d_{iy}^\dagger d_{iy} - d^\dagger_{iz} d_{iz})^{1/2}$, with $M=1$ in the present case. A $1/M$ expansion up to the quadratic order of the Holstein-Primakoff bosons $d_y$ and $d_z$ followed by a Bogoliuobov transformation allows us to extract the ground state energy [@Xiao2009]. The detailed derivations are presented in the Supplemental Material [@supplemental]. Numerically diagonalizing the quadratic boson Hamiltonians on a square network consisting of $100\times 100$ cluster unit cells gives the energies of BC and PL which can be used to extract the regimes of BC and PL with finite stiffness.

The results are summarized in the shaded regimes in Fig. \[Fig:PD\_noring\]. The blue/green shaded regions in Fig. \[Fig:PD\_noring\] represent the stable BC/PL regimes in which the stiffness of BC/PL are finite. Surprisingly, we find that the BC and PL have *different* stiffness. In particular, there is a parameter range over which the BC order is stable, while the PL order is not due to negative stiffness (see supplementa material [@supplemental]). This represents a fluctuation mechanism to select the BC AFM order.

We find a much wider parameter regime where both BC and PL have positive stiffness. The flavor-wave theory also suggests that the energy splitting between these two orders are negligible 