we find a tendency toward ${\Gamma }_{5u}$ antiferro-octupole ordering.

Figure 1

(a) Crystal structure of $\mathrm{Np}{\mathrm{O}}_2$. $a$ is the lattice constant. (b) eight-site cluster (gray spheres) on the fcc lattice taken in our numerical calculation.

Figure 2

Fourth-order processes with respect to $p\text{−}f$ hoppings. (a) Electrons doubly occupy a Np site. Electrons occupy (b) two or (c) one O site simultaneously, and then exchange. Electrons occupy (d) two or (e) one O site simultaneously, and then return to the initial sites. The solid and dashed lines in (b)–(e) distinguish paths for different electrons.

Figure 3

Phase diagrams obtained from the multipole correlation functions. Here 3gF denotes the ${\Gamma }_{3g}$ moment with $\mathbf{q}=(0,0,0)$, and so on. The antiferro (AF) ordering vector is $\mathbf{q}=(0,0,1)$ in units of $2\pi ∕a$ for all the AF phases. (a) Phase diagram for $U_p=0$. (b) Phase diagram for $U_p=0$ in a magnified scale around $(U^{\prime }-J)∕({ϵ}_p-{ϵ}_f+U^{\prime }-J)=0.145$ and $\left(pf\pi \right)=0$. (c) Phase diagram for $U_p=\infty$.