FIG. 5: Shown is the low-energy spectrum for 4 × 4 × 2 spin-1/2 moments that experience the following set of Heisenberg exchange coupling constants: J ‖ 1 = 0, J ⊥ 1 > 0, and J ‖ 2 = 0 . 3 J ⊥ 1 = J ⊥ 2 . The Hund’s rule exchange coupling is set to J 0 = − 1 . 35 J ⊥ 1 . with the strength of the Hund’s rule coupling in the presence of off-diagonal frustration. The following Heisenberg exchange coupling constants were chosen: J ‖ 1 = 0, J ⊥ 1 > 0, and J ‖ 2 = 0 . 3 J ⊥ 1 = J ⊥ 2 . A hidden ferromagnetic state is then expected in the absence of Hund’s rule coupling, while a cSDW is expected when Hund’s rule in obeyed. In the former case displayed by Fig. 4, notice the coincidence between the linear spin-wave approximation about hidden-order, Eq. (12) and Fig. 2, and the present exact-diagonalization results. It indicates that the classical prediction of long-range hidden magnetic order holds true for spin-1/2 moments per iron orbital. Figure 6 displays the spectrum at strong Hund’s rule coupling, − J$_{0}$ = 15 J ⊥ 1 , on the other hand. The bulk of the groundstate energy is due to the Hund’s rule coupling, which is given by 16 J$_{0}$ = − 240 J ⊥ 1 when Hund’s rule is obeyed. Notice the low-energy spin-1 state at the cSDW wave number ( π/a, 0), as well as the softening of the spin-1 excited state at the wave number ( π/a, π/a ) that characterizes the N´ eel state. 14 -36.4 -36.2 -36 -35.8 -35.6 -35.4 -35.2 -35 -34.8 (1,1) (0,0) (1,0) (2,0) (2,1) (2,2) ENERGY (J 1 ) MOMENTUM ( /2a) TRANSITION FROM COMMENSURATE SDW INTO HIDDEN FERROMAGNET S = 0 S = 1 S = 2