FIG. 3: The spectral weight of observable (+) spinwaves at the quantum critical point that sepa- rates hidden ferromagnetic (N´ eel) order from a cSDW is shown in units of ¯ h . The following set of Heisenberg exchange coupling constants are used: J ‖ ( ⊥ ) 1 = 0, J ⊥ ( ‖ ) 1 > 0, and J ‖ 2 = 0 . 3 J ⊥ ( ‖ ) 1 = J ⊥ 2 . The critical Hund’s rule exchange coupling is set to the value listed in Table I. Also shown are contours over the Brillouin zone for such spinwaves at a low fixed excitation energy. quadratically with wavenumber. Notice, however, the mild depression in the top spectrum that is centered at the cSDW wave number k$_{cSDW}$ = ( π/a, 0). The bottom graph in Fig. 2 shows how it softens completely to zero energy at the critical value for Hund’s rule coupling, − J$_{0}$$_{c}$ = 0 . 8 J ⊥ ( ‖ ) 1 , in which case the hidden ferromagnetic (N´ eel) state transits into the cSDW state by Table I. These special zero-energy modes result from the collapse of the frequency Ω$_{+}$( k$_{cSDW}$ ) to zero there. Evaluating (Ω$_{+}$Ω$_{−}$) 1 / 2 at k$_{cSDW}$ from the expressions for the frequencies that are listed below Eq. (10) yields the spin gap there: The former also implies a divergent spectral weight (11) for such spinwaves at the quantum Δ$_{cSDW}$ = 2 s [(4 J ⊥ 2 − J$_{0}$$_{c}$ )( J$_{0}$ − J$_{0}$$_{c}$ )] 1 / $^{2}$. (14) 11  0  2  4  6  8  10 (0,0) ( ,0) ( , ) (0,0) TRUE SPECTRAL WEIGHT (+) MOMENTUM (1/a) - /a 0 /a - /a 0 /a k y k$_{x}$