10 FIG. 8: (Color online) h vs J$_{3}$ along the scan B ( − 4 ≤ J$_{3}$ ≤ 0 and J$_{2}$ = 1 . 5). J$_{1}$ = − 1. k = 0 and l = 1. FIG. 7: (Color online) (a) l vs J$_{2}$ and (b) h vs J$_{2}$ along the scan A ( − 4 ≤ J$_{2}$ ≤ 0 and J$_{3}$ = − 0 . 4). k = 0. J$_{1}$ = − 1. ( h = 0 , k = 0 , l = 1)]. We find that the relation of h vs J$_{3}$ is well described by The value of h is independent of J$_{2}$ . For example, when h = 0 . 2, we have J$_{3}$ = − 1 . 236. The plot of l vs J$_{2}$ (which is not shown here) indicates that l remains constant ( l = 1) for − 4 ≤ J$_{3}$ ≤ 0, where J$_{2}$ = 1 . 5. We find that the relation of l vs J$_{2}$ is well described by In the Scan C, we choose J$_{2}$ = − 0 . 4 and J$_{1}$ = − 1. J$_{3}$ is changed as a parameter between -3.5 and 0. We make a plot of h vs J$_{3}$ and l vs J$_{3}$ in Figs. 9(a) and (b). As shown in Fig. 9(a), the value of h decreases with increasing J$_{3}$ , showing an abrupt decrease at J$_{3}$ = − 2 . 5 (which is denoted by J$_{2}$J$_{3}$ = J 2 $_{1}$) [the phase boundary (first-order) between the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2) and the helical order along the a axis]. The value of h decreases with further increasing J$_{3}$ , following Eq.(23), and reduces to zero at J$_{3}$ = − 1 [the phase boundary (second-order) between the helical order along the a axis and the phase with ( h = 0 , k = 0 , l = 1)]. In Fig. 9(b), the value of l increases with increasing J$_{3}$ . It shows an abrupt change from l = 1 / 2 to l = 1 at J$_{3}$ = − 2 . 5 at the phase boundary (first-order) between the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2) and the helical state along the a axis. for − 2 . 5 ≤ J$_{2}$ ≤ − 1 (the helical phase along the c axis). For example, when l = 0 . 8, we have J$_{2}$ = − 1 . 24 as shown in Fig. 7(a). We note that l is equal to 1 at J$_{2}$ = − 1 [the phase boundary(second order) between the helical phase along the c axis and the phase with ( h = 0 , k = 0 , l = 1)] and remain unchanged ( l = 1) for − 1 ≤ J$_{2}$ ≤ 0. We also make a plot of h vs J$_{2}$ in Fig. 7(b). The value of h undergoes a sudden change from h = 0 . 5 to 0 at J$_{2}$ = − 2 . 5 (at the phase boundary between the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2) and the helical order along the c axis) and remains constant ( h = 0) for − 2 . 5 ≤ J$_{2}$ ≤ 0. Note that there is no change of h at J$_{2}$ = − 1. Here we show the contour plot of J ( h, 0 , l ) for the typ- ical points in the ( J$_{2}$ , J$_{3}$ ) phase diagram (see Fig. 10). The maximum points in the countour plot of J ( h, 0 , l )) correspond to the magnetic Bragg peaks in the ( h, 0 , l ) plane. In the Scan B (see Fig. 6), we choose J$_{2}$ = 1 . 5 and J$_{1}$ = − 1. J$_{3}$ is changed as a parameter between -4.0 and 0. We make a plot of h vs J$_{2}$ in Fig. 8. The value of h decreases with increasing J$_{3}$ and reduces to zero at J$_{3}$ = − 1 [the phase boundary (second-order) between the helical order along the a axis and the phase with 2. Scan B 3. Scan C cos( πh ) = − J$_{1}$ J$_{3}$ . (23) cos( πl ) = − J$_{1}$ J$_{2}$ , (22) VIII. CONTOUR PLOT OF ( h, 0 , l ) FOR TYPICAL POINTS IN THE ( J$_{2}$, J$_{3}$ ) PHASE DIAGRAM