9 FIG. 6: (Color online) Scans A, B, and C in the phase diagram ( J$_{2}$ , J$_{3}$ ) to determine the location of the magnetic Bragg peaks in the ( h, k, l ) reciprocal space. J$_{1}$ = − 1. Note that J$_{2}$ = − 1 . 68469 and J$_{3}$ = − 0 . 537 for β -MnO$_{2}$ (denoted by an open circle). B. Boundaries between ordered phases 1. The phase boundary between the spin order with the phase ( h = 1 / 2 , k = 0 , and l = 1 / 2 ) and the helical spin order along the c axis The difference of J ( h, k, l ) between the helical spin or- der along the c axis and the phase ( h = 1 / 2 , k = 0 , l = 1 / 2) is given by So the helical ordered phase is energetically favorable when with the condition of | J$_{1}$/J$_{2}$ | < 1. When J$_{3}$ < 0, this inequality can be rewritten as or 2. The phase boundary between the spin order with ( h = 0 , k = 0 , l = 1 ) and the spin order with ( h = 1 / 2 , k = 0 , l = 1 / 2 ) Then the helical order is stable for J$_{3}$ < − 1. This in- equality is satisfied for the condition of the helical order The difference of J ( h, k, l ) between the ( h = 0 , k = 0 , l = 1) and the phase ( h = 1 / 2 , k = 0 , l = 1 / 2) is given by All four ordered phases merge only at the multicritical point at ( J$_{2}$ = − 1 and J$_{3}$ = − 1), where the line given by 2 J$_{1}$ = J$_{2}$ + J$_{3}$ and the curve J$_{2}$J$_{3}$ = J 2 1 intersect with other (see the phase diagram of J$_{3}$ vs J$_{2}$ ). Note that J$_{1}$ = − 1. As will be shown later, the line (2 J$_{1}$ = J$_{2}$ + J$_{3}$ ) is of the first-order. The lines denoted by J$_{2}$ = − 1 and J$_{3}$ = − 1 are of the second-order. (1) Using the program of finding maximum in J ( q ) for each point ( J$_{2}$ , J$_{3}$ ) in the phase digram, we find the local maximum point ( h, 0 , l ) in the h - l plane. The location of these points is plotted as a function of J$_{2}$ when J$_{3}$ is fixed and as a function of J$_{3}$ when J$_{2}$ is fixed. (2) Using the program of finding maximum in J ( q ) for each point ( J$_{2}$ , J$_{3}$ ) in the phase diagram, we find the local maximum point ( h, k, 0) in the h - k plane. These points are plotted as a function of J$_{2}$ when J$_{3}$ is fixed and as a function of J$_{3}$ when J$_{2}$ is fixed. When the phase ( h = 1 / 2 , k = 0 , l = 1 / 2) is energetically favor- able. When the phase h = 0, k = 0, l = 1) is energetically favorable. The difference of J ( h, k, l ) between the ( h = 0 , k = 0 , l = 1) and the helical order along the a axis is given by In the Scan A (see Fig. 6), we choose J$_{3}$ = − 0 . 4 and J$_{1}$ = − 1. J$_{2}$ is changed as a parameter between − 4 . 0 and 0. We make a plot of l vs J$_{2}$ in Fig. 7(a). The discontinuity in l vs J$_{2}$ occurs at J$_{2}$ = − 2 . 5 [the phase boundary (first order) between the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2) and the helical order along the c axis], where the relation J$_{2}$J$_{3}$ = J 2 1 is satisfied ( J$_{1}$ = − 1). J$_{max}$ ( h = 0 , k = 0 , l ) − J ( h = 1 / 2 , k = 0 , l = 1 / 2) = 4 J$_{2}$ − 4 J 2 1 J$_{3}$ . J$_{2}$ > J 2 1 J$_{3}$ , J$_{2}$J$_{3}$ < J 2 $_{1}$, (19) p$_{2}$p$_{3}$ < 1 . J ( h = 1 / 2 , k = 0 , l = 1 / 2) − J ( h = 0 , k = 0 , l = 1) = − 4 J$_{2}$ − 4 J$_{3}$ + 8 J$_{1}$. 2 J$_{1}$ > J$_{2}$ + J$_{3}$, (20) 2 J$_{1}$ < J$_{2}$ + J$_{3}$. (21) J$_{max}$ ( h, k = 0 , l = 1) − J ( h = 0 , k = 0 , l = 1) = − 4 J$_{3}$ ( J$_{1}$ J$_{3}$ − 1) $^{2}$. 3. The phase boundary between the helical order along the a axis and the phase ( h = k = 1 / 2 , and l = 1 ) 1. Scan A | J$_{1}$/J$_{3}$ | < 1 . C. Multicritical point ( J$_{2}$ = − 1 and J$_{3}$ = − 1 ) and phase transition lines