content_image/357903.jpg content_image/357901.jpg
<img src='content_image/357903.jpg'>

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: 

<img src='content_image/357901.jpg'>

The former also implies a divergent spectral weight (11) for such spinwaves at the quantum 

