Ordered spin structures of $\beta$-MnO$_{2}$ (rutile-type) systems with competing exchange interactions: numerical approach using equi-energy contour plot

Using numerical calculations of equi-energy contour plot of the Fourier transform of the spin Hamiltonian, we study the magnetic phase diagram ($J_{2}$ and $J_{3}$) of the rutile type $\beta$-MnO$_{2}$, where $J_{1}$ ($<0$) is fixed and is the antiferromagnetic interaction along the diagonal direction, $J_{2}$ is the interaction along the c axis, and $J_{3}$ is the interaction along the a axis. The magnetic phase diagram consists of the multricritical point (the intersection $J_{2}J_{3}=J_{1}^{2}$ and $J_{2}+J_{3}=2J_{1}$), the helical order along the c axis, the $(h=1/2,k=0,l=1/2)$ phase, the helical order along the a axis, and the phase $(h=0,k=0,l=1)$. The shift of the location of the magnetic Bragg peak in the $(h,0,l)$ reciprocal lattice plane is examined with the change of $J_{2}$ and $J_{3}$ in the phase diagram. The shift is discontinuous on the first-order phase transition, and is continuous on the second-order phase transition. The detail of our magnetic phase diagram is rather different from that reported by Yoshimori.