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Pauli principle then implies that spin s = ¯ h/ 2 moments exists on both the 3 d$_{xz}$ and the 3 d$_{yz}$ orbitals in the limit of weak Hund’s rule coupling. Henceforth, we shall assume that only two spin-1/2 orbitals, a and b , exist per iron atom in parent compounds to ferro-pnictide high- T$_{c}$ superconductors. This is consistent with the ordered moment of 2 µ$_{B}$ obtained by DFT calculations for antiferromagnetic groundstates of ferro-pnictide parent compounds. 4 , 5 , 6 Let us next fix notation by assuming unique diagonal and unique off-diagonal Heisenberg ex- change coupling constants over the nearest neighbor and the next-nearest neighbor links of the square lattice of iron atoms: 

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The author and Rezayi have recently shown that strong enough off-diagonal frustration, J ⊥ 2 > 0, can lead to either hidden ferromagnetic order or to hidden N´ eel order among spin- 1/2 moments that are governed by the J$_{0}$ - J$_{1}$ - J$_{2}$ model Hamiltonian 10 (1). Figure 1 displays the two types of hidden order, both of which show no net magnetic moment. Proximity of the collinear antiferromagnetic state, which exists at J$_{2}$ > | J$_{1}$ | / 2 when Hund’s rule is enforced, to such hidden order can therefore account for the low ordered moment that is observed by elastic neutron diffraction in the cSDW state that exists at low temperature in undoped ferro-pnictide materials. 2 Balancing the energy of such hidden-order groundstates in the classical limit, s → ∞ , against that of conventional magnetic order leads to critical values for the Hund’s rule coupling that are listed in Table I. True magnetic order becomes unstable to hidden magnetic order at weaker Hund’s-rule coupling, − J$_{0}$ < − J$_{0}$$_{c}$ . A true magnetically ordered groundstate therefore first becomes unstable to the hidden-order groundstate that has the larger of the two critical values for the Hund’s rule coupling strength, − J$_{0}$$_{c}$ . By Table I, for example, the true cSDW state becomes unstable to hidden ferromagnetic order at weak Hund’s rule coupling only if the off-diagonal nearest-neighbor Heisenberg exchange is larger than the corresponding diagonal exchange: J ⊥ 1 > J ‖ $_{1}$. 

Linear Spin-Wave Theory. Study of the classical limit, s → ∞ , therefore indicates that extremely low ordered moments are possible at weak enough Hund’s rule coupling, J$_{0}$ < 0, when off-diagonal frustration exists: J ⊥ 2 > 0. The nature of spin-wave excitations about the hidden magnetic order that is shown in Fig. 1 can also be obtained from the classical limit. In such case, each spin-1/2 moment obeys the following dynamical equation for precession: 

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