bers have been detected in a ferro-pnictide superconductor . 15 These disperse up in energy in an anisotropic fashion that is consistent with the presence of some degree of cSDW order. In this paper, we continue to explore the hidden magnetic order that is predicted to exist in multi-orbital Heisenberg models over the square lattice when the diagonal and the off- diagonal exchange coupling constants are sufficiently different. 10 Two spin-1/2 orbitals per iron atom are assumed, which implies spin-1 iron moments if Hund’s rule is obeyed. A quan- tum critical point 16 is identified within the linear spin-wave approximation that separates hidden ferromagnetic or hidden N´ eel order at weak Hund’s rule coupling from a cSDW at strong Hund’s rule coupling. Numerical exact diagonalizations of the two-orbital Heisenberg model over a 4 × 4 lattice indicate that the moment associated with cSDW order decays strongly as it enters hidden magnetic order through the putative quantum critical point. The linear spin-wave approximation, on the other hand, finds observable low-energy spin- wave excitations at the quantum critical point that disperse linearly from the cSDW wave numbers in an anisotropic fashion. A spin gap begins to grow there upon entering hidden magnetic order. Observable low-energy spin-wave excitations due to hidden order that are centered at zero momentum are also predicted by the linear spin-wave approximation, but these have very small spectral weight. Fits of the spin-wave spectrum observed by inelastic neutron scattering in ferro-pnictide materials 14 , 15 to the critical linear spin-wave spectrum are achieved. They notably account for the dispersion of the low-energy spinwaves centered at the cSDW wave numbers and for the local maximum that exists at the wavenumber associated with N´ eel order. A critical Hund’s rule coupling of order 100 meV is extracted from these fits. Below we discuss useful theoretical aspects of the Heisenberg model for ferro-pnictide materials 7 before we go on to analyze it. Recent low-temperature measurements of the optical conductivity in the cSDW phase of parent compounds to ferro-pnictide high- T$_{c}$ superconductors find a strong suppression of the integrated kinetic energy for the conduction electrons compared to that predicted by electronic band-structure calculations. 17 A low carrier density in the cSDW phase of ferro- pnictide materials is consistent with quantum-oscillation experiments 18 , 19 and with angle- resolved photoemission studies 20 that find evidence for tiny 2D Fermi-surface “spots”, each 3 II. MULTI-ORBITAL HEISENBERG MODEL