FIG. 19: (Color online) Spin structure for the helical order along the a axis with ( h$_{max}$ = 2/3, 0, 1). J$_{1}$ = − 1, and J$_{3}$ = − 2 . 0. FIG. 20: (Color online) (a) Crystal structure of β -MnO$_{2}$. ∠ A$_{1}$O$_{4}$B$_{1}$ = ∠ C$_{1}$O$_{1}$B$_{1}$ = 130 . 62 $^{◦}$. ∠ A$_{1}$O$_{4}$A$_{2}$ = 111 . 60 $^{◦}$. ∠ B$_{1}$O$_{3}$D$_{1}$ = 98 . 77 $^{◦}$. The distance (in the units of ˚ A ) is denoted by numerical value. (b) Distorted octahedra of β -MnO$_{2}$. ∠ O$_{1}$B$_{1}$O$_{4}$ = 98 . 77 $^{◦}$, ∠ O$_{1}$B$_{1}$O$_{3}$ = 81 . 225 $^{◦}$, ∠ O$_{5}$O$_{1}$O$_{6}$ = 89 . 590 $^{◦}$, ∠ O$_{1}$O$_{6}$O$_{5}$ = 45 . 205 $^{◦}$, and ∠ O$_{1}$B$_{1}$O$_{5}$ = ∠ O$_{4}$B$_{1}$O$_{5}$ = 90 . 0 $^{◦}$. The distance (in the units of ˚ A ) is denoted by numerical value. X. DISCUSSION: NATURE OF EXCHANGE INTERACTIONS IN THE RUTILE-TYPE β -MnO$_{2}$ A. Overview B. Origin of the direct exchange interaction J$_{2}$ x axis is not perpendicular to the y axis (98 . 78 $^{◦}$, 81 . 22 $^{◦}$). We show that the magnetic phase diagram with J$_{1}$ = − 1 is uniquely determined by the combination of J$_{2}$ and J$_{3}$ , where the sign of J$_{3}$ is mainly negative and the sign of J$_{2}$ is changed between negative to positive. For β - MnO$_{2}$, we have p$_{2}$ = 1 . 68469 and as p$_{3}$ = 0 . 537, leading to J$_{2}$ = − 1 . 68469 and J$_{3}$ = − 0 . 537 when J$_{1}$ = − 1. This point ( J$_{2}$ , J$_{3}$ ) for β -MnO$_{2}$ is in the helical order along the c axis, as shown in the magnetic phase diagram (Fig. 6). In Fig. 20(a), we show the structure of β -MnO$_{2}$. Figure 20(b) shows the distorted octahedron where one Mn 4+ ion (cation) at the point B$_{1}$, is surrounded by six O 2 − ions (anion) at the points O$_{1}$, O$_{2}$, O$_{3}$, O$_{4}$, O$_{5}$, and O$_{6}$. The cation-cation separation (A$_{1}$C$_{1}$, B$_{1}$D$_{1}$) along a [001] axis is considerably smaller (2 . 871 ˚ A ). Because of the distorted octahedron formed by O 2 − ions in the vicin- ity of Mn 4+ ion, the ground orbital state of Mn 4+ ion (3 d $^{3}$, L = 3 and S = 3 / 2) is split into the t$_{2}$$_{g}$ ( dϵ ) level [lower energy level, triple degenerate; d ( xy ), d ( yz ), and d ( zx ) states] and the e$_{g}$ ( dγ ) level [upper energy, double degenerate; d (3 z 2 − r $^{2}$), d ( x 2 − y $^{2}$))]. The t$_{2}$$_{g}$ state is occupied by three electrons with spin up state | + 〉 . The ϵ$_{g}$ state is empty. As a result, the ground state is orbital singlet, indicating that the orbital angular momentum is quenched. In β -MnO$_{2}$, the e$_{g}$ electrons are responsible for the metallic conduction, while the localized t$_{2}$$_{g}$ elec- trons are responsible for the magnetism. In Fig. 20(b), the (3 z 2 − r $^{2}$) orbital axis at the point B$_{1}$ coincides with O$_{5}$B$_{1}$O$_{6}$ (the z axis). The d ( x 2 − y $^{2}$) axes coincide with O$_{3}$B$_{1}$O$_{4}$ (the x axis) and the O$_{1}$B$_{1}$O$_{2}$ (the y axis). The x and y axes are perpendicular to the z axis. However, the We consider the exchange interaction J$_{2}$ between the points B$_{1}$ and D$_{1}$ (or between the points A$_{1}$ and C$_{1}$). The distance B$_{1}$D$_{1}$ (A$_{1}$C$_{1}$) is 2 . 871 ˚ A . In Fig. 20(b), one d ( xy ) orbital ( t$_{2}$$_{g}$ ) from the point B$_{1}$ bisects the angle ∠ O$_{1}$B$_{1}$O$_{3}$ and meets at the middle point of the edge O$_{1}$O$_{3}$. The other d ( xy ) orbital from the point D$_{1}$ (the center of neighboring octahedron) bisects the angle ∠ O$_{1}$D$_{1}$O$_{3}$ and meets at the middle point of the edge 14